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UDGRP Winter 2022

Table of Contents
UDGRP Winter 2022 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 2022 organised by the Math Club. The projects start from Saturday, 7th December, 2022. The topics being considered this time are presented below.

Category Theory #

Category theory originally arose as methods of formalising certain facts that hold for entire classes of mathematical structures, so as to make life easier and prove theorems at once for a large class of objects. In the last 50 or so years, it has however become a very diverse and useful branch of mathematics unto itself. We plan to have a brisk walk through the basic areas of category theory as well as look at how such abstractions actually help in solving very relevant problems in real life.

  • Primary References:
  • Group Leaders:
    • Aaratrick Basu (B2)
    • Aprameya Girish Hebbar (B3)
  • Brief Sketch of Project:
    • Introduction to Categories, Functors and Natural Transformations: Why do category theory? Basic definitions and examples. Is there something higher than a category?
    • Representabilty and the Yoneda lemma: Representable functors and examples. The Yoneda lemma. Using the Yoneda lemma. Universal elements.
    • Adjoint functors. Basic definitions and examples. Units and Counits. Adjunctions via initial objects.
    • Some applications. What is Compositionality? Profunctors and Monoidal categories.

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 (B1)
  • Pragalbh Kumar Awasthi (B1)
  • Shriyaa Srivastava (B1)

The project is officially completed.

Galois Theory #

In the nineteenth century, two very young mathematicians, Neils Abel and Evariste Galois, answered a question about which the mathematical world had been curious for centuries. It is a question about the central activity of algebra, solving equations, particularly the question: Is there a formula for solving fifth degree polynomial equations? In this reading project, we will provide a modernized approach towards solving this question and discover the many uses of this approach and how it plays a central role in Algebra.

  • Primary References:
  • Group Leaders:
    • Ishaan Bhadoo (B2)
    • Varun Balasubramaniam (B2)
  • Brief Sketch of Project:
    • Basic Algebra (Groups, Rings)
    • Algebraic extensions: degree, Splitting fields and normal extensions, Algebraic closure.
    • Ruler and Compass constructions
    • Separability and Galois Correspondence
    • Notion of Solvable Groups
    • Solvability by radicals.
    • The Fundamental Theorem of Algebra
    • Transcendence of \(\pi\)

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 (B1)
  • Aditya Garg (B1)

The project is officially completed.

Generating Functions #

Generating functions are a useful tool which have a wide range of applicability. They can be used to find recurrence relations, sometimes even the exact form of a sequence. They can be used to get asymptotic formulae for the sequence when the exact form is too difficult to calculate, as in the case of The Prime number Theorem. Generating functions can also be used to calculate averages and other statistical properties of a sequence. This winter, we’ll try and learn some of these applications of Generating functions.

  • Primary References:
  • Group Leaders:
    • Aditya Prabhu (B2)
    • Pavan K Srinivasan (B2)
  • Brief Sketch of Project:
    • Introduction to generating functions with recurrences
    • Analytic v/s Formal theory of series
    • Cards, Decks and Hands
    • Using generating functions to find out some properties of partitions of \(n\), determining which permutations can be \(k^\textsf{th}\) powers of other permutations etc.

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.

  • Anshuman Agrawal (B1)
  • Dhruba Sarkar (B1)
  • Nikhil Nagaria (B1)

The project is officially completed.

Markov Chain and Mixing Time #

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. Time permitting, we will look at Card Shuffling chains, which are an interpretation of Random Walks on the Symmetric Group.

  • Pre-Requisites:
    • A first course in Probability Theory and Linear Algebra will be sufficient.
  • Primary Reference:
  • Group Leaders:
    • Deepta Basak (B2)
    • Saraswata Sensarma (B2)
  • Brief Sketch of Project:
    • Using the classical examples of Gambler’s Ruin and The Coupon Collector’s Problem, we will introduce the idea of a Finite Markov Chain. We shall talk about certain characteristics of a Markov Chain like Irreducibility, Aperiodicity and Time Reversibility.
    • Next, we shall talk about the existence and uniqueness of the Stationary Distribution of a Markov Chain.
    • Next, we proceed to look at some further examples of Markov Chains that come up throughout probability and statistics.
    • A crucial result that we shall discuss at this point is that under some mild conditions, Markov Chains do, infact, converge to their Stationary Distributions. We define total variation distance and mixing time, the key tools for quantifying that convergence.
    • If there is time, we might look at some more applications like 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.

  • Aditya Garg (B1)
  • Anasmit Pal (B1)
  • Doshi Malav Dhaval (B1)
  • Hrishik Koley (B1)
  • Ruchira Mukherjee (B1)
  • Sankha Subhra Chakraborty (B1)
  • Shriyaa Srivastava (B1)

The project is officially completed.

Probabilistic Method #

We will introduce The Probabilistic Method as a tool for tackling problems in Graph Theory, Ramsey Theory, etc. A brief introduction to Graph Theory and Ramsey Theory will be given. We will discuss some Combinatorial Number Theory, along with some probabilistic techniques. We will talk about the Linearity of Expectation and show its use as a powerful tool in proving some results in Graph Theory and several other topics. We might discuss results like LLN ( WLLN & SLLN) and CLT ,then move forward into Probabilistic Number Theory ,especially the Probability Theory for Additive Functions. Finally, we will look into Probabilistic Analysis of Algorithms and Randomized Algorithms

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.

  • Anasmit Pal (B1)
  • Hrishik Koley (B1)
  • Malav Dhaval Doshi (B1)
  • Ruchira Mukherjee (B1)

The project is officially completed.

Topology and Topological Groups #

Topology arises in most of the field of mathematics quite naturally, and the study of topological spaces is a prime task in mathematics. Topology attempts to generalize geometry to apply it to a vast number of settings. This is going to a primary introduction to topology and we will also see what happens when we add more structure to the space such as topological groups. If time permits we will give a brief motivation for Algebraic Topology.

  • Primary Reference:
  • Group Leaders:
    • Soumya Dasgupta (B2)
    • Trishan Mondal (B2)
  • Brief Sketch of Project:
    • We will start off with introduction to topological spaces, and discuss about bases, and continuous function and homeomorphisms on topological spaces.
    • We will also need some tools from group theory, and give a brief idea about matrix groups, which we will need later when we come to Topological groups.
    • Next task would be to cover important topics such as connectedness, path-connectedness and compactness.
    • We will discuss about ways to construct new topological spaces from existing spaces (quotient topology, connected sums, adjunction).
    • Finally we will discuss about topological groups, and see how the matrix groups are just geometric objects in disguise.
    • If time permits we will give a motivation for Algebraic Topology.
  • Notes: Available here.

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.

  • Pragalbh Kumar Awasthi (B1)

The project is officially completed.

Related

UDGRP Winter 2023
9 Projects | 4 Weeks | 9 Groups | 13 Leaders | 19 Participants