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Picture Courtesy: Dhrubajyoti Das
Details About Our Speakers and Their Respective Talks
[Speaker 15: Akshay C Kharade]
Date & Day : 19 th September, Friday
Time: 3 p.m.
Venue: Madhava Hall
Title: Exploring Group Cohomology
Abstract: This presentation offers an overview of Group Cohomology and its key applications. We will delve into the group extension problem and explore how the theory extends to profinite groups. Additionally, we will discuss the classification of k-forms of algebraic groups and the existence of Frobenius-stable B-N pair, particularly for finite groups of Lie type. If time permits I will introduce Brauer groups, which play a crucial role in classifying central simple algebras.
Through this presentation, we aim to highlight the significance and versatility of Group Cohomology in various math topics.
[Speaker 14: Dr. Nirjan Biswas]
Date & Day : 5 th September, Friday
Time: 4 p.m.
Venue: Madhava Hall
Title: Strict Faber Krahn type inequality under polarization and fractional strict monotonicity over annuli
Abstract: Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that $B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in (1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with $p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let $\lambda ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-\Delta _p)^s$ in $B\setminus \overline{B'}$ with the homogeneous nonlocal Dirichlet boundary conditions. We see that $\lambda ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict monotonicity, we discuss a strict Faber-Krahn type inequality for $\lambda _{p,q}^s(\cdot )$ under polarization. Polarization is the simplest symmetrization of functions on $\mathbb{R}^d$.
[Speaker 13: Kaustabh Mondal ]
Date & Day : 21st August, Thursday
Time: 5 p.m.
Venue: Madhava Hall
Title: Introduction to Adic Space 2/2
Abstract: Continuing our discussion on adic spaces, we will define the structure sheaf on the space of continuous valuations on a Huber pair. We will describe a few standard examples of adic spaces, such as the classical rigid analytic open and closed discs. We will mostly focus on the open adic unit disc over Z_p and its analytic points, which lead to the definition of an analytic adic space.
[Speaker 12: Kaustabh Mondal ]
Date & Day : 14 th August, Thursday
Time: 5 p.m.
Venue: Madhava Hall
Title: Introduction to Adic Spaces 1/2
Abstract: The category of adic spaces can be thought of as a unification of both the category of formal schemes and the category of rigid analytic spaces. In this talk, we will define adic spaces and their structure sheaf. Parallel to the rigid open (and closed) unit disk, we will discuss the examples of the adic open (and closed) unit disk. Finally, we will see that there is a natural generic fibre functor from adic spaces over Z_p to adic spaces over Q_p, in contrast to the case of formal schemes over Z_p, due to the lack of a generic point in Spf Z_p.
[Speaker 11: Dr. Prashant Arote ]
[Speaker 11: Dr. Prashant Arote ]
Date & Day : 4 th July, Friday
Time: 4 p.m.
Venue: Seminar Hall 41
Title: The Representation Theory of SL_2(\mathbb{F}_q): A Model Case for Deligne-Lusztig Theory
Abstract: Deligne-Lusztig theory provides a geometric framework for constructing and understanding the irreducible representations of finite groups of Lie type, fundamentally linking algebraic geometry and representation theory. The group SL_2(\mathbb{F}_q) serves as a natural and instructive base case for this broader theory. In this talk, we will examine the structure and representation theory of SL_2(\mathbb{F}_q), focusing on the classification of its irreducible representations and the interplay between character theory and algebraic constructions. We will outline how key aspects of the Deligne-Lusztig approach manifest in this low-rank example, including the role of algebraic varieties and étale cohomology in producing representations. Through this lens, SL_2(\mathbb{F}_q) not only provides concrete insight into abstract concepts but also illustrates how Deligne-Lusztig theory generalizes to higher-rank groups and more intricate settings within the representation theory of finite groups of Lie type.
Date & Day : 20th June, Friday
Time: 4 p.m.
Venue: Seminar Hall 41
Title: Strichartz Estimates on Waveguide Manifolds
Abstract: In this talk, we will discuss Strichartz estimates on waveguide manifolds and their application to solving the Schrödinger equation. A waveguide can be roughly seen as a space that lies between the Euclidean case and the torus case, because it is a product of both. This means we can combine techniques used for the Euclidean case and for the torus case when we work on waveguides. Our main aim is to show how these combined methods help us solve the Schrödinger equation in this setting.
[Speaker 9: Dr. Abhishek Sarkar]
[Speaker 9: Dr. Abhishek Sarkar]
Date & Day : 4th April,2025, Wednesday
Time: 4 p.m.
Venue: Seminar Hall 41
Title of the talk : Algebra of differential operators
Abstract: In this talk, we explore differential operators from both geometric and algebraic perspectives. Beginning with the classical setting of differential operators on smooth manifolds and their connection to vector fields, we extend to Weyl algebras and their generalizations over commutative algebras through the lens of derivations. A key structural insight is the realization of these algebras as almost commutative filtered algebras, which bridges their symbolic and operator-theoretic aspects. This framework naturally leads to the theory of D-modules and their connection to the moduli space of vector bundles with flat connections. The talk offers an accessible introduction for students of geometry, algebra and mathematical physics.
[Speaker 8: Dr. Srijonee Shabnam Chaudhury]
[Speaker 8: Dr. Srijonee Shabnam Chaudhury]
Date & Day: 22nd May,2025, Thursday
Time: 3 p.m.
Location: Madhava Hall
Title of the talk : The Level Of Some Algebraic Objects
Abstract : In this expository talk, we first introduce the basic notion of an invariant called the ‘level’ over some algebraic objects (mainly over some fields and rings). After that, we try to discuss all the important progress in the study of finding the upper bound of this invariant over local and global fields and over their maximal orders. Finally, we discuss the scope of further research on this topic.
[Speaker 7: Dr. Shubham Jaiswal]
[Speaker 7: Dr. Shubham Jaiswal]
Date & Day : 8 th May, 2025, Thursday
Time: 4 p.m.
Online
Title of the talk: Minimal Generating Sets of Splitting Field & Cluster Towers.
Abstract : For an irreducible polynomial, when the Galois group is not the whole symmetric group, it implicitly says there is an interrelationship among the roots. A famous result of Galois says that for a prime degree polynomial, the splitting field is generated by two roots when the Galois group is solvable. In this talk we discuss our recent work where we want to understand how big or how small the subset of roots be, so that it generates the splitting field. By a minimal generating set we mean a generating set for which no proper subset is a generating set. We connect the concept of minimal generating sets to the concept of cluster towers.
[Speaker 6: Paramananda Das]
[Speaker 6: Paramananda Das]
Date & Day: 18th April,2025, Friday.
Time: 4 p.m.
Venue: Madhava Hall.
Title: Fractional Laplacian, Sobolev Spaces, and More
Abstract: In this talk, we will spend most of our time understanding the terms given in the title, their properties, and their usage. Finally, I will discuss my recent work arxiv.org/abs/2410.03233 . This is a joint work with Dr. Mousomi Bhakta and Dr. Debdip Ganguly.
[Speaker 5 : Dr. Darshan Nasit]
[Speaker 5 : Dr. Darshan Nasit]
Date and day : 3rd April, 2025.
Time: 4 p.m.
Venue: Madhava Hall.
Title : Dimension of the space of vector-valued cusp forms
Abstract: In this talk, we will begin by defining vector-valued cusp forms of integral weight with respect to a congruence subgroup. We will then review the classification of irreducible representations of \( \text{SL}_2(\mathbb{F}_q) \) for any odd prime \( q \geq 5 \). Finally, we will show that the dimension of the space of cusp forms with respect to the full modular group is equal to the multiplicity of the corresponding representation in the space of classical cusp forms for the principal congruence subgroup of level \( q \).
[Speaker 4 : Ayon Roy]
[Speaker 4 : Ayon Roy]
Date and day: 22nd March, Saturday
Time: 3 p.m.
Venue: Madhava Hall.
Title: Recent Developments around Word Maps on Algebraic Groups
Abstract: The surjectivity of word maps over groups has been a rapidly growing area of research in recent years. In this talk, we will explore the structure of the image of various word maps over linear algebraic groups. We will begin with a motivation for studying word maps, inspired by Ore's conjecture. Rather than focusing on proofs, the talk will primarily highlight recent developments in this field, explaining them step by step along with the underlying motivations. Key topics to be introduced include Almost Laws, Thom's phenomenon, and Borel's Dominance Theorem.
Prerequisites: Group theory and basic knowledge of affine variety.
[Speaker 3: Aakash Gupta]
[Speaker 3: Aakash Gupta]
Date and day : 7th March, Friday.
Time: 4 p.m.
Venue: Madhava Hall.
Title: Understanding the Tychonoff Theorem through Moore-Smith Sequences.
Abstract: One of the most important properties which makes Metric Spaces much simpler or elegant is that they are First Countable Spaces, it helps us to characterize a few complicated properties using sequences.
But what if our space is not First Countable? Do we have any analogue for general topological spaces.Thanks to E. H. Moore and Herman L. Smith, in 1922 they generalized the concept of sequences from the set of natural numbers to arbitrary directed sets.
In this talk we will learn about this generalization, and see how they replace the role of sequences in general topological spaces and finally we attempt to give a proof of the famous Tychonoff Theorem using this generalization.
Prerequisites: A first basic undergraduate course in Point Set Topology.
[Speaker 2: Dhruv Bhasin]
[Speaker 2: Dhruv Bhasin]
Date and day: 20th February, Thursday.
Time: 4 p.m.
Venue: Madhava Hall.
Title: The World of Union-Closed Families of Sets.
Abstract: Intersection-closed structures appear everywhere in mathematics: collections of subgroups of a group, ideals of a ring, independent sets of a graph, etc. are all closed under intersections. The dual of intersection-closed families, namely, union-closed families are widely studied in combinatorics. In this talk, I will give a brief history and survey about the literature on union-closed families of sets and talk about my work on union-closed families: understanding an operation on union-closed families of sets (called closure) and understanding interactions of union-closed families with cubical complexes and with coding theory.
[Speaker 1: Sutirtha Datta]
[Speaker 1: Sutirtha Datta]
Date and day : 4th Feb, 2025
Time: 4:00 p.m.
Venue: Madhava Hall
Title: Ping Pong with Groups
Abstract: When is a subgroup of a group free? When can we detect free groups effectively? This talk introduces an elementary method to explore these questions. If time permits, we will also discuss Tits' alternative.
Prerequisites: Basic Group Theory
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