This is the home page of the UDGRP Winter 2025 organised by the Math Club. The projects start from Monday, 24th November, 2025. The topics being considered this time are presented below.
Introduction to Probabilistic Methods#
Probabilistic Methods was a branch mastered by Paul Erdős. He used to often refer to it as the subject that makes proofs beautiful and elegant or, to state it in his terms, proofs from “The Book”. The methods not only provides elegant existence proofs but also quantitative bounds, often improving upon what is achievable through purely constructive arguments. Roughly speaking, the method works as follows: You are trying to prove that a structure with certain desired properties exists, one defines an appropriate probability space of structures and then shows that the desired properties hold in this space with positive probability. We would initially go through Asymptotics, which will be helpful throughout the course. Mostly, this is a subject that helps you to connect every existing branch of Mathematics to Probability.
- Primary Reference:
- The Probabilistic Method – Noga Alon and Joel Spencer
- Group Leader:
- Subhojit Maji (B2)
- Topics to be covered:
- Probability Foundations
- Asymptotic Methods
- Graph Theory Applications
- Number Theory Applications
- Techniques in the Probabilistic Method
The following people, apart from the group leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions. Students marked (*) also gave presentations during the project.
- Arkaraj Mukherjee (B1)
- * Ayush Mandal (B1)
- * Anurag Bhunia (B1)
- Arkadeep Das (B1)
- Kanak Sanghvi (B2)
- Shankha Suvra Dam (B3)
- Waiz Arman (B1)
Exploring Random Walks#
The beauty of random walks is that it is deceptively simple to set up, but beneath lies and exciting world of surprising applications and magnificent structures. Motivating it by a couple of simple examples, we would at first try to unravel some of its magnificent properties, then would establish its relation with electrical networks. After that we would end the session either by explicitly constructing Brownian motion from random walks, or by generalizing the concept of random walks to the model of random polymers.
- Primary References:
- Random Walk Notes (Leiden University) – Hollander et al.
- Markov Chains and Mixing Times – David A. Levin, Yuval Peres and Elizabeth L. Wilmer
- Random Polymers – Frank den Hollander
- Probability and Measure – Billingsly
- Group Leaders:
- Aritrabha Majumdar (B3)
- Samadrita Bhattacharya (B3)
- Topics to be covered:
- Motivation and definition
- Reflection principle
- Arcsine law
- Stopping times and game systems
- Ruin problem and extension to higher dimensions
- LLN, large deviation, recurrence and transience, relation to Brownian motion
- Probabilistic interpretation of voltage
- Current and resistance, rewriting Kirchoff’s current and voltage law
- Rayleigh’s Monotonicity theorem
- Escape probabilities
- Skokhord Embedding / Introduction to Random Polymers
The following people, apart from the group leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions. Students marked (*) also gave presentations during the project.
- Anurag Bhunia (B1)
- Adrija Chatterjee (B3)
- * Lokesh Korada (B1)
- Shankha Suvra Dam (B3)
Tilings: Algebraic & Probabilistic Expedition#
We will introduce the mathematics of tilings of the plane and higher dimensional shapes from both an algebraic and a probabilistic perspective. Tilings takes an interesting question that serves as a perfect meeting ground for geometry, group theory, combinatorics, representation theory, and probability. On the algebraic side, we study how symmetric groups classify periodic tilings and explore substitution tilings such as the Penrose tiling. On the probabilistic side, we investigate random tilings, domino models, and the striking emergence of global order from local randomness, such as the arctic circle phenomenon. Expect to start off by seeing a lot of beautiful structures, and slowly delve into it’s connections to algebra, probability, and, if time permits, topology.
- Primary References:
- Tilings and Patterns – Branko Grünbaum, G. C. Shephard
- Lectures on Random Lozenge Tilings – Vadim Gorin
- The Symmetries of Things – John Conway, Heidi Burgiel, Chaim Goodman-Strauss
- Aperiodic Order: Volume 1 – Michael Baake, Uwe Grimm
- Group Leader:
- Hrishik Koley (M1)
- Topics to be covered:
- Definitions and basic examples of tilings: monohedral, periodic, and aperiodic tilings
- Symmetry groups of the plane
- Classification of regular and semi-regular tilings
- Substitution rules and aperiodic tilings
- Random domino tilings of finite regions
- The dimer model and uniform random tilings
- Limit shapes and the Arctic Circle Theorem
The following people, apart from the group leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions. Students marked (*) also gave presentations during the project.
- Anurag Bhunia (B1)
- * Arkaraj Mukherjee (B1)
- * Karthikeya Sharma (B1)
- * Kinshuk Banik (B1)
Computing & Logic#
Functional programming is a paradigm that is rising in importance in contemporary times. Modern functional programming has a history deeply linked with that of mathematical logic, and provides a powerful alternative to the imperative approach. In this project, we will explore ways of thinking about abstractions in computer science, particularly from this functional point of view. We shall cover topics in propositional logic, lambda calculus, type theory and general functional programming.
- Primary References:
- Program = Proof – S. Mimram
- Basic Proof Theory – A. S. Troelstra and H. Schwichtenberg
- Lectures on the Curry-Howard Isomorphism – M. Sørensen and Paweł Urzyczyn
- Group Leaders:
- Siddharto Das (B2)
- V Sai Prabhav (B2)
- Topics to be covered:
- Propositional Logic
- Untyped lambda calculus
- Simply typed lambda calculus
- Topics in type theory
- Using Haskell
- State and mutability
- If time permits –
- Grammars
The following people, apart from the group leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions. Students marked (*) also gave presentations during the project.
- Shankha Suvra Dam (B3)
Analytic Number Theory#
Analytic number theory investigates the distribution of prime numbers through methods drawn from analysis. The subject demonstrates how classical questions in number theory can be approached using powerful analytic techniques, often leading to striking and unexpected results. This course provides an introduction to some of the central ideas and results in the field, while also pointing toward deeper connections such as those surrounding the Riemann Hypothesis. The course is designed to balance rigor with accessibility, introducing fundamental analytic methods in number theory to students with prior exposure to elementary number theory and real analysis.
- Primary References:
- Introduction to Analytic Number Theory – T. M. Apostol
- An Introduction to the Theory of Numbers – G. H. Hardy and E. M. Wright
- The Theory of the Riemann Zeta-Function – E. C. Titchmarsh
- Group Leader:
- Kanak Sanghvi (B2)
- Topics to be covered:
- Average orders of arithmetic functions
- Chebyshev’s functions
- Finite abelian groups and Dirichlet characters
- An elementary proof of the Prime Number Theorem
- Dirichlet L-functions and Dirichlet’s theorem
- If time permits –
- Gauss sums, Euler products and Connections to the Riemann Hypothesis
The following people, apart from the group leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions. Students marked (*) also gave presentations during the project.
- * Anjan Majhi (B2)
- * Arkadeep Das (B1)
- Ojas Somannavar (B1)
- * Vardhan Yogesh Katekar (B1)
Introduction to Functional Analysis#
Functional Analysis plays a central role in modern mathematics, bridging the gap between linear algebra, topology, and analysis. In this course, we will explore fundamental concepts that form the backbone of this field, with an emphasis on spaces of functions and their geometric structure.
- Primary References:
- Functional Analysis: An Elementary Introduction – Markus Haase
- Mathematical Analysis – Giaquinta, Modica
- Group Leader:
- Ramdas Singh (B2)
- Topics to be covered:
- Inner product spaces and orthogonality
- Normed spaces
- Sequences as vector spaces
- Metric spaces and convergence
- Closure, continuity, and compactness, and basic topology
- Hilbert spaces and Banach spaces
- The Riemann integral and Lebesgue spaces
The following people, apart from the group leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions. Students marked (*) also gave presentations during the project.
- Anjan Majhi (B2)
- Ojas Somannavar (B1)
- Chethan Gowda N S (B1)