Talks given by students during the year 2026 are listed below.

Fall 2026

EP2616: Beyond Integer dimensions: A Motivation Towards Fractal Geometry

Speaker: Sarbartha Sarkar (M.Math 2026-28)
Abstract: What does it mean for a set to have a dimension? Smooth curves are 1-dimensional, smooth surfaces are 2-dimensional and solids are 3-dimensional. But in the nineteenth and early twentieth century mathematicians discovered objects for which this intuition is no longer enough. We get to meet some of these “mathematical monsters”, including the Weierstrass function, the Cantor set and the Koch snowflake, and see why ordinary notions of length and size fail for them. Using the idea of scaling, we then arrive at Hausdorff measure and Hausdorff dimension, and briefly see how these ideas led to Fractal geometry and how it connects with Geometric measure theory.
Video: https://www.youtube.com/watch?v=_injH3XZWt0
Notes: Fractal Geometry - Sarbartha Sarkar
Date and Time: Friday (October 9, 2026), 2:15 PM
Venue: Room 008, CR Rao Building, Ground Floor

EP2615: The magic of knots: Learn to tie your shoe laces better

Speaker: Yuvraj Singh Rajpurohit (M.Math 2026-28)
Abstract: This talk is intended to be a basic introduction to knot theory. I will be giving intuition as well as the rigorous definitions and some proofs. Subject to time, a number of knot invariants will also be discussed, especially the unknotting number and the genus of a knot. Towards the end I will get to shoe lace tying and how it can be related to knot theory and I will talk about the “knot theory” behind the granny knots and the reed knots. The aim is to introduce something different in an easy way to everyone.
Pre-requisites: Knowing a bit of point-set topology would help.
Video: https://www.youtube.com/watch?v=tr3ytBkWHaU
Date and Time: Wednesday (October 7, 2026), 11:20 AM
Venue: Room 008, CR Rao Building, Ground Floor

EP2614: An introduction to Simplicial Homology

Speaker: Anant Kerur (M.Math 2028)
Abstract: This talk introduces the concept of simplicial homology groups as an alternative way to categorize surfaces. We start by introducing the concept of simplicial complexes and simplicial maps, as a way to simplify working on surfaces in general. We then define certain operations on these complexes, namely orientation and the boundary operator, and using these we define the simplicial homology group of a complex. We will briefly go over some simple constructions of these groups. On defining homology groups we will observe certain properties given by the aforementioned operations, and use them to define simplicial approximations. Using all of these tools we will prove the topological invariance of homology groups. Time permitting, we may introduce the concept of CW complexes as a way to make the calculation of homology groups easier.
Video: https://www.youtube.com/watch?v=uJJAJnJ_Z0g
Date and Time: Saturday (October 3, 2026), 10:45 AM
Venue: Room 520, SN Bose Building, Fifth Floor

EP2613: Ax - Grothendieck Theorems

Speaker: Arkapriyo Hore (Mastermath), Universität Münster
Abstract: The Ax-Grothendieck theorem says that an injective polynomial map $\mathbb{A}^n \to \mathbb{A}^n$ over an algebraically closed field is necessarily surjective, reminiscent of the corresponding fact from finite-dimensional linear algebra. In this talk, we will focus on a proof based on the philosophy of reducing an infinite problem to a finite one. We will briefly overview a proof of the theorem for $\mathbb{C}$ using complex analysis and then present a proof of the theorem using finite fields. If time permits, we will see another application of the same philosophy to group actions on affine space.
Video: https://www.youtube.com/watch?v=uqKY6tLh3eE
Notes: Ax-Grothendieck - Arkopriyo Hore
Date and Time: Wednesday (30th September), 4:15 pm
Venue: Room 008, CR RAO Building, Ground Floor

EP2612: Measure theoretic interpretation of chaos in 1 dimensional dynamical systems

Speaker: Ayan Pal (M.Math 1st year), ISI Kolkata
Abstract: In this walk we will be illustrating the role of multimodal interval maps and Li–Yorke pairs in the study of one-dimensional dynamical systems. After introducing the notions of distal, asymptotic and Li–Yorke pairs, we discuss how these concepts provide a measure-theoretic perspective on chaos. We focus on results of Bruin and Jiménez López showing that, for smooth multimodal maps, nontrivial asymptotic pairs are negligible from the viewpoint of Lebesgue measure, while under suitable topological mixing assumptions Li–Yorke pairs are in fact typical. These results demonstrate that although large measurable scrambled sets cannot exist, chaotic behaviour between pairs of points may occur for almost every pair, revealing a striking connection between topological dynamics, ergodic theory and interval maps.
Video: https://www.youtube.com/watch?v=DCflhzgRp3o
Date and Time: Monday (28th September), 4:15 pm
Venue: 008, CR Rao Bhavan, Ground floor

EP2611: Optimization for Machine Learning

Speaker: Vishal Vikram Parth (M.Math 1st year), ISI Delhi
Abstract: This talk presents a broad overview of optimization for machine learning, bridging theoretical foundations with practical algorithms. We begin with the basic geometry of optimization: global and local optima, gradients and Hessians, stationarity, first-order necessary conditions, and the classification of stationary points via curvature. Convexity, L-smoothness, and strong convexity are introduced as key structural assumptions, along with their implications for convergence and global optimality. We then derive gradient descent from the steepest-descent direction and discuss step-size selection, termination criteria, stochastic and mini-batch gradients, and the empirical loss setting. The talk covers acceleration and adaptation methods, including momentum, Nesterov accelerated gradient, AdaGrad, RMSProp, and Adam, as well as regularization techniques such as L2, Adam with L2, and AdamW. Additional topics include learning-rate schedules, Newton’s method, and logistic regression. Finally, we turn to constrained optimization, Lagrangian duality, KKT conditions, and their application to hard- and soft-margin support vector machines, including the dual formulation and the kernel trick. The goal is to provide a unified view of the optimization tools that underlie modern machine learning.
Pre-requisites: Recommended that attendees already have familiarity with the subject; run as a crash course in cohort with Co-Labs.
Video: (Not recorded)
Notes: Optimization - Vishal Vikram Parth
Date and Time: Saturday (19th September), starts at 5:30 p.m.
Venue: Online lecture

EP2610: Fundamental Groups: An Introduction to Algebraic Topology

Speaker: Muhammad Shameel (M.Math 1st year), ISI Kolkata
Abstract: This talk introduces the fundamental group as an introduction to algebraic topology. We begin with motivating problems in topology like: is there a map from B^n to B^n without any fixed points? Is there a retraction from B^n to S^n? Is R^n ≅ R^m for m ≠ n? These problems are hard to tackle using just point-set topology; using fundamental groups we can resolve these questions for 2 dimensions. After defining the fundamental group, we develop covering spaces to calculate the fundamental group, listing key properties of covering spaces and discussing its relation to the fundamental group and its subgroups. We then discuss further properties of the fundamental group, including its behavior under homotopy equivalence and the extent to which it captures the topology of a space. We look at a more analytical way to calculate the fundamental group using van Kampen’s theorem, which computes the fundamental group of a space from those of simpler open pieces covering it, and we conclude by applying it to determine the fundamental groups of all orientable surfaces.
Video: https://youtube.com/@mathclubisik?si=u4UNRylmk_vbDEJP
Date and Time: Friday (18th September), starts at 2:30 p.m.
Venue: Room 008, CR Rao Bhavan, Ground Floor

EP2609: ‘The Random Variable is Roughly Its Expectation’ — The Kahn-Kalai Conjectures

Speaker: Atmadeep Sengupta (M.Stat 1st year), ISI Kolkata
Abstract: Let X be a finite set and Xp denote its random subset, where each element is included independently with probability p. Consider any increasing property of Xp (e.g., the containment of arithmetic progressions of length r, or the containment of consecutive integers). For |X| = n and p = p(n), a value p̂ is defined as the threshold if p = o(p̂) implies P(Xp has the property) → 0, and p = ω(p̂) implies P(Xp has the property) → 1. The first Kahn-Kalai conjecture proposes an elegantly computable bound for this threshold. The second Kahn-Kalai conjecture focuses on a special case for random graphs, suggesting an even stronger bound. In this talk, we will explore the importance and broader implications of both conjectures. Additionally, we will walk through a short proof of the first conjecture — due to Tran and Vu — and examine exactly why this proof does not imply the second.
Pre-requisites: Probability 1 & 2, Basic Graph Theory (recommended, but not compulsory), and a healthy dose of patience to bear with the speaker!
References:
Video: https://www.youtube.com/live/uKR5JFws9pA?si=dZmZCJksGa3BOOh_
Notes: Kahn_Kalai_Conjecture - Atmaadeep Sengupta
Date and Time: Tuesday (17th September), starts at 4:15 p.m.
Venue: Room 008, CR Rao Bhavan, Ground Floor

EP2608: Regression Techniques

Speaker: Lavanis A
Abstract: Linear regression is one of the fundamental statistical tools for predictive analysis, providing a framework for understanding relationships between variables and making predictions based on historical data. This work introduces the key concepts and assumptions underlying linear regression and examines situations in which it is an appropriate modelling technique. The discussion begins with simple linear regression, including estimation and inference for model parameters, and extends to multiple linear regression. Analysis of variance (ANOVA) is introduced to assess the significance of regression models, followed by measures of goodness of fit, including R² and adjusted R². The limitations of linear regression are then examined, particularly in situations where the response variable is binary or the assumptions of the linear model are not appropriate. This motivates an introduction to logistic regression as a classification framework for binary outcomes. The logistic regression model is developed through the estimation and interpretation of its parameters, with emphasis on modelling the probability of an outcome taking the value 0 or 1. Overall, the work provides a foundation in regression-based predictive modelling, covering both continuous and binary response settings.
Pre-requisites: Not specified.
Video: (Not mentioned)
Notes: Linear_Regression - Lavanis Aravindam
Date and Time: Wednesday (16th September), starts at 6:00 p.m.
Venue: Not specified.

EP2607: What is Haar Measure?

Speaker: Srijeet Bhattacharjee (M.Math 1st year), ISI Kolkata
Abstract: The Lebesgue measure on R has more properties than an abstract measure. Specifically it has certain approximation properties and is translation invariant, which makes the measure compatible with both the topology and group structure on R. We would investigate Haar measure which is a generalization of these two properties in the setting of an abstract topological group. This makes it an indispensable tool in Harmonic analysis and probability theory.
Pre-requisites: Measure theory, group theory, topology (only basic ideas, even Metric spaces suffices).
Video: https://www.youtube.com/watch?v=ptHjE8JInlc
Notes: Haar_Measure - Srijeet Bhattacharjee
Date and Time: Tuesday (15th September), starts at 4:15 p.m.
Venue: Room 008, CR Rao Bhavan, Ground Floor

EP2606: From Random Walks to Heat Kernels: An Introduction to Markov Chains

Speaker: Aritrabha Majumdar (M.Math 1st year)
Abstract: Markov chains provide a fundamental framework for describing stochastic systems whose future evolution depends on the present state but not on the past. This talk develops an elementary introduction to both discrete-time and continuous-time Markov chains, with particular emphasis on the connections between the two settings. We begin with transition matrices, the Chapman–Kolmogorov equations, simple random walks, stationary distributions, and convergence to equilibrium in finite state spaces. We then pass to continuous time by introducing exponential holding times and jump rates, leading naturally to the infinitesimal generator and the Kolmogorov equations. Finally, we interpret the transition kernel as a matrix exponential, connecting Markov-chain evolution with semigroup theory and the heat equation. The goal is to develop a unified picture of Markov chains in which transition matrices, generators, equilibrium distributions, and heat kernels arise as different descriptions of the same underlying stochastic evolution.
Pre-requisites: Idea of conditional probability, expectation, variance, discrete and continuous random variable (and their pmf and pdf), linear algebra, notion of matrix exponential, notion of convergence in metric spaces.
Video: https://www.youtube.com/watch?v=X7CVbTMtClU
Notes: Intro to Markov Chains - Aritrabha Majumdar
Date and Time: Saturday (22nd August), 12 noon to 1:30 p.m.
Venue: 520, SN Bose 5th floor

EP2605: Dirichlet’s Theorem on Primes in Arithmetic Progression

Speaker: Kaustubh Dubey (M.Math 1st year)
Abstract: In 1837, Peter Dirchelet established that any arithmetic progression of the form $a$, $a+q$, $a+2q$,… where $\gcd(a,q)=1$ contains infinitely many primes. The results established by Dirchelet laid the groundwork for various fields of mathematics including algebraic and analytic number theory. Apart from that, the tools developed led to various algorithms which have applications in modern signal processing and theoretical computer science.to name a few.
In this talk, I shall cover the major results established by Dirchelet in proving his theorem.
Pre-requisites: Group Theory
Video: (Recorded but corrupted)
Notes: Dirichlet’s theorem - Kaustubh Dineshkumar Dubey
Date and Time: Saturday (22nd August), 11 a.m. to 12 noon
Venue: 520, SN Bose 5th floor

EP2604: Smallest Volume Hyperbolic Link Complement with n Components

Speaker: Shashank Gupta, IISER Mohali
Abstract: In 2001, Cao and Meyerhoff proved that the smallest volume hyperbolic link complement with one cusp is the figure-8 knot complement. Agol posed a generalisation of this as a question: what is the smallest volume hyperbolic link complement with n components? In his thesis, Venzke proposed an answer to Agol’s question, arguing that the volume for n ≥ 11 is beaten by the (n−1)-fold cyclic cover of the Whitehead link complement. But this was without proof until Kaiser–Purcell–Rollins proved Venzke’s statement for n ≥ 60. In this talk, I will discuss the original conjecture of Agol, which states that minimally twisted chain links are the smallest volume hyperbolic n-cusped manifolds. The Kaiser–Purcell–Rollins paper provides computational evidence that they are beaten by the Whitehead cyclic cover, but this is not a complete proof showing that the chain link is globally minimal volume among all n-cusped manifolds in these ranges. Even though most data support this numerically, a complete classification or proof is still open for some of these small n. Also, a complete classification of minimal volume n-cusped manifolds for 3 ≤ n ≤ 59 needs to be given. I will discuss all the possibilities related to this open problem. The talk will start with some preliminaries from hyperbolic knot theory and then lead to this open problem. It may be useful for people who want to work in the vast domain of 3- and 4-dimensional geometry and topology.
Pre-requisites: Basic knowledge of knot theory, Hyperbolic Geometry.
Video: (Not recorded)
Date and Time: Friday (14th August), 4:30 PM to 5:45 PM.
Venue: 008, CR Rao Bhavan

EP2603: Can we transmit information, through an adversarial medium, with certainty?

Speaker: Sauparna Kar (M. Stat, 2027)
Abstract: Suppose you are grounded at home for getting bad grades. Now it is the World Cup season and you are an Argentina fan. You need to know the result of the match against England. For that, you ask a friend to cycle past your home and tell you the result. But the catch is that your parents are keeping a strict lookout for such shenanigans. How can you overcome such a situation while not making things worse for yourself? We will look at such kinds of problems in the classical as well as the quantum domain.
Pre-requisites: VM-1 and VM-2, Group Theory (basic definitions, stabilizer, normalizer), Analysis 1 and 2, and some ideas from Analysis 3. Postulates of Quantum Mechanics will be introduced in the talk.
Target Audience: B. Stat, 2nd year onwards
Video: (Not Recorded)
Date and Time: Saturday, 8th August 2026, 6:00 PM - 7:30 PM
Venue: Online (Google Meet)

Spring 2026

EP2602: A probabilistic proof of Cayley’s Theorem using Poisson Branching process

Speaker: Ahan Chakraborty (B. Stat, 2027)
Abstract: Cayley’s formula establishes that the number of distinct labeled trees on $n$ vertices is exactly $n^{n-2}$. While classical proofs traditionally rely on combinatorial bijections with Prufer codes, double counting, or algebraic graph theory approaches, this talk presents an alternative probabilistic proof by modeling tree generation through a Galton-Watson branching process with a Poisson offspring distribution.
Pre-requisites: All pre-requisites are covered in B1, so B1 students can attend as well.
Video: (Not Recorded)
Date and Time: Sunday, 19th April 2026, 6:00 PM - 7:30 PM
Venue: Online (Google Meet)

EP2601: RSA Cryptography and Shor’s Algorithm

Speaker: Srijeet Bhattacharjee (B. Stat, 2026)
Abstract: We will learn the working of RSA cryptography, one of the first public key encryption algorithms, connect the difficulty of decryption to the famous P vs NP problem, and learn Shor’s quantum algorithm that can be used in principle to decrypt RSA code in polynomial time.
Pre-requisites: All pre-requisites are covered in B1, so B1 students can attend as well.
Video: (Not Recorded)
Date and Time: Thursday, 12th March 2026, 4:15 PM - 5:45 PM
Venue: Room 520, 5th Floor, SN Bose Bhavan