UDGRP Summer 2022

This is the home page of the UDGRP Summer 2022 organised by the Math Club. The projects start from Wednesday, 1^st June 2022. The topics being considered this time are presented below.

Introduction to Commutative Algebra #

Commutative Algebra is the study of modules over commutative rings. Apart from being a marvelous subject on its own, it is the backbone of Algebraic Geometry and Algebraic Number Theory. This reading project will begin with basic concepts of groups and rings, and assisted by a myriad of examples and research articles, it will culminate in the famous Lasker-Noether Theorem.

• Pre-Requisites: Basic Set Theory
• Primary References:
  • Commutative Algebra, Volume 1 – Oscar Zariski & Pierre Samuels
  • Introduction to Commutative Algebra – M. F. Atiyah & I. G. MacDonald
• Group Leader: Saheb Mohapatra (B3)
• Topics Covered:
  • Basic Group and Ring Theory
  • Localisation
  • Noetherian Rings and Modules
  • Hilbert Basis Theorem
  • Lasker-Noether Theorem

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.

• Shubhrojyoti Dhara (B2)
• Soumya Dasgupta (B2)
• Aaratrick Basu (B2)

The project is officially completed.

Introduction to Measure theory #

A measure is a generalization of geometrical measurements (length, area, volume) and other basic ideas like mass and probability. These seemingly disparate ideas have a lot in common and may frequently be considered as a single mathematical construct. One of the main goals of Lebesgue’s measure theory is to create a fundamental framework for doing integration that works well with limits and admits a large class of functions that Riemann’s integration theory does not cover. It turned out that it also provided new ways of thinking about measuring objects, which is useful in many other areas of math, like probability theory. Measure theory is a fundamental base to many advanced topics in Analysis, like Functional analysis, Harmonic Analysis and PDEs.

• Pre-Requisites: Set theory, Real Analysis.
• Primary Reference:
  • An Introduction to Measure Theory – Terence Tao
• Group Leaders:
  • Aprameya Girish Hebbar (B3)
  • Sanchayan Bhowal (B3)
  • Sarvesh Ravichandran Iyer (SRF)
• Topics Covered:
  • Elementary and Jordan measure.
  • Lebesgue measure and its properties.
  • The Lebesgue Integral on \(\mathbb{R}^d\) and its properties.
  • Sigma algebras, abstract measure spaces.
  • Abstract Lebesgue Integral, convergence theorems.

The following people, other than the leaders, have completed the reading project by going through, in detail, all the assigned parts and participating in the discussions.

• Ishaan Bhadoo (B2)
• Sambit Mishra (B3)
• Saraswata Sensarma (B2)
• Prognadipto Majumder (B2)

The project is officially completed.

An Introduction to Algebraic Topology #

Algebraic Topology consists of a vast and powerful theory that attempts to answer the fundamental questions of topology using algebraic invariants of topological spaces. A study of these invariants requires a new language, known as category theory to understand the notions of how these algebraic structures interact with their topological counterparts. We look at one of the most fundamental invariants: Homotopy. We see how homotopy acts as a functor between categories, and introduce the fundamental group. We then proceed to study the celebrated Seifert - Van Kampen Theorem, which allows us to compute the fundamental group of a vast number of topological spaces. Finally, we shall look further into the theory of covering spaces in an attempt to see the beautiful Galois-like correspondence between the subgroups of a fundamental group and the unique covering spaces of the space.

• Pre-Requisites: Metric Spaces, Elementary Point Set Topology, Elementary Real Analysis, Groups, Normal Subgroups, The Isomorphism Theorems.
• Primary References:
  • Topology – James R. Munkres
  • Introduction to Topological Manifolds – John M. Lee
  • An Introduction To Algebraic Topology – Joseph J. Rotman
• Group Leader: Srigyan Nandi (B3)
• Topics Covered:
  • Notions of Elementary Point Set Topology: Connectedness and Compactness.
  • Homotopy And The Fundamental Group, Simply Connected Spaces, Covering Spaces, The Fundamental Group of A Circle.
  • Free Groups, Free Products, Free Abelian Groups, The Seifert - Van Kampen Theorem, Fundamental Groups of Wedged Spaces, Fundamental Groups of Graphs.
  • Covering Spaces, Groups Acting On Topological Spaces, Deck Transformation Groups, Semi-simply connected spaces, Universal Covers, The Classification of Covering Spaces.

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

• Aratrick Basu (B2)
• Eeshan Pandey (B2)
• Gautham Ravuri (B3)
• Soumya Dasgupta (B2)

The project is officially completed.

Representation of Finite Groups #

The prototypical example of a group is \( \textsf{Aut}(X) \), where \(X\) is some object (\(X\) can just be a set without any structure). Since a group \(G\) is a very abstract object, to extract information from it, we map it to something more concrete, namely, \( \textsf{Aut}(X) \). So, a representation of a group is any map from \(G\) to \( \textsf{Aut}(X) \) that respects the group structures. We can linearize this map naturally if our objective \(X\) is a vector space over some field \(k\) (we assume \(\textsf{char}(k) = 0\) for this project). In this project, we study linear representations of finite groups.

• Pre-Requisites: Linear Algebra
• Primary References:
  • Representation Theory of Finite Groups – Benjamin Steinberg
  • Linear Representation of Finite Groups – Jean-Pierre Serre
  • Abstract Algebra – David Steven Dummit & Richard M. Foote
• Group Leader: Rahul Mazumder (B3)
• Topics Covered:
  • Group Theory
  • Group Representations: definitions, examples, permutation representations, Mashke’s Theorem, Tensor Product of modules over commutative rings
  • Character Theory
  • Burnside’s \(pq\) Theorem

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

• Varun Balasubramanian (B2)
• Ishaan Bhadoo (B2)

The project is officially completed.

Analytic Number Theory #

Consider the function on natural numbers which gives the number of distinct prime factors. You’ll notice that the function takes huge dips on large prime numbers and this makes it hard to study the behaviour of number of distinct prime factors. Analytic Number Theory is all about studying behaviour of such “analytic” functions and how to solve the problems as above.

• Pre-Requisites: Elementary Number Theory, Algebra, Real Analysis
• Primary Reference:
  • An Introduction to Analytic Number Theory – Tom M. Apostol
• Group Leader: Shreyash Kharat (B3)
• Topics Covered:
  • Mobius Inversion Formula
  • Dirichlet Product and Inverse
  • General Analytic Functions
  • Generalized Convolutions
  • Bell Series
  • Averages of Analytic Functions
  • Chebyshev’s functions
  • Prime Number Theorem

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

• Rahil Miraj (B3)
• Saraswata Sensarma (B2)
• Shubhrojyoti Dhara (B2)
• Trishan Mondal (B2)

The project is officially completed.

Machine Learning #

Suppose you are given a sample of 3 variable data (2 predictors and 1 response) from a population and you fit a linear model on it and it gives good results with respect to errors. What if the population prediction data is spread around the surface of a cuboid and the sample you got only belonged to one of the faces of the cuboid? Your linear model won’t be useful to draw out inferences on the population data. What if it is spread around the surface of one half of a sphere, or if it is clustered? In this project, we will deal with various distributions and types of data in real world problems, like numerical data, image data, text data, audio data, etc.

• Pre-Requisites: Probability 1 & 2, Statistics 1, Linear Algebra, Real Analysis
• Primary Reference:
  • An Introduction to Statistical Learning with Applications using R – Gareth James, Daniela Witten, Trevor Hastie & Robert Tibshirani
• Group Leader: Shreyash Kharat (B3)
• Topics Covered:
  • Linear and Logistic Regression
  • K Nearest Neighbour Model
  • Decision Tree
  • Sampling and Model Evaluation (Stratified, Random, Cluster and Accuracy, \(\textsf{R}^2\), Precision, Recall, F1, F0.5 respectively)
  • Ensemble and Boosting Techniques
  • Support Vector Machines
  • Neural Networks: Dense, Convolutional, Recurrent.
  • Computer Vision and Natural Language Processing

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

• Bikram Halder (B2)
• ASV Arun (B2)

The project is officially completed.

For any other queries, feel free to contact us via mail at mathclub.isibang@gmail.com. For updates on future events, do not forget to join our mailing list.