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Student Talks 2023

Table of Contents

All expository talks given by students of ISI Bangalore in the year 2023 at our Math Club are listed below.

Fall 2023

EP2314: Thresholds and Expectation Thresholds #

  • Speaker: Bikram Halder (B. Math, 2024)
  • Abstract: Many problems in probabilistic combinatorics boil down to computing the thresholds for increasing properties, a certain value that indicates the drastic change of a structure possessing that property. In 2006, Kahn and Kalai conjectured that for any non-trivial increasing property on a finite set, its threshold is never far from its “expectation threshold”, a natural (and often easy to compute) lower bound on the threshold. A positive answer to this conjecture allows one to narrow down the location of thresholds to a tiny window. In particular, this easily implies several difficult results, such as thresholds for perfect hypergraph matching (known as Shamir’s problem, solved by Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). In this talk, I will introduce the Kahn-Kalai conjecture with some motivating examples, and then briefly discuss the recent resolution by Huy Tuan Pham and Jinyoung Park, along with some previous work on the topic.
  • References:
  • Notes: Available here.
  • Video: Available here.
  • Date and Time: Tuesday, 26th December 2023, 8:00 PM - 10:00 PM

EP2313: Free Groups and Foldings #

  • Speaker: Aditya Prabhu (B. Math, 2024)
  • Abstract: Every group can be realized as a quotient of a free group, a group which is generated by a set of elements without any non-trivial relations. The Nielsen-Schreier theorem tells us that subgroups of free groups are free. But given a subgroup generated by some elements, can you tell me the rank of this subgroup? What about its generators? Its index? In general, solving this problem algebraically is quite a task. However, John Stallings came up with an ingenious technique in 1983 which tackles these problems geometrically by performing a simple “fold” on a graph. This talk intends to cover some applications of this technique.
  • Pre-requisites: Knowledge of group theory is required. Some algebraic topology will help see the connections much better, but is not mandatory by any means.
  • References:
  • Notes: Available here.
  • Video: Available here.
  • Date and Time: Sunday, 29th October 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2312: Critical Percolation on Non-Amenable groups #

  • Speaker: Ishaan Bhadoo (B. Math, 2024)
  • Abstract: Percolation models were initially introduced by Broadbent and Hammersley in 1956. One specific area of importance is Bernoulli percolation, a process that involves tossing a biased coin (say, \(\textsf{Ber}(p)\)) for each edge of a graph and removing edges if the outcome is tails. When we execute such a process on an infinite graph, many natural questions arise. For instance, one may inquire about the values of \(p\) for which we are guaranteed to have an infinite connected component in the graph. Although these processes were traditionally studied on Euclidean lattices, substantial progress has also been made in comprehending percolation in general settings with different background geometries.
    Many questions related to percolation on general graphs were initially introduced in the highly celebrated paper by Benjamini and Schramm, aptly titled “Percolation Beyond \(\mathbb{Z}^d\), Many Questions And a Few Answers”. In our talk, we will present numerous classical results uncovered by Benjamini, Schramm, and their co-authors in the 1990s. We will then shift our focus to the recent advances in the field by Tom Hutchcroft and co-authors, which address many questions originally raised by Benjamini and Schramm. In particular, we will discuss the uniqueness and non-uniqueness phases of transitive graphs. We will then divert our attention to critical percolation and discuss Hutchcroft’s amazing new proof, demonstrating that percolation dies at criticality for graphs exhibiting exponential growth.
  • Notes: Available here.
  • Video: Available here.
  • Date and Time: Sunday, 8th October 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2311: Analytic Continuation of the Riemann Zeta Function and the Prime Number Theorem #

  • Speaker: Amit Kumar Basistha (B. Math, 2025)
  • Abstract: The Prime Number Theorem (PNT) is a fundamental result in number theory that describes the distribution of prime numbers. The PNT has been studied extensively and has numerous proofs, many involving Complex Analysis and the behavior of the Riemann Zeta function. The main goal of this talk will be to present a proof of the theorem. While doing so we will also explore the beautiful world of analytic continuation of functions in the complex plane, particularly of the Gamma and Zeta function. We will also discuss the growth and behavior of the Zeta function near the line \(\textsf{Re}(s)=1\).Through the talk we will try to convey the idea how two seemingly unrelated areas of mathematics: Number Theory which is discrete at its heart and Complex Analysis can come together to give such powerful results.
  • References:
  • Video: Available here.
  • Date and Time: Saturday, 7th October 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2310: Representations of the symmetric and alternating groups #

  • Speaker: Pavan K Srinivasan (B. Math, 2024)
  • Abstract: The theory of representations of the symmetric group has a wide variety of potential applications, ranging from quantum chemistry to the study of symmetric functions.In this talk, we begin by introducing some notions of finite group representations including some basic results about characters of groups. We will use these tools to find combinatorial criteria for the irreducibility of representations of symmetric and alternating groups using the Robinson-Schensted-Knuth (RSK) Algorithm. We will also see how irreducible representations of \(S_n\) behave when restricted to particular subgroups, and see elegant methods to compute character values of these representations.
  • Video: [TBU]
  • Date and Time: Monday, 2nd October 2023, 11:30 AM - 1:30 PM
  • Venue: G-26 Classroom, Academic Building

EP2309: Estimation in Curie-Weiss Tensor Potts model #

  • Speaker: Sanchayan Bhowal (M. Math, 2025)
  • Abstract: Spin glass models are a fundamental class of models extensively studied in statistical physics. The celebrated Ising model and its generalization, the Potts model, are classical examples of the same. In this talk, we introduce a further generalization involving the concept of peer interactions, called the tensor Curie Weiss model. Much like the other spin glass systems, it exhibits phase transition. We will then derive the limiting distribution of the maximum likelihood estimates of the parameters and the magnetization vector. We will describe the limiting distribution for a certain subset of the parameter space. If time permits, we would extend our analysis to the remaining space. This talk will primarily be based on the paper arXiv:2307.01052.
  • Pre-requisites: Familiarity with MLEs and convergence laws in probability is recommended.
  • Notes: Available here.
  • Video: Available here.
  • Date and Time: Sunday, 1st October 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2308: A Proof of Cayley’s Formula via Poisson Branching Process #

  • Speaker: Bikram Halder (B. Math, 2024)
  • Abstract: In this talk, we will explore an alternative proof of Cayley’s Formula, which determines the number of labeled trees in a graph with \(n\) vertices as \(n^{n-2}\). While the commonly taught Prufer code approach and other proofs rely on combinatorial techniques, I’ll introduce a fresh perspective through a probabilistic proof employing the Poisson Branching Process.
  • Notes: Available here.
  • References:
  • Video: Available here.
  • Date and Time: Saturday, 30th September 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2307: On the extrema of some log-correlated fields #

  • Speaker: Saraswata Sensarma (B. Math, 2024)
  • Abstract: The last decade has seen significant growth in log-correlated fields (LCFs) within probability theory. In this lecture, we plan to give a gentle introduction to the behavior of their maxima. We will begin with some simple examples of LCFs, and demonstrate the striking similarity of the results obtained. This ubiquity would hint at the presence of an underlying unified treatment of these situations, which is precisely the theory of LCFs.
    To begin with, we will concern ourselves with Branching Random Walks, the simplest and the most prototypical example of LCFs. We will discuss results concerning expectation and distribution of the maximum of the Branching Random Walk, and how they compare to the extrema of iid random variables. Next, we will consider the discrete Gaussian free field, and in particular the 2-dimensional discrete Gaussian free field (2d-DGFF). Using the results obtained for Branching Random Walks, we will derive the expected maximum of the 2d-DGFF. This would involve some comparison inequalities (Sudakov-Fernique inequality and Slepian’s Lemma) for Gaussian Processes. Finally, we will look at some more examples of LCFs based on eigenvalues of randomly generated unitary matrices (CUE) and the Riemann zeta function, and some conjectures pertaining to them.
  • References:
    • Most of the material on Branching Random Walks and Gaussian Free Fields is covered in the lecture notes.
    • Talk by Ofer Zeitouni at ICTS: This covers most of the materials on models based on Random Matrices and the Riemann Zeta function. Zeitouni also suggests many references in this talk.
  • Video: Available here.
  • Date and Time: Sunday, 3rd September 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2306: Projective Geometry #

  • Speaker: Shriyaa Srivastava (B. Math, 2025)
  • Abstract: Standing at the end of a long corridor, the parallel lines seem to go somewhere… and meet? Euclidean geometry doesn’t allow this… but what if we did? Further, what if we dismissed concrete notions of lengths and angles? Projective geometry studies geometric properties that are invariant under projective transformations. Math jargon aside, this means that we “see” objects from a different “perspective” / “point of view”. We shall understand the nature of these transformations and build familiarity with cross-ratios, leading to a proof of Pascal’s theorem. If time permits, we shall also look at parallel and central projections.
  • References:
    • Geometric Transformation III – I. M. Yaglom
    • Projective Geometry – Milivoje Lukić (IMO Compendium)
  • Video: Available here.
  • Date and Time: Saturday, 2nd September 2023, 11:00 AM - 12:30 PM
  • Venue: G-26 Classroom, Academic Building

EP2305: What is so special about \(\mathbb{Z}\left[ \frac{1+\sqrt{-19}}2 \right]\) ? #

  • Speaker: Ruchira Mukherjee (B. Math, 2025)
  • Abstract: Every Euclidean domain is a Principal Ideal domain but not the other way round. A classic counterexample is \(\mathbb{Z}\left[ \frac{1+\sqrt{-19}}2 \right]\). Are there more? If so, how are they similar? The talk shall mainly introduce what class numbers are and how they help in determining whether the ring of integers of a number field is a PID, along with a general flow leading to the generalized Riemann Hypothesis (where there are numbers, there is the Riemann Hypothesis!).
  • References:
  • Video: Available here.
  • Date and Time: Saturday, 19th August 2023, 11:00 AM - 12:30 PM
  • Venue: G-26 Classroom, Academic Building

EP2304: On a proof of the Mednykh’s Formula via Dijkgraff-Witten Theory on 1 + 1 Cobordisms #

  • Speaker: Prognadipto Majumder (B. Math, 2024)
  • Abstract: We will discuss the Djikgraff-Witten approach to Topological Quantum Field Theories which are essentially representations of the Cobordism Category. We would only dwell on the simplest non trivial dimensions (1 + 1) and see how the classification of 2-D surfaces leads to an elegant result like the Mednykh’s Formula. We will try to motivate how something that was conceived as a toy model for quantum theory is leading to “brave new mathematics” and might (???) provide some understanding of classification of manifolds and structure of higher categories.
  • Pre-requisites: Some familiarity with the basics of group representations might be useful.No knowledge of category theory will be assumed but familiarity with definition of categories and functors would be helpful.
  • Date and Time: Sunday, 13th August 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building

EP2303: Hyperbolic Geometry, Hyperbolic Surfaces and Fuchsian Groups #

  • Speaker: Aaratrick Basu (B. Math, 2024)
  • Abstract: We introduce and motivate hyperbolic geometry as a Riemannian geometry in 2 dimensions. We then discuss the group of orientation preserving isometries, \(PSL_2(\mathbb{R})\). This leads naturally to Fuchsian groups, interesting objects in their own right. We finish by discussing some elementary results on surfaces.
  • References:
  • Notes: Available here.
  • Video: Available here.
  • Date and Time: Saturday, 12th August 2023, 11:00 AM - 12:30 PM
  • Venue: 2nd Floor Auditorium, Academic Building
Spring 2023

EP2302: Michael Keane’s proof of the SLLN #

  • Speaker: Sanchayan Bhowal (B. Math, 2023)
  • Abstract: Weak law of large numbers is generally taught in undergraduate probability courses. However, the proof of strong law of large numbers is generally omitted. The original proof involved manipulation of moments. However, we will look at Keane’s proof of the strong law. Time to get stronger!
  • References:
  • Date and Time: Sunday, 2nd April 2023, 11:00 AM - 12:00 PM
  • Venue: G-26 Classroom, Academic Building

EP2301: Introduction to Ising Model #

  • Speaker: Sarvesh Ravichandran Iyer (SRF)
  • Abstract: An Introduction to Ising Models and some discussions (pre-requisites for Hugo Duminil-Copin’s talk at ICTS, Bangalore)
  • Video: Available here.
  • Date and Time: Sunday, 8th January 2023, 10:30 AM - 12:30 PM
  • Venue: G-26 Classroom, Academic Building