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Lecture Series 2022

Table of Contents

LS2205: Topological Groups and Their Representations #

Number of Talks: 7

Target

In the following talks, we will discuss the representation theory of topological groups, building from the familiar theory for finite groups. We begin by introducing representations of finite groups and character theory for finite groups. We then look at topological groups and their elementary properties, also looking at some of their elementary representations. Following this, the Haar measure will be introduced along with the existence and uniqueness of the Invariant Haar Mean on compact groups. We then proceed to look at amenable groups, and more specifically compact groups, and study their representations using the Haar mean, to obtain familiar results, such as those in the theory of finite groups. We finally proceed to discuss the Peter-Weyl Theorems.

Schedule

  • Talk 1: Introduction To The Representation Theory Of Finite Groups
    Speaker: Rahul Mazumdar (B. Math, 2023)
    Abstract: In this talk, we will define representations, provide some examples and discuss some fundamental results like Schur’s lemma and Mashcke’s theorem.
    Date and Time: Saturday, 8th October 2022, 11:00 AM - 12:30 PM

  • Talk 2: Character Theory For Finite Groups
    Speaker: Rahul Mazumdar (B. Math, 2023)
    Abstract: In this talk, we will discuss characters of representations and see how they encode information about representations. The primary aim of this talk will be to cover Schur’s Orthogonality Relations.
    Date and Time: Sunday, 9th October 2022, 11:00 AM - 12:30 PM

  • Talk 3: Introduction to Topological Groups
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: We introduce topological groups and look at some of their fundamental properties. We proceed to topological homomorphisms and isomorphisms and look at some consequences of the separation axioms induced due to the topology on these groups.
    Date and Time: Saturday, 15th October 2022, 11:00 AM - 12:30 PM

  • Talk 4: Representations of Topological Groups
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: We introduce the notion of a continuous representation, after which we proceed to discuss the one-dimensional representations of some topological groups, in particular those of \(\mathbb R\) and \(\mathbb{S}^1\). We introduce the matrix exponential and then discuss finite dimensional representations of topological groups. We introduce compact groups and discuss some representations of compact groups.
    Date and Time: Sunday, 16th October 2022, 11:00 AM - 12:30 PM

  • Talk 5: Haar Measures on Groups
    Speaker: Aprameya Girish Hebbar (B. Math, 2023)
    Abstract: In this talk, we will discuss Haar measures, which are measures on locally compact topological groups. We’ll talk about the existence and uniqueness of such measures. Some explicit Haar measure constructions for various important classes of groups will be explored. We’ll wrap up with a brief look at modular functions and unimodular groups.
    Date and Time: Friday, 21st October 2022, 5:30 PM - 7:00 PM

  • Talk 6: Compact and Amenable Groups
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: We will introduce amenable groups, and using the Haar Mean, we will see how some familiar results from the representation theory of finite groups can be extended to amenable groups, such as Schur’s lemma. Following this, we will look at the character theory for amenable groups.
    Date and Time: TBA

  • Talk 7: The Peter-Weyl Theorems
    Speaker: Rahul Mazumdar (B. Math, 2023)
    Abstract: We will look at the Peter-Weyl Theorems, also taking a look at class functions. We will finally introduce square integrable class functions for compact groups.
    Date and Time: TBA

Venue: G-26 Classroom, Academic Building
Videos: Available here.

LS2204: Certain Inequality Estimate of Trace Class Hyponormal Operator #

Number of Talks: 2

  • Speaker: Sagar Ghosh (M. Math, 2023)
  • Abstract: I will briefly introduce the Hyponormal Operator, Some Properties, and the definition of the Trace Class Operator. I will also try to give a brief review of the Different Norms of Operator and their relations. And then I will move to the Berger-Shaw Theorem (the main topic of my lecture). If time permits, I will introduce Putnam’s Inequality.
  • Pre-requisites: Familiarity with Basic Functional Analysis and Complex Analysis will be sufficient.
  • Date and Time:
    • Saturday, 1st October 2022, 10:30 AM - 12:00 PM
    • Sunday, 2nd October 2022, 10:30 AM - 12:00 PM
  • Venue: G-26 Classroom, Academic Building

LS2203: Convex Sets and Convex Functions: A Primer #

  • Speaker: Sarvesh Ravichandran Iyer (SRF)
  • Abstract: Beginning from convex sets and convex functions, we will prove Krasnoselskii’s theorem , discuss some of the regularity results around convex sets and functions, and have some glimpses into Mixed Volumes and Minkowski’s Inequality.
  • Notes: Available here
  • Date & Time:
    • Saturday, 24th September 2022, 2:00 PM - 4:00 PM
    • Sunday, 25th September 2022, 2:00 PM - 4:00 PM
  • Venue: G-26 Classroom, Academic Building
  • Videos: Available here.

LS2102: Metric Spaces, Isometry Group and Arzelà-Ascoli Theorem #

Number of Talks: 2

  • Talk 1: Metric Spaces and Isometry Group
    Speaker: Soumya Dasgupta (B. Math, 2024)
    Abstract: This talk is going to be a brief introduction to metric spaces. We will look at some examples and interrelation between metric spaces and normed linear spaces, and isometry group of a metric space. Finally, we will classify the isometry group of \(\mathbb{R}^n\) under the \(L^p\) norm.
    Date and Time: Saturday, 10th September 2022, 10:30 AM - 12:30 PM

  • Talk 2: Arzelà-Ascoli Theorem
    Speaker: Prognadipto Majumder (B. Math, 2024)
    Abstract: Brief review of notions of separation, dense sets in metric spaces, and Bolzano Weierstrass theorem. Introduction of the notion of convergence of functions and viewing the space of functions as a metric space. Equicontinuity and Arzelà-Ascoli Theorem (main result of the talk).
    Date and Time: Sunday, 11th September 2022, 10:30 AM - 12:30 PM

Venue: G-26 Classroom, Academic Building
Videos: Available here.

LS2101: Matrix Groups #

Number of Talks: 2

  • Talk 1: The Topology Of Matrix Groups
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: We will introduce Matrix groups in this talk. We will discuss various topological properties of matrix groups, like connectedness and simple connectedness. Some special groups of low order are also discussed. We motivate the definition of the Lie Group.
    Date and Time: Saturday, 3rd September 2022, 11:00 AM - 12:30 PM

  • Talk 2: Matrix Groups, and Lie Algebras
    Speaker: Aprameya Girish Hebbar (B. Math, 2024)
    Abstract: We will start by introducing exponentiation and logarithm of matrices and explore its properties. We will discuss Lie algebras and the properties of exponentiation map on the Lie algebra of a matrix group.
    Date and Time: Sunday, 4th September 2022, 11:00 AM - 12:30 PM

Venue: G-26 Classroom, Academic Building