We cordially invite you to the in-house symposium on March 28 and 29, 2019. This symposium features lectures by postdocs and senior research students. The lectures will be in LH-1, Department of Mathematics, IISc.

The schedule is as follows.

### Day 1: Thursday, March 28 2019.

 Time Speaker 10.45 am-11.15 am Lecture 1 Bikramaditya Sahu 11.15 am-11.45 am Lecture 2 Projesh Nath Choudhury TEA BREAK 12.00 noon-12.30pm Lecture 3 Sumana Hatui 12.30 pm-1.00 pm Lecture 4 Mamta Balodi LUNCH BREAK 2.30 pm-3.00 pm Lecture 5 Safdar Quddus 3.00 pm-3.30 pm Lecture 6 Samarpita Ray 3.30 pm-4.00 pm Lecture 7 Asha Dond TEA BREAK 4.15 pm-4.45 pm Lecture 8 Gouranga Mallik 4.45 pm-5.15 pm Lecture 9 Arun Maiti 5.15 pm-5.45 pm Lecture 10 Srijan Sarkar HIGH TEA

### Day 2: Friday, March 29 2019.

 Time Speaker 9.45 am- 10.15 am Lecture 1 Sanjoy Kumar Jhawar 10.15 am-10.45 am Lecture 2 Somnath Pradhan 10.45 am-11.15 am Lecture 3 Sneh Bala Sinha TEA BREAK 11.30 am-12.00 noon Lecture 4 Abhash Kumar Jha 12.00 noon-12.30 pm Lecture 5 K Hariram 12.30 pm-1.00 pm Lecture 6 Pramath Anamby LUNCH BREAK 2.30 pm- 3.00 pm Lecture 7 Ritwik Pal 3.00 pm- 3.30 pm Lecture 8 Soumitra Ghara 3.30 pm- 4.00 pm Lecture 9 Debmalya Sain TEA BREAK 4.15 pm-4.45 pm Lecture 10 Surjit Kumar 4.45 pm-5.15 pm Lecture 11 Gopal Datt 5.15 pm-5.45 pm Lecture 12 Anwoy Maitra HIGH TEA

## Abstracts

### Day 1: Thursday, March 28 2019.

#### Lecture 1

Title: Blocking sets of certain line sets in PG(2,q)

Consider the Desarguesian projective plane PG(2,q), where q is a prime power. Given a non-empty subset L of the line set of PG(2,q), an L-blocking set is a subset B of the point set of PG(2,q) such that every line of L contains at least one point of B. In this talk, we discuss the minimum size L-blocking sets of the line sets L, where L is defined with respect to an irreducible conic in PG(2,q).

#### Lecture 2

Title: Distance matrices of trees: invariants, old and new

Speaker: Projesh Nath Choudhury

In 1971, Graham and Pollak showed that if $D_T$ is the distance matrix of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$, not $T$. This independence from the tree structure has been verified for many different variants of weighted bi-directed trees. In my talk (over an arbitrary commutative ring):

1. I will present a general setting which strictly subsumes every known variant, and where we show that $\det(D_T)$ – as well as another graph invariant, the cofactor-sum – depends only on the edge-data, not the tree-structure.

2. More generally – even in the original unweighted setting – we strengthen the state-of-the-art, by computing the minors of $D_T$ where one removes rows and columns indexed by equal-sized sets of pendant nodes. (In fact we go beyond pendant nodes.)

3. We explain why our result is the “most general possible”, in that allowing greater freedom in the parameters leads to depends on the tree-structure.

We will discuss related results for arbitrary strongly connected graphs, including a third, novel invariant. If time permits, a formula for $D_T^{-1}$ will be presented for trees $T$, whose special case answers an open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which extends to our general setting a result of Graham-Lovasz (Advances in Math. 1978). (Joint with Apoorva Khare.)

#### Lecture 3

Title: Finite $p$-groups having Schur multiplier of maximum order

Speaker: Sumana Hatui

The concept of Schur multiplier of a group G was introduced by Schur in 1904 in his studying of projective representation of groups. The Schur multiplier $M(G)$ of group $G$ is the second homology group $H_2(G; \mathbb Z)$ with integral coecients, where $\mathbb Z$ is regarded as trivial $G$-module. In 2009, Niroomand gives an upper bound on the order of $M(G)$ for non-abelian $p$-groups $G$ of order $p^n$ having derived subgroup of order $p^k$, which is the following $|M(G)| = p^{\frac{1}{2} (n+ k-2)(n-k-1) +1 }$

In this talk I will discuss about the p-groups G for which $|M(G)|$ attains this maximum bound.

#### Lecture 4

Title: Hopf-cyclic cohomology of $H$-categories

Speaker: Mamta Balodi

Connes introduced cyclic cohomology of algebras as an extension of de Rham homology of manifolds in the noncommutative setup. Later a general framework for cyclic theory of algebras on which a Hopf algebra $H$ acts/coacts was provided by Hajac, Khalkhali, Rangipour and Sommerhäuser. This general setup is referred to as the Hopf-cyclic theory. The cyclic cohomology of linear categories was defined by McCarthy. In this talk, we will discuss the Hopf-cyclic cohomology of an $H$-category. An $H$-category may be seen as an “$H$-module algebra with several objects” in the sense of Mitchell. By extending Connes’ original construction of cyclic cohomology, we will interpret the cocycles and the coboundaries as characters of differential graded $H$-categories equipped with closed graded traces. This is joint work with Abhishek Banerjee.

#### Lecture 5

Title: Group action on some non-commutative spaces

Speaker: Safdar Quddus

Some of the natural/famous (classical)group actions on classical manifolds do extend to the associated non-commutative spaces. We shall talk about these spaces and study their Hochschild and (periodic) cyclic (co)homology.

#### Lecture 6

Title: On entwined modules over linear categories

Speaker: Samarpita Ray

In classical Hopf algebra theory, one of the main objects of interest has been modules and comodules of a Hopf algebra with some compatibility condition. These are commonly known as relative Hopf modules. Relative Hopf modules were futher generalized to the widely studied Doi-Hopf modules, which are modules of an algebra and comodules of a coalgebra with a compatibility condition controlled by a bialgebra. However, lately it was shown that the background bialgebra is redundant given that some “entwining” conditions are imposed between the algebra and the colagebra. Brzezinski and Majid introduced the notion of entwining structures and it was realized that entwined modules provide a very clear formalism for understanding Doi-Hopf modules. In this work, we introduce a categorical generalization for entwining structures and study Frobenius and separability conditions for functors on entwined modules. (Jointly with Dr Mamta Balodi and Prof. Abhishek Banerjee)

#### Lecture 7

Title: Stabilized finite element methods for convection-diffusion problem

Speaker: Asha Dond

The standard Galerkin finite element methods fail to provide a stable and non-oscillatory solution for the convection-dominated diffusion problems. We develop patch-wise local projection stabilized conforming and nonconforming finite element methods for the convection-diffusion problems. It is a composition of the standard Galerkin finite element method, the patch-wise local projection stabilization and weakly imposed Dirichlet boundary conditions on the discrete solution. We study a priori and a posteriori error analysis for this patch-wise local projection stabilization. The numerical experiments confirm the efficiency of the proposed stabilization technique and validate the theoretical convergence rates. This is joint work with Thirupathi Gudi.

#### Lecture 8

Title: Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

Speaker: Gouranga Mallik

I will discuss a unified framework for goal-oriented a posteriori estimation covering in particular higher-order conforming, nonconforming, and discontinuous Galerkin finite element methods. This is a joint work with Martin Vohralik and Soleiman Yousef. The considered problem is a model linear second-order elliptic equation with inhomogeneous Dirichlet and Neumann boundary conditions and the quantity of interest is given by an arbitrary functional composed of a volumetric weighted mean value (source) term and a surface weighted mean (Dirichlet boundary) flux term. We specifically do not request the primal and dual discrete problems to be resolved exactly, allowing for inexact solves. Our estimates are based on $\boldsymbol{ H}({\rm div})$-conforming flux reconstructions and $H^1$-conforming potential reconstructions and provide a guaranteed upper bound on the goal error. The overall estimator is split into components corresponding to the primal and dual discretization and algebraic errors, which are then used to prescribe efficient stopping criteria for the employed iterative algebraic solvers.

#### Lecture 9

Title: Loop product on level homology

Speaker: Arun Maiti

In the early 21st century D. Sullivan introduced loop product and coproduct on homology of free loop space of a closed Riemannian manifold $M$. In this talk we will see how the loop product and coproduct on level homology can be used to answer some of the questions about closed geodesics on $M.$

#### Lecture 10

Title: Factorizations of Contractions

Speaker: Srijan Sarkar

The celebrated Sz.-Nagy and Foias theorem asserts that every pure contraction is unitarily equivalent to an operator of the form $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ where $\mathcal{Q}$ is a $M_z^*$-invariant subspace of a $\mathcal{D}$-valued Hardy space $H^2_{\mathcal{D}}(\mathbb{D})$, for some Hilbert space $\mathcal{D}$.

On the other hand, the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exist a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_{\Phi}, M_{\Psi})$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where $\Phi(z)=(P+zP^{\perp})U^*\quad \text{and} \quad \Psi(z)=U(P^{\perp}+zP) \quad \quad (z \in \mathbb{D}).$

In this context, it is natural to ask whether similar factorization results hold true for pure contractions. In this talk we will answer this question. More particularly, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$. Then $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exist $\mathcal{B}(\mathcal{D})$-valued polynomials $\varphi$ and $\psi$ of degree $\leq 1$ such that $\mathcal{Q}$ is a joint $(M_{\varphi}^*, M_{\psi}^*)$-invariant subspace and

$P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\varphi \psi}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\psi \varphi}|_{\mathcal{Q}} \ and\ (T_1, T_2) \cong (P_{\mathcal{Q}} M_{\varphi}|_{\mathcal{Q}}, P_{\mathcal{Q}} M_{\psi}|_{\mathcal{Q}}).$ Moreover, there exist a Hilbert space $\mathcal{E}$ and an isometry $V \in \mathcal{B}(\mathcal{D}; \mathcal{E})$ such that

$\varphi(z) = V^* \Phi(z) V \ and \ \psi(z) = V^* \Psi(z) V \quad \quad (z \in \mathbb{D}),$ where the pair $(\Phi, \Psi)$, as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on $H^2_{\mathcal{E}}(\mathbb{D})$. As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions. This is a joint work with Jaydeb Sarkar and Bata Krishna Das.

### Day 2: Friday, March 29 2019.

#### Lecture 1

Title: Percolation in enhanced random connection model.

Speaker: Sanjoy Kumar Jhawar

We study phase transition and percolation at criticality for the enhanced random connection model (eRCM). The model is an extension of the random connection model (RCM). The RCM is a random graph whose vertex set is a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. The vertices at $x,y\in \mathcal{P}\_{\lambda}$ are connected with probability $g(|x-y|)$ independent of everything else, where $g:[0,\infty) \to [0,1]$ and $| \cdot |$ is the Euclidean norm. The eRCM is obtained by considering two points to be neighbours if there is an edge between them in the RCM or the edges emanating from them in the RCM intersect. We derive conditions on $g$ so that the eRCM exhibits a phase transition, that is, there exists a $\lambda_c\in (0,\infty)$ such that for $\lambda > \lambda_c$ there exists an infinite connected component in the graph and for $\lambda < \lambda_c$ no percolation occurs. We derive a condition on $g$ under which no percolation occurs at criticality.

#### Lecture 2

Title: Risk-sensitive ergodic control of reflected diffusion processes in orthant.

We study risk-sensitive ergodic control problem for controlled diffusion processes in the nonnegative orthant. We consider ergodic cost evaluation criteria. Under certain assumptions we first establish the existence of a solution of the corresponding HJB equation. In addition, we completely characterize the optimal control in the space of stationary Markov controls.

#### Lecture 3

Title: Transcendence of generalized Euler-Lehmer constants.

Speaker: Sneh Bala Sinha

In this article, we study the arithmetic properties of generalized Euler–Lehmer constants. We show that these infinite family of numbers are transcendental with at most one exception.

#### Lecture 4

Title: Fundamental Fourier coefficients of Siegel modular forms of half-integral weight.

Speaker: Abhash Kumar Jha

A Siegel modular form has a Fourier series expansion and the Fourier coefficients are supported on the set of symmetric, positive definite and half-integral matrices. It is natural to ask the following question; does there exists a proper subset of the set of symmetric, positive definite and half-integral matrices which determines the Siegel modular forms. In this talk we shall give a survey of recent developments on this topic and give an affirmative answer to this question in the case of Siegel modular forms of half-integral weight.

#### Lecture 5

Title: Bounds on sup-norm of Siegel modular forms.

Speaker: Hariram krishna

In this talk we discuss the sup-norm problem in the context of Siegel cusp forms. For the analogous case of elliptic modular forms, optimal bounds have been found in the 90’s and numerous results on level aspects, spanning the past 2 decades. Recently there are results for Siegel cusp form that are Saito-Kurokawa lifts. We wish to provide a reasonable bound for a generic Siegel cusp form in terms of its weight for a fixed arbitrary genus. This work uses an analogue of Peterson trace formula for getting bounds on Fourier coefficients and a counting argument for matrices in the index set of the Fourier expansion for which contribution is significant, and computations with the corresponding Bergman kernel.

#### Lecture 6

Title: Fourier coefficients determining Hermitian cusp forms.

Speaker: Pramath Anamby

Recognition results for modular forms has been a very useful theme in the theory. We know that the Sturm’s bound, which applies quite generally to a wide class of modular forms, says that two modular forms are equal if (in a suitable sense) their ‘first’ few Fourier coefficients agree. Moreover, the classical multiplicity-one result for elliptic newforms of integral weight 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 $f_1=f_2.$

However, when one moves to higher dimensions, say, to the spaces of Siegel modular forms, Hermitian modular forms etc, the situation is drastically different But one can still ask the question whether a certain subset, especially one which consists of an arithmetically interesting set of Fourier coefficients.

In this talk we prove 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.

#### Lecture 7

Title: On signs of Hecke eigenvalues of modular forms.

Speaker: Ritwik Pal.

Eigenvalues of Hecke eigenforms are of considerable interest to number theorists, in particular their distribution (e.g. with respect to Sato-Tate measure) , their magnitude (Ramanujan-Petersson conjecture) and more recently study of their signs have been the focus of intensive research. Whereas the first two topics above are classical with a long history, the aspect about signs has become very popular now a days and indeed some striking results concerning them have been proved. In this talk, I will briefly introduce modular forms (elliptic modular form and Siegel modular form of genus 2) and the Hecke operators (certain linear operators) acting on them. I will discuss the history of known results about the signs of Hecke eigenvalues of certain sets of modular forms, called Hecke eigenforms. Finally I will state a recent result of mine (with prof. Soumya Das) in this line of research.

#### Lecture 8

Title: A construction of homogeneous operators on domains in $\mathbb C^{ {m}}$

Speaker: Soumitra Ghara

Starting from a scalar valued positive definite kernel $K$ on a domain $\Omega$ in $C^m$, we construct a new kernel $\mathbb K$ on $\Omega$ taking values in $m\times m$ complex matrices. We then obtain a realization of the Hilbert space $(\mathcal H, \mathbb K)$ determined by the kernel $\mathbb K$. Finally we show that if the multiplication tuple on $(\mathcal H, K)$ is homogeneous with respect to the group $\rm{Aut} (\Omega)$, then so is the multiplication tuple on $(\mathcal H, \mathbb K).$ This is a joint work with Gadadhar Misra.

#### Lecture 9

Title: A study of the norm attainment set of a bounded linear operator between Banach spaces.

Speaker: Debmalya Sain

In this talk I would try to present an overview of the operator norm attain- ment problem in the context of Hilbert spaces and Banach spaces. As we will

see, the concepts of Birkhoff-James orthogonality and semi-inner-products (s.i.p) play an important role in characterizing the norm attainment set of a bounded linear operator between Banach spaces. I would also discuss the Birkhoff-James orthogonality of bounded linear operators between Banach spaces and some of its applications towards obtaining characterizations of Euclidean spaces. Finally, as a natural continuation of the discussed topics, I would like to present some interesting open problems in the geometry of Minkowski spaces.

#### Lecture 10

Title: Von Neumann’s inequality for operator-valued multishifts

Speaker: Surjit Kumar

The von Neumann’s inequality says that if $T$ is a contraction on a Hilbert space $\mathcal H$, then $\|p(T)\| \leq \sup_{|z|<1} |p(z)|$ for every polynomial $p$. Generalizing this result, Sz.-Nagy proved that every contraction has a unitary dilation. Later Ando extended this result and showed that every pair of commuting contractions dilates to a pair of commuting unitaries. Surprisingly, it fails for a $d$-tuple of commuting contractions with $d \geq 3$. Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights dilates to a tuple of commuting unitaries, and hence satisfies von Neumann’s inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In particular, we show that if $A$ and $B$ are commuting contractive $d$-tuples of operators such that $B$ satisfies the matrix-version of von Neumann’s inequality and $(1, \ldots, 1)$ is in the algebraic spectrum of $B$, then the tensor product $A \otimes B$ satisfies the von Neumann’s inequality if and only if $A$ satisfies the von Neumann’s inequality. We also exhibit several families of operator-valued multishifts for which the von Neumann’s inequality always holds.

This is a joint work with Rajeev Gupta and Shailesh Trivedi.

#### Lecture 11

Title: Meromorphically normal families and a meromorphic Montel-Carathéodory theorem

Speaker: Gopal Datt

Normality is a notion of sequential compactness in the space of holomorphic functions, and more generally, holomorphic mappings with values in general complex spaces. In this talk, normality and related notions such as quasi-normality and meromorphic normality in one and higher dimensions will be discussed. We shall also discuss some sufficient conditions of meromorphic normality for families of meromorphic mappings taking values in a complex projective space. As a consequence of these sufficient conditions we shall, finally, see a meromorphic version of the Montel-Carath[é]{}odory theorem.

#### Lecture 12

Title: The continuous extension of complex geodesics

Speaker: Anwoy Maitra

In this talk we will give a quick introduction to the Kobayashi (pseudo) distance, which is an intrinsic, biholomorphically invariant distance on every complex manifold, and which is very useful in complex analysis. A natural object of study is isometries with respect to the Kobayashi distance. Of particular importance are holomorphic isometries between the unit disk and a given complex manifold; these are called complex geodesics. An important question is whether complex geodesics (assuming they exist) extend continuously (or in a more regular manner) to the boundary of the disk. A famous result by Lempert shows that convex domains admit complex geodesics. We discuss a few known results on the boundary-regularity question, state a conjecture, and then present a new result for convex domains.

Acknowledgements: Thanks to Hassain M for his help in this compilation.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 17 Sep 2019