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UDGRP Summer 2023

Table of Contents
UGDRP SUMMER 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 Summer 2023 organised by the Math Club. The projects start from Saturday, 10th June, 2023. The topics being considered this time are presented below.

Applied Stochastic Processes #

The study of Markov Chains is a lively and central part of modern probability theory. It has ties to several other mathematical areas as well, such as Algebraic Combinatorics, Graph Theory and Representation Theory to name a few. In this group reading project, we are going to introduce Markov Chains and look at a lot of classical (and useful) examples. Our primary focus will be on the mixing of Markov Chains and computation of mixing times. We will spend some time dealing with coupling and Strong Stationary times, which will help us give upper bounds to the mixing times of some Markov Chains. We will also discuss some methods to give lower bounds to the mixing time. The second part of the project would concern other types of Markov Chains, mainly Shuffling Chains and Random Walk on Graphs, on both finite and countable state spaces. We will also deal with the Algebraic Treatment of Markov Chains, which would give further insight regarding the mixing of a large class of Chains.

  • References:

    • Markov Chains and Mixing Times – Levine, Peres, Wilmer
    • Random Walks and Heat Kernels – Martin T. Barlow
    • Analysis on Graphs – Alexander Grigor\('\)yan

  • Group Leader:

    • Ishaan Bhadoo (B3)
    • Saraswata Sensarma (B3)
    • Sarvesh Ravichadran Iyer (Ph.D)

  • Topics Covered:

    • [TBA]

The following people, apart from the group leader, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions.

  • Malav Dhaval Doshi (B2)
  • Ria Agarwal (B2)
  • Rishi Somani (B3)

The project is officially completed.

Geometric Group Theory #

Guillermo Moreno said, “Groups, like men, will be defined by their actions.” Geometric Group Theory is all about studying actions of groups on Geometric Objects. By looking at such actions and the inherent geometry of the space being acted on, we can gain information about the structure of the group. For example, free groups are precisely the ones that admit a “free” action on a tree. This can be then used to very easily prove the Nielsen-Schreier theorem; that the subgroups of free groups are free. Over this summer, we will try to learn many interesting things such as Groups acting freely and non-freely on trees, Free groups and Folding, The Ping-Pong Lemma, etc.

  • References:
    • Office Hours with a Geometric Group Theorist – Matt Clay & Daniel Margalit
    • Trees – Jean-Pierre Serre
    • Topics in Geometric Group Theory – Pierre de La Harpe
  • Group Leader: Aditya Vinay Prabhu (B3)
  • Topics Covered:
    • Basics of Group Theory
    • Free Groups and Group Presentations
    • Cayley Graphs
    • Group Actions on Graphs
    • Nielsen-Schreier Theorem and it’s proof via Group actions on trees
    • Congruence Subgroups and the presentation for \(SL_2(\mathbb{Z})\) using it’s action on Farey trees

The following people, apart from the group leader, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions.

  • Dhruba Sarkar (B2)
  • Nikhil Nagaria (B2)
  • Rinkiny Ghatak (B2)

The project is officially completed.

Representation Theory #

In this project, we study groups from a different setting than usual, using linear algebra. Vector spaces have a lot of structure, and we exploit them using “representations” to study groups. This leads to a lot of surprising results, a major one being Burnside’s theorem: “every group of order is simple”. We will see a proof of this in this project, seeing how number theory and representation theory interact to provide us with this result. Given time, we might also see applications of representation theory in probability, studying Diaconis’ work on card shuffling.

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.

  • Amit Kumar Basistha (B2)
  • Pragalbh Kumar Awasthi (B2)

The project is officially completed.

Introduction to Algebraic Topology #

The aim of learning algebraic topology is to develop algebraic invariants to study topological spaces.

The goal of the project would be to start with basic point set topology and build necessary machineries required in algebraic topology. Algebraic topology can be approached in various ways. Not surprisingly enough the initial stems of algebraic topology were mostly motivated from analysis. Fulton’s book on algebraic topology focuses more on this approach. The other way would be to take a more classical approach of building stronger algebraic background such as homology (followed by cohomology) theory to study algebraic topology, done in Rotman’s book.

We will mostly focus on homotopy theory and learn tools such as the Seifert-Van Kampen theorem and Covering Spaces to calculate the fundamental group of a space. We will try to delve deeper into Covering Spaces as its applications are not just restricted to algebraic topology. In fact if time permits we would definitely try to go over Grothendieck’s formulation of fundamental groups.

  • References:
  • Group Leader: Soumya Dasgupta (B3)
  • Topics Covered:
    • Fundamental groups and higher homotopy groups
    • Fiber and cofiber sequences
    • Computation of fundamental groups and some applications of \(\mathbb{\pi_{1}(S^{1})} \cong \Z\)
    • Seifert van-Kampen theorem for fundamental groupoids

The following people, apart from the group leader, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions.

  • Pragalbh Kumar Awasthi (B2)

The project is officially completed.

Algebraic Geometry #

This UDGRP will be an introduction to algebraic geometry, with a focus on plane curves. Algebraic geometry occupies a central role in modern mathematics; it was born out of the study of the curves and surfaces defined by polynomial equations (so called algebraic curves and surfaces), but scarcely any domains of math have been left untouched by it. Algebraic geometry originated with Descarte’s forays into what is now called analytic geometry. It quickly proceeded to become one of the central pursuits of mathematicians, attracting minds as illustrious as Newton, Fermat, and Euler. Solving a system of polynomial equations corresponds to finding the intersections of the curves they define, and a stream of results in and around the 18th century culminated in what is called Bezout’s theorem; two curves defined by degree m and degree n polynomials intersect in exactly mn points, provided care is taken to define their intersection properly. In particular, the curves taken should lie in projective space, and the desire to simplify the study of curves also serves as excellent motivation for projective space. We shall study, and prove Bezout’s theorem in this UDGRP. Many problems in mathematics that involve space of some kind are easy to solve in limited regions of the space, but not throughout it. Mathematicians then divide the space into many tiny parts, solve the problem for each part, and piece the solutions together. You will get your first taste of this when you learn about Max Noether’s theorem, which lets us solve the problem of determining when a curve can be defined by given sets of equations by examining the question at points where the curves described by the sets of equations intersect. Then we shall generalize the notion of algebraic curves enabling us to study their properties without reference to any surrounding space; we shall study curves independent of the space they are embedded in. Doing so will allow us to talk about blowing up: a powerful geometric technique to correct imperfections (singularities) of curves, that has since spread as far as the study of differential equations.

Finally, we shall turn our attention to functions whose domains are algebraic curves. In particular, we shall examine rational functions, but the only rational functions with no poles anywhere on the algebraic curve they are defined on, are the constant functions. So we shall be forced to allow functions with poles, and in particular we shall examine, given a set of prescribed locations where we are willing to accept poles, the size of the space of functions with poles at only those points, ending with what is probably the most celebrated theorem in the study of algebraic curves; the Riemann Roch theorem.

  • References:
  • Group Leader: Eeshan Pandey (B3)
  • Topics Covered:
    • [TBA]

The following people, apart from the group leader, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions.

  • Ruchira Mukherjee (B2)
  • Sankha Subhra Chakraborty (B2)

The project is officially completed.

Introduction to Differential Geometry #

We will start with a simple question: How would three ants know if they are walking on a cylindrical bottle or a ball? The answer involves concepts such as curvature, classes of surfaces, and geodesics. In the field of Differential Geometry, we study the geometric properties of various curves and surfaces using analytical techniques. Dealing with properties like the curvature for curves in plane and space, we can determine the local (and global) nature of the given curve. In this group reading project, we will explore the fundamentals of differential geometry and learn how to describe and analyze curves and surfaces in different ways. We will start by going through basic concepts of multivariable calculus and learn how to use tools such as partial derivatives, gradients, divergence, curl, and integrals to study functions of several variables. We will then introduce the concepts of curves and arc lengths, and see how to measure the curvature and torsion of a curve in space. We will also investigate the global properties of curves such as the Frenet-Serret formulas and the four-vertex theorem.

Next, we will move on to surfaces and learn how to define and classify them using local and global parameters. We will see how to compute the first and second fundamental forms of a surface, which encode the metric and curvature information of the surface. We will also learn how to find the normal and principal curvatures of a surface, and how they relate to the Gaussian curvature and the Gauss map. In addition to that, we will introduce the notion of geodesics, which are the shortest paths on a surface.

Finally, if time permits, we will cover topics such as Minimal Surfaces and the Gauss-Bonnet theorem, which relates the total curvature of a surface to its topological characteristics.

  • Pre-Requisites: Basic Analysis and Linear Algebra
  • References:
  • Group Leaders:
    • Devansh Kamra (B3)
    • R Gnanananda Shreyas (B3)
  • Topics Covered:
    • Point set Topology on \(\mathbb{R}^{n}\)
    • Continuity and Differentiability on \(\mathbb{R}^{n}\)
    • Chain rule for Multivariable Functions
    • Directional Derivatives
    • Clairout’s and Hermann Schwartz Theorem
    • Total derivatives and reduction of Total derivatives
    • Mean Value theorem in \(\mathbb{R}^{n}\)
    • Taylor series for Multivariable Functions
    • Inverse and Implicit Function Theorem
    • Riemann Integration for Multivariable Functions
    • Fubini’s Theorem
    • Level Curves
    • Reparametrization of Curves
    • Arc Length
    • Curvature and Torsion
    • Frenet-Serret equations
    • Plane Curves, Space Curves, Simple Closed Curves
    • Isoperimetric Inequality
    • The Four Vertex Theorem
    • Planes and Normals
    • Surface Integrals
    • Green’s Theorem
    • Divergence Theorem and Strokes’s Theorem on \(\mathbb{R}^{n}\)

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.

  • Anasmit Pal (B2)
  • Anit Das (B2)
  • Anshuman Agarwal (B2)
  • Shriya Srivastava (B2)

The project is officially completed.

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