Karambir Das

Limit Laws of Random Graphs using Stein’s method

Abstract. Charles Stein devised a groundbreaking criterion for a random variable (r.v.) to be a Gaussian random variable in 1972. How a random variable obeys this criteria would determine how close this r.v. is from the Gaussian r.v. in some suitable metric space. This leads to Stein’s bound and people tailored this bound to obtain convergence to Gaussian r.v. (CLT) i various fields of probability. My talk will focus on how this Stein’s bound can be applied in random graphs models. First, I will present how we proved CLT in Erdős–Rényi multiplex model, based on a joint work with Bhaswar Bhattacharya et al. Finally, we will explore more on this method in Random Geometric Graphs. This will be based on an ongoing work with Srikanth Iyer.

Sourish Parag Maniyar

The story of component sizes in random graphs.

Abstract. During the late 1950s and early 1960s, Paul Erdős and Alfred Rényi published a series of papers which essentially led to the genesis of what we today call Random Graph Theory. They studied the phase transition displayed by the size of the largest connected component, i.e. how its size changes with respect to the connection probability between vertices. In the subsequent years, many results were published that explored the nature of the component sizes in more detail. In this talk, we will trace this development, introducing key objects of interest and highlighting some of the central results along the way.

Mainak Ghosh

Bertini type theorems over fields and discrete valuation rings.

Abstract. The classical Bertini theorem states that, given a smooth projective variety $X$ over an algebraically closed field, almost all hypersurface sections of $X$ are also smooth. Given this, some natural questions come to mind. Does the base field have to be algebraically closed? Does the base even have to be a field? Can we replace smoothness with some other properties? In this talk, we will try to address some of these questions. We will discuss several Bertini-type theorems over fields (both finite and infinite). For finite fields, we will follow Poonen’s closed point sieve method. We will also discuss some results over discrete valuation rings. This talk is based on joint work with Prof. Amalendu Krishna.

Anantha Krishna B

On Haar Measure on Reductive Groups.

Abstract. Let $G$ be a connected reductive group defined over a local non-Archimedean field $k$. Haar measures on $G$ are left-invariant Radon measures that play a crucial role in harmonic analysis on $G$. In this talk, we discuss the construction of the Haar measure on $G$ introduced by Benedict H. Gross and Wee Teck Gan. We also briefly discuss the role of this particular measure in the formal degree conjecture, which relates a generalized notion of the degree of square-integrable irreducible representations of $G$ to values of certain $L$-functions.

Ritabrata Das

The Weil Representation and the Metaplectic Group.

Abstract. The Weil representation arises naturally in the study of the relationship between the symplectic and Heisenberg groups. It was introduced by André Weil in the 1960s in connection with his work on theta functions and quadratic forms. For a fixed central character, the Stone–von Neumann theorem ensures a unique irreducible representation of the Heisenberg group associated with a symplectic vector space over a local field. The symplectic group acts on this representation only projectively, and resolving this obstruction leads to the construction of the metaplectic double cover of the symplectic group and the Weil representation. In this talk, I will outline the construction of this cover, describe the Weil representation in the Schrödinger model, and briefly indicate its connections to theta functions and the theta correspondence.

Golam Mostafa Mondal

Approximation and Extension of CR Functions on CR Submanifolds of $\mathbb{C}^{n}$.

Abstract. A domain $\Omega \subset \mathbb{C}^{n}$ is Runge if every holomorphic function on $\Omega$ can be uniformly approximated on compact subsets by holomorphic polynomials. In the case of the complex plane, Runge domains are exactly the simply connected ones, but no such characterization exists in higher dimensions. In this talk, I will describe a class of Runge-type domains contained in certain CR submanifolds of $\mathbb{C}^{n}$. In particular, I will present a result showing that every real-analytic CR function on domains in certain CR submanifolds, which are invariant under certain holomorphic dynamics, can be uniformly approximated by holomorphic polynomials. I will also discuss some applications of this result, including a theorem on the holomorphic extension of CR functions on such domains. This is an ongoing project with Sanjoy Chatterjee

Ritvik Saharan

Geodesic divergence and quasi-redirection in certain Riemannian manifolds.

Abstract. We use the notion of quasi-redirection which was developed by Rafi and Qing to generalize the notion of Gromov hyperbolicity to a broader class of metric spaces and it in turn, induces its own bordification ( the quasi-redirecting boundary). We then show that if $(M,g)$ is a complete Riemannian manifold diffeomorphic to $\mathbb{R}^{2}$ that either $(M,g)$ is Gromov hyperbolic (iff $M$ is biholomorphic to $\mathbb{D}$) or $(M,g)$ shows at most linear divergence for geodesics (iff $(M,g)$ is biholomorphic to $C$). We note that this rigidity falls apart in higher than $3$ as one can define a Riemannian metric on $\mathbb{R}^{n}$ by considering connected sums of $S^{n}$ which makes it quasi-isometric to any locally finite one-ended graph $G$. Finally if time permits, we show that quasi-redirection is symmetric if $(M,g)$ is simply connected and has no conjugate points and briefly go through quasi-redirection in simply connected non-compact homogeneous $3$-manifolds.

Ajay Kumar Nair

Mapping class group dynamics on $PSL(2, \mathbb{C})$-character varieties.

Abstract. Let $F_{n}$ be the free group on n letters and $Out(F_{n})$ be the group of outer automorphisms of $F_{n}$. The $PSL(2,\mathbb{C})$-character variety of $F_{n}$, denoted by $\chi(F_{n})$, is defined as the quotient of $Hom(F_{n},PSL(2,\mathbb{C}))$ via the following relation: $\rho_{1} \sim \rho_{2}$ iff the closures of the orbits of $\rho_{1}$ and $\rho_{2}$ intersect. There is a natural action of $Out(F_{n})$ on $\chi(F_{n})$. Minsky defined a domain of discontinuity for this action by defining primitive-stable representations. Interpreting $F_{n}$ as the fundamental group of some punctured surface $\Sigma$, we define the character variety of $\Sigma$ as the character variety $\chi(F_{n})$ of $F_{n}$. Now, by the Dehn-Nielsen-Baer-Epstein theorem for punctured surfaces, we see that the mapping class group of $\Sigma$, denoted by $MCG(\Sigma)$, is a subgroup of $Out(F_{n})$. We consider the action of $MCG(\Sigma)$ on $\chi(F_{n})$. Following Minsky, we define simple-stable representations, which form a domain of discontinuity for this action. We prove that the holonomy of hyperbolic cone surfaces with cone angles less than $\pi$ is simple-stable. We also prove that the holonomy of hyperbolic cone surfaces with exactly one cone point is primitive-stable,answering a question of Minsky.

Ajay Prajapati

$p$-adic $L$-functions and trivial zero conjectures.

Abstract. $p$-adic $L$-functions and trivial zero conjectures Bloch-Kato conjecture is one of the fundamental open problems in number theory which relates special values of complex $L$-function to arithmetic objects of a $p$-adic geometric Galois representation $V$. Birch and Swinnerton-Dyer conjecture is such an example. Under suitable conditions,we can attach a $p$-adic analytic function to $V$ (called its $p$-adic $L$-function) which is characterized by interpolation of special values of its complex $L$-function. At some points the $p$-adic $L$-function have extra zeros which are introduced in the interpolation process.These are called trivial zeros.Relationship between $p$-adic $L$-value and complex $L$-value at such a point is very important from the perspective of both Iwasawa theory and Bloch-Kato conjectures. In our talk, we will see examples of $p$-adic $L$-functions, phenomenon of trivial zeros, and conjectures about them.

Ashutosh Jangle

Non-commutative Iwasawa Theory.

Abstract. For an odd prime $p$, one can formulate the Iwasawa main conjecture for an admissible $p$-adic Lie group $G$. This conjecture is called the Non-commutative main conjecture of Iwasawa theory. It was proven by Kakde and Ritter-Weiss independently around the same time. We aim to give an introduction to the non-commutative main conjecture. We would also give an outline of the proof due to Kakde.

Aniruddha Seal

Weak Galerkin method for time-fractional mobile/immobile diffusion equation.

Abstract. This presentation investigates a multidimensional time-fractional mobile/immobile diffusion equation involving the Caputo derivative of order $\alpha \in (0,1)$. This type of models are commonly used to describe anomalous transport processes in porous and heterogeneous media. The singularity of the time-fractional derivative near the initial point $t = 0$ poses significant computational challenges. To tackle these, we develop a hybrid numerical scheme that combines the weak Galerkin method for spatial discretization, the backward Euler method for the classical time derivative, and the non-uniform $L1$ approximation for the fractional term. We perform rigorous stability and error analyses in suitable norm to establish the reliability of the method. Numerical experiments are included in support of the theoretical findings and to illustrate the method’s effectiveness.

Pratibha Shakya

A $C^{0}$-IP method for the approximation of nondivergence form elliptic equations with Cordes coefficients.

Abstract. Nondivergence form elliptic equations with discontinuous coefficients do not generally possess a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. In this talk, we will consider a $C^{0}$-IP method for a class of these problems which satisfy the Cordes condition. We derive a priori error bounds for the problems. The theoretical results are illustrated by numerical experiments.

Susmita Das

Invariant subspaces of the compressions of the Hardy shift on some parametric spaces.

Abstract. We will discuss on the class of operators $S_{\alpha,\beta}$ obtained by compressing the Hardy shift on the parametric spaces $H_{\alpha,\beta}^{2}$ corresponding to the pair $\{\alpha,\beta\}$ satisfying $|\alpha|^{2}+|\beta|^{2} = 1$. We show, for nonzero $\alpha,\beta$, each $S_{\alpha,\beta}$ is indeed a shift $M_{z}$ on some analytic reproducing kernel Hilbert space and present a complete classification of their invariant subspaces. While all such invariant subspaces $\mathcal{M}$ are cyclic, we show, unlike other classical shifts, they may not be generated by their corresponding wandering subspaces $(\mathcal{M} \ominus S_{\alpha,\beta} \mathcal{M})$. We provide a necessary and sufficient condition along this line and show, for a certain class of $\alpha, \beta$, there exist $S_{\alpha,\beta}$-invariant subspaces $\mathcal{M}$ such that $M̸ \neq [\mathcal{M} \ominus S_{\alpha,\beta}\mathcal{M}]_{S_{\alpha,\beta}}$.