Student Talks 2024

Table of Contents

All expository talks given by students of ISI Bangalore in the year 2024 at our Math Club are listed below. #

Fall 2024

EP2418: Unifying Forms and Homology: An Introduction to de Rham’s theorem and Poincáre Duality #

Speaker: Anasmit Pal (B.Math, 3rd year)
Abstract: Differential topology and geometry owe much of their foundational structure to de Rham’s theorem and Poincaré duality. These results bridge the gap between de Rham cohomology and cohomology with compact support and singular cohomology, offering profound insights into the topology of smooth manifolds. In this talk, we will begin by recalling the essentials of differential forms, both in Euclidean spaces and on manifolds, and explore the idea of integration on manifolds. We’ll also touch on Stokes’ theorem which is a broader generalization of the one we study in multivariable calculus.
The core of the presentation will delve into different cohomology theories, such as smooth singular cohomology and cohomology with compact support. We will discuss the role of the Mayer-Vietoris sequence in these contexts, laying the groundwork for the main focus: a statement and sketch of the proof of de Rham’s theorem. This will naturally lead into a discussion of Poincaré duality for de Rham cohomology, accompanied by examples and key applications.
Notes: [TBU]
Video: [TBU]
Date and Time: Thursday, 24th October 2024, 6:00 PM - 7:30 PM
Venue: G-26 classroom, Academic Building

EP2417: Percolation and Kesten’s Theorem #

Speaker: Malav Dhaval Doshi (B.Math, 3rd year)
Abstract: This talk is intended to introduce the idea of Bernoulli percolation. Starting of with basic inequalities, we will cover Russo’s formula and introduce notion of critical probability, ergodicity, sub and super critical phases. We will prove some results regarding the sub-critical and super-critical phases concluding with Kesten’s theorem for \(\mathbb{Z}^2\) lattice.
Notes: [TBU]
Video: [TBU]
Date and Time: Saturday, 5th October 2024, 11:00 AM - 12:30 PM
Venue: G-26 classroom, Academic Building

EP2416: Trapped or Free - Understanding Random Walk Dynamics #

Speaker: Aritrabha Majumdar (B. Math, 2nd year)
Abstract: The objective of this talk is to investigate the recurrence and transience of random walks, concentrating on essential probabilistic techniques and criteria to distinguish between these behaviours. The talk will examine foundational concepts and their implications for understanding the long-term properties of random walks in various dimensions.
Pre-requisites: Knowledge of basic probability theory.
Notes: [TBU]
Video: [TBU]
Date and Time: Wednesday, 2nd October 2024, 9:00 PM - 10:30 PM
Venue: G-26 classroom, Academic Building

EP2415: Zeeman’s Collapsibility Conjecture #

Speaker: Pragalbh Kumar Awasthi (B. Math, 3rd year)
Abstract: The Dunce Hat is the simplest example of a non-collapsible polyhedron which is contractible. However, its product with the unit interval acquires enough room that it becomes collapsible. In 1963, E.C. Zeeman conjectured that the same holds for any contractible polyhedron, and the problem remains open to this date.
Formal 3-deformations of 2-polyhedra correspond to certain operations on the presentations of their fundamental groups. After a survey of results from PL topology and results related to Zeeman’s conjecture, we shall start exploring what this correspondence is and prove that the conjecture for a certain type of 2-polyhedrons implies Andrews-Curtis Conjecture with Stabilizations.
We shall then look at a few swindle-type arguments and how Mazur swindle can be used to prove Topological Schoenflies Theorem-a generalization of Jordan Curve Theorem. Finally, making use of this, we shall see a proof of the fact that Zeeman’s conjecture on collapsibility implies the famous Poincare Conjecture.
Pre-requisites: Basic algebraic topology will be assumed in proofs of some results. However, a sizable portion of the talk will be accessible to people with little background in the subject.
Notes: [TBU]
Video: [TBU]
Date and Time: Sunday, 28th September 2024, 11:00 AM - 12:30 PM
Venue: G-26 classroom, Academic Building

EP2414: On Topological Groups and Pontryagin Duality #

Speaker: Dhruba Sarkar (B. Math, 3rd year)
Abstract: Pontryagin Duality is an analogue of Fourier Transformation for Locally Compact Abelian (LCA) groups, with underlying topological structure. In this talk, we will introduce topological groups and look at some of its properties. In particular, we shall talk about Locally Compact groups and look at a version of Open Mapping Theorem for LCA groups. We will then look at the proof of Pontryagin duality by showing it for compact, discrete, compactly generated and LCA groups. We will end the talk with some applications of Pontryagin duality and a version of Structure Theorem on LCA groups.
Pre-requisites: Basic knowledge of Topology
Notes: [TBU]
Video: [TBU]
Date and Time: Sunday, 22nd September 2024, 11:00 AM - 12:30 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2413: An Elementary Proof of the Prime Number Theorem #

Speaker: Korou Khundrakpam (B. Math, 2nd year)
Abstract: The distribution of prime numbers, particularly the Prime Number Theorem, has fascinated mathematicians for centuries, with key milestones marked by advances made by notable mathematicians throughout the course of history. In this talk, we will explore Chebyshev’s contributions to the problem, mainly his work that laid the foundation for estimating the density of prime numbers. We will then delve into the elegance and significance of Selberg’s “elementary” proof of the Prime Number Theorem, highlighting how it revolutionized our understanding of the asymptotic behavior of primes. By connecting these pivotal results, we will gain insight into how modern analytic and elementary techniques contribute to unraveling the mysteries of prime number distribution.
Pre-requisites: Basic knowledge of Number Theory.
Notes: [TBU]
Video: [TBU]
Date and Time: Saturday, 21st September 2024, 11:00 AM - 12:30 PM
Venue: G-26 Classroom, Academic Building

EP2412: Discriminant of Polynomials through the Lens of Galois Theory #

Speaker: Anasmit Pal (B. Math, 3rd year)
Abstract: This talk explores the intricate relationship between discriminants and Galois groups of polynomials over a given field (mainly over \(\mathbb{Q}\)). We begin by establishing foundational Galois theoretic concepts and theorems. Then, we delve into diverse methods for computing discriminants, including:
Vandermonde matrices and Newton’s Identities
Resultants of Polynomials
Norms of finite field extensions

We will also demonstrate how to compute Galois groups by analyzing cycle types of automorphisms within \(S_n\). This talk will be a very basic one with the pre-requisite being field theory, and will include lots of interesting examples.

Notes: [TBU]
Video: Available here
Date and Time: Sunday, 1st September 2024, 11:00 AM - 1:00 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2411: Introduction to Modules of Differentials #

Speaker: Muhammed K.M. (M. Math, 2nd year)
Abstract: We will start by defining an algebra over a commutative ring. We will discuss some of its properties, recalling tensor product and localization of algebras. We will then introduce the central topic of our talk,module of relative differentials,discussing some of its properties. We will also introduce exact sequences related to differentials. If time permits,we will also talk about base change of relative differentials.
Pre-requisites: Basic knowledge of ring theory is expected. Some knowledge of tensors and localisation of rings will be helpful,but not at all necessary.
Notes: [TBU]
Video: Available here
Date and Time: Saturday, 31st August 2024, 3:00 PM - 4:00 PM
Venue: G-26 Classroom, Academic Building

EP2410: Besicovitch’s Covering Theorem: A Geometric Approach #

Speaker: Saptarshi Halder (M. Math, 2nd year)
Abstract: We start by defining the Radon measure in n-dimensional Euclidean space. We derive the Vitali’s Covering Theorem. However we observe some control issues there which is our main motivation to study Besicovitch’s covering theorem. The crucial geometric difference is that Vitali’s Covering Theorem provides a cover of enlarged balls, whereas Besicovitch’s Covering Theorem yields a cover out of the original balls, at the price of a certain controlled amount of overlap. These covering theorems will enable us to define and study Maximal functions which we encounter frequently in the field of mathematical analysis.
Pre-requisites: Set theory and basic knowledge of general measure theory would help but not at all compulsory.
Notes: [TBU]
Video: Available here
Date and Time: Sunday, 25th August 2024, 10:30 AM - 12:30 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2409: Tribone Tilings: Honeycombs to Benzels #

Speaker: Hrishik Koley (B. Math, 3rd year)
Abstract: Tilings have been a very interesting field of algebraic combinatorics, since it’s dawn, and yet has a plethora of unanswered questions, partly owing to the fact that new shapes and new tiles never cease to arise and being considered. From such wide classes of shapes and tiles, Conway and Lagarias considered and showed that certain roughly triangular regions in the hexagonal grid or honeycombs cannot be tiled by shapes Thurston later dubbed tribones. However, in 2021, Propp introduced a two-parameter family of roughly hexagonal regions in the hexagonal grid called benzels and showed that a tiling by tribones exists under certain constraints on these parameters.
Pre-requisites: Some knowledge of graph theory, group theory, and basic topology might help, but I would anyways introduce all the needed concepts.
Notes: Available here
Video: [TBU]
Date and Time: Saturday, 24th August 2024, 11:30 AM - 01:00 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2408: A Glimpse into Geometric Group Theory: Folding and the Ping-Pong #

Speaker: Rinkiny Ghatak (B.Math, 3rd year)
Abstract: Geometric group theory looks at groups through the geometric objects that they act on (graphs, topological spaces, manifolds, etc.), and the geometric properties of those objects help us understand algebraic properties of the group. In the first half of the talk, I will talk about how we can “fold” graphs to study subgroups of free groups, which is otherwise a difficult task with pure algebraic tools. In the second half, we will see the ping-pong lemma, where we identify free groups by seeing if they “play ping-pong” or not. We will see a very nice application in the study of the subgroups of \(PSL(2,\mathbb{C})\) by looking at their action on the Riemann sphere.
Prerequisites: Basic notions of group theory would help. Having some knowledge of algebraic topology will help see more connections, but it is not at all necessary.
Notes: [TBU]
Video: Available here
Date and Time: Sunday, 18th August 2024, 11:00 AM - 01:00 PM
Venue: G-26 Classroom, Academic Building

EP2407: Homological Methods in Number Theory #

Speaker: Amit Kumar Basistha (B. Math, 3rd year)
Abstract: The application of Homological Methods in Algebraic Number Theory, particularly in Class Field Theory, gained popularity around the 1950s and has propelled enormous advancement in the subject. In this talk, I will introduce some notions of Group Cohomology and how they are applicable in Number Theory. In particular, I will introduce the notion of ‘Abstract Class Formation’ and define the reciprocity isomorphism and its functorial properties. Finally I will present a nice characterization of the Norm Groups of a Local Field.
Pre-requisites: Knowledge of Abstract Algebra and some point set topology will be assumed. Further the content of the talk on Homological Algebra by Sankha Subhra Chakraborty will be assumed.
Notes: Available here
Video: Available here
Date and Time: Sunday, 11th August 2024, 11:00 AM - 01:00 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2406: Introduction to Homological Algebra and Derived Functors #

Speaker: Sankha Subhra Chakraborty (B. Math, 3rd year)
Abstract: When a left or right exact functor is applied to a short exact sequence, exactness is only preserved at one end. Derived functors provide a systematic method to continue this sequence on the other end. Most of the talk will include introducing and explaining the setup for talking about these derived functors, including exactness, chain complexes, chain homotopy, and resolutions.
Pre-requisites: Knowledge of elementary linear algebra concepts such as linear maps, kernel, image etc will be helpful for following the talk.
Notes: Available here
Video: Available here
Date and Time: Saturday, 10th August 2024, 10:30 AM - 12:30 PM
Venue: 2nd Floor Auditorium, Academic Building

Spring 2024

EP2405: Chaotic Dynamics on the Riemann Sphere #

Speaker: Saptarshi Halder (M. Math, 1st year)
Abstract: We give an elementary introduction to the holomorphic dynamics on the Riemann Sphere. We outline some basic concepts regarding this. Next we introduce the notion of Fatou and Julia sets. We see some easy examples. If time permits we will introduce the Mandelbrot set.
Pre-requisites: Basic Real and Complex analysis, Set theory.
Video: Available here
Date and Time: Tuesday, 2nd April 2024, 11:00 AM - 1:00 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2404: Bernstein’s problem for minimal surfaces in \(\mathbb{R}^3\) #

Speaker: Devansh Kamra (B. Math, 2024)
Abstract: Bernstein’s problem poses the question that if the graph of a scalar function on \(\mathbb{R}^{n-1}\) is a minimal surface, then whether the function is linear or not. This result holds if and only if \(n \leq 8\). We will solve the problem for \(n = 3\) after introducing the notion of minimal surfaces and comparing various equivalent definitions, along with developing a necessary condition for the graph of a scalar function to be a minimal surface.
References: [TBU]
Notes: [TBU]
Video: Available here.
Date and Time: Thursday, 28th March 2024, 9:00 PM - 10:30 PM
Venue: G-26 Classroom, Academic Building

EP2403: Exact Sequences of Modules and Lazard’s Theorem #

Speaker: Sankha Subhra Chakraborty (B. Math, 2025)
Abstract: This talk will serve as an introduction to the theory of exact sequences of modules. We will look at Hom and tensor, and discuss their exactness as functors to motivate the need for defining projective and flat modules. We will look at some characterisations of projective modules and discuss the hierarchy of free, projective and flat modules. We will introduce and discuss direct limits of directed systems of modules, and we shall conclude the talk with a theorem of Lazard, namely, the characterisation of flat modules as direct limit of finitely generated free modules.
References: [TBU]
Notes: [TBU]
Video: Available here.
Date and Time: Sunday, 3rd March 2024, 11:00 AM - 1:00 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2402: Introduction to Non-Commutative Spaces #

Speaker: Saqib Mushtaq Shah (M. Math, 2025)
Abstract: We will first demonstrate that commutative (unital) \(C^*\)-algebras are dual to compact Hausdorff spaces through the Gelfand representation. Drawing inspiration from the commutative case, we attempt to construct duals (referred to as noncommutative spaces) of noncommutative \(C^*\)-algebras using various methods of generalization. We prove some basic properties and will establish the relationship between these noncommutative spaces.
References: (See the notes)
Notes: Available here.
Video: Available here.
Date and Time: Sunday, 21st January 2024, 11:00 AM - 1:00 PM
Venue: 2nd Floor Auditorium, Academic Building

EP2401: A Gist of Optimal Transport #

Speaker: Malav Dhaval Doshi (B. Math, 2025)
Abstract: In this talk we will explore the roots of optimal transport. Initiating with Monge formation we will delve into the more generalized formation of the optimal transport problem given by Kantorivich. We would then look into the dual formation of the both and an introduction to Legendre Transforms. The core focus of this talk would be to prove the result that rather than optimizing the dual on every pair of functions it suffices to optimize it over the convex pairs of the function. Though it assumes very few things but still it is a cornerstone result of the topic!!
References:
- Topics in Optimal Transport – Cedric Villani
- Lecture Series on Optimal Transport – Brittany Hamfeldt
Video: Available here.
Date and Time: Sunday, 14th January 2024, 3:00 PM - 5:00 PM
Venue: G-26 Classroom, Academic Building