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Amrinder Kaur
Let L_{G}(s) denote the L-function attached to a particular form G on GL(6). We establish an upper bound for the mean square estimate of Rankin–Selberg L-function L_{G×G}(s) on the critical line. This result is then utilized to give an asymptotic formula for the discrete sum of coefficients of L_{G×G}(s).
Anand Chitrao
Let p be a prime. The main objects in this talk will be finite dimensionl representations of the absolute Galois group of a p-adic number field in vector spaces over a (possibly different) p-adic number field. These 'p-adic' representations are more difficult to understand than their 'complex' counterparts, i.e., representations in vector spaces over the complex numbers. Examples of such representations include the Tate module of an Elliptic curve at a prime p (after inverting p). In this talk, we will see a method to classify such representations. This method involves considering representations of a simpler group Γ ≃ Zp* with an extra operator φ over a more complicated field.
Ashok Kumar
In this talk, we will introduce the Funk-Finsler structure in the unit disc of the hyperbolic Klein model and infer that it is a Randers metric. Further, we compute certain geometric quantities of this Funk-Finsler metric, namely, the Riemann curvature, Ricci curvature and the flag curvature in the Klein unit disc. Finally, we show that this metric space is a Cartan-Hadamard manifold.
Arkadeepta Roy
The Gauss-Bonnet theorem is a cornerstone of differential geometry, capturing a deep and elegant relationship between local geometric quantities and global topological invariants. For a compact, oriented Riemannian 2-manifold, the classical theorem asserts that the total Gaussian curvature is proportional to the Euler characteristic, intertwining curvature with topology in a way that is both surprising and powerful.
In this talk, we will first revisit the classical version of the theorem, emphasizing the geometric intuition and analytic underpinnings behind its proof via differential forms and the exterior derivative. We will then discuss generalizations to manifolds with boundary, as well as higher-dimensional analogues such as the Chern-Gauss-Bonnet theorem, where curvature forms and characteristic classes come into play.
Bina Jha
Given a finite interval and some subset of Z/p^{nu}Z for ν ≥ 1, Large sieve inequality for prime powers, due to A. Selberg provides an estimate for the cardinality of the number of integers from given interval whose reduction mod p^{\nu} does not falls in any of the subsets of residues mod p^{\nu}, {\nu} ≥ 1 for each prime p of our interest. The proof for this inequality includes the use of mulplicative functions of two variables, however in this talk we give a simple proof avoiding the multiplicative functions and by extending the method originally used by A. Selberg for the special case of this inequality involving only sieving by primes.
Dilip Kumar Sahoo
Multiple zeta values have been studied extensively in recent years as their appearance in many branches of mathematics and mathematical physics. In this talk we will prove that these values are appeared as a series in the coefficients of Laurent series expansion of one variable Beta function function B(s, 1/2). As a consequence, we are able to calculate the values of these series recursively.
Lehmer's problem and inertia degrees of primes
Lehmer's problem and inertia degrees of primes
Gorekh Prasad
Mahler measure of an algebraic integer α, denoted by M_α, is the product of all the conjugates of α that lies outside the unit circle of the complex plane. One of the longstanding open problems related to the Mahler measure is the Lehmer's problem, which asks for an absolute constant c>1 such that M_α ≥ c for any nonzero algebraic integer α which is not a root of unity. Though this problem has been verified for various classes of algebraic integers, including the class of nonreciprocal algebraic integers, the general case remains open. In this talk, after giving an overview of Lehmer's problem, we shall discuss our results on Lehmer's problem under various splitting conditions of primes. This is a joint work with Shanta Laishram.
Muskan Bansal
Let E be an elliptic curve defined over ℚ having split multiplication reduction at a fixed rational prime p > 3. Denote by L(E/ℚ,s) the Hasse-Weil L-function of $E/ℚ $ and assume that ord_{s=1} L(E/ℚ ,s) = 1. In this case, using Kato's reciprocity law and the results of Kolyvagin and Gross-Zagier, we get two elements in the Bloch-Kato Selmer group. In this talk, we discuss Perrin-Riou's main conjecture which relates these two elements via a logarithm map.
Categorical semantics of compositional reinforcement learning
Categorical semantics of compositional reinforcement learning
Onkar Kale
Reinforcement learning (RL) systems often struggle with scalability, interpretability, and safety in complex tasks. In this talk, we will discuss a categorical framework for compositional reinforcement learning, where tasks are modeled as objects in a category of Markov Decision Processes (MDPs). Using categorical constructs like pullbacks and pushouts, we formalize how tasks can be decomposed, recombined, and verified in a modular fashion.
Rational functions on Projective line and Bilinear forms
Rational functions on Projective line and Bilinear forms
Rakesh Pawar
I will discuss how to relate rational functions on the Projective line over a field k up to 'algebraic deformations' with symmetric matrices/bilinear forms over k. I will mention some recent related results (joint with Frederic Deglise) and related open questions at the end.
Rithwik M R
For a given finite abelian group G, we will introduce the analogue of Fourier series on G, establish the concept of Fourier bias, state key results on Fourier bias of a multiplicative subgroup of G. Further, we will use the above results to establish the existence of bases of a given order for Z/pZ.
Rohit Pokhrel
The Fargues-Fontaine curve has been a fundamental object of study since its discovery in 2010, particularly through the work of Laurent Fargues and Peter Scholze, whose results show that it is sufficient to formulate the Geometric Langlands program over a p-adic field on the moduli space of vector bundles over the Fargues-Fontaine curve. In this talk, we will discuss perfectoid fields and their tilting, which is a functor that maps a perfectoid field to a perfectoid field of characteristic~p. This functor is not an equivalence of categories; in fact, there are multiple perfectoid fields that are tilts to a single perfectoid field. We will see that all these perfectoid fields can be viewed as points of a projective scheme called the Fargues-Fontaine curve, which is analogous to the open unit disc {z ∈ ℂ | |z|<1} in ℂ . If time permits, we will also discuss our recent work in multivariate setups, which suggests the existence of a projective scheme analogous to the polydisc {(z_1,...,z_n) ∈ℂ^{n} | |z_i|<1, 1≤i≤ n\} in ℂ^{n} , for n ≥ 1 .
Optimal transport and P.D.E
Optimal transport and P.D.E
Saumyajit Das
Brief overview of Optimal Transport theory and how Monge-Ampere is related to Optimal transport.
Characterization of constant curvature spaces
Characterization of constant curvature spaces
Shubham Yadav
In this talk, I will present the Killing-Hopf Theorem, which states that complete, simply connected Riemannian manifolds of constant sectional curvature are precisely the Euclidean space, the n-sphere, and hyperbolic space.
Selvam V
In this talk, we will discuss the Bloch-Kato Conjecture. Understanding the arithmetic nature of special values of complex L-functions associated to algebraic varieties, motives, or automorphic representations over global fields, as well as connecting these L-values to the orders of related algebraic structures like Chow groups or Selmer groups, is a central problem in number theory. BSD Conjecture is an example and Bloch-Kato Conjecture is a generalization. I will give an overview of the required theory of Geometric Galois representations , Bloch-Kato Selmer groups and L-functions. We will discuss the level-1 conjectures required for the Bloch-Kato Conjecture to be stated. Finally I will state the Conjecture and discuss its implications in special cases. We will restrict to characteristic 0 case.
Sudip Karmakar
In number theory, we study how numbers can be broken into prime factors. In number fields, this kind of factorization may fail, and the class number helps measure how far we are from unique factorization. An important result tells us that, for any number field, the class number is always finite. In this talk, we will explain what this means, introduce key ideas such as ideal class groups and Minkowski’s bound, and give an outline why the class number must be finite.
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