Ajay Nair

When is a homotopy equivalence of surfaces homotopic to a homeomorphism?

Abstract. The Borel Conjecture states that any homotopy equivalence between aspherical closed manifolds is homotopic to a homeomorphism. The results like Mostow-Rigidity and Dehn-Nielsen-Baer theorem affirm this statement in certain special cases. We look at this conjecture for the case of surfaces (may not be closed). We characterize the homotopy equivalence of surfaces with the help of Goldman Bracket, which is a Lie Algebra structure on the set of free homotopy classes of closed curves on the surface.

Abhishek Khetan

Improved Upper Bounds for Trifferent Codes

Abstract. A ternary code of blocklength $n$ is any collection of $n$-length strings over a $3$-letter alphabet. The elements in a code are called codewords. A ternary code is said to be trifferent if for any three distinct codewords we can find an index where the three codewords are pairwise different. A long standing open problem in coding theory is to obtain optimal bounds for the size of a trifferent code. A classical 'pruning argument' shows that any trifferent code of blocklength $n$ cannot have more than $2 \times (1.5)^n$ codewords. Recent advances improved the constant term to $0.6$. In a joint work with Siddharth Bhandari we obtain an upper bound of $n^{-2/5} \times (1.5)^n$.

Chandan Pradhan

Existence of Isometric Embedding of $S_q^m$ into $S_p^n$: The Singular Solution.

Abstract. In this talk, we will discuss a fundamental problem regarding the isometric embedding of finite-dimensional Schatten classes. This discussion is based on a portion of my collaborative work published in "J. Funct. Anal. 282 (2022), no. 1, Paper No. 109281".

Karambir Das

The realm of Random Geometric Graphs

Abstract. Random Geometric Graphs (R.G.G.) have gained popularity after their initial introduction by Gilbert in 1961. It has varied applications in network systems, virus spreading, epidemiology and many more. Imagine points are scattered around on a plane, and there is an edge between them if they are within some fixed distance $r$. We call it the Geometric Graph. When the underlying points are random, we call it R.G.G. Asymptotic behaviour of random functionals, such as the number of vertices of fixed degree, number of connected components, etc, are important to study, which would be the main focus of this talk.

Jnaneshwar Baslingker

Log-concavity of top rows of Young diagrams.

Abstract. An open problem in combinatorics, due to Chen, states that the distribution of the longest increasing sub-sequence of a random permutation is log-concave. Using RSK correspondence and the connection to Meixner ensemble, we show that the distribution of top row of Young diagrams under Poissonized Plancherel measure is log-concave, which is a Poissonized version of Chen's conjecture. This is an ongoing joint work with Manjunath Krishnapur and Mokshay Madiman.

Purba Banarjee

The Math in Math Finance

Abstract. In this talk we delve into the mathematical modelling of stock prices, starting from the discrete case of binomial modelling to the continuous version of a Geometric Brownian motion. We derive the famous Black-Scholes pricing equation for valuation of financial derivatives through a risk-neutral argument. We prove the Girsanov Theorem that ensures a change of measure from the real-world probability measure to the risk-neutral measure.

Debaprasanna Kar

Weighted boundary limits of the Kobayashi--Fuks metric on h-extendible domains

Abstract. The Ricci curvature of the Bergman metric on a bounded domain $D \subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1} g_D)$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_D$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a K\"ahler metric on $D$ known as the Kobayashi--Fuks metric. We will study the boundary behavior of the Kobayashi--Fuks metric on the class of h-extendible domains. In particular we derive the non-tangential boundary asymptotics of the Kobayashi--Fuks metric and its Riemannian volume element by the help of some maximal domain functions and then using their stability results on h-extendible local models.

Digjoy Paul

On $q,t$- Catalan polynomials

Abstract. Consider an even number of people seated at a round table. How many ways can simultaneous handshakes occur without anyone's arms crossing? These are the classical Catalan numbers named after Belgian mathematician Eugene Catalan. The $n$-th Catalan number also counts the plane trees on $n+1$ vertices, Dyck paths (specific lattice paths) of length $2n$, etc. In 2003, James Haglund and Mark Haiman introduced a two-parameter family of polynomials, known as $q,t$-Catalan polynomials , defined as a 'weighted' sum of all Dyck paths of length $2n$. It is still an open problem to give a combinatorial proof of its symmetry in $q$ and $t$. In this talk, we also sketch how q,t-Catalan polynomials connect to Algebra, particularly Representation theory.

Hiranya Dey

On the order sequence of a group

Abstract. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We will discuss some recent results about this poset. This is based on a joint work with Peter Cameron.

Vineeth Chintala

Two problems in Graph colouring.

Abstract. This is an expository talk, introducing some fundamental problems in Graph colouring.

Agniva Chatterjee

Laplace transforms of Hardy-space functions on some (weakly) convex domains.

Abstract. The Polya-Ehrenpreis-Martineau theorem characterizes entire functions of exponential type that occur as (Fourier-)Laplace transforms of analytic functionals supported on a compact convex set in ${\mathbb C}^n$. Such Paley--Weiner-type results have also been obtained for analytic functionals induced by (holomorphic) Hardy-space functions on convex domains. In this case, the range of the Laplace transform has been completely characterized as a weighted Bergman space of entire functions in two different cases: for arbitrary convex domains in $\mathbb C$ by Lutsenko-Yumulmukhametov, and for strongly convex domains in ${\mathbb C}^n$ by Lindholm. The common feature of these two classes of domains is the $L^2$-boundedness of the (Cauchy-)Leray transform, which appears to be deeply connected to the Laplace transform. We extend the above-mentioned Paley--Wiener results to a class of (weakly) convex Reinhardt domains in ${\mathbb C}^2$. The Leray transform was previously studied on this class by Barrett--Lanzani.

Ritvik Saharan

Ribbon Concordance of Knots is a partial order.

Abstract. In this talk, we shall go through Ian Agol's proof of a conjecture by Gordon which held that ribbon concordances are partially ordered. We first note that a concordance between knots $K_0$ and $K_1$ is a smooth embedding of an annulus $C = e( S^1 \times [0,1] )$ in $S^3$ with $e( S^1 \times \{0\} )$ being isotopic to $K_0$ and $e(S^1 \times \{1\} )$ being isotopic to $K_1$ . Now we project it to $[0,1]$ and if this turns out to be a Morse function with critical points having an index of $0$ or $1$ only, then we say that $C$ is a ribbon concordance from $K_1$ to $K_0$ which may also be written as $K_1 \geq K_0$ . By considering the representation varieties of the knot groups to $SO(n)$ and the relations between them, the partial ordering of ribbon concordance shall be shown.

Arnab Pal

Convergence of an Adaptive Finite Element Method for a General Second Order Elliptic PDE

Abstract. In this presentation, we delve into the convergence properties of Adaptive Finite Element Methods (AFEM) applied to partial differential equations (PDEs) of the form: \begin{align} \mathcal{L}u:=-\nabla \cdot(A\nabla u)+ b\cdot\nabla u +cu &= f\ \text{in} \ \Omega\\ u &=0 \ \text{on} \ \partial\Omega, \end{align} subject to appropriate assumptions on the coefficients $A$, $b$, $c$, and the term $f$. We assume the solution to be $H^2$-regular while the dual problem is only $H^1$-regular.

Initially, we focus on confirming triangulation and employing Lagrange finite elements to construct finite element spaces. Following this, we discuss the well-posedness of the discrete problem and derive an a priori error estimate. Subsequently, we will talk about the AFEM framework for the given PDE. We explore a posteriori error estimates, analyzing both reliability and efficiency aspects. Next, using techniques such as estimator reduction and quasi-orthogonality, we establish the convergence of AFEM for the PDE. Theoretical findings are illustrated by numerical experiments.