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UDGRP Winter 2023

Table of Contents
UDGRP Winter 2023 HAS BEEN OFFICIALLY CONCLUDED. FIND BELOW THE TOPICS COVERED IN EACH GROUP AND LIST OF PEOPLE WHO COMPLETED THE PROJECT.

This is the home page of the UDGRP Winter 2023 organised by the Math Club. The projects start from Saturday, 29th November, 2023. The topics being considered this time are presented below.

Knot Theory #

In the 19th century, Lord Kelvin made a failed attempt to classify chemical elements by considering them as knots in luminiferous aether. However, topologists of the early 20th century, like Max Dehn, J. W. Alexander and others, picked up on the topic and enriched the Theory of Knots. In recent years, Knot Theory has seen its applications in a wide range of subjects, like Quantum Field Theory, Organic Chemistry, and even in the study DNA and polymers. In this project, we will take a look at the basics of Knot Theory, colourability of knots, some knot invariants and some algebraic structures on knots.

  • Primary References:

  • Group Leader:

    • Dhruba Sarkar (B2)

  • Topics Covered:

    • What is homeomorphism, embedding, ambient isotopy?
    • Definition of Knot
    • Reidemeister Theorem and Reidemeister Moves
    • Knot Invariants - Knot Coloring, Alexander Polynomial, Kauffhart Polynomial, Jones Polynomial etc.
    • Knot Group
    • Links and Braids

    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.

  • * Yuvraj Singh Rajpurohit (B1)

Random Graphs #

This project aims to provide a fundamental exploration of random graphs, a fundamental concept lying in the intersection of graph theory, probability, and combinatorics, that is widely used in network science. We will introduce students to the basic graph theory and probabilistic techniques used in the study of random graphs. Both classical and modern approaches will be covered, with a focus on understanding the properties and applications of random graphs in computer science and network science.

This course aims to provide students with a solid foundation in probabilistic tools for studying random graphs and a taste of their fascinating real-world applications. Students interested in computer science will find it more interesting as it will open up further directions in network science and combinatorics.

  • Primary References:
  • Group Leaders:
    • Bikram Halder (B3)
    • Hrishik Koley (B2)
  • Topics Covered:
    • Basic concepts and definitions related to graphs.
    • Basic probabilistic methods: LLNs, CLT, 1st & 2nd moment methods, convergence of random variables, basic branching process, Chernoff & Cramer bounds, Matringales, concentration inequalities and their use in graphs.
    • Asymptotics: Miscellaneous inequalities, asymptotic notations.
    • Models of random graphs: Overview of 2 models of random graphs, namely, the Erdős–Rényi model \((G_{n,p})\) and the Gilbert model \((G_{n,m})\), which are closely related.
    • Properties of random graphs: Study various properties of random graphs such as subgraph count, degree distribution, giant component and phase transition as well as various thresholds and sharp thresholds related to them.
    • Kahn-Kalai Conjecture
    • Basic Idea of Preferential Attachment Model
    • Applications: Discussion of applications of random graphs in network

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.

  • * Adrija Chatterjee (B1)

Galois Theory #

Galois theory is an important branch of abstract algebra, with a rich history and many applications. The central object of study in Galois theory is the structure of field extensions, which arise naturally in many mathematical contexts, such as the solutions of polynomial equations. The theory was developed by Évariste Galois and Niels Abel in the 19th century to resolve the fundamental question: “Can every polynomial equation be solved by radicals?”. Galois Theory provides a bridge betweeen Field Theory and Group Theory which is given by the Fundamental Theorem of Galois Theory. In this reading project we will learn about Field Extensions, Galois Groups and other algebraic structures and will try to motivate how these concepts can be applied to study polynomial equations.

  • Primary References:
  • Group Leaders:
    • Amit Kumar Basistha (B2)
    • Gandharv Sawant (B2)
  • Topics Covered:
    • Basic Group Theory
    • Field Extensions
    • Normality and Separability
    • Ruler and Compass Constructions
    • Field Automorphisms
    • Fundamental Theorem of Galois Theory

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.

  • Arkapriyo Hore (B1)
  • Anjali Elizabeth Paul (B2)

Commutative Algebra #

Commutative Algebra is the subject that provides the basic tools and structure required to study Algebraic Number Theory and Algebraic Geometry. Studying commutative rings allows one to build up the notions that are taken for granted in elementary number theory from the roots, and this is helpful because not only does it reveal the bare minimum required to achieve a result, but also demonstrates the limits of one’s assumptions. In this project we aim to cover basic concepts such as prime and maximal ideals, special rings such as Unique Factorization Domains and Principal Ideal Domains, the Spectrum of a ring, Noetherian Rings, the ascending chain condition, and exact sequences. We shall also try to showcase some classical theorems such as Cayley-Hamilton theorem over commutative rings, Hilbert’s Basis Theorem and Nakayama’s Lemma.

  • Primary References:
  • Group Leaders:
    • Sankha Subhra Chakraborty (B2)
    • Tejas Varma (B2)
  • Topics Covered:
    • Prime and Maximal Ideals
    • Operations on Ideals
    • Spectrum of a Ring
    • Radical of an Ideal
    • Local Rings
    • Modules
    • Cayley-Hamilton theorem
    • Nakayama’s lemma
    • Noetherian Rings
    • Hilbert Basis 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.

  • Kalmanje Avyaktha Achar (B1)
  • Arkapriyo Hore (B1)
  • Anjali Elizabeth Paul (B2)

Algebraic Topology #

One of the main goals in algebraic topology is to find algebraic invariants which characterize spaces to a reasonable extent. As we cover basic point-set topolgy, we shall explore a few interesting spaces which will serve as important counterexamples later. We shall also discuss the construction of cell complexes, the invariance of the Euler characteristic of a surface and cover two proofs of the classification theorem of closed surfaces.

We shall then study the linking properties of the result of cutting n-twisted bands, and define winding numbers to give a proof of Brower’s fixed point theorem in \(\mathbb{R}^2\). These will serve as the motivation for defining a key invariant called the fundamental group of a space. Using this, we will prove many results including Borzuk Ulam theorem and the Ham Sandwich theorem. We may also discuss some applications of the van Kampen theorem.

  • Primary References:
  • Group Leaders:
    • Anasmit Pal (B2)
    • Pragalbh Kumar Awasthi (B2)
  • Topics Covered:
    • Basics of Point-Set Topology
    • Cell Complexes
    • The Euler Characteristic
    • Classification of Closed Surfaces
    • Linking and Winding Numbers
    • Brouwer’s Fixed Point Theorem
    • The Fundamental Group
    • Borsuk Ulam theorem and Ham Sandwich 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.

  • Arya Biswas (B1)
  • Anant Umesh Kerur (B1)

Measure Theory #

We have all heard that the Cantor set has “length 0”. But what is this length? We resolve this by defining the notion of the “measure” of a set, which reduces to length for intervals. Defining this measure for certain “Borel subsets” of R we also obtain the “uniform distribution” which is an integral part of Probability Theory. Developing this notion in a more general setting we can also use this to define “Area” and 2-dimensions and “Volume” in 3 (and higher) dimensions. After developing the so-called Lebesgue measure, we use it to define a more adequate notion of integration than the Riemann Integral. Because of treating measure in a more general setting, this development immediately gives the integral in higher dimensions once we have a measure on \(\mathbb{R}^{d}\). The main facts we will prove about the Lebesgue integral are: The Lebesgue Dominated Convergence Theorem (DCT), and the Monotone Convergence Theorem (MCT).

  • Primary References:
  • Group Leaders:
    • Anshuman Agrawal (B2)
    • Bhavesh Pandya (B2)
  • Topics Covered:
    • Riemann Integration and Convergence of function (point wise and uniform).
    • Algebra, Sigma Algebra, their properties. Measure, finite, Sigma finite, semi finite measures.
    • Outer measure, Caratheodory’s Theorem(1st), Pre Measure, Caratheodory-Hann Extension Theorem.
    • Measurable Sets.
    • Lebesgue Measure, Measurable functions.
    • Lebesgue Integration: Integration on L+, simple functions, approximating measurable functions by Simple functions.
    • Integration on \(\mathbb{R}, \mathbb{C}\).
    • Fatou’s Lemma, Monotone Convergence Theorem, Dominated Convergence 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.

  • Sarbartha Sarkar (B1)
  • Aditya Garg (B2)
  • Nikhil Nagaria (B2)
  • Pragalbh Kumar Awasthi (B2)

Random Walks, Stochastic Processes, and Brownian Motion #

Probability theory involves analyzing event behavior over a significant time span. This UDGRP aims to explore fundamental concepts and the roots of random processes without delving into technical details, offering an intuitive understanding. We will begin by discussing important results such as the Law of Large Numbers and the Central Limit Theorem. Then, we’ll examine examples of random processes, with a primary focus on connecting Brownian Motion to a simple random walk on the real line. The project will also explore Brownian Motion’s properties, including its application in analyzing the Empirical Distribution Function, hitting times, arc sine law, drift, and Markov processes. If time allows, we may introduce Time Series, with a focus on the Kalman Filter and Optimal Transport.

While there are no prerequisites for this UDGRP, having knowledge of results like the Law of Large Numbers, the Central Limit Theorem (excluding proofs), standard distributions, and covariance/correlation can be beneficial.

  • Primary References:
    • Stochastic Processes – Sheldon M. Ross
    • Probability Theory and its applications, Vol. I & Vol. II – William Feller
    • Analysis of Time Series – Christopher Chatfield
    • Topics in Optimal Transport – Cedric Villani
  • Group Leaders:
    • Malav Dhaval Doshi (B2)
  • Topics Covered:
    • SLLN, CLT, Some Standard Distributions and their properties and Covariance/Correlation (Overview i.e modulo proofs)
    • Introduction to Random Walk and Stochastic Process
    • Poisson Process and deriving some essential results related to it.
    • Relation of Brownian motion with Random Walk and equivalent definition of brownian motion.
    • Introduction to brownian bridge
    • Analyse “Empirical Distribution Function” using Brownian Motion, LLN and CLT.
    • Analyze the hitting times and variations of Brownian motion.
    • Brownian motion having a drift and (semi)markov processes.

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.

  • * Aritrabha Majumdar (B1)
  • * Samadrita Bhattacharya (B1)
  • * Shiva Sanjay Semwal (B1)
  • Muhammed Shameel K V (B1)

Combinatorial Game Theory #

Combinatorial games are played between two players who make alternating moves with no hidden information or chance elements. In this project, we will study various such games and build general techniques to find strategies and analyse positions which arise in them. Often, this involves cleverly assigning values to quantify the advantage a player has in a given position and then studying the arithmetic of these values. Typically, these values behave quite differently from ordinary numbers, and understanding their structure will be one of our main goals.

We shall begin with a discussion of various types of games, their combinations and equivalences and give solutions for some specific games. Next, we will specialize to the study of impartial games where allowed moves depend only on the position and not the player. We will prove the Sprague-Grundy theorem which states that all such games are equivalent to a nim game. We will also study various partizan games including the game of Hackenbush and prove the colon principle used to simplify positions in it. We will also spend some time on nim games on graphs and a combinatorial game theoretic analysis of chess endgames. If time permits, we may also study infinitesimal games and temperature theory.

  • Primary References:
  • Group Leaders:
    • Nikhil Nagaria (B2)
    • Pragalbh Kumar Awasthi (B2)
  • Topics Covered:
    • Basics of Combinatorial Game Theory
    • Some Simple Combinatorial Games
    • Several Impartial Games
    • Impartial Hackenbush: Fusion and Colon Principle
    • Sprague-Grundy Theorem
    • Hackenbush: The Purple Mountain and the Green Jungle
    • Nim on Graphs
    • A Combinatorial Game Theoretic Analysis of Chess Endgames

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.

  • * Yuvraj Singh Rajpurohit (B1)
  • * Daibik Barik (B1)
  • * Arya Biswas (B1)
  • Shiva Sanjay Semwal (B1)
  • Suryansh Shailesh Shirbhate (B1)

Analytic Number Theory #

Analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet’s 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet’s theorem on arithmetic progressions. It is well known for its results on prime numbers in multiplicative number theory (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring’s problem).

  • Primary References:
  • Group Leaders:
    • Aishik Sarkar (B2)
  • Topics Covered:
    • Convolutions
    • Various number theoretic functions
    • Asymptotic formulae
    • Chebysev’s functions and relations equivalent to the prime number theorem
    • Hensel’s lemma
    • Dirchlet’s theorem of primes in an arithmetic progression

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.

  • Korou Khundrakpam (B1)

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