This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.
The focus of the first part is when the domain is the complex plane. In this case, given a finite cyclic configuration of points $P \subset \partial\mathbb{H}^3=\mathbb{CP}^1$, we construct a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ that is asymptotic to a twisted ideal polygon with ideal vertices contained in $P$. Moreover, we prove that given any ideal twisted polygon in $\mathbb{H}^3$ with rotational symmetry, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.
In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $PSL_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely marked and bordered surfaces (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (PSL_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$.
In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.
This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.
Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.
We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.
The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.
For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).
This thesis consists of two parts. In the first part, we introduce coupled K¨ahler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Amp`ere vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the K¨ahlerness of the symplectic form which arises in the moment map interpretation of Calabi-Yang-Mills equations.
A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.
In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.
In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.
The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss the structures of the pairs of commuting $C_0$-semigroups of isometries in generality as well as under certain additional assumptions like double commutativity or dual double commutativity.
The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space
$\mathcal E$ is defined as
\begin{equation}
(S_t^\mathcal E f)(x) = \begin{cases}
f(x-t) &\text{if } x\ge t,\\
0 & \text{otherwise,}
\end{cases}
\end{equation}
for $f\in L^2(\mathbb R_+,\mathcal E).$
Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for
a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining
the factorizations of the right-shift-semigroup.
Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete
description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal
E=\mathcal V_1\mathcal V_2$, (such a pair is called as a factorization of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite
dimensional Hilbert space.
Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.
This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:
We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.
Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.
We then talk about multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.
Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.
This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems. The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and L1 data in a circular oscillating domain. As the source term we are considering is in L1, we will examine the renormalized solutions. The second problem we will investigate is an optimal control problem governed by a second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the oscillations occur in $m$ directions, with $1< m < n$. We will explore the asymptotic behavior of this problem by homogenizing the corresponding optimality systems.
A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.
In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.
In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.
This thesis focuses on the study of correlations in multispecies totally and partially asymmetric exclusion processes (TASEPs and PASEPs). We study various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP, and multispecies TASEP on a ring with multiple copies of each particle. The primary goal of this thesis is to understand the two-point correlations of adjacent particles in these processes. The details of the results are as follows:
We first discuss the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlation of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.
Next, we prove a formula for the correlation of adjacent particles on the first two sites in a multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the Ferrari-Martin multiline process described above.
We then talk about the multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlation between adjacent particles on the last two sites in a multispecies B-TASEP. To solve this conjecture, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make some progress towards the above conjecture.
Finally, we discuss a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson (Trans. AMS., 2017) to this case and prove formulas for two-point correlations and relate them to the TASEP speed process.
Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.
This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.
Second, we discuss a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.
Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.
Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.
This thesis consists of two parts. In the first part, we introduce coupled Kähler-Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima-Lichnerowicz-type theorem as another obstruction. Using the Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of the Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Ampère vortex equation. We establish a correspondence result between Gieseker stability and the existence of almost Hermitian-Yang-Mills metric in a particular case. We also investigate the Kählerness of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations.
This talk will comprehensively examine the homogenization of partial differential equations (PDEs) and optimal
control problems with oscillating coefficients in oscillating domains. We will focus on two specific problems.
The first is the homogenization of a second-order elliptic PDE with strong contrasting diffusivity and $L^1$
data in a circular oscillating domain. As the source term we are considering is in $L^1$, we will examine the
renormalized solutions. The second problem we will investigate is an optimal control problem governed by a
second-order semi-linear PDE in an $n$-dimensional domain with a highly oscillating boundary, where the
oscillations occur in $m$ directions, with $1<m<n$
. We will explore the asymptotic behavior of this problem by
homogenizing the corresponding optimality systems.
The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermat’s Last Theorem.
We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.
In the first part of the talk we will discuss the main statement of local class field theory and discuss the statement of Local Langlands correspondence for $GL_2(K)$, where $K$ is a non-archimedean local field. In the process, we will also introduce all the objects in the statement of correspondence. We will then discuss a brief sketch of the proof of the main statement of local class field theory.
Convection dominated fluid flow problems show spurious oscillations when solved using the usual Galerkin finite element method (FEM). To suppress these un-physical solutions we use various stabilization methods. In this thesis, we discuss the Local Projection Stabilization (LPS) methods for the Oseen problem.
This thesis mainly focuses on three different finite element methods each serving a purpose of its own. First, we discuss the a priori analysis of the Oseen problem using the Crouzeix-Raviart (CR1) FEM. The CR1/P0 pair is a well-known choice for solving mixed problems like the Oseen equations since it satisfies the discrete inf-sup condition. Moreover, the CR1 elements are easy to implement and offer a smaller stencil compared with conforming linear elements (in the LPS setting). We also discuss the CR1/CR1 pair for the Oseen problem to achieve a higher order of convergence.
Second, we discuss the a posteriori analysis for the Oseen problem using the CR1/P0 pair using a dual norm approach. We define an error estimator and prove that it is reliable and discuss an efficiency estimate that depends on the diffusion coefficient.
Next, we focus on formulating an LPS scheme that can provide globally divergence free velocity. To achieve this, we use the $H(div;\Omega)$ conforming Raviart-Thomas (${\rm RT}^k$) space of order $k \geq 1$. We show a strong stability result under the SUPG norm by enriching the ${\rm RT}^k$ space using tangential bubbles. We also discuss the a priori error analysis for this method.
Finally, we develop a hybrid high order (HHO) method for the Oseen problem under a generalized local projection setting. These methods are known to allow general polygonal meshes. We show that the method is stable under a “SUPG-like” norm and prove a priori error estimates for the same.
Modelling price variation has always been of interest, from options pricing to risk management. It has been observed that the high-frequency financial market is highly volatile, and the volatility is rough. Moreover, we have the Zumbach effect, which means that past trends in the price process convey important information on future volatility. Microscopic price models based on the univariate quadratic Hawkes process can capture the Zumbach effect and the rough volatility behaviour at the macroscopic scale. But they fail to capture the asymmetry in the upward and downward movement of the price process. Thus, to incorporate asymmetry in price movement at micro-scale and rough volatility and the Zumbach effect at macro-scale, we introduce the bivariate Modified-quadratic Hawkes process for upward and downward price movement. After suitable scaling and shifting, we show that the limit of the price process in the Skorokhod topology behaves as so-called Super-Heston-rough model with the Zumbach effect.
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras. The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$. We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic” generalization of the partial sum property in root systems: every positive root is an ordered sum of simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak faces of the set of weights, and their complete classification for arbitrary $V$.
The primary goal of this dissertation is to establish bounds for the sup-norm of the Bergman kernel of Siegel modular forms. Upper and lower bounds for them are studied in the weight as well as level aspect. We get the optimal bound in the weight aspect for degree 2 Siegel modular forms of weight $k$ and show that the maximum size of the sup-norm $k^{9/2}$. For higher degrees, a somewhat weaker result is provided. Under the Resnikoff-Saldana conjecture (refined with dependence on the weight), which provides the best possible bounds on Fourier coefficients of Siegel cusp forms, our bounds become optimal. Further, the amplification technique is employed to improve the generic sup-norm bound for an individual Hecke eigen-forms however, with the sup-norm being taken over a compact set of the Siegel’s fundamental domain instead. In the level aspect, the variation in sup-norm of the Bergman kernel for congruent subgroups $\Gamma_0^2(p)$ are studied and bounds for them are provided. We further consider this problem for the case of Saito-Kurokawa lifts and obtain suitable results.
The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermat’s Last Theorem.
We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras. The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$. We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic” generalization of the partial sum property in root systems: every positive root is an ordered sum of simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak faces of the set of weights, and their complete classification for arbitrary $V$.
In the first part of the talk we will discuss the main statement of local class field theory and sketch a proof of it. We will then discuss the statement of local Langlands correspondence for $GL_2(K)$, where $K$ is a non archimedian local field. In the process we will also introduce all the objects that go in the statement of the correspondence.
This thesis has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasi-analyticity. In contrast, the second part reflects a different mathematical theme, focusing on the classical problem of $L^p$ boundedness of spherical maximal function on the Heisenberg group.
The highlights of the first part are as follows: An uncertainty principle due to Ingham (proved initially on $\mathbb{R}$) investigates the best possible decay admissible for the Fourier transform of a function that vanishes on a nonempty open set. One way to establish such a result is to use a theorem of Chernoff (proved originally on $\mathbb{R}^n$), which provides a sufficient condition for a smooth function to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this part of this thesis, we aim to prove various analogues of theorems of Ingham and Chernoff in different contexts such as the Heisenberg group, Hermite and special Hermite expansions, rank one Riemannian symmetric spaces, and Euclidean space with Dunkl setting. More precisely, we prove various analogues of Chernoff’s theorem for the full Laplacian on the Heisenberg group, Hermite and special Hermite operators, Laplace-Beltrami operators on rank one symmetric spaces of both compact and non-compact type, and Dunkl Laplacian. The main idea is to reduce the situation to the radial case by employing appropriate spherical means or spherical harmonics and then to apply Chernoff type theorems to the radial parts of the operators indicated above. Using those Chernoff type theorems, we then show several analogues of Ingham’s theorem for the spectral projections associated with those aforementioned operators. Furthermore, we provide examples of compactly supported functions with Ingham type decay in their spectral projections, demonstrating the sharpness of Ingham’s theorem in all of the relevant contexts mentioned above.
In the second part of this thesis, we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
Modelling price variation has always been of interest, from options pricing to risk management. It has been observed that the high-frequency financial market is highly volatile, and the volatility is rough. Moreover, we have the Zumbach effect, which means that past trends in the price process convey important information on future volatility. Microscopic price models based on the univariate quadratic Hawkes (hereafter QHawkes) process can capture the Zumbach effect and the rough volatility behaviour at the macroscopic scale. But they fail to capture the asymmetry in the upward and downward movement of the price process. Thus, to incorporate asymmetry in price movement at micro-scale and rough volatility and the Zumbach effect at macroscale, we introduce the bivariate Modified-QHawkes process for upward and downward price movement. After suitable scaling and shifting, we show that the limit of the price process in the Skorokhod topology behaves as so-called Super-Heston-rough model with the Zumbach effect.
Non-malleable codes (NMCs) are coding schemes that help in protecting crypto-systems under tampering attacks, where the adversary tampers the device storing the secret and observes additional input-output behavior on the crypto-system. NMCs give a guarantee that such adversarial tampering of the encoding of the secret will lead to a tampered secret, which is either same as the original or completely independent of it, thus giving no additional information to the adversary. The specific tampering model that we consider in this work, called the “split-state tampering model”, allows the adversary to tamper two parts of the codeword arbitrarily, but independent of each other. Leakage resilient secret sharing schemes help a party, called a dealer, to share his secret message amongst n parties in such a way that any $t$ of these parties can combine their shares to recover the secret, but the secret remains hidden from an adversary corrupting $< t$ parties to get their complete shares and additionally getting some bounded bits of leakage from the shares of the remaining parties.
For both these primitives, whether you store the non-malleable encoding of a message on some tamper-prone system or the parties store shares of the secret on a leakage-prone system, it is important to build schemes that output codewords/shares that are of optimal length and do not introduce too much redundancy into the codewords/shares. This is, in particular, captured by the rate of the schemes, which is the ratio of the message length to the codeword length/largest share length. This thesis explores the question of building these primitives with optimal rates.
The focus of this talk will be on taking you through the journey of non-malleable codes culminating in our near-optimal NMCs with a rate of 1/3.
This talk has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasi-analyticity. On the other hand, the second part reflects a different mathematical theme, focusing on the classical problem of $L^p$ boundedness of spherical maximal function on the Heisenberg group.
The highlights of the first part are as follows: An uncertainty principle due to Ingham (proved initially on $\mathbb{R}$) investigates the best possible decay admissible for the Fourier transform of a function that vanishes on a nonempty open set. One way to establish such a result is to use a theorem of Chernoff (proved originally on $\mathbb{R}^n$), which provides a sufficient condition for a smooth function to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this part of this talk, we plan to discuss various analogues of Chernoff’s theorem for the full Laplacian on the Heisenberg group, Hermite, and special Hermite operators, Laplace-Beltrami operators on rank one symmetric spaces of both compact and non-compact type, and Dunkl Laplacian. Using those Chernoff type theorems, we then show several analogues of Ingham’s theorem for the spectral projections associated with those aforementioned operators. Furthermore, we provide examples of compactly supported functions with Ingham type decay in their spectral projections, demonstrating the sharpness of Ingham’s theorem in all of the relevant contexts mentioned above.
In this second part of this talk, we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
Several critical physical properties of a material are controlled by its geometric construction. Therefore, analyzing the effect of a material’s geometric structure can help to improve some of its beneficial physical properties and reduce unwanted behavior. This leads to the study of boundary value problems in complex domains such as perforated domain, thin domain, junctions of the thin domain of different configuration, domain with rapidly oscillating boundary, networks domain, etc.
This talk will discuss various homogenization problems posed on high oscillating domains. We discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.), where the oscillatory part is made of two materials with high contrasting conductivities. Thus the low contrast material acts as near insulation in-between the conducting materials. Mathematically this leads to the study of degenerate elliptic PDE at the limiting scale. We also briefly explain another interesting article (see, ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.), where the oscillations are on the curved interface with general cost functional.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its construction as it is the main tool in our analysis.
The second part of this talk will briefly discuss the boundary optimal control problems subject to Laplacian and Stokes systems.
In the third part of the talk, we will discuss the homogenization of optimal control problems subject to a elliptic variational form with high contrast diffusivity coefficients. The interesting result is the difference in the limit behavior of the optimal control problem, which depends on the control’s action, whether it is on the conductive part or insulating part. In both cases, we derive the \two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete-scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We obtain the homogenized equation for the state, but the two-scale separation of the cost functional remains as an open question.
The study of the optimal control problems governed by partial differential equations(PDEs) have been a significant research area in applied mathematics and its allied areas. The optimal control problem consists of finding a control variable that minimizes a cost functional subject to a PDE. In this talk, I will present finite element analysis of Dirichlet boundary optimal control problems governed by certain PDEs. This talk will be divided into four parts.
In the first part, we discuss the Dirichlet boundary control problem, its physical interpretation, mathematical formulation, and some approaches (numerical) to solve it.
In the second part, we study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and co-state variables. We propose a finite element-based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. We present the analysis for $L^2$ cost functional, but this analysis can also be extended to the gradient cost functional problem. A priori error estimates of optimal order in the energy norm are derived up to the regularity of the solution.
In the third part, we discuss the Dirichlet boundary optimal control problem governed by the Stokes equation. We develop a finite element discretization by using $\mathbf{P}_1$ elements (in the fine mesh) for the velocity and control variable and $P_0$ elements (in the coarse mesh) for the pressure variable. We present a posteriori error estimators for the error in the state, co-state, and control variables. As a continuation of the second part, we extend our ideas to the linear parabolic equation in the last part of the presentation. The space discretization of the state and co-state variables is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. We use $H^1$-conforming 3D finite elements for the control variable. We present the error estimates of state, adjoint state, and control.
The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product
theorem [Crelle 1911]
, which implies that absolutely monotonic
functions (i.e., power series with nonnegative coefficients) preserve
positivity on matrices of all dimensions. A famous result of Schoenberg
and of Rudin [Duke Math. J. 1942, 1959]
shows the converse: there are
no other such functions.
Motivated by modern applications, Guillot and Rajaratnam
[Trans. Amer. Math. Soc. 2015]
classified the entrywise positivity
preservers in all dimensions, which act only on the off-diagonal entries.
These two results are at “opposite ends”, and in both cases the preservers
have to be absolutely monotonic.
The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
The talk will begin by discussing connections between metric geometry and positivity, also via positive definite functions. Following this, we present Schoenberg’s motivations in studying entrywise positivity preservers, followed by classical variants for matrices with entries in other real and complex domains. Then we shall see the result due to Guillot and Rajaratnam on preservers acting only on the off-diagonal entries, touching upon the modern motivation behind it. This is followed by its generalization in the thesis. Finally, we present the (remaining) main results in the thesis, and conclude with some of the proofs.
In this talk, we will see an interplay between hermitian metrics and singular Riemann surface foliations. It will be divided into three parts. The first part of the talk is about the study of curvature properties of complete Kahler metrics on non-pseudoconvex domains. Examples of such metrics were constructed by Grauert in 1956, who showed that it is possible to construct complete Kahler metrics on the complement of complex analytic sets in a domain of holomorphy. In particular, he gave an explicit example of a complete Kahler metric (the Grauert metric) on $\mathbb{C}^n \setminus {0}$. We will confine ourselves to the study of such complete Kahler metrics. We will make some observations about the holomorphic sectional curvature of such metrics in two prototype cases, namely (i) $\mathbb{C}^n \setminus {0}$, $n>1$, and (ii) $(B^N)\setminus A$, where $A$ is an affine subspace. We will also study complete Kahler metrics using Grauert’s construction on the complement of a principal divisor in a domain of holomorphy and show that there is an intrinsic continuity in the construction of this metric, i.e., we can choose this metric in a continuous fashion if the corresponding principal divisors vary continuously in an appropriate topology.
The second part of the talk deals with Verjovsky’s modulus of uniformization that arises in the study of the leaf-wise Poincare metric on a hyperbolic singular Riemann surface lamination. This is a function defined away from the singular locus. One viewpoint is to think of this as a domain functional. Adopting this view, we will show that it varies continuously when the domains vary continuously in the Hausdorff sense. We will also give an analogue of the classical Domain Bloch constant by D. Minda for hyperbolic singular Riemann surface laminations.
In the last part of the talk, we will discuss a parametrized version of the Mattei-Moussu theorem, namely a holomorphic family of holomorphic foliations in $\mathbb{C}^2$ with an isolated singular point at the origin in the Siegel domain are holomorphically equivalent if and only if the holonomy maps of the horizontal separatrix of the corresponding foliations are holomorphically conjugate.
In this thesis, we analyse certain dynamically interesting measures arising in holomorphic dynamics beyond the classical framework of maps. We will consider measures associated with semigroups and, more generally, with meromorphic correspondences, that are invariant in a specific sense. Our results are of two different flavours. The first type of results deal with potential-theoretic properties of the measures associated with certain polynomial semigroups, while the second type of results are about recurrence phenomena in the dynamics of meromorphic correspondences. The unifying features of these results are the use of the formalism of correspondences in their proofs, and the fact that the measures that we consider are those that describe the asymptotic distribution of the iterated inverse images of a generic point.
The first class of results involve giving a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh–Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which we can describe explicitly given a choice of a set of generators. In particular, we generalize the classical result of Brolin to certain finitely generated polynomial semigroups. To do so, we establish the continuity of the logarithmic potential for the Dinh–Sibony measure, which might also be of independent interest. Thereafter, we use the $F$-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.
The second class of results involves meromorphic correspondences. These are, loosely speaking, multi-valued analogues of meromorphic maps. We shall present an analogue of the Poincare recurrence theorem for meromorphic correspondences with respect to the measures alluded to above. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result on the invariance properties of the supports of the measures mentioned, and, as a corollary, give a geometric description of the support of such a measure.
We prove Hardy’s inequalities for the fractional power of Grushin operator $\mathcal{G}$ which is chased via two different approaches. In the first approach, we first prove Hardy’s inequality for the generalized sublaplacian. We first find Cowling–Haagerup type of formula for the fractional sublaplacian and then using the modified heat kernel, we find integral representations of the fractional generalized sublaplacian. Then we derive Hardy’s inequality for generalized sublaplacian. Finally using the spherical harmonics, applying Hardy’s inequality for individual components, we derive Hardy’s inequality for Grushin operator. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\mathbb{R}^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\mathbb{R}^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\mathcal{G}_s f$ in $L^p(\mathbb{R}^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy–Littlewood–Sobolev inequality for the Grushin operator.
Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\mathbb{R}^n)$. We find a relation between the boundedness of sublaplacian multipliers $m(\tilde{\mathcal{L}})$ on polarised Heisenberg group $\mathbb{H}^n_{pol}$ and the boundedness of Hermite multipliers $m(\mathcal{H})$ on modulation spaces $M^{p,q}(\mathbb{R}^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe those conditions on multipliers are more than required restrictive. We improve the results for the special case $p=q$ of the modulation spaces $M^{p,q}(\mathbb{R}^n)$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}(\mathbb{R}^n)$ and the boundedness of Fourier multipliers on torus $\mathbb{T}^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr"odinger equation related to Hermite on modulation spaces.
This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First, we have focused on the geometry of these two domains. The symmetrized bidisc, a non-homogeneous domain, is partitioned into a collection of orbits under the action of its automorphism group. We investigate the properties of these orbits and pick out some necessary properties so that the symmetrized bidisc can be characterized up to biholomorphic equivalence. As a consequence, among other things, we have given a new defining condition of the symmetrized bidisc and we have found that a biholomorphic copy of the symmetrized bidisc defined by E. Cartan. This work on the symmetrized bidisc also helps us to develop a characterization of the bidisc. Being a homogeneous domain, the bidisc’s automorphism group does not reveal much about its geometry. Using the ideas from the work on the symmetrized bidisc, we have identified a subgroup of the automorphism group of the bidisc and observed the corresponding orbits under the action of this subgroup. We have identified some properties of these orbits which are sufficient to characterize the bidisc up to biholomorphic equivalence.
Turning to operator theoretic work on the domains, we have focused mainly on the Schur Agler class class on the bidisc and the symmetrized bidisc. Each element of the Schur Agler class on these domains has a nice representation in terms of a unitary operator, called the realization formula. We have generalized the ideas developed in the context of the bidisc and the symmetrized bidisc and applied it to the Nevanlinna problem and the interpolating sequences. It turns out, our generalization works for a number of domains, such as annulus and multiply connected domains, not only the bidisc and the symmetrized bidisc.
This thesis is devoted to the study of nodal sets of random functions. The random functions and the specific aspect of their nodal set that we study fall into two broad categories: nodal component count of Gaussian Laplace eigenfunctions and volume of the nodal set of centered stationary Gaussian processes (SGPs) on $\mathbb{R}^d$, $d \geq 1$.
Gaussian Laplace eigenfunctions: Nazarov–Sodin pioneered the study of nodal component count for Gaussian Laplace eigenfunctions; they investigated this for random spherical harmonics (RSH) on the two-dimensional sphere $S^2$ and established exponential concentration for their nodal component count. An analogous result for arithmetic random waves (ARW) on the $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$, was established soon after by Rozenshein.
We establish concentration results for the nodal component count in the following three instances: monochromatic random waves (MRW) on growing Euclidean balls in $\R^2$; RSH and ARW, on geodesic balls whose radius is slightly larger than the Planck scale, in $S^2$ and $\mathbb{T}^2$ respectively. While the works of Nazarov–Sodin heavily inspire our results and their proofs, some effort and a subtler treatment are required to adapt and execute their ideas in our situation.
Stationary Gaussian processes: The study of the volume of nodal sets of centered SGPs on $\mathbb{R}^d$ is classical; starting with Kac and Rice’s works, several studies were devoted to understanding the nodal volume of Gaussian processes. When $d = 1$, under somewhat strong regularity assumptions on the spectral measure, the following results were established for the zero count on growing intervals: variance asymptotics, central limit theorem and exponential concentration.
For smooth centered SGPs on $\mathbb{R}^d$, we study the unlikely event of overcrowding of the nodal set in a region; this is the event that the volume of the nodal set in a region is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for probability of the overcrowding event. We first obtain overcrowding estimates for the zero count of SGPs on $\mathbb{R}$, we then deal with the overcrowding question in higher dimensions in the following way. Crofton’s formula gives the nodal set’s volume in terms of the number of intersections of the nodal set with all lines in $\mathbb{R}^d$. We discretize this formula to get a more workable version of it and, in a sense, reduce this higher dimensional overcrowding problem to the one-dimensional case.
Several critical physical properties of a material are controlled by its geometric construction. Therefore, analyzing the effect of a material’s geometric structure can help to improve some of its beneficial physical properties and reduce unwanted behavior. This leads to the study of boundary value problems in complex domains such as perforated domain, thin domain, junctions of the thin domain of different configuration, domain with rapidly oscillating boundary, networks domain, etc.
In this thesis colloquium, we will discuss various homogenization problems posed on high oscillating domains. We discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.), where the oscillatory part is made of two materials with high contrasting conductivities. Thus the low contrast material acts as near insulation in-between the conducting materials. Mathematically this leads to the study of degenerate elliptic PDE at the limiting scale. We also briefly explain another interesting article (see, ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.), where the oscillations are on the curved interface with general cost functional. Due to time constraints, we may not discuss other chapters of the thesis.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its construction as it is the main tool in our analysis.
The second part of the talk will be homogenizing optimal control problems subject to the considered PDEs. The interesting result is the difference in the limit behavior of the optimal control problem, which depends on the control’s action, whether it is on the conductive part or insulating part. In both cases, we derive the two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete-scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We do obtain the homogenized equation for the state, but the two-scale separation of the cost functional remains as an open question.
Non-malleable codes (NMCs) are coding schemes that help in protecting crypto-systems under tampering attacks, where the adversary tampers the device storing the secret and observes additional input-output behavior on the crypto-system. NMCs give a guarantee that such adversarial tampering of the encoding of the secret will lead to a tampered secret, which is either same as the original or completely independent of it, thus giving no additional information to the adversary. Leakage resilient secret sharing schemes help a party, called a dealer, to share his secret message amongst $n$ parties in such a way that any $t$ of these parties can combine their shares to recover the secret, but the secret remains hidden from an adversary corrupting $< t$ parties to get their complete shares and additionally getting some bounded bits of leakage from the shares of the remaining parties.
For both these primitives, whether you store the non-malleable encoding of a message on some tamper-prone system or the parties store shares of the secret on a leakage-prone system, it is important to build schemes that output codewords/shares that are of optimal length and do not introduce too much redundancy into the codewords/shares. This is, in particular, captured by the rate of the schemes, which is the ratio of the message length to the codeword length/largest share length. The research goal of the thesis is to improve the state of art on rates of these schemes and get near-optimal/optimal rates.
In this talk, I will specifically focus on leakage resilient secret sharing schemes, describe the leakage model, and take you through the state of the art on their rates. Finally, I will present a recent construction of an optimal (constant) rate, leakage resilient secret sharing scheme in the so-called “joint and adaptive leakage model” where leakage queries can be made adaptively and jointly on multiple shares.
The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product
theorem [Crelle 1911]
, which implies that absolutely monotonic
functions (i.e., power series with nonnegative coefficients) preserve
positivity on matrices of all dimensions. A famous result of Schoenberg
and of Rudin [Duke Math. J. 1942, 1959]
shows the converse: there are
no other such functions.
Motivated by modern applications, Guillot and Rajaratnam
[Trans. Amer. Math. Soc. 2015]
classified the entrywise positivity
preservers in all dimensions, which act only on the off-diagonal entries.
These two results are at “opposite ends”, and in both cases the preservers
have to be absolutely monotonic.
The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
The talk will begin by discussing connections between metric geometry and positivity, also via positive definite functions. Following this, we present Schoenberg’s motivations in studying entrywise positivity preservers, followed by classical variants for matrices with entries in other real and complex domains. Then we shall see the result due to Guillot and Rajaratnam on preservers acting only on the off-diagonal entries, touching upon the modern motivation behind it. This is followed by its generalization in the thesis. Finally, we present the (remaining) main results in the thesis, and conclude with some of the proofs.
The study of the optimal control problems governed by partial differential equations (PDEs) have been a significant research area in the applied mathematics and its allied areas. The optimal control problem consists of finding a control variable that minimizes a cost functional subject to a PDE. In this talk, I will present finite element analysis of Dirichlet boundary optimal control problems governed by certain PDEs. This talk will be divided into three parts.
In the first part, we study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Poisson equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and co-state variables. We propose a finite element-based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. We present the analysis for $L^2$ cost functional, but this analysis can also be extended to the gradient cost functional problem. A priori error estimates of optimal order in the energy norm are derived up to the regularity of the solution.
In the second part, we discuss the Dirichlet boundary optimal control problem governed by the Stokes equation. We develop a finite element discretization by using $\mathbf{P}_1$ elements (in the fine mesh) for the velocity and control variable and $P_0$ elements (in the coarse mesh) for the pressure variable. We present a new a posteriori error estimator for the control error. This estimator generalizes the standard residual type estimator of the unconstrained Dirichlet boundary control problems by adding terms at the contact boundary that address the non-linearity. We sketch out the proof of the estimator’s reliability and efficiency.
As a continuation of the first part, we extend our ideas to the linear parabolic equation in the third part of this presentation. The space discretization of the state and co-state variables is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. We use $H^1$-conforming 3D finite elements for the control variable. We present a sketch to demonstrate the existence and uniqueness of the solution; and the error estimates of state, adjoint state, and control.
In this talk, we will see an interplay between hermitian metrics and singular Riemann surface foliations. It will be divided into three parts. The first part of the talk is about the study of curvature properties of complete Kahler metrics on non-pseudoconvex domains. Examples of such metrics were constructed by Grauert in 1956 who showed that it is possible to construct complete Kahler metrics on the complement of complex analytic sets in a domain of holomorphy. In particular, he gave an explicit example of a complete Kahler metric (the Grauert metric) on $\mathbb{C}^n \setminus {0}$. We will confine ourselves to the study of such complete Kahler metrics. We will make some observations about the holomorphic sectional curvature of such metrics in two prototype cases, namely (i) $\mathbb{C}^n \setminus {0}$, $n>1$, and (ii) $(B^N)\setminus A$, where $A$ is an affine subspace. We will also study complete Kahler metrics using Grauert’s construction on the complement of a principal divisor in a domain of holomorphy and show that there is an intrinsic continuity in the construction of this metric, i.e., we can choose this metric in a continuous fashion if the corresponding principal divisors vary continuously in an appropriate topology.
The second part of the talk deals with Verjovsky’s modulus of uniformization that arises in the study of the leaf-wise Poincare metric on a hyperbolic singular Riemann surface lamination. This is a function defined away from the singular locus. One viewpoint is to think of this as a domain functional. Adopting this view, we will show that it varies continuously when the domains vary continuously in the Hausdorff sense. We will also give an analogue of the classical Domain Bloch constant by D. Minda for hyperbolic singular Riemann surface laminations.
In the last part of the talk, we will discuss a parametrized version of the Mattei-Moussu theorem namely, a holomorphic family of holomorphic foliations in $\mathbb{C}^2$ with an isolated singular point at the origin in the Siegel domain are holomorphically equivalent if and only if the holonomy maps of the horizontal separatrix of the corresponding foliations are holomorphically conjugate.
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances.
Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency.
Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were know n only for a specific set of utility functions. We complement the algorithmic results by proving that the fair rent division problem (under genral utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search).
Our work respectively addresses fair division of rent, cake (divisible), and discrete (indivisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among $n$ agents can be efficiently computed using only the valuations of $n-1$ agents. The $n$th (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.
In this talk, we shall focus on certain dynamically interesting measures arising in holomorphic dynamics beyond the classical framework of maps. We will consider measures associated with semigroups and, more generally, with meromorphic correspondences, that are invariant in a specific sense. Our results are of two different flavours. The first type of results deal with potential-theoretic properties of the measures associated with certain polynomial semigroups, while the second type of results are about recurrence phenomena in the dynamics of meromorphic correspondences. The unifying features of these results are the use of the formalism of correspondences in their proofs, and the fact that the measures that we consider are those that describe the asymptotic distribution of the iterated inverse images of a generic point.
The first class of results involve giving a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh–Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which we can describe explicitly given a choice of a set of generators. In particular, we generalize the classical result of Brolin to certain finitely generated polynomial semigroups. To do so, we establish the continuity of the logarithmic potential for the Dinh–Sibony measure, whose proof will be sketched. If time permits, we will discuss bounds on the capacity and diameter of the Julia sets of such semigroups, for which we use the $F$-functional of Mhaskar and Saff.
The second class of results involves meromorphic correspondences. These are, loosely speaking, multi-valued analogues of meromorphic maps. We shall present an analogue of the Poincare recurrence theorem for meromorphic correspondences with respect to the measures alluded to above. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. If time permits, we shall also discuss a result on the invariance properties of the supports of the measures mentioned.
This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First, we have focused on the geometry of these two domains. The symmetrized bidisc, a non-homogeneous domain, is partitioned into a collection of orbits under the action of its automorphism group. We investigate the properties of these orbits and pick out some necessary properties so that the symmetrized bidisc can be characterized up to biholomorphic equivalence. As a consequence, among other things, we have given a new defining condition of the symmetrized bidisc and we have found that a biholomorphic copy of the symmetrized bidisc defined by E. Cartan. This work on the symmetrized bidisc also helps us to develop a characterization of the bidisc. Being a homogeneous domain, the bidisc’s automorphism group does not reveal much about its geometry. Using the ideas from the work on the symmetrized bidisc, we have identified a subgroup of the automorphism group of the bidisc and observed the corresponding orbits under the action of this subgroup. We have identified some properties of these orbits which are sufficient to characterize the bidisc up to biholomorphic equivalence.
Turning to operator theoretic work on the domains, we have focused mainly on the Schur Agler class class on the bidisc and the symmetrized bidisc. Each element of the Schur Agler class on these domains has a nice representation in terms of a unitary operator, called the realization formula. We have generalized the ideas developed in the context of the bidisc and the symmetrized bidisc and applied it to the Nevanlinna problem and the interpolating sequences. It turns out, our generalization works for a number of domains, such as annulus and multiply connected domains, not only the bidisc and the symmetrized bidisc.
In this talk, we focus on random graphs with a given degree sequence. In the first part, we look at uniformly chosen trees from the set of trees with a given child sequence. A non-negative sequence of integers $(c_1,c_2,\dots,c_l)$ with sum $l-1$ is a child sequence for a rooted tree $t$ on $l$ nodes, if for some ordering $v_1,v_2,\dots,v_l$ of the nodes of $t$, $v_i$ has exactly $c_i$ many children in $t$. We consider for each $n$, a child sequence $\mathbf{c}^{(n)}$, with sum $n-1$, and let $\mathbf{t}_n$ be the random tree having the uniform distribution on the set of all plane trees with $n$ vertices, which has $\mathbf{c}^{(n)}$ as their child sequence. Under the assumption that a finite number of vertices of $\mathbf{t}_n$ has large degrees, we show that the scaling limit of $\mathbf{t}_n$ is the Inhomogeneous Continuum Random Tree (ICRT), in the Gromov-Hausdorff topology. This generalizes a result of Broutin and Marckert from 2012, where they show the scaling limit to be the Brownian Continuum Random Tree (BCRT), under the assumption that no vertex in $\mathbf{t}_n$ has large degree.
In the second part, we look at vacant sets left by random walks on random graphs via simulations. Cerný, Teixeira and Windisch (2011) proved that for random $d$-regular graphs, there is a number $u_*$, such that if a random walk is run up to time $un$ with $u<u_*$, $n$ being the total number of nodes in the graph, a giant component of linear size, in the subgraph spanned by the nodes yet unvisited by the random walk, emerges. Whereas, if the random walk is tun up to time un with $u>u_*$, the size of the largest component, of the subgraph spanned by nodes yet unvisited by the walk, is $\text{o}(n)$. With the help of simulations, we try to look for such a phase transition for supercritical configuration models, with heavy-tailed degrees.
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances.
Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency.
Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were know n only for a specific set of utility functions. We complement the algorithmic results by proving that the fair rent division problem (under genral utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search).
Our work respectively addresses fair division of rent, cake (divisible), and discrete (indivisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among $n$ agents can be efficiently computed using only the valuations of $n-1$ agents. The $n$th (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.
This thesis is devoted to the study of nodal sets of random functions. The random functions and the specific aspect of their nodal set that we study fall into two broad categories: nodal component count of Gaussian Laplace eigenfunctions and volume of the nodal set of centered stationary Gaussian processes (SGPs) on $\mathbb{R}^d$, $d \geq 1$.
Gaussian Laplace eigenfunctions: Nazarov–Sodin pioneered the study of nodal component count for Gaussian Laplace eigenfunctions; they investigated this for random spherical harmonics (RSH) on the two-dimensional sphere $S^2$ and established exponential concentration for their nodal component count. An analogous result for arithmetic random waves (ARW) on the $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$, was established soon after by Rozenshein.
We establish concentration results for the nodal component count in the following three instances: monochromatic random waves (MRW) on growing Euclidean balls in $\R^2$; RSH and ARW, on geodesic balls whose radius is slightly larger than the Planck scale, in $S^2$ and $\mathbb{T}^2$ respectively. While the works of Nazarov–Sodin heavily inspire our results and their proofs, some effort and a subtler treatment are required to adapt and execute their ideas in our situation.
Stationary Gaussian processes: The study of the volume of nodal sets of centered SGPs on $\mathbb{R}^d$ is classical; starting with Kac and Rice’s works, several studies were devoted to understanding the nodal volume of Gaussian processes. When $d = 1$, under somewhat strong regularity assumptions on the spectral measure, the following results were established for the zero count on growing intervals: variance asymptotics, central limit theorem and exponential concentration.
For smooth centered SGPs on $\mathbb{R}^d$, we study the unlikely event of overcrowding of the nodal set in a region; this is the event that the volume of the nodal set in a region is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for probability of the overcrowding event. We first obtain overcrowding estimates for the zero count of SGPs on $\mathbb{R}$, we then deal with the overcrowding question in higher dimensions in the following way. Crofton’s formula gives the nodal set’s volume in terms of the number of intersections of the nodal set with all lines in $\mathbb{R}^d$. We discretize this formula to get a more workable version of it and, in a sense, reduce this higher dimensional overcrowding problem to the one-dimensional case.
For a commuting $d$-tuple of operators $\boldsymbol T=(T_1, \ldots , T_d)$ defined on a complex separable Hilbert space $\mathcal{H}$, let $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$ be the $d \times d$ block operator $\big (\big ( \big [ T_j^*,T_i] \big )\big )$ of commutators: $[T_j^*,T_i] := T_j^* T_i - T_i T_j^*$. We define an operator on the Hilbert space $\mathcal{H}$, to be designated the determinant operator, corresponding to the block operator $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$. We show that if the $d$-tuple is cyclic, the determinant operator is positive and the compression of a fixed set of words in $T_j^*$ and $T_i$ – to a nested sequence of finite dimensional subspaces increasing to $\mathcal{H}$ – does not grow very rapidly, then the trace of the determinant of the operator $\big (\big ( \big [ T_j^*,T_i] \big )\big )$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a certain small class of commuting $d$-tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
Let $\Omega$ be an irreducible classical bounded symmetric domain of rank $r$ in $\mathbb{C}^d$. Let $\mathbb{K}$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $\mathbb{K}$ consisting of linear transformations acts naturally on any $d$-tuple $\mathbf{T}$ of commuting bounded linear operators by the rule: \begin{equation} k \cdot \mathbf{T} = \big( k_1(T_1, \dots, T_d), \dots, k_d(T_1, \dots, T_d) \big), \ k \in \mathbb{K}, \end{equation} where $k_1(\mathbf{z}), \dots, k_d(\mathbf{z})$ are linear polynomials. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\mathbf{T}$ is $\mathbb{K}$-homogeneous. We realize a certain class of $\mathbb{K}$-homogeneous $d$-tuples $\mathbf{T}$ as a $d$-tuple of multiplication by the coordinate functions $z_1, \dots, z_d$ on a reproducing kernel Hilbert space $\mathcal{H}_K$. (The Hilbert space $\mathcal{H}_K$ consisting of holomorphic functions defined on $\Omega$, with $K$ as reproducing kernel.) Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(\Omega)$. For an irreducible bounded symmetric domain $\Omega$ of rank 2, an explicit description of the operator $\sum_{i=1}^d T_i^* T_i$ is given. Based on this formula, a conjecture giving the form of this operator in any rank $r \geq 1$ was made. This conjecture was recently verified by H. Upmeier.
This thesis studies the mixing times for three random walk models. Specifically these are the random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by 3-cycles of the form $(i, n − 1, n)$ and $(i, n, n − 1)$. We call this the transpose top-2 with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $(n − 3/2)\log (n)$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group Bn generated by the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 \leq i \leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n \log (n)$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 \leq i < n$ also has a cutoff at $(n − 1/2)\log (n)$.
The random walk on the complete monomial group: Let $G_1 \subseteq \cdots \subseteq G_n \subseteq \cdots$ be a sequence of finite groups with $|G_1| > 2$. We study the properties of a random walk on the complete monomial group $G_n \wr S_n$ (wreath product of $G_n$ with $S_n$) generated by the elements of the form $(e,\dots, e, g; id)$ and $(e,\dots, e, g^{−1}, e,\dots, e, g; (i, n))$ for $g \in G_n$, $1 \leq i < n$. We call this the warp-transpose top with random shuffle on $G_n \wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n \log (n) + (1/2) n \log(|G_n| − 1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n \log (n)$ if $|G_n| = o( n^\delta )$ for all $\delta > 0$.
The first part of this talk deals with identifying and proving the scaling limit of a uniform tree with given child sequence. A non-negative sequence of integers $\mathbf{c}=(c_1, c_2, …, c_l)$ with sum $l-1$ is called a child sequence for a rooted tree $t$ on $l$ nodes, if for some ordering $v_1, v_2,…, v_l$ of the nodes, $v_i$ has exactly $c_i$ many children. Consider for each $n$, a child sequence $\mathbf{c}^n$ with sum $n-1$, and let $\mathbf{t}_n$ be the plane tree with $n$ nodes, which is uniformly distributed over the set of all plane trees having $\mathbf{c}^n$ as their child sequence. Broutin and Marckert (2012) prove that under certain assumptions on $\mathbf{c}^n$, the scaling limit of $\mathbf{t}_n$, suitably normalized, is the Brownian Continuum Random Tree (BCRT). We consider a more general setting, where a finite number of vertices of $\mathbf{t}_n$ are allowed to have large degrees. We prove that the scaling limit of $\mathbf{t}_n$ in this regime is the Inhomogeneous Continuum Random Tree (ICRT), in the Gromov-Hausdorff sense.
In the second part, we look at vacant sets left by random walks on random graphs via simulations. Cerny, Teixeira and Windisch (2011) proved that for random $d$-regular graphs, there is a number $u_{\star}$, such that if a random walk is run up to time $un$ with $u<u_{\star}$, $n$ being the total number of nodes in the graph, a giant component of size $\text{O}(n)$ of the subgraph spanned by the vacant nodes i.e. the nodes that are not visited by the random walk, is seen. Whereas if the random walk is run up to time $un$ with $u>u_{\star}$, the size of the largest component of the subgraph spanned by the vacant nodes becomes $\text{o}(n)$. With the help of simulations, we try to investigate whether there is such a phenomenon for supercritical configuration models with heavy-tailed degrees.
This thesis studies the mixing times for three random walk models. Specifically, these are the random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $\left(n-\frac{3}{2}\right)\log n$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n\log n$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i< n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.
The random walk on the complete monomial group: Let $G_1\subseteq\cdots\subseteq G_n \subseteq\cdots $ be a sequence of finite groups with $|G_1|>2$. We study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(e,\dots,e,g;$id$)$ and $(e,\dots,e,g^{-1},e,\dots,e,g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n+\frac{1}{2}n\log (|G_n|-1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n\log n$ if $|G_n|=o(n^{\delta})$ for all $\delta>0$.
Let $\boldsymbol T=(T_1, \ldots , T_d)$ be $d$ -tuple of commuting operators on a Hilbert space $\mathcal{H}$. Assume that $\boldsymbol T$ is hyponormal, that is, $\big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ] :=\big (\big ( \big [ T_j^*,T_i] \big )\big )$ acting on the $d$-fold direct sum of the Hilbert space $\mathcal{H}$ is non-negative definite. The commutator $[T_j^*,T_i]$, $1\leq i,j \leq d$, of a finitely cyclic and hyponormal $d$-tuple is not necessarily compact and therefore the question of finding trace inequalities for such a $d$-tuple does not arise. A generalization of the Berger-Shaw theorem for commuting tuple $\boldsymbol T$ of hyponormal operators was obtained by Douglas and Yan decades ago. We discuss several examples of this generalization in an attempt to understand if the crucial hypothesis{\rm in their theorem requiring the Krull dimension of the Hilbert module over the polynomial ring defined by the map $p\to p(\boldsymbol T)$, $p\in \mathbb C[\boldsymbol z]$, is optimal or not. Indeed, we find examples $\boldsymbol T$ to show that there a large class operators for which $\text{trace}[T_j^*,T_i]$, $1\leq j,i \leq d$, is finite but the $d$-tuple is not finitely polynomially cyclic, which is one of the hypothesis of the Douglas-Yan theorem. We also introduce the weaker notion of “projectively hyponormal operators” and show that the Douglas-Yan theorem remains valid even under this weaker hypothesis. However, one might look for a function of $ \big [\big [ \boldsymbol T^*, \boldsymbol T \big ] \big ]$ which may be in trace class. For this, we define an operator valued determinant of a $d\times d$-block operator $\boldsymbol B := \big (\big ( B_{i j} \big ) \big )$ by the formula
\begin{equation} \text{dEt}\big (\boldsymbol{B}\big ):=\sum_{\sigma, \tau \in \mathfrak S_d} \text{sgn}(\sigma)B_{\tau(1),\sigma(\tau(1))}B_{\tau(2),\sigma(\tau(2))},\ldots, B_{\tau(d),\sigma(\tau(d))}. \end{equation}
It is then natural to investigate the properties of the operator
$\mbox{dEt}\big (\big [\big [ \boldsymbol T^*, \boldsymbol T \big ]\big ] \big ),$
in this case, $B_{i j} = [T_j^*,T_i]$.
Indeed, we show that the operator dEt equals the generalized commutator
$\text{GC} \big (\boldsymbol T^*, \boldsymbol T \big )$ introduced earlier by
Helton and Howe. Among other things, we find a trace inequality for the operator
$\mbox{dEt}\big (\big [\big [ \boldsymbol T^*, \boldsymbol T \big ]\big] \big ),$
after imposing certain growth and cyclicity condition on the operator $\boldsymbol T$, namely,
\begin{equation} \text{trace} \big( {\rm dEt} \big( [[ \boldsymbol{T}^*, \boldsymbol{T} ]] \big) \big) \leq m \vartheta d! \prod_{i=1}^{d} |T_i|^2 \end{equation}
for some $\vartheta \geq 1.$ We give explicit examples illustrating the abstract inequality.
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$-filtration.
The systematic study of determinantal processes began with the work of Macchi (1975), and since then it has appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths), and physics (fermions, repulsion arising in quantum physics). The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a separable Hilbert space. Let $H$ and K be two finite-dimensional subspaces of a Hilbert space, and $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons (2003) showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalues of Wishart matrix ensembles $W(N, N)$ and $W(N-1, N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M, N)$ has the same distribution as the directed last-passage time $G(M, N)$ on $\mathbb{Z}^2$ with the i.i.d. exponential weight. This was recently used by Basu and Ganguly to obtain stochastic domination between $G(N, N)$ and $G(N-1, N+1)$. Similar connections are also known between the largest eigenvalue of the Meixner ensemble and the directed last-passage time on $\mathbb{Z}^2$ with the i.i.d. geometric weight. We prove another stochastic domination result, which combined with Lyons’ result, gives the stochastic domination between the eigenvalue processes of Meixner ensembles $M(N, N)$ and $M(N-1, N+1)$.
In 1976, E.M. Stein proved $L^p$ bounds for spherical maximal function on Euclidean space. The lacunary case was dealt on later by C.P. Calderon in 1979. In a recent paper, M. Lacey has proved sparse bound for these functions and $L^p$ bounds will follow immediately as a result.
In this talk, we will look at various maximal functions corresponding to spherical averages and find sparse bounds for those functions. We will also observe some weighted and unweighted estimates that will follow as a consequences.
First, we will show sparse bound for lacunary spherical maximal function on Heisenberg group . Next we move on to full spherical maximal function. Then we study lacunary maximal function corresponding to the spherical average on product of Heisenberg groups. Finally, we will revisit generalized spherical averages on Euclidean space and prove sparse bounds for the related maximal functions.
This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about a problem on certain holomorphic differential operators on the space of Jacobi forms.
In the first part we will briefly discuss how the statistics of signs of newforms determine them (work of Matomaki-Soundararajan-Kowalski) and then introduce certain ‘Linnik-type’ problems (the original problem was concerning the size of the smallest prime in an arithmetic progression in terms of the modulus) which ask for the size of the first negative eigenvalue (in terms of the analytic conductor) of various types modular forms, which has seen a lot of recent interest. Also specifically we will discuss the problem in the context of Yoshida lifts (a certain subspace of the Siegel modular forms), where in the thesis, we have improved upon the previously known result on this topic significantly. We will prove that the smallest $n$ with $\lambda(n)<0$ satisfy $n < Q_{F}^{1/2-2\theta+\epsilon}$, where $Q_{F}$ is the analytic conductor of a Yoshida lift $F$ and $0<\theta <1/4$ is some constant. The crucial point is establishing a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level.
We will focus on a similar question concerning the first negative Fourier coefficient of a Hilbert newform. If ${C(\mathfrak{m})}_{\mathfrak{m}}$ denotes the Fourier coefficients of a Hilbert newform $f$, then we show that the smallest among the norms of ideals $\mathfrak{m}$ such that $ C(\mathfrak{m})<0$, is bounded by $Q_{f}^{9/20+\epsilon}$ when the weight vector of $f$ is even and $Q_{f}^{1/2+\epsilon}$ otherwise. This improves the previously known result on this problem significantly. Here we would show how to use certain ‘good’ Hecke relations among the eigenvalues and some standard tools from analytic number theory to achieve our goal.
Finally we would talk about the statistical distribution of the signs of the Fourier coefficients of a Hilbert newform and essentially prove that asymptotically, half of them are positive and half negative. This was a breakthrough result of Matomaki-Radziwill for elliptic modular forms, and our results are inspired by those. The proof hinges on establishing some of their machinery of averages multiplicative functions to the number field setting.
In the second part of the talk we will introduce Jacobi forms and certain differential operators indexed by $\{D_{v}\}_{0}^{2m}$ that maps the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the direct sum of the differential operators $D_{v}$ for $v={1,2,…,2m}$ maps $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. Inspired by certain conjectures of Hashimoto on theta series, S. Bocherer raised the question whether any of the differential operators be removed from that map while preserving the injectivity. In the case of even weights S. Das and B. Ramakrishnan show that it is possible to remove the last operator. In the talk we will discuss the case of the odd weights and prove a similar result. The crucial step (and the main difference from the even weight case) in the proof is to establish that a certain tuple of congruent theta series is a vector valued modular form and finding the automorphy of the Wronskian of this tuple of theta series.
This work has two parts. The first part contains the study of phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogeneous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y\mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid \cdot \mid$ is the Euclidean norm. In the inhomogeneous version of the model, points of $\mathcal{P}_{\lambda}$ is endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability
\begin{equation} \left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right) \end{equation}
for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.
In the second part we consider an inhomogeneous random connection model on a $d$ -dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of isolated vertices converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.
Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let $e$ be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for $p=2$. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{r})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are NOT isomorphic for $r \geq 2e+2$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{r}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{r}))]$ are NOT isomorphic for $r\geq4$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s well-known big-constant/small-constant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms, having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$ have the same eigenvalues of the $p$-th Hecke operator $T_p$ for almost all primes $p$, then $f_1=f_2$.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$.
We expect similar applications. We also discuss few results on the square-free Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, \mathbb{Z})$ then we consider its Saito–Kurokawa lift $F_g$ and a certain ‘restricted’ $L^2$-norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau’))$ ($(\tau,z,z,\tau’)$ in Siegel’s upper half-plane of degree 2) to $H \times H$ is its restriction to $z=0$, which we denote by $F|_{z=0}$. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$-norms of such pullbacks to central values of $L$-functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$ $L$-functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$ with the norm of its pullback provides a measure of concentration of $F_g$ along $z=0$. We recall certain conjectures pertaining to the size of the’mass’. We use the amplification method to improve the currently known bound for $N(F_g)$.
We study risk-sensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.
First, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddle-point equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.
Subsequently, we study risk-sensitive ergodic control problem for controlled reflecting diffusion processes in the non-negative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.
Then we study risk-sensitive zero-sum/nonzero-sum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the non-negative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding Hamilton-Jacobi-Isaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.
Finally, we study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra. We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$ admits a $p$-filtration.
This work has two parts. The first part contains the study of phase transition and percolation at criticality for three planar random graph models, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y \mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid\cdot\mid$ is the Euclidean norm. In the inhomogenous version of the model, points of $\mathcal{P}_{\lambda}$ are endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability
\begin{equation} \left(1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)\right) \end{equation}
for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.
In the second part we consider an inhomogeneous random connection model on a $d$-dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of vertices of degree $j$ converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.
Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let $e$ be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.
In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.
Construction: For $r\geq 1$ the construction of irreducible representations of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work for $p=2$. In this case we give a construction of all irreducible representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$ and for $r \geq 4e+2$ with Char$(O)=0$.
Representation Growth: For a rigid group $G$, it is well known that the abscissa of convergence $\alpha(G)$ of the representation zeta function of $G$ gives precise information about its representation growth. Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$ for either $p > 2$ or Char$(O)=0$. We complete these results by proving that $\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.
Group Algebras: The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and $\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel, for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$ and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{2m})]$ and $\mathbb{C}[SL_2(F_q[t]/(t^{2m}))]$ are NOT isomorphic for $m > e$. As a corollary we obtain that the group algebras $\mathbb{C}[SL_2(\mathbb{Z}/2^{2m}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{2m}))]$ are NOT isomorphic for $m>1$.
Branching Laws: We give a description of the branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for $p=2$. In this case, we again show that many results for $p=2$ are quite different from the case $p > 2$.
We prove Hardy’s inequalities for the fractional power of Grushin operator $\mathcal{G}$ which is chased via two different approaches. In the first approach, we first prove Hardy’s inequality for the generalized sublaplacian. We first find Cowling–Haagerup type of formula for the fractional sublaplacian and then using the modified heat kernel, we find integral representations of the fractional generalized sublaplacian. Then we derive Hardy’s inequality for generalized sublaplacian. Finally using the spherical harmonics, applying Hardy’s inequality for individual components, we derive Hardy’s inequality for Grushin operator. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\mathbb{R}^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\mathbb{R}^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\mathcal{G}_s f$ in $L^p(\mathbb{R}^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy–Littlewood–Sobolev inequality for the Grushin operator.
Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\mathbb{R}^n)$. We find a relation between the boundedness of sublaplacian multipliers $m(\tilde{\mathcal{L}})$ on polarised Heisenberg group $\mathbb{H}^n_{pol}$ and the boundedness of Hermite multipliers $m(\mathcal{H})$ on modulation spaces $M^{p,q}(\mathbb{R}^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe those conditions on multipliers are more than required restrictive. We improve the results for the special case $p=q$ of the modulation spaces $M^{p,q}(\mathbb{R}^n)$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}(\mathbb{R}^n)$ and the boundedness of Fourier multipliers on torus $\mathbb{T}^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr"odinger equation related to Hermite on modulation spaces.
This talk would have two parts. In the first part, we will discuss some topics which can be classified as ‘Linnik-type’ problems (the motivation being his original question about locating the first prime in an arithmetic progression) in the context of Hecke eigenvalues of modular forms on various groups, and then talk about the distribution of their signs. In the second part we will discuss differential operators on modular forms, and then talk about their applications to questions about Jacobi forms.
It is well-known that the sequence of Hecke eigenvalues mentioned above are often real, and has infinitely many sign changes. First part of the talk would discuss the problem of estimating the location of the first such sign change in the context of Hecke eigenvalues of Yoshida lifts (a certain subspace of the Siegel modular forms) and Fourier coefficients of Hilbert modular forms. We show how to improve the previously best known results on this topic significantly.
The crucial inputs behind these would be to establish a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level for treating Yoshida lifts; and exploiting Hecke relations along with generalising related results due to K. Soundararajan, K. Matomaki et al. for the case of Hilbert modular forms. In both cases we measure the location of the eigenvalues or Fourier coefficients in terms of an analytic object called the ‘analytic conductor’, which would be introduced during the talk. Moreover in the case of Hilbert modular forms, we will also discuss quantitative results about distribution of positive and negative Hecke eigenvalues. The proof depends on establishing a certain result on a particular types of multiplicative functions on the set of integral ideals of a totally real number field.
In the second part of the talk, we will introduce the space of Jacobi forms and certain results due to J. Kramer and, briefly, a conjecture due to Hashimoto on theta series attached to quaternion algebras to motivate the results to follow. The (partial) solution of this conjecture by Arakawa and B"ocherer transfers the question to one about differential operators on Jacobi forms, and we would report on previously known and new results on this topic.
The heart of the second part of the talk would focus on the question about the differential operators on Jacobi forms. It is well known that certain differential operators ${D_{v}}_{0}^{2m}$ map the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the sum of the differential operators $D_{v}$ for $v={1,2,…2m}$ map $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. The question alluded to above boils down to investigate whether one can omit certain differential operators from the list above, maintaining the injective property. In this regard, we would discuss results of Arakawa–B"ocherer, Das–Ramakrishnan, and finally our results. The main point would be to establish automorphy of the Wronskian of a certain tuple of congruent theta series of weight 3/2.
We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:
Estimates for holomorphic images of subsets in convex domains: Consider the following problem: given domains $\Omega_1\varsubsetneq \mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in \Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that $f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that are at a distance of at least $r$ from the complement of $\Omega_1$. This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 = \Omega_2$ being the unit disk) which gives a sharp lower bound of the latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$ are convex domains. In doing so, we make crucial use of the Kobayashi pseudodistance.
Upper bounds for the Kobayashi metric: We provide new upper bounds for the Kobayashi metric on bounded convex domains in $\mathbb{C}^n$. This bears relation to Graham’s well-known big-constant/small-constant bounds from above and below on convex domains. Graham’s upper bounds are frequently not sharp. Our estimates improve these bounds.
The continuous extension of Kobayashi isometries: We provide a new result in this direction that is based on the properties of convex domains viewed as distance spaces (equipped with the Kobayashi distance). Specifically, we sharpen certain techniques introduced recently by A. Zimmer and extend a result of his to a wider class of convex domains having lower boundary regularity. In particular, all complex geodesics into any such convex domain are shown to extend continuously to the unit circle.
A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$: We introduce and study a property that we call “visibility with respect to the Kobayashi distance”, which is an analogue of the notion of uniform visibility in CAT(0) spaces. It abstracts an important and characteristic property of Gromov hyperbolic spaces. We call domains satisfying this newly-introduced property “visibility domains”. Bharali–Zimmer recently introduced a class of domains called Goldilocks domains, which are visibility domains, and proved for Goldilocks domains a wide range of properties. We show that visibility domains form a proper superclass of the Goldilocks domains. We do so by constructing a family of domains that are visibility domains but not Goldilocks domains. We also show that visibility domains enjoy many of the properties shown to hold for Goldilocks domains.
Wolff–Denjoy-type theorems for visibility domains: To emphasise the point that many of the results shown to hold for Goldilocks domains can actually be extended to visibility domains, we prove two Wolff–Denjoy-type theorems for taut visibility domains, with one of them being a generalization of a similar result for Goldilocks domains. We also provide a corollary to one of these results to demonstrate the sheer diversity of domains to which the Wolff–Denjoy phenomenon extends.
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ Fourier coefficients. These type of results are known as ‘recognition results’ and have been a useful theme in the theory of modular forms,having lots of applications. As an example we would recall the Sturm’s bound (which applies quite generally to a wide class of modular forms), which says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. As another example we would mention the classical multiplicity-one result for elliptic new forms of integral weight, which says that if two such forms $f_1,f_2$
have the same eigenvalues of the $p$-th Hecke operator $T_p$
for almost all primes $p$
, then $f_1=f_2$
.
The heart of the first part of the talk would concentrate on Hermitian cusp forms of degree $2$. These objects have a Fourier expansion indexed by certain matrices of size $2$ over an imaginary quadratic field. We show that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square–free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which will also be discussed. This result was established by A. Saha in the context of Siegel modular forms – and played a crucial role (among others) in the automorphic transfer from $GSp(4)$ to $GL(4)$. We expect similar applications. We also discuss few results on the square–free Fourier coefficients of elliptic cusp forms.
In the second part of the talk we introduce Saito–Kurokawa lifts: these are certain Siegel modular forms lifted from classical elliptic modular forms on the upper half plane $H$. If $g$ is such an elliptic modular form of integral weight $k$ on $SL(2, Z)$ then we consider its Saito–Kurokawa lift $F_g$
and a certain ‘restricted’ $L^2$
-norm, which we denote by $N(F_g)$ (and which we refer to as the ‘mass’), associated with it.
Pullback of a Siegel modular form $F((\tau,z,z,\tau'))$
($(\tau,z,z,\tau')$
in Siegel’s upper half-plane of degree 2) to $H \times H$
is its restriction to $z=0$
, which we denote by $F\|_{z=0}$
. Deep conjectures of Ikeda (also known as ‘conjectures on the periods of automorphic forms’) relate the $L^2$
-norms of such pullbacks to central values of $L$-functions for all degrees.
In fact, when a Siegel modular form arises as a Saito–Kurokawa lift (say $F=F_g$
), results of Ichino relate the mass of the pullbacks to the central values of certain $GL(3) \times GL(2)$
$L$
-functions. Moreover, it has been observed that comparison of the (normalized) norm of $F_g$
with the norm of its pullback provides a measure of concentration of $F_g$
along $z=0$. We recall certain conjectures pertaining to the size of the ‘mass’. We use the amplification method to improve the currently known bound for $N(F_g)$
.
An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchell, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an H-category will denote an “H-module algebra with several objects” and a co-H-category will denote an “H-comodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. We study the cohomology in such module categories. In particular, we consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants. We will develop these cohomology theories in a manner similar to the “H-finite cohomology” obtained by Guedenon and the cohomology of relative Hopf modules studied by Caenepeel and Guedenon respectively. This is one of the two thesis problems which we plan to discuss in detail.
If time permits, we will also give a brief presentation of the other thesis project. In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” ($\mathbb{F}_1$) has been developed. It is in this context that monoids became topologically and geometrically relevant objects of study. In our work, we abstract out the topological characteristics of the prime spectrum of a commutative monoid, endowed with the Zariski topology is homeomorphic to the spectrum of a ring i.e., it is a spectral space. Spectral spaces, introduced by Hochster, are widely studied in the literature. We use ideals and modules over monoids to present many such spectral spaces. We introduce closure operations on monoids and obtain natural classes of spectral spaces using finite type closure operations. In the process, various closure operations like integral, saturation, Frobenius and tight closures are introduced for monoids. We study their persistence and localization properties in detail. Next, we make a study of valuation on monoids and prove that the collection of all valuation monoids having the same group completion forms a spectral space. We also prove that the valuation spectrum of any monoid gives a spectral space. Finally, we prove that the collection of continuous valuations on a topological monoid whose topology is determined by any finitely generated ideal also gives a spectral space.
We study risk-sensitive stochastic optimal control and differential game problems. These problems arise in many applications including heavy traffic analysis of queueing networks, communication networks, and manufacturing systems.
First, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in $\mathbb{R}^{d}$. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddle-point equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationary Markov strategies.
Subsequently, we study risk-sensitive ergodic control problem for controlled reflecting diffusion processes in the non-negative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This, in turn, leads to the complete characterization of optimal controls in the space of stationary Markov controls.
Then we study risk-sensitive zero-sum/nonzero-sum stochastic differential games on the infinite horizon, where the state is a controlled reflecting diffusion in the non-negative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding Hamilton-Jacobi-Isaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies.
Finally, we study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criteria, where the state space is a controlled diffusion process in $\mathbb{R}^{d}.$ Under certain conditions, we establish the existence of a Nash equilibrium in stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.
The systematic study of determinantal processes began with the work of Macchi (1975), and since then they have appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths, measures on Young diagrams), and physics (fermions). A particularly interesting and well-known example of a discrete determinantal process is the Uniform spanning tree (UST) on finite graphs. We shall describe UST on complete graphs and complete bipartite graphs—in these cases it is possible to make explicit computations that yield some special cases of Aldous’ result on CRT.
The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a Hilbert space of functions on a given set. Let $H$ and $K$ are two finite dimensional subspaces of a Hilbert space, and let $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and also provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalue of Wishart matrix ensembles $W(N,N)$ and $W(N-1,N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M,N)$ has the same distribution as the directed last-passage time $G(M,N)$ on $Z^2$ with i.i.d. exponential weights. We, thus, obtain stochastic domination between $G(N,N)$ and $G(N-1,N+1)$ - answering a question of Riddhipratim Basu. Similar connections are also known between the largest eigenvalue of Meixner ensemble and directed last-passage time on $Z^2$ with i.i.d. geometric weights. We prove a stochastic domination result which combined with the Lyons’ result gives the stochastic domination between Meixner ensemble $M(N,N)$ and $M(N-1,N+1)$.
The main aim of this thesis is to explain the of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.
To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show the convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Caratheodory rigidity constant converges to 1 near smooth boundary points.
Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain.
We also study the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.
Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.
Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesqui-analytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are non-negative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a non-negative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.
For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n-1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the push-forward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the push-forward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued non-negative kernel $K$ is quasi-invariant, then $K^{(1,1)}$ is also a quasi-invariant kernel.
In the last twenty years, several notions of what is called the algebraic geometry over the “field with one element” has been developed. One of the simplest approaches to this is via the theory of monoid schemes. The concept of a monoid scheme itself goes back to Kato and was further developed by Deitmar and by Connes, Consani and Marcolli. The idea is to replace prime spectra of commutative rings, which are the building blocks of ordinary schemes, by prime spectra of commutative pointed monoids. In our work, we focus mostly on abstracting out the topological characteristics of the prime spectrum of a commutative pointed monoid. This helps to obtain several classes of topological spaces which are homeomorphic to the the prime spectrum of a monoid. Such spaces are widely studied and are called spectral spaces. They were introduced by M. Hochster. We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations like integral, saturation, Frobenius and tight closures on monoids. We prove that the collection of all continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.
The main aim of this thesis is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point.
To estimate the higher order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Caratheodory metric on planar domains and in the process, we show convergence of the Szego and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Caratheodory rigidity constant converges to 1 near smooth boundary points.
Next on the line is a conformal metric defined using holomorphic quadratic differentials. This was done by T. Sugawa and we will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain.
We also study the Hurwitz metric that was introduced by D. Minda. It’s construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.
Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.
Let $K$ be a bounded domain and $K:\Omega \times \Omega \to \mathbb{C}$ be a sesqui-analytic function. We show that if $\alpha,\beta>0$ be such that the functions $K^{\alpha}$ and $K^{\beta}$, defined on $\Omega\times\Omega$, are non-negative definite kernels, then the $M_m(\mathbb{C})$ valued function $K^{(\alpha,\beta)} := K^{\alpha+\beta}(\partial_i\bar{\partial}_j\log K)_{i,j=1}^m$ is also a non-negative definite kernel on $\Omega\times\Omega$. Then we find a realization of the Hilbert space $(H,K^{(\alpha,\beta)})$ determined by the kernel $K^{(\alpha, \beta)}$ in terms of the tensor product $(H, K^{\alpha})\otimes (H, K^{\beta})$.
For two reproducing kernel Hilbert modules $(H,K_1)$ and $(H,K_2)$, let $A_n, n\geq 0$, be the submodules of the Hilbert module $(H, K_1)\otimes (H, K_2)$ consisting of functions vanishing to order $n$ on the diagonal set $\Delta:= \{ (z,z):z\in \Omega \}$. Setting $S_0=A_0^\perp, S_n=A_{n-1}\ominus A_{n}, n\geq 1$, leads to a natural decomposition of $(H, K_1)\otimes (H, K_2)$ into an infinite direct sum $\oplus_{n=0}^{\infty} S_n$. A theorem of Aronszajn shows that the module $S_0$ is isometrically isomorphic to the push-forward of the module $(H,K_1K_2)$ under the map $\iota:\Omega\to \Omega\times\Omega$, where $\iota(z)=(z,z), z\in \Omega$. We prove that if $K_1=K^{\alpha}$ and $K_2=K^{\beta}$, then the module $S_1$ is isometrically isomorphic to the push-forward of the module $(H,K^{(\alpha, \beta)})$ under the map $\iota$. We also show that if a scalar valued non-negative kernel $K$ is quasi-invariant, then $K^{(1,1)}$ is also a quasi-invariant kernel.
We consider a finite version of the one-dimensional Toom model with closed boundaries. Each site is occupied either by a particle of type 0 or of type 1, where the total number of particles of type 0 and type 1 are fixed to be n_0 and n_1 respectively. We call this an (n_0,n_1)-system. The dynamics are as follows: the leftmost particle in a block can exchange its position with the leftmost particle of the block to its right.
In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)-system and (n_0, 1)-system, using an enriched two-dimensional model.
In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function $f : O \to E$, where $O$ is an open subset of the complex plane and $E$ a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.
For $\Omega_1$ a subset of the Riemann sphere, we consider the space of locally holomorphic maps of $\Omega_1$ into $E$ vanishing at infinity if infinity belongs to $\Omega_1$, denoted by $P(\Omega_1,E)$. For two complementary subsets $\Omega_1$ and $\Omega_2$ of the Riemann sphere we prove that given two locally convex topological vector spaces $E$ and $F$ in separating duality, under some general conditions, we can define a separating duality between $P(\Omega_1,E)$ and $P(\Omega_2,F)$.
Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, non-sparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such non-sparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.
We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.
Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.
We also use our VAR modelling methods to predict time delays in many-variable systems. Using simulated data from non-linear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.
We apply the above methods to the following real experimental data:
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.
In addition to the above methods, we apply non-parametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed point-process and real time-series data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.
Obtaining a sparse representation of high dimensional data is often the first step towards its further analysis. Conventional Vector Autoregressive (VAR) modelling methods applied to such data results in noisy, non-sparse solutions with a too many spurious coefficients. Computing auxiliary quantities such as the Power Spectrum, Coherence and Granger Causality (GC) from such non-sparse models is slow and gives wrong results. Thresholding the distorted values of these quantities as per some criterion, statistical or otherwise, does not alleviate the problem.
We propose two sparse Vector Autoregressive (VAR) modelling methods that work well for high dimensional time series data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments using simulated data, our methods show consistently higher accuracy compared to other contemporary methods in recovering the true sparse model. The relative absence of spurious coefficients in our models permits more accurate, stable and efficient evaluation of auxiliary quantities. Our VAR modelling methods are capable of computing Conditional Granger Causality (CGC) in datasets consisting of tens of thousands of variables with a speed and accuracy that far exceeds the capabilities of existing methods.
Using the Conditional Granger Causality computed from our models as a proxy for the weight of the edges in a network, we use community detection algorithms to simultaneously obtain both local and global functional connectivity patterns and community structures in large networks.
We also use our VAR modelling methods to predict time delays in many-variable systems. Using simulated data from non-linear delay differential equations, we compare our methods with commonly used delay prediction techniques and show that our methods yield more accurate results.
We apply the above methods to the following real experimental data:
Application to the Hela gene interaction dataset: The network obtained by applying our methods to this dataset yields results that are at least as good as those from a specialized method for analysing gene interaction. This demonstrates that our methods can be applied to any time series data for which VAR modelling is valid.
In addition to the above methods, we apply non-parametric Granger Causality analysis (originally developed by A. Nedungadi, G. Rangarajan et al) to mixed point-process and real time-series data. Extending the computations to Conditional GC and by increasing the efficiency of the original computer code, we can compute the Conditional GC spectrum in systems consisting of hundreds of variables in a relatively short period. Further, combining this with VAR modelling provides an alternate faster route to compute the significance level of each element of the GC and CGC matrices. We use these techniques to analyse mixed Spike Train and LFP data from monkey electrocorticography (ECoG) recordings during a behavioural task. Interpretation of the results of the analysis is an ongoing collaboration.
The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3-dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain. Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priori error estimates to show the optimal convergence of the method with respect to the regularity, for this we have enriched the finite element space with element-wise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem. Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results here are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution ˜uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and red-refinement of a tetrahedron. We present conclusions and possible extensions for the future works.
In this talk, we present a portion of the paper “Sur certains espaces de fonctions holomorphes.I.” by Alexandre Grothendieck. For a function $f: O \to E$, where $O$ is an open subset of the complex plane and $E$ a locally convex topological vector space, we define two notions: holomorphicity and weak derivability. We discuss some properties of the holomorphic functions and see the condition under which these two notions coincide.
For $\Omega_1$ a subset of the Riemann sphere, we consider the space of locally holomorphic maps of $\Omega_1$ into $E$ vanishing at infinity if infinity belongs to $\Omega_1$, denoted by $P(\Omega_1,E)$. For two complementary subsets $\Omega_1$ and $\Omega_2$ of the Riemann sphere we prove that given two locally convex topological vector spaces $E$ and $F$ in separating duality, under some general conditions, we can define a separating duality between $P(\Omega_1,E)$ and $P(\Omega_2,F)$.
We consider a finite version of the one-dimensional Toom model with closed boundaries. Each site is occupied either by a particle of type 0 or of type 1, where the total number of particles of type 0 and type 1 are fixed to be n_0 and n_1 respectively. We call this an (n_0, n_1)-system. The dynamics are as follows: the leftmost particle in a block can exchange its position with the leftmost particle of the block to its right.
In this thesis, we have shown the following. Firstly, we have proven a conjecture about the density of 1’s in a system with arbitrary number of 0’s and 1’s. Secondly, we have made progress on a conjecture for the nonequilibrium partition function. In particular, we have given an alternate proof of the conjecture for the (1, n_1)-system and (n_0, 1)-system, using an enriched two-dimensional model.
The classification of homogeneous scalar weighted shifts is known. Recently, Koranyi obtained a large class of inequivalent irreducible homogeneous bi-lateral 2-by-2 block shifts. We construct two distinct classes of examples not in the list of Koranyi. It is then shown that these new examples of irreducible homogeneous bi-lateral 2-by-2 block shifts, together with the ones found earlier by Koranyi, account for every unitarily inequivalent irreducible homogeneous bi-lateral 2-by-2 block shift.
In this talk we will discuss an analytic model theory for pure hyper-contractions (introduced by J. Agler) which is analogous to Sz.Nagy-Foias model theory for contractions. We then proceed to study analytic model theory for doubly commuting n-tuples of operators and analyze the structure of joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely characterize the doubly commuting quotient modules of a large class of reproducing kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the wandering subspace property of commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces $H_k$ on the unit ball in $\mathbb{C}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1},…,M_{z_n})$ can be described in terms of suitable $H_k$-inner functions. We also prove that $H_k$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogeneous polynomials as an application. Along the way we also prove a refinement of a result of Arveson on the uniqueness of the minimal dilations of pure row contractions.
The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3-dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogenous boundary data on a 2D domain.
Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. In this talk, we first present a quadratic finite element method for three dimensional ellipticobstacle problem which is optimally convergent (with respect to the regularity). We derive a priorierror estimates to show the optimal convergence of the method with respect to the regularity, forthis we have enriched the finite element space with element-wise bubble functions. Further, aposteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment. Next, we discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem.Using the localized behavior of DG methods, we derive a priori and a posteriori error estimates forlinear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions.We consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. We then proposea new and simpler residual based a posteriori error estimator for finite element approximationof the elliptic obstacle problem. The results here are two fold. Firstly, we address the influenceof the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution uh of the discrete solution uh which satisfies the exact boundaryconditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyse them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, we discuss a uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we discuss the ideas for nodes to element connectivity and red-refinement of a tetrahedron. We present conclusions and possible extensions for the future works.
Homogenization of boundary value problems posed on rough domains has paramount importance in real life problems. Materials with oscillating (rough) boundary are used in many industrial applications like micro strip radiator and nano technologies, biological systems, fractal-type constructions, etc. In this talk, we will be focusing on homogenization of optimal control problems. We will begin with homogenization of a boundary control problem on an oscillating pillar type domain. Then, we will consider a time-dependent control problem posed on a little more general domain called branched structure domain. Asymptotic analysis of this interior control problem will be explained. Next, we will present a generalized unfolding operator that we have developed for a general oscillatory domain. Using this unfolding operator, we study the homogenization of a non-linear elliptic problem on this general highly oscillatory domain. Also, we analyse an optimal control problem on a circular oscillating domain with the assistance of this operator. Finally, we consider a non-linear optimal control problem on the above mentioned general oscillatory domain and study the asymptotic behaviour.
The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:
Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.
When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:
INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$-point interpolant based on Pick’s approach and involving the study of rational inner functions.
Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$-to-$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.
INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$-point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.
The Pick–Nevanlinna interpolation problem in its fullest generality is as follows:
Given domains $D_1$, $D_2$ in complex Euclidean spaces, and a set ${(z_i,w_i): 1\leq i\leq N}\subset D_1\times D_2$, where $z_i$ are distinct and $N$ is a positive integer $\geq 2$, find necessary and sufficient conditions for the existence of a holomorphic map $F$ from $D_1$ into $D_2$ such that $F(z_i) = w_i$, $1\leq N$.
When such a map $F$ exists, we say that $F$ is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem – which we shall study in this thesis – have been of lasting interest:
INTERPOLATION FROM THE POLYDISC TO THE UNIT DISC: This is the case $D_1 = D^n$ and $D_2 = D$, where $D$ denotes the open unit disc in the complex plane and $n$ is a positive integer. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case $n=1$. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for $n\geq 2$, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur–Agler class. This is notable because when $n = 2$ the latter result completely solves the problem for the case $D_1 = D^2$, $D_2 = D$. However, Pick’s approach can also be effective for $n\geq 2$. In this thesis, we give an alternative characterization for the existence of a $3$-point interpolant based on Pick’s approach and involving the study of rational inner functions.
Cole, Lewis and Wermer lifted Sarason’s approach to uniform algebras – leading to a characterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of $(N\times N)$ matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a $D^n$-to-$D$ interpolant in terms of the positivity of a family of matrices parametrized by a class of polynomials.
INTERPOLATION FORM THE UNIT DISC TO THE SPECTRAL UNIT BALL: This is the case $D_1 = D$ and $D_2$ is the set of all $(n\times n)$ matrices with spectral radius less than $1$. The interest in this arises from problems in Control Theory. Bercovici, Fois and Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc – leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any $n$ and $N=2$. We shall present a necessary condition for the existence of a $3$-point interpolant. This we shall achieve by modifying Pick’s approach and applying the aforementioned result due to Bharali.
In this talk we will discuss an analytic model theory for pure hyper-contractions (introduced by J. Agler) which is analogous to Sz.Nagy-Foias model theory for contractions. We then proceed to study analytic model theory for doubly commuting n-tuples of operators and analyze the structure of joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely characterize the doubly commuting quotient modules of a large class of reproducing kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the wandering subspace property of commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces H_k on the unit ball in $\mathbb{C}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1},…,M_{z_n})$ can be described in terms of suitable H_k-inner functions. We also prove that H_k-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogeneous polynomials as an application. Along the way we also prove a refinement of a result of Arveson on the uniqueness of the minimal dilations of pure row contractions.
It is known that the characteristic function $\theta_T$ of a homogeneous contraction $T$ with an associated representation $\pi$ is of the form \begin{equation} \theta_T(a) = \sigma_{L}(\phi_a)^* \theta(0) \sigma_{R}(\phi_a), \end{equation}
where, $\sigma_{L}$
and $\sigma_{R}$
are projective representation of the
Mobius group Mob with a common multiplier. We give another proof
of the “product formula”.
Also, we prove that the projective representations $\sigma_L$
and
$\sigma_R$
for a class of multiplication operators, the two
representations $\sigma_{R}$
and $\sigma_{L}$
are unitarily equivalent to
certain known pair of representations $\sigma_{\lambda + 1}$
and
$\sigma_{\lambda - 1},$
respectively. These are described explicitly.
Let $G$ be either (i) the direct product of $n$-copies of the
bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic
automorphism group of the polydisc $\mathbb D^n.$
A commuting tuple of bounded operators $\mathsf{T} = (T_1, T_2,\ldots
,T_n)$
is said to be $G$-homogeneous if the joint spectrum of $\mathsf{T}$
lies in $\overline{\mathbb{D}}^n$
and $\varphi(\mathsf{T}),$
defined using
the usual functional calculus, is unitarily equivalent with $\mathsf{T}$
for all $\varphi \in G.$
We show that a commuting tuple $\mathsf{T}$
in the Cowen-Douglas class of
rank $1$ is $G$ - homogeneous if and only if it is unitarily equivalent
to the tuple of the multiplication operators on either the reproducing
kernel Hilbert space with reproducing kernel $\prod_{i = 1}^{n}
\frac{1}{(1 - z_{i}\overline{w}\_{i})^{\lambda_i}}$ or $\prod_{i = 1}^{n}
\frac{1}{(1 - z_{i}\overline{w}\_{i})^{\lambda}},$
where $\lambda,$
$\lambda_i$
, $1 \leq i \leq n,$
are positive real numbers, according as
$G$ is as in (i) or (ii).
Let $\mathsf T:=(T_1, \ldots ,T_{n-1})$
be a $G$-homogeneous $(n-1)$-tuple
of rank $1$ Cowen-Douglas class, where $G$ is the the direct product of
$n-1$-copies of the bi-holomorphic automorphism group of the disc. Let
$\hat{T}$
be an irreducible homogeneous (with respect to the
bi-holomorphic group of automorphisms of the disc) operator in the
Cowen-Douglas class on the disc of rank $2$. We show that every
irreducible $G$ - homogeneous operator, $G$ as in (i), of rank $2$ must be
of the form
\begin{equation}
(T_1\otimes I_{\widehat{H}},\ldots , T_{n-1}\otimes
I_{\widehat{H}}, I_H \otimes \hat{T}).
\end{equation}
We also show that if $G$ is chosen to be the group as in (ii), then there are no irreducible $G$- homogeneous operators of rank $2.$
Homogenization of boundary value problems posed on rough domains has paramount importance in real life problems. Materials with oscillating (rough) boundary are used in many industrial applications like micro strip radiator and nano technologies, biological systems, fractal-type constructions, etc. In this talk, we will be focusing on homogenization of optimal control problems. We will begin with homogenization of a boundary control problem on an oscillating pillar type domain. Then, we will consider a time-dependent control problem posed on a little more general domain called branched structure domain. Asymptotic analysis of this interior control problem will be explained. Next, we will present a generalized unfolding operator that we have developed for a general oscillatory domain. Using this unfolding operator, we study the homogenization of a non-linear elliptic problem on this general highly oscillatory domain. Also, we analyse an optimal control problem on a circular oscillating domain with the assistance of this operator. Finally, we consider a non-linear optimal control problem on the above mentioned general oscillatory domain and study the asymptotic behaviour.
We study asymptotic analysis (homogenization) of second-order partial differential equations(PDEs) posed on an oscillating domain. In general, the motivation for studying problems defined on oscillating domains, come from the need to understand flow in channels with rough boundary, heat transmission in winglets, jet engins and so on. There are various methods developed to study homogenization problems namely; multi-scale expansion, oscillating test function method, compensated compactness, two-scale convergence, block-wave method, method of unfolding etc.
In this thesis, we consider a two dimensional oscillating domain (comb shape
type) $\Omega_{\epsilon}$
consists of a fixed bottom region $\Omega^-$
and an oscillatory
(rugose) upper region $\Omega_{\epsilon}^{+}$. We introduce an optimal control problems in
$\Omega_{\epsilon}$
for the Laplacian operator. There are mainly two types of optimal
control problems; namely distributed control andboundary control. For distributed control
problems in the oscillatingdomain, one can put control on the oscillating part or on the fixed
part and similarly for boundary control problem (control on the oscillatingboundary or on the
fixed part the boundary). Considering controls on theoscillating part is more interesting and
challenging than putting control on fixed part of the domain. Our main aim is to characterize
the controlsand study the limiting analysis (as $\epsilon \to 0$
) of the optimalsolution.
In the thesis, we consider all the four cases, namely distributed and boundary controls both
on the oscilalting part and away from the oscillating part. Since, controls on the oscillating
part is more exciting, in this talk, we present the details of two sections. First we consider
distributed optimal control problem, where the control is supported on the oscillating part
$Omega_{\epsilon}^{+}$
with periodic controls and with Neumann condition on the oscillating
boundary $\gamma_{\epsilon}$
. Secondly, we introduce boundary optimal control
problem, control applied through Neumann boundary condition on the oscillating boundary
$\gamma_{\epsilon}$
with suitable scaling parameters. We characterize the optimal control
using unfolding and boundary unfolding operators and study limiting analysis. In the limit, we
obtain two limit problems according to the scaling parameters and we observe that limit
optimal control problem has three control namely; a distributed control, a boundary control
and an interface control.
A pair of commuting bounded operators $(S,P)$ acting on a
Hilbert space, is
called a $\Gamma$
-contraction, if it has the symmetrised bidisc
\begin{equation}
\Gamma=\{ (z_1+z_2,z_1z_2):|z_1| \leq 1,|z_2| \leq 1 \}\subseteq \mathbb{C}^2
\end{equation}
as a spectral set. For every $\Gamma$-contraction $(S,P)$
, the operator equation
$S-S^*P=D_PFD_P$ has a unique solution $F$ with numerical radius, $w(F)$ no greater than one,
where $D_P$ is the positive square root of $(I-P^*P)$
. This unique operator is called the
fundamental operator of $(S,P)$. This thesis constructs an explicit normal boundary dilation for a
$\Gamma$-contraction. A triple of commuting bounded operators $(A,B,P)$
acting on a
Hilbert space with the closure of the tetrablock
\begin{equation}
E=\{(a_{11},a_{22},\det A):
A=\begin{pmatrix} a_{11} & a_{12} \\
a_{21} & a_{22} \end{pmatrix}\text{ with }\lVert A \rVert
<1\}\subseteq\mathbb{C}^3
\end{equation}
as a spectral set, is called a tetrablock contraction. Every tetrablock contraction
possesses two fundamental operators and these are the unique solutions of
\begin{equation}
A-B^*P=D_PF_1D_P, \ \text{ and } \ B-A^*P=D_PF_2D_P.
\end{equation}
Moreover, $w(F_1)$ and $w(F_2)$ are no greater than one. This thesis also constructs an explicit
normal boundary dilation for a tetrablock contraction. In these constructions, the
fundamental operators play a pivotal role. Both the dilations in the symmetrized bidisc and in the
tetrablock are proved to be minimal. But unlike the one variable case, uniqueness of minimal
dilations usually does not hold good in several variables, e.g., Ando’s dilation is not unique.
However, we show that the dilations are unique under a certain natural condition. In view of the
abundance of operators and their complicated structure, a basic problem in operator
theory is to find nice functional models and complete sets of unitary invariants. We develop a
functional model theory for a special class of triples of commuting bounded operators
associated with the tetrablock. We also find a set of complete sort of unitary invariants for
this special class. Along the way, we find a Beurling-Lax-Halmos type of result for a triple
of multiplication operators acting on vector-valuedHardy spaces. In both the model theory
and unitary invariance,fundamental operators play a fundamental role. This thesis
answers the question when two operators $F$ and $G$ with $w(F)$ and $w(G)$ no greater than one,
are admissible as fundamental operators, in other words, when there exists a
$\Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and
$G$ is the fundamental operator of $(S^*,P^*)$
. This thesis also answers a
similar question in the tetrablock setting.
In the 1980s, Goldman introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F. This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra.
In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.
We use these results to identify the quotient of the Goldman Lie algebra of a non-closed surface by its center as a sub-algebra of the first Hochschild cohomology of the fundamental group.
Using hyperbolic geometry, we prove a special case of a theorem of Chas, namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them.
We also construct infinitely many pairs of length equivalent curves in an= y hyperbolic surface F of finite type. Our construction shows that given a self-intersecting geodesic x of F and any self-intersection point P of x, we get a sequence of such pairs.
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let $X_1,X_2,...,X_k$
be i.i.d. matrices of size $n \times n$ whose entries
are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$
, where each
$Y_i = X_i$ or $X_i^{-1}$
. We show that the eigenvalues form a determinantal
point process. The case where $k=2$, $Y_1=X_1,Y_2=X_2^{-1}$
was derived
earlier by Krishnapur. The case where $Y_i =X_i$
for all $i=1,2,...,n$
, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.
Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the contemporary applications are much far and wide. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis. We plan to start with an illustrative example of limiting analysis in 1-D for a second order elliptic partial differential equation. We will also see some classical results in the case of periodic composite materials and oscillating boundary domain. The emphasis will be on the computational importance of homogenization in numerics by the introduction of correctors. In the second part of the talk, we will see a study on optimal control problems posed in a domain with highly oscillating boundary. We will consider periodic controls in the oscillating part of the domain with a model problem of Laplacian and try to understand their optimality and asymptotic behavior.
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let $X_1,X_2,...,X_k$
be i.i.d matrices of size nxn whose entries
are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form $Y_1.Y_2....Y_n$
, where each
$Y_i = X_i or (X_i)^{-1}$
. We show that the eigenvalues form a determinantal
point process. The case where k=2, $Y_1=X_1,Y_2=X_2^{-1}$
was derived
earlier by Krishnapur. The case where $Y_i =X_i$
for all $i=1,2,...,n$
, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.
The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.
This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:
(i) We introduce the weight of a group which has a presentation with
number of relations is at most the number of generators.
We prove that the number of vertices of any crystallization of a connected
closed 3-manifold $M$ is at least the weight of the
fundamental group of $M$. This lower bound is sharp for the 3-manifolds
$\mathbb{R P}^3$
, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$
,
$S^{\hspace{.2mm}2} \times S^1$
, $\TPSS$
and $S^{\hspace{.2mm}3}/Q_8$
,
where $Q_8$
is the quaternion group. Moreover,
there is a unique such vertex minimal crystallization in each of these
seven cases. We also construct crystallizations of
$L(kq-1,q)$ with $4(q+k-1)$ vertices for $q \geq 3$
, $k \geq 2$
and
$L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent
result of Swartz,
our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.
(ii) We present an algorithm to find certain types of crystallizations of
$3$-manifolds from a given presentation $\langle S \mid R \rangle$
with
$\#S=\#R=2$
. We generalize this algorithm for presentations with three
generators and certain class of relations.
This gives us crystallizations of closed connected orientable 3-manifolds
having fundamental groups $\langle x_1,x_2,x_3 \mid
x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$
with $4(m+n+k-3)+ 2\delta_n^2 + 2
\delta_k^2$
vertices for $m\geq 3$
and $m \geq n \geq k \geq 2$
, where
$\delta_i^j$
is the Kronecker delta.
If $n=2$ or $k\geq 3$
and $m \geq 4$
then these crystallizations
are vertex-minimal for all the known cases.
(iii) We found a minimal crystallization of the standard pl K3 surface.
This minimal crystallization is a ‘simple crystallization’.
Using this, we present minimal crystallizations of all simply connected pl
$4$-manifolds of “standard” type, i.e., all the connected sums of
$\mathbb{CP}^2$
, $S^2 \times S^2$
, and the K3 surface. In particular, we
found minimal crystallizations of a pair of 4-manifolds which are
homeomorphic
but not pl-homeomorphic.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$
. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(d\sigma)$
, where $d\sigma$
is the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$
. Then
It follows that $\widehat{fd\sigma}\in L^p(\mathbb{R}^n)$
for all
$p>2n/(n-1)$
. This result can be extended to compactly
supported measure on $(n-1)$-dimensional manifolds with
appropriate assumptions on the curvature. Similar results are
known for measures supported in lower dimensional manifolds in
$\mathbb{R}^n$
under appropriate curvature conditions. However, the
picture for fractal measures is far from complete. This thesis is
a contribution to the study of asymptotic properties of the
Fourier transform of measures supported in sets of fractal
dimension $0<\alpha<n$
for $p\leq 2n/\alpha$
.
In 2004, Agranovsky and Narayanan proved that if $\mu$ is a
measure supported in a $C^1$
-manifold of dimension $d<n$
, then
$\widehat{fd\mu}\notin L^p(\mathbb{R}^n)$
for $1\leq p\leq \frac{2n}{d}$
. We
prove that the Fourier transform of a measure $\mu_E$ supported in
a set $E$ of fractal dimension $\alpha$ does not belong to
$L^p(\mathbb{R}^n)$
for $p\leq 2n/\alpha$
. We also study $L^p$
-asymptotics
of the Fourier transform of fractal measures $\mu_E$ under
appropriate conditions on $E$ and give quantitative versions of
the above statement by obtaining lower and upper bounds for the
following:
\(\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|\xi|\leq L}|\widehat{fd\mu_E}(\xi)|^pd\xi,\) \(\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{L\leq |\xi|\leq 2L}|\widehat{fd\mu_E}(\xi)|^pd\xi.\)
This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:
(i) We introduce the weight of a group which has a presentation with
number of relations is at most the number of generators.
We prove that the number of vertices of any crystallization of a connected
closed 3-manifold $M$ is at least the weight of the
fundamental group of $M$. This lower bound is sharp for the 3-manifolds
$\mathbb{R P}^3$
, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$
,
$S^{\hspace{.2mm}2} \times S^1$
, $\TPSS$
and $S^{\hspace{.2mm}3}/Q_8$
,
where $Q_8$
is the quaternion group. Moreover,
there is a unique such vertex minimal crystallization in each of these
seven cases. We also construct crystallizations of
$L(kq-1,q)$ with $4(q+k-1)$ vertices for $q \geq 3$
, $k \geq 2$
and
$L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent
result of Swartz,
our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.
(ii) We present an algorithm to find certain types of crystallizations of
$3$-manifolds from a given presentation $\langle S \mid R \rangle$
with
$\#S=\#R=2$
. We generalize this algorithm for presentations with three
generators and certain class of relations.
This gives us crystallizations of closed connected orientable 3-manifolds
having fundamental groups $\langle x_1,x_2,x_3 \mid
x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$
with $4(m+n+k-3)+ 2\delta_n^2 + 2
\delta_k^2$
vertices for $m\geq 3$
and $m \geq n \geq k \geq 2$
, where
$\delta_i^j$
is the Kronecker delta.
If $n=2$ or $k\geq 3$
and $m \geq 4$
then these crystallizations
are vertex-minimal for all the known cases.
(iii) We found a minimal crystallization of the standard pl K3 surface.
This minimal crystallization is a ‘simple crystallization’.
Using this, we present minimal crystallizations of all simply connected pl
$4$-manifolds of “standard” type, i.e., all the connected sums of
$\mathbb{CP}^2$
, $S^2 \times S^2$
, and the K3 surface. In particular, we
found minimal crystallizations of a pair of 4-manifolds which are
homeomorphic
but not pl-homeomorphic.
In this talk we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the L^p boundedness of operators M which are known to be bounded on L^2: We obtain sufficient conditions on the kernel of the operator M so that it satisfies weighted L^p estimates. As an application we prove L^p boundedness of Hermite pseudo-multipliers.
Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the Gromov-Hausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result - more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple
$(X, d, μ)$
, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.
Using this result, we prove that the Deligne-Mumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudo-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov’s compactness theorem for pseudo-holomorphic curves is an important tool in symplectic topology, its applicability is limited due to the non-existence of a general method to construct pseudo-holomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many way of mechanising are: (1) generating results using Automated Theorem Provers, (2) Interactive theorem proving in a Proof Assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in First Order Logic using Braid Groups. This is achieved by developing a set of Axioms whose canonical model is the Infinite Braid Group. This renders the problem of distinguishing Knots and Links, amenable to implementation in First Order Logic based Automated Theorem provers. We further state and prove results pertaining to Models of Braid Axioms.
The second part of the thesis deals with formalising Knot Theory in Higher Order Logic using the Isabelle Proof Assistant. We formulate equivalence of Links in Higher Order Logic. We obtain a construction of Kauffman Bracket in the Isabelle Proof Assistant. We further obtain a machine checked proof of invariance of Kauffman Bracket.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility.
Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
Unlike the partial differential equations, the solutions of variational inequalities exhibit singularities even when the data is smooth due to the existence of free boundaries. Therefore the numerical procedure of these problems based on uniform refinement becomes inefficient due to the loss of the order of convergence. A popular remedy to enhance the efficiency of the numerical method is to use adaptive finite element methods based on computable a posteriori error bounds. Discontinuous Galerkin methods play a very important role in the local mesh adaptive refinement techniques.
The main focus in this thesis has been on the derivation of reliable and efficient error bounds for the discontinuous Galerkin methods applied to elliptic variational inequalities. The variational inequalities can be split into two kinds, namely, inequalities of the first kind and the second kind. We study an elliptic obstacle problem and a Signorini contact problem in the category of the first kind, while the frictional plate contact problem in the category of the fourth order variational inequalities of second kind. The mathematical analysis of error estimation in this class of problems crucially depends on a suitable nonlinear smoothing function that enriches the smoothness of the numerical solution. Another remarkable advantage of discontinuous Galerkin methods has been realized in the applications to higher order problems. Numerical experiments support the theoretical results and exhibit optimal convergence.
This thesis investigates two different types of problems in multi-variable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.
I. We introduce the defect sequence for a contractive tuple of
Hilbert space operators and investigate its properties. We show that there
are upper bounds for the defect dimensions. The upper bounds are
different in the non-commutative and in the commutative case. The tuples
for which these upper bounds are obtained are called maximal contractive
tuples. We show that the creation operator tuple on the full Fock space
and the co-ordinate multipliers on the
Drury-Arveson space are maximal. We also show that if M is an
invariant subspace under the creation operator tuple on the full Fock
space, then the restriction is always maximal. But the situation is
starkly different for co-invariant subspaces. A characterization for a
contractive tuple to be maximal is obtained. We define the notion of
maximality for a submodule of the Drury-Arveson module on the
d-dimensional unit ball. For $d=1$
, it is shown that every submodule of the
Hardy module over the unit disc is maximal. But for $d>2$
, we prove that any
homogeneous submodule or a submodule generated by polynomials is not
maximal. We obtain a characterization of maximal submodules of the
Drury-Arveson module. We also study pure tuples and see how the defect
dimensions play a role in their irreducibility.
II. We investigate the following question : Let $(T_1, ....., T_n)$
be a
commuting $n$-tuple of bounded linear operators on a Hilbert space $H$. Does
there exist a generating wandering subspace $W$ for $(T_1, ....., T_n)$
? We
got some affirmative answers for the doubly commuting
invariant subspaces of the Bergman space and the Dirichlet space over the
unit polydisc. We show that for any doubly commuting invariant subspace
of the Bergman space or the Dirichlet space over polydisc, the tuple
consisting of restrictions of co-ordinate multiplication operators
always possesses a generating wandering subspace.
Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the Gromov-Hausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result - more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple
$(X, d, μ)$
, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.
Using this result, we prove that the Deligne-Mumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudo-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov’s compactness theorem for pseudo-holomorphic curves is an important tool in symplectic topology, its applicability is limited due to the non-existence of a general method to construct pseudo-holomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.