We consider the computation of n-variate polynomials over a field F via a sequence of arithmetic operations such as additions, subtractions, multiplications, divisions, etc. It has been known for at five decades now that a random n-variate polynomial of degree n is hard to compute. Yet not a single explicit polynomial is provably known to be hard to compute (although we have a lot of good candidates). In this talk we will first describe this problem and its relationship to the P vs NP problem. We will then describe several partial results on this problem, both old and new, along with a more general approach/framework that ties together most of these partial results.

We shall discuss a theorem of Bernstein published in 1975 about the number of common solutions of n complex polynomials in n variables in terms of the mixed volumes of their Newton polytopes. This is a far reaching generalisation of the Fundamental Theorem of Algebra and Bezout’s Theorem about intersections of plane algebraic curves. If time permits, we shall sketch a proof of Bernstein’s theorem using Hilbert functions of monomial ideals in polynomial rings.

I will give a gentle historical (and ongoing) account of matrix positivity and of operations that preserve it. This is a classical question studied for much of the past century, including by Schur, Polya-Szego, Schoenberg, Kahane, Loewner, and Rudin. It continues to be pursued actively, for both theoretical reasons as well as applications to high-dimensional covariance estimation. I will end with some recent joint work with Terence Tao (UCLA).

The entire talk should be accessible given a basic understanding of linear algebra/matrices and one-variable calculus. That said, I will occasionally insert technical details for the more advanced audience. For example: this journey connects many seemingly distant mathematical topics, from Schur (products and complements), to spheres and Gram matrices, to Toeplitz and Hankel matrices, to rank one updates and Rayleigh quotients, to Cauchy-Binet and Jacobi-Trudi identities, back full circle to Schur (polynomials).

In this talk we discuss a formulation of Quantum Theory of Dark matter and discuss some operators on Hilbert spaces of singular measures.

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 $\theta_T(a) = \sigma_{L}(\phi_a)^{*} \theta(0) \sigma_{R}(\phi_a),$ where, $\sigma_{L}$ and $\sigma_{R}$ are projective representation of the M"{o}bius group M"{o}b 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 $(T_1\otimes I_{\widehat{H}},\ldots , T_{n-1}\otimes I_{\widehat{H}}, I_H \otimes \hat{T}).$

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 $2$.

In this talk the interplay between the combinatorial structures of finite simple graphs and various homological invariants like regularity, depth etc. of related algebraic objects shall be discussed. Some open problems, recent developments and ongoing projects shall be discussed. In particular some new techniques developed in my thesis to study Castelnuovo-Mumford regularity of algebraic objects related to graphs shall be discussed in some details.

The aim of this talk is to give an overview of some recent results in two interconnected areas:

**a) Random discrete structures:** One major conjecture in probabilistic
combinatorics, formulated by statistical physicists using non-rigorous
arguments and enormous simulations in the early 2000s, is as follows: for a
wide array of random graph models on $n$ vertices and degree
exponent $\tau>3$, typical distance both within maximal components
in the critical regime as well as on the minimal spanning tree on the giant
component in the supercritical regime scale like
$n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}$. In other words, the degree
exponent determines the universality class the random graph belongs to. The
mathematical machinery available at the time was insufficient for providing
a rigorous justification of this conjecture.

More generally, recent research has provided strong evidence to believe that several objects, including (i) components under critical percolation, (ii) the vacant set left by a random walk, and (iii) the minimal spanning tree, constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff sense, and these limiting objects are universal under some general assumptions. We will discuss recent developments in a larger program aimed at a complete resolution of these conjectures.

**b) Stochastic geometry:** In contrast, less precise results are known in the
case of spatial systems. We discuss a recent result concerning the length
of spatial minimal spanning trees that answers a question raised by Kesten
and Lee in the 90’s, the proof of which relies on a variation of Stein’s
method and a quantification of a classical argument in percolation theory.

Based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.

The notion of a weakly proregular sequence in a commutative ring was first formally introduced by Alonso-Jeremias-Lipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel and Porta-Shaul-Yekutieli: a precise definition of this notion will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. Every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A.

In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.

Dualizing complexes were first introduced in commutative algebra and algebraic geometry by Grothendieck and play a fundamental role in Serre-Grothendieck duality theory for schemes. The notion of a dualizing complex was extended to noncommutative ring theory by Yekutieli. There are existence theorems for dualizing complexes in the noncommutative context, due to Van den Bergh, Wu, Zhang, and Yekutieli amongst others.

Most considerations of dualizing complexes over noncommutative rings are for algebras defined over fields. There are technical difficulties involved in extending this theory to algebras defined over more general commutative base rings. In this talk, we will describe these challenges and how to get around them. Time permitting, we will end by presenting an existence theorem for dualizing complexes in this more general setting.

The material described in this talk is work in progress, carried out jointly with Amnon Yekutieli.

The field of geodynamics deals with the large scale forces shaping the Earth. Computational geodynamics, which uses numerical modeling, is one of the most important tools to understand the mechanisms within the deep Earth. With the help of these numerical models we can address some of the outstanding questions regarding the processes operating within the Earth’s interior and their control on shaping the surface of the planet. Much of Earth’s surface observations such as gravity anomalies, plate motions, dynamic topography, lithosphere stress field, owe their origin to convection within the Earth’s mantle. While we understand the basic nature of such flow in the mantle, a lot remains unexplained, including the complex rheology of the deep mantle and how this density driven convective flow couples with the shallow surface. In this talk I will discuss how my group is using numerical modeling to understand the influence of the deep mantle on surface observations.

For a finite abelian group $G$ with $|G| = n$, the Davenport Constant $DA(G)$ is defined to be the least integer $k$ such that any sequence $S$ with length $k$ of elements in $G$ has a non-empty $A$ weighted zero-sum subsequence. For certain sets $A$, we already know the precise value of constant corresponding to the cyclic group $\mathbb{Z} / n \mathbb{Z}$. But for different group $G$ and $A$, the precise value of it is still an open question. We try to find out bounds for these combinatorial invariant for random set $A$. We got few results in this connection. In this talk I would like to present those results and discuss about an extremal problem related to this combinatorial invariant.

The study of weighted inequalities in Classical Harmonic
Analysis started in 70’s, when B. Muckenhoupt characterised in 1972 the
weights $w$ for which the Hardy–Littlewood maximal function is bounded in
$L^p(w)$. At that time the question about how the operator depended on the
constant associated with $w$, which we denote by $[w]_{A_p}$, was not
considered (i.e., *quantitative estimates*) were not investigated.

From the beginning of 2000’s, a great activity has been carried out in order to obtain the sharp dependence for singular integral operators, reaching the solution of the so-called $A_2$ conjecture by T. P. H"ytonen.

In this talk we consider operators with homogeneous singular kernels, on which we assume smoothness conditions that are weaker than the standard ones (this is why they are called rough). The first qualitative weighted estimates are due to J. Duoandikoetxea and J. L. Rubio de Francia. For the norm of these operators in the space $L^2(w)$ we obtain a quantitative estimate which is quadratic in the constant $[w]_{A_2}$.

The results are based on a classical decomposition of the rough operators as a sum of other operators with a smoother kernel, for which a quantitative reelaboration of a dyadic decomposition proposed by M. T. Lacey is applied.

We will overview as well the most recent advances, mainly associated with quantitative estimates for these rough singular integrals. In particular, Coifman–Fefferman type inequalities (which are new even in their qualitative version), weighted $A_p$-$A_{\infty}$ inequalities and a quantitative version of weak $(1,1)$ estimates will be shown.

Conformal blocks are refined invariants of tensor product of representations of a Lie algebra that give a special class of vector bundles on the moduli space of curves. In this talk, I will introduce conformal blocks and explore connections to questions in algebraic geometry and representation theory. I will also focus on some ``strange” dualities in representation theory and how they give equalities of divisor classes on the moduli space of curves.

The Riemann-Roch theorem is fundamental to algebraic geometry. In 2006, Baker and Norine discovered an analogue of the Riemann-Roch theorem for graphs. In fact, this theorem is not a mere analogue but has concrete relations with its algebro-geometric counterpart. Since its conception this topic has been explored in different directions, two significant directions are i. Connections to topics in discrete geometry and commutative algebra ii. As a tool to studying linear series on algebraic curves. We will provide a glimpse of these developments. Topics in commutative algebra such as Alexander duality and minimal free resolutions will make an appearance. This talk is based on my dissertation and joint work with i. Bernd Sturmfels, ii. Frank-Olaf Schreyer and John Wilmes and iii. an ongoing work with Alex Fink.

The Hadamard product of two matrices is formed by multiplying corresponding entries, and the Schur product theorem states that this operation preserves positive semidefiniteness.

It follows immediately that every analytic function with non-negative Maclaurin coefficients, when applied entrywise, preserves positive semidefiniteness for matrices of any order. The converse is due to Schoenberg: a function which preserves positive semidefiniteness for matrices of arbitrary order is necessarily analytic and has non-negative Maclaurin coefficients.

For matrices of fixed order, the situation is more interesting. This talk will present recent work which shows the existence of polynomials with negative leading term which preserve positive semidefiniteness, and characterises precisely how large this term may be. (Joint work with D. Guillot, A. Khare and M. Putinar.)

Commutators of singular integral operators with BMO functions were introduced in the seventies by Coifman-Rochberg and Weiss. These operators are very interesting for many reasons, one of them being the fact that they are more singular than Calderon-Zygmundm operators. In this lecture we plan to give several reasons showing the ``bad” behavior of these operators.

The goal of this talk is to present an algorithm which takes a compact square complex belonging to a special class as input and decides whether its fundamental group splits as a free product. The special class is built by attaching tubes to finite graphs in such a way that they satisfy a nonpositive curvature condition. This construction gives rise to a rich class of complexes, including, but not limited to, closed surfaces of positive genus. The algorithm can be used to deduce the celebrated Stallings theorem for this special class, as also the well known Grushko decomposition theorem.

A group is cyclic iff its subgroup lattice is distributive. Ore’s generalized one direction of this result. We will discuss a dual version of Ore’s result, for any boolean interval of finite groups under the assumption that the dual Euler totient of the interval is nonzero. We conjecture that the dual Euler totient is always nonzero for boolean intervals. We will discuss some techniques which may be helpful in proving it. We first see that dual Euler totient of an interval of finite groups is the Mobius invariant (upto a sign) of its coset poset P. Next in the boolean group complemented case, we prove that P is Cohen-Macaulay, using the existence of an explicit EL-labeling. We then see that nontrivial betti number of the order complex is nonzero, and so is the dual Euler totient.

Quasi-algebras were introduced as algebras in a monoidal category. Since the associativity constraints in these categories are allowed to be nontrivial, the class of quasi-algebras contains various important examples of non-associative algebras like the octonions and other Cayley algebras. The diamond lemma is a reduction method used in algebra. The original diamond lemma was stated in graph theory by Newman which was later generalized to associative algebras by Bergman. In this talk, we will see the analog of this lemma for the group graded quasi-algebras with some interesting examples like octonion algebra and generalized octonions

Vortex streets are a common feature of fluid flows at high Reynolds numbers and their study is now well developed for incompressible fluids. Much less is known, however, about compressible vortex streets. A fundamental reason appears to be the inapplicability of the point vortex model to compressible flows. In this talk, we discuss point vortices in the context of weakly compressible flows and elaborate on the problems involved. We then adopt the hollow vortex model where each vortex is modelled as a finite-area constant pressure region with non-zero circulation. For weakly compressible flows steady hollow vortex solutions are well known to be candidates for the leading order solution in a perturbative Rayleigh-Jansen expansion of a compressible flow. Here we give details of that expansion based on the vortex street solutions of Crowdy & Green (2012). Physical properties of the compressible vortex streets are described. Our approach uses the Imai-Lamla method coupled with analytic function theory and conformal mapping. (Joint work with Darren Crowdy)

The formalism of an “abelian category’’ is meant to axiomatize the operations of linear algebra. From there, the notion of “derived category’’ as the category of complexes “upto quasi-isomorphisms’’ is natural, motivated in part by topology. The formalism of t-structures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example). In this talk we want to discuss a notion of punctual (=”point-wise’’) gluing of t-structures which is possible in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to useful constructions in the motivic world.

In this talk, I would continue dealing with Sabra shell model of Turbulence and study one of the important questions for fluid flow problems namely, finding controls which are capable of preserving the invariant quantities of the flow. Controls are designed in the feedback form such that resultant controlled flow will preserve certain physical properties of the state such as enstrophy, helicity. We use the theory of nonlinear semigroups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space under consideration.

Graph-partitioning problems are a central topic of research in the study of algorithms and complexity theory. They are of interest to theoreticians with connections to error correcting codes, sampling algorithms, metric embeddings, among others, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications. One of the central problems studied in this field is the sparsest cut problem, where we want to compute the cut which has the least ratio of number of edges cut to size of smaller side of the cut. This ratio is known as the expansion of the cut. In this talk, I will talk about higher order variants of expansion (i.e. notions of expansion corresponding to partitioning the graph into more than two pieces, etc.), and how they relate to the graph’s eigenvalues. The proofs will also show how to use the graph’s eigenvectors to compute partitions satisfying these bounds. Based on joint works with Prasad Raghavendra, Prasad Tetali and Santosh Vempala.

In this lecture I am going to present control problems associated with shell models of Turbulence. Shell models of turbulence are simplified caricatures of equations of fluid mechanics in wave-vector representation. They exhibit anomalous scaling and local non-linear interactions in wave number space. We would like to study control problem related to one such widely accepted shell model of turbulence known as sabra shell model. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the ow near a priori known state. We derive the optimal controls in terms of the solution of adjoint equation for corresponding linearised problems.

Understanding the origins of intermittency in turbulence remains one of the most fundamental challenges in applied mathematics. In recent years there has been a fresh attempt to understand this problem through the development of the method of Fourier decimation. In this talk, we will review these recent results and analyse the role of precise numerical simulations in understanding the mathematics of the Navier-Stokes and Euler equations.

It is generally well known that there is an innate notion of things like isomorphisms or epimorphisms. This allows us to talk about isomorphisms or epimorphisms of various objects: groups, rings, algebras,etc. In other words, “isomorphism” is really a categorical notion.However, it is not so well known that finiteness itself is alsocategorical. In this talk, we will discuss how finiteness applies tovarious categories. This will allow usto see finite sets, finite dimensional vector spaces, finitely generated algebras and compact sets as manifestations of the same basic idea.

Consider the infinite Ginibre ensemble (the distributional limit of the eigenvalues of nxn random matrices with i.i.d. standard complex Gaussian entries) in the complex plane. For a bounded set U, let H_r(U) denote the probability (hole probability) that no points of infinite Ginibre ensemble fall in the region rU. We study the asymptotic behavior of H_r(U) as r–>\infty. Under certain conditions on U we show that \log H_r(U)=C_U.r^4 (1+o(1)) as r–> \infty. Using potential theory, we give an explicit formula for C_U in terms of the minimum logarithmic energy of the set with a quadratic external field. We calculate C_U explicitly for some special sets such as the annulus, cardioid, ellipse, equilateral triangle and half disk.

Moreover, we generalize the above hole probability results for a class of determinantal point processes in the complex plane.

In algebraic geometry the concept of height pairing (a particular example of linking numbers) of algebraic cycles lies at the confluence of arithmetic, Hodge theory and topology. In a series of two talks, I will explain the notion of Beilinson’s height pairing for cycles homologous to zero. This will bring into picture the notion of Arakelov/arithmetic intersection theory. I will give sufficient background of this theory and provide examples. Finally, I will talk about my recent work with Dr. Jose Ignacio Burgos, about a generalization of Beilinson’s height pairing for higher algebraic cycles.

In this talk, I shall consider an abstract Cauchy problem for a class of impulsive sub-diffusion equation. Existence and regularity of solution of the problem shall be established via eigenfunction expansion. Further, I shall establish the approximate controllability of the problem by applying unique continuation property via internal control acts on a sub-domain.

It was shown by Basu, Sidoravicius and Sly that a TASEP starting with the step initial condition, i.e., with one particle each at every nonpositive site of $\mathbb{Z}$ and no particle at positive sites, with a slow bond at the origin where a particle jumping from the origin jumps at a smaller rate $r < 1$, has an asympototic current which is strictly less than 1/4. Here we study the limiting measure of the TASEP with a slow bond. The distribution of regular TASEP started with the step initial condition converges to the invariant product Bernoulli measure with density 1/2. The slowdown due to the slow bond implies that there is a long range effect near the origin where the region to the right of origin is sparser and there is a traffic jam to the left of the slow bond with particle density higher than a half. However, the distribution becomes close to a product Bernoulli measure as one moves far away from the origin, albeit with a different density ? < 1/2 to the right of the origin and ?’ > 1/2 to the left of the origin. This answers a question due to Liggett. The proof uses the correspondence between TASEP and directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times, and the geometric properties of the maximal paths there.

A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multiscale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed at the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods.

In this talk, diffusion and reaction of several mobile chemical species are considered in the pore space of a heterogeneous porous medium. The reactions amongst the species are modelled via mass action kinetics and the modelling leads to a system of multispecies diffusion; reaction equations (coupled semi-linear partial differential equations) at the micro scale where the highly nonlinear reaction rate terms are present at the right hand sides of the system of PDEs, cf. [2]. The existence of a unique positive global weak solution is shown with the help of a Lyapunov functional, Schaefer’s fixed point theorem and maximal Lp-regularity, cf. [2, 3]. Finally, with the help of periodic homogenization and two-scale convergence we upscale the model from the micro scale to the macro scale, e.g. [1, 3]. Some numerical simulations will also be shown in this talk, however for the purpose of illustration, we restrict ourselves to some relatively simple 2- dimensional situations.

As an extension to the previous model, we consider the mixture of two fluids. For such models, a system of Stokes-Cahn-Hilliard equations will be considered at the micro scale in a perforated porous medium. We first explain the periodic setting of the model and the existence results. At the end homogenization of the model will be shown using some extension theorems on Sobolev spaces, two-scale convergence and periodic unfolding.

As big data sets have become more common, there has been significant interest in finding and understanding patterns in them. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will present recent Falconer type theorems, established by myself and my collaborators, for a wide class of finite point configurations in any dimension. The techniques we used come from analysis and geometric measure theory, and the key step was to obtain bounds on multilinear analogues of generalized Radon transforms.

Let H denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on R2 is in L2(H,μ), where μ is the Masur-Veech measure on H, and give applications to bounding error terms for counting problems for saddle connections. We will review classical results in the Geometry of Numbers which anticipate this result. This is joint work with Yitwah Cheung and Howard Masur.

We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi’s conjecture. This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.