Number Theory Seminar
Speaker: Arijit Chakraborty, Department of Mathematics, UCSD
Host: Keshav Aggarwal
Title: Counting Number Fields by Discriminant: Power-Saving Error Terms and New Directions
Time, day and date: 10:00:00 AM – 10:50:00 AM, Friday, August 14
Venue: Online (https://us06web.zoom.us/j/82517817235?pwd=afanPYlw2EJOEv7Xl0BipJKkm6jBKz.1)
Meeting ID: 825 1781 7235
Passcode: p8fAMa
Abstract: Malle's conjecture predicts the asymptotic growth of number fields with prescribed Galois groups when ordered by discriminant. In this talk, I will discuss my work on obtaining power-saving error terms for counting (C_2 \wr H)-extensions of an arbitrary number field, under mild assumptions on (H), and briefly highlight the main ideas behind the result. I will conclude with ongoing directions involving more general wreath products and Malle's conjecture for non-concentrated groups, including joint work on degree-(16 (D_8)-extensions.

Geometry and Topology seminar
Speaker: Devadatta Hegde
Host: Sudarshan Gurjar
Title: An introduction to equivariant cohomology through an example
Time, day and date: 11:30:00 AM – 12:30:00 PM, Friday, August 14
Venue: Room 215
Abstract: Equivariant cohomology was introduced by A. Borel in the late 1950s. It has since found applications in many parts of mathematics, including, more recently, automorphic forms. In this talk, we give an introduction to this topic by using it to prove the classical result in enumerative geometry that a generic cubic surface contains exactly 27 lines.

Talk
Speaker: Hariom Sharma, IIT Bombay
Host: Ravi Raghunathan
Title: Symplectic Model for Zelevinsky Modules of GL(n, D)
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, August 14
Venue: Ramanujan Hall
Abstract: Let D be a quaternion division algebra over a non-Archimedean local field F of characteristic zero. Let m = (Δ₁,..., Δₖ) be a well-ordered multisegment, and let π(m) = Z(Δ₁) × ··· × Z(Δₖ) be the associated Zelevinsky module of GL(n, D). In this talk, we study the existence and uniqueness of symplectic models for Zelevinsky modules. Under the assumption that no representation occurring in the cuspidal support of m admits a symplectic model, we prove that π(m) admits a symplectic model if and only if every segment in m has even length. We further show that, whenever such a model exists, it is unique. As a consequence, if the unique irreducible quotient Z(m) of π(m) admits a symplectic model, then every segment in m has even length.

Analysis seminar 
Speaker: Sheetal Wankhede, SRM University
Host: Prachi Mahajan
Title: Univalent harmonic functions and their applications to special functions
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, August 14
Venue: Online (https://meet.google.com/ezq-gmpb-esb)
Abstract: The theory of univalent functions has been a central area of research in geometric function theory due to its deep connections with complex analysis and its wide range of applications. This work investigates integral operators and coefficient conditions for analytic, harmonic, and logharmonic mappings, with particular emphasis on their geometric properties such as univalence, close-to-convexity, and starlikeness.

The first part of the study focuses on a Cesàro-type integral transform defined through Hornich operations. Sufficient conditions are established on the parameters of the operator to guarantee univalence in the analytic and harmonic settings. These results are further extended to logharmonic mappings, where new univalence criteria are obtained and a connection between harmonic and non-vanishing logharmonic functions is revealed.

The second part of the work develops new monotone coefficient conditions that ensure the univalence and close to-convexity of harmonic mappings. These criteria generalize several classical results from the analytic setting and provide effective tools for studying harmonic functions. As an application, the obtained conditions are employed to investigate the geometric behavior of harmonic mappings associated with Gaussian hypergeometric functions. Furthermore, new coefficient conditions characterizing starlikeness are derived and applied to shifted Gaussian hypergeometric functions.

Students' Seminar
Speaker: Aditya Khambete, IIT Bombay
Host: Suman Kumar Sahoo
Title: An elementary proof of the Strong Law of Large Numbers
Time, day and date: 5:15:00 PM – 6:00:00 PM, Friday, August 14
Venue: Room 114
Abstract: Most of you are already familiar with the Strong Law of Large Numbers. But have you ever wondered about how to prove it? The Strong Law of Large Numbers is a fundamental result in probability theory, asserting that normalized partial sums converge almost surely to the expected value. In this talk, I present Etemadi's elementary proof for identically distributed, pairwise independent random variables with finite first moment. The talk would be accessible to anyone who has completed a probability course.