UDGRP Summer 2025

This is the home page of the UDGRP Summer 2025 co-organised by the Math Club, ISI Bangalore and Math Club, ISI Kolkata. The projects start from Monday, 19^th May, 2025. The topic being considered this time are presented below.

An Invitation to High Dimensional Probability #

High Dimensional Probability (HDP) is a branch of probability which deals with large dimensional random vectors. The techniques are well suited for the study of models with a large number of random inputs which are roughly independent - so that the space of parameters is high dimensional. Such models arise quite frequently in data science, randomized algorithms, signal processing, and quantum information theory.

    There are four essential themes of high dimensional probability: Concentration, Suprema, Universality and Phase transitions. In this project, we wish to explore the first and second themes in considerable detail, while touching upon the latter two briefly.

• References:
  • An Introduction to High Dimensional Probability – Roman Vershynin
  • High Dimensional Probability – Ramon van Handel
  • Modern Discrete Probability – Sebastian Roch
  • Probability in High dimensions – Joel A. Tropp
• Group Leader:
  • Saraswata Sensarma (M.Math 1st year, ISI Kolkata)
• Topics to be Covered:
  • Concentration Inequalities: Markov and Chebychev inequalities, moment methods, Cramer-Chernoff bounds, Hoeffding and Bernstein inequalities, martingale methods - Azuma-Hoeffding, Efron-Stein and bounded difference inequalities, entropy bounds.
    • Applications: random graphs, random matrices, data science, signal processing, and asymptotic convex geometry.
  • Supremum: Upper bounds to supremum-direct method, covering method, chaining and generic chaining, Dudley’s inequality
    • Applications: singular values of random matrices, convex analysis
  • Universality and Phase transitions: Lindeberg exchange method, central limit theorem, Phase transitions will be studied via examples involving random graphs and percolation theory.

Random Walk on Graphs #

This summer reading course introduces the theory of random walks on graphs — a special class of Markov chains — and explores its powerful connections and applications. Starting with the basic definition of a random walk on a graph, we will see how key questions (e.g., expected return times) can be answered elegantly by interpreting graphs as electrical networks. This perspective leads to simple, intuitive methods for computing hitting probabilities and resistances.

    By the end of the course, participants will understand both the probabilistic and analytic techniques behind random walks, and see how these tools solve a wide range of problems in combinatorics, probability, and network theory.

    Note: Basic tools like moment methods, bounded difference inequalities will be covered by Saraswata in his High Dimensional Probability course.

• References:
  • Random Walks and Heat Kernels on Graphs – Martin T. Barlow
  • Markov Chains and Mixing Times – David A. Levin, Yuval Peres and Elizabeth L. Wilmer
  • Reversible Markov Chains and Random Walks on Graphs – David J. Aldous and James Allen Fill
• Group Leader:
  • Bikram Halder (M.Math 1st year, ISI Kolkata)
• Topics to be Covered:
  • Random Walk Basics: Definitions, transition probabilities, recurrence vs. transience, and examples.
  • Electrical Network Analogy: Using resistance networks to compute random walk properties like hitting times and return probabilities.
  • Applications to Percolation on Trees: How random walks illuminate the structure of infinite trees and critical phenomena in percolation.
  • Counting Spanning Trees: Employing Green’s functions and the matrix-tree theorem to count spanning trees in finite graphs.
  • Other Topics (time permitting): Mixing times — how quickly random walks converge to equilibrium — and heat kernel estimates for diffusion.

Representation Theory & Symmetric Functions #

Symmetry is everywhere – from the shapes of molecules to the roots of equations. But how do we quantify symmetry mathematically? This question is answered by \textbf{representation theory}, where abstract groups like permutations and rotations are transformed into concrete matrices and combinatorial objects. Vector spaces and matrices, having a lot of structure, are exploited to study groups by using “representations”.     We will emphasize on studying the representation theory of the symmetric group, where we will prove that its irreducible representations are in bijection with integer partitions and construct them explicitly via Specht modules.     We will also explore how the representation-theoretic perspective leads naturally to symmetric functions – which plays a key role in modern combinatorics. We will finally go on to examine combinatorial manifestations through the Robinson-Schensted-Knuth correspondence and Schur functions.

• References:
  • Representation Theory: A First Course – William Fulton and Joe Harris
  • The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions – Bruce E. Sagan
  • Linear Representations of Finite Groups – J.P. Serre
• Group Leaders:
  • Anasmit Pal (B.Math 3rd Year, ISI Bangalore)
  • Hrishik Koley (B.Math 3rd Year, ISI Bangalore)
• Topics to be Covered:
  • Basics of Group Theory
  • Representations of Finite Groups
  • Character Theory
  • Induced Representations, Group Algebras, Real Representations
  • Young Diagrams, and Frobenius Character Formula
  • Symmetric Functions, and Robinson-Schensted-Knuth (RSK) Correspondence
  • Other Topics (time permitting): Murnaghan-Nakayama Rule — Quasisymmetric Functions — Knuth Equivalence — Littlewood-Richardson Rule

Galois Theory #

Galois theory as the name suggests was introduced by Galois for studying roots of a polynomial. The main problem was to characterize polynomial equations that are solvable by radicals. The Galois theory which was developed by Galois is quite different from what we study now, it’s the works of Emil Artin which will be our main focus. We will need a basic understanding of Group Theory, Rings and Modules, though all these topics that may sound unfamiliar are actually quite intuitive. In fact one can easily get a hold of prerequisites while reading field theory, which is the start of Galois Theory. We definitely look at the Fundamental Theorem of Galois theory, which gives bijection between intermediate field extensions $k \subseteq L \subseteq K$ and subgroups of the Galois group of the field extensions $K$ over $k$, whenever the extension is a finite Galois extension. The Fundamental Theorem of Galois Theory will resolve Galois’ dream, we will give a precise characterization of polynomials that are solvable by radicals. Other applications include the most elementary proof of the Fundamental Theorem of Algebra that actually uses algebra. But Galois theory is not just about all these, it has deep connections with other fields of mathematics such as covering spaces. There are also other aspects that one can look at such as Galois theory for infinite algebraic extensions or one can also look at more practical examples coming from Algebraic Geometry where the field extensions are not necessarily algebraic.

• References:
  • Field and Galois Theory – Patrick Morandi
  • Galois Theory – Ian Stewart
• Group Leader:
  • Soumya Dasgupta (M.Math 1st Year, ISI Kolkata)
• Topics Covered:
  • Fundamental theorem of Galois theory
  • Some particular field extensions such as cyclic and cyclotomic
  • Kummer extensions
  • Hilbert theorem 90
  • Other Topics (time permitting): Aspects to Galois theory in topology or non algebraic extensions

Analytic Number Theory #

The works of Euler, Dirichlet and Riemann revolutionised the study of number theory by using relatively modern analysis to study the millenniums old subject, and gave rise to what is known today as analytic number theory. The subject has the speciality that though there are many questions that can be explained to a complete layman, the solutions require some of the most sophisticated mathematical structures.

    This reading course will be the first steps in learning the vast subject, and we would see some of the beautiful results, while stating the present-day best known result in related topic.

• References:
  • Introduction to Analytic Number Theory – Tom. M Apostol
  • Motivational Lectures – Present Day Analytic Number Theorists
• Group Leader:
  • Srijeet Bhattacharjee (B.Stat 2nd Year, ISI Kolkata)
• Topics Covered:
  • Arithmetical functions, Dirichlet Convolution, average orders.
  • Dirichlet Characters, Gauss Sums, Primes in an arithmetic progression
  • Dirichlet Series, Euler products
  • Other Topic (time permitting): Riemann zeta function — other L-functions — Grand Riemann Hypothesis — Sieve theory and distribution of primes — Probabilistic number theory — Equidistribution problems