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[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]
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