What is ...? | Spring 2024 · Math Club, ISI Bangalore

All the videos are available in the playlist here.

WI2401: What is ?(x)? #

Speaker: Aaratrick Basu (B. Math, 2024)
Abstract: We introduce the function \(?(x)\) defined by Minkowski in 1904, and studied quite well since. It is a continuous function on \([0,1]\) which has very interesting properties relating to dynamical systems, number theory, integral transforms, and is a rich object as a function itself via analysis of continued fractions and convergents. We give a whirlwind tour of some of these aspects, presenting only the main definitions and results. No prerequisites beyond first-year knowledge of analysis and algebra will be assumed formally, but we do demand accepting certain technical facts on faith.
Video: Available here.
Date and Time: Saturday, 9th March 2024, 3:30 PM - 4:30 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2402: What is Graphon? #

Speaker: Bikram Halder (B. Math, 2024)
Abstract: Graphons, short for graph functions, are limiting objects of large finite graphs with respect to the cut-metric, which provides a powerful analytic framework for analyzing large graphs. In this talk, we will introduce graphons, motivate them, discuss how they complete the space of finite graphs. We will look at some results that connect the the finite world of graphs with the continuous world of graphons.
Video: Available here.
Date and Time: Sunday, 10th March 2024, 11:30 AM - 12:30 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2403: What is Shannon Entropy? #

Speaker: Hrishik Koley (B. Math, 2025)
Abstract: Entropy is a concept that is most commonly associated with a state of disorder. The concept of information entropy was introduced by Claude Shannon in his 1948 paper A Mathematical Theory of Communication. It gives us an idea of the information content of the process. We will introduce Shannon entropy as a fundamental limit for how much we can compress a source, without risking distortion or loss. We will be introducing this concept of Shannon entropy and look at it’s various characterizations. If time permits, we may also take a look at Alfréd Rényi’s extension of Shannon entropy, famously known as, the Rényi entropy.
Video: Available here.
Date and Time: Saturday, 16th March 2024, 3:00 PM - 4:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2404: What is Cohomlogy? #

Speaker: Trishan Mondal (B. Math, 2024)
Abstract: In this talk we will briefly introduce the (ordinary)cohomology theory, Eilenberg-steenrod axioms. Then we will look into the cohomology ring of a topological spaces and it’s application in Hopf-algebra, \(H\)-spaces, how cohomology operations motivates a construction of loop spaces which gives us EHP sequence. Then we will try to look at Serre finiteness conditions via cohomology operations.
Video: Available here.
Date and Time: Sunday, 17th March 2024, 11:30 AM - 12:30 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2405: What is Stable Homotopy? #

Speaker: Soumya Dasgupta (B. Math, 2024)
Abstract: In an introductory course on algebraic topology, one encounters fundamental groups as robust topological invariants. Homotopy groups naturally extend this notion. The functor \([(\mathbb{S}^n,e_1), - ] : \mathsf{Top}_{*} \to \mathsf{Grp}\), particularly for \(n > 1\), becomes a functor from pointed spaces to abelian groups, i.e., \([(\mathbb{S}^n,e_1), - ] : \mathsf{Top}_{*} \to \mathsf{Ab}\). However, unlike homology groups, homotopy groups lack a suspension isomorphism theorem. Fruedenthal’s work established that the suspension functor induces an isomorphism under certain conditions, paving the way for stable homotopy theory. Though computing stable homotopy groups for even spheres remains formidable, their significance in various geometric and topological theories continues to motivate exploration of this challenging area of study. We will talk about Pontryagin-Thom construction, and conclude that classification of framed manifolds upto framed cobordism reduces to the computation of stable homotopy groups of spheres.
Video: Available here.
Date and Time: Sunday, 17th March 2024, 4:00 PM - 5:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2406: What is Mixing Time? #

Speaker: Deepta Basak (B. Math, 2024)
Abstract: We start with a concise introduction to Markov Chains, focusing on stationary distributions and the conditions necessary for their existence and uniqueness. Our primary interest lies in understanding the long-term behavior of finite Markov Chains, which leads us to define the Mixing Time. This measures the time required for a Markov Chain to get “close” to its stationary distribution. Moving forward, we introduce several eigenvalue techniques and explore the connections between the eigenvalues of transition matrices and the Mixing Time. Additionally, we examine some upper and lower bounds on the Mixing Time in terms of the Relaxation Time. Throughout our discussion, we will use the Random Walk on the \(n\)-dimensional Hypercube as a central example to illustrate these concepts effectively.
Video: Available here.
Date and Time: Saturday, 23rd March 2024, 3:00 PM - 4:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2407: What is Coupling? #

Speaker: Saraswata Sensarma (B. Math, 2024)
Abstract: In this talk, we give brief introduction to the notion of coupling of probability measures. Coupling is an extremely versatile tool, which finds applications in almost all facets of probability theory. In the first half of the talk, we will explore its use in studying mixing in finite Markov chains. In the second half, we introduce the notion of stochastic domination and analyze it through the lens of coupling. We then extend this notion to posets, which naturally leads us to correlation inequalities like Harris’ inequality and FKG inequality. These find extensive applications in percolation theory and the Ising model, and the random cluster model in general.
Video: Available here
Date and Time: Sunday, 24th March 2024, 10:30 AM - 11:30 AM
Venue: 2nd Floor Auditorium, Academic Building

WI2408: What is Cut-Off? #

Speaker: Sanchayan Bhowal (M. Math, 2025)
Abstract: I will introduce the cutoff phenomenon in Markov Chains. I will discuss some examples where this is observed like random walk in hypercube and card shuffling. Finally I’ll discuss the characterization of cutoff under some special cases like birth death chains. We will also look into famous examples by Aldous and Pak where the characterization falls short.
Video: Available here.
Date and Time: Sunday, 24th March 2024, 12:00 PM - 1:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2409: What is Graph Cohomology? #

Speaker: Prognadipto Majumdar (B. Math, 2024)
Abstract: We’ll define models of graph complexes and explore basic topological properties of graphs. We’ll move on to an application of a certain model of graph cohomolgy which directly gives us the Chromatic Polynomial. If time permits, we may look at some connections to TQFTs.
Video: [TBU]
Date and Time: Sunday, 31st March 2024, 11:30 AM - 12:30 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2410: What is Ideal Class Group? #

Speaker: Shubhrojyoti Dhara (B. Math, 2024)
Abstract: The Ideal Class groups measure to a certain extent how far a certain ring of integers is, from being a PID. We explore its relations to the equivalence classes of positive definite quadratic forms. We advance to the proof of FLT of Kummer for special primes \(p \nmid h_{\mathcal{O}_{\mathscr{F}}}\). Lastly, we will introduce Adeles and Idele Class Groups.
Video: (Not Recorded)
Date and Time: Sunday, 31st March 2024, 4:00 PM - 5:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2411: What is Poincaré Recurrence Theorem? #

Speaker: Devansh Kamra (B. Math, 2024)
Abstract: Poincaré’s Recurrence Theorem is a celebrated result in Ergodic Theory, which states that a dynamical system under certain constraints is bound to return arbitrarily close to its initial stage after a finite amount of time. This result has found widespread applications in the study of dynamical systems. We would be discussing this result from a measure-theoretic viewpoint, building upon the necessary notions before developing a neat proof of the theorem, along with some applications.
Video: Available here.
Date and Time: Saturday, 6th April 2024, 4:00 PM - 5:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI2412: What is Knot Theory? #

Speaker: Dhruba Sarkar (B. Math, 2025)
Abstract: In this talk, we will try to provide a mathematical formulation of our usual intuition of knots. We will talk about knot equivalence and how we can associate numbers, polynomials, topological and algebraic structure on knots to differentiate between them. In particular, we will look at Knot Colorability and how we use it to differentiate between knots.
Video: Available here.
Date and Time: Sunday, 7th April 2024, 4:00 PM - 5:00 PM
Venue: 2nd Floor Auditorium, Academic Building

WI24GSL: What is Bennequin’s Inequality? #

Speaker: Balarka Sen (B. Math, 2021)
Abstract: A contact structure on a 3-manifold is a nowhere-integrable plane field on the manifold. Of particular interest are contact structures which are “tight”, in which case the geometry of the contact manifold is sufficiently complicated. We focus our discussion to a semi-local study of contact structures near embedded surfaces (possibly, with boundary) in the manifold. As a consequence, we deduce the Thurston-Bennequin inequality for Legendrian knots in tight contact manifolds. We discuss some modest applications, and time permitting, indicate several ambitious ones.
Pre-requisites: Strong background in differential forms (eg, Frobenius integrability theorem).
Video: Available here.
Date and Time: Monday, 8th April 2024, 9:00 PM - 10:00 PM
Venue: Online (Google Meet)

Banner Design Credits: Ruchira Mukherjee (B.Math, 2025)