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UDGRP Summer 2021

Table of Contents
UGDRP SUMMER 2021 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 2021 organised by the Math Club. The projects start from Saturday, 24th July 2021. The topics being considered this time are presented below.

Introduction to Commutative Algebra with motivating examples #

Commutative Algebra forms the backbone of modern algebra and associated topics. A working knowledge of the same is indeed important and very useful for most mathematicians, may the be Algebraists, Number Theorists, Discrete Mathematicians, Lie Theorists, Differential Geometers or Analyst. This reading project is intended to give you a flavor of the same at a very basic level. Starting with a revision of the theory of commutative rings and \(R-\)modules, the reading shall slowly enter into the beginnings of Commutative Algebra. Throughout the course, many motivating examples from Geometry, Number Theory and so on shall be presented to develop interest and purpose of the things discussed.

  • Pre-Requisites: A working knowledge of Groups and Vector Spaces, basic concepts of Rings, though it shall be revised.
  • Primary References:
  • Group Leaders:
    • Samarth M. Bhat (B3)
    • Soham Ghosh (B3)
  • Topics Covered:
    • Basics of Ring and Fields
    • Domains and Polynomial Rings
    • Properties of Ideals and their operations
    • Norms of Ideals
    • Noetherian Rings and Chain Conditions
    • Basics of Commutative Algebra
    • * Module Theory
    • * Integral Extensions, Valuations and Dedekind Domains
    • * Basics of (Affine) Algebraic Geometry

The following people, other than the leaders, have completed the reading project by going through, in details, all the assigned parts and participating in the discussions and classes which occurred. They have also presented in these classes time to time. Students marked (*) did the * topic above. They also did projects on some topics, mentioned in brackets beside their names.

  • (*) Balaji Subramonium P. (B3) (Noether Normalization Lemma and Hilbert’s Nullstellensatz)
  • Mainak Samanta (B2) (Finite Norm Property of Integral Domains)
  • Saheb Mohapatra (B2) (Euclidean Domains with Unique Quotient and Remainder)

The project is officially completed.

Logic and Automated Theorem Proving #

This is a two part reading group. The first part would be interested to venture into the field of mathematical logic and axiomatic mathematics. This would show that sometimes, things which are discounted as “technical” or “just for language” are actually very beautiful in their own rights and appreciable at a pretty elementary level. The second part would be interested to learn about the Lean Language and how it can be implemented for automated theorem proving. The course shall be decided as per interested.

  • Pre-Requisites: Nothing as such, though a basic course in sets and computer science would be helpful.
  • Primary References:
  • Group Leader: Anupam Nayak (B3)
  • Topics Covered:
    • First Order Logic
    • Naive Set Theory
    • ZFC and Axiomatic Set Theory
    • Computability and Recursion Theory
    • Incompleteness Theorems
    • Background for ATP
    • * Model Theory, Non-standard analysis and constructive logic

The following people, other than the leader, have completed the reading project by going through, in details, all the assigned parts and participating in the discussions and classes which occurred. They have also presented in these classes time to time. Students marked (*) did the * topic above.

  • Srigyan Nandi (B2)
  • (*) Pranav Krishna (B3)

The project is officially completed.

Introduction to Knot Theory #

Knot theory is a branch of mathematics which is perhaps the most “visual” yet fascinating. A considerably less sought after branch, it is one of the key elements of what is called “Geometric Topology”. We shall be looking at knots, links, their invariants, classifications, associated polynomials and algebraic structures. We shall also have a look at other aspects of geometric topology and how they can be applied to Knot Theory. The main reading should be accessible even to those not very familiar with Point-Set Topology, though a familiarity would almost surely be helpful.

  • Pre-Requisites: Basic knowledge of groups and their actions, metric spaces. Topology will be helpful, though not necessary.
  • Primary References:
  • Group Leaders:
    • Snehinh Sen (B3)
    • Harshul Khanna (B3)
  • Topics Covered:
    • Elements of Point-Set and Algebraic Topology. (As in Munkres)
    • Knots and their nomenclature.
    • Knot Invariants.
    • Introduction to Manifolds.
    • Surfaces, their properties and classification.
    • Knot Surfaces and Torus Knots.
    • Polynomials on Knots.
    • A look at knots and statistical mechanics (project by Srigyan Nandi (B2)).

The following people, other than the leaders, have completed the reading project by going through, in details, all the assigned parts and participating in the discussions and classes which occurred. They have also presented in these classes time to time.

  • Akhilesh Mondal (B3)
  • Harshul Khanna (B3)
  • Praveen Kumar (B3)
  • Srigyan Nandi (B2) (Also did a project on Knots and Statistical Mechanics)
  • Sonali Priyadarshini Behara (B3)

The project is officially completed.

Matrix Groups and a look at their Representation Theory #

Matrix Groups are very special. Not only are they groups but also have an intrinsic vector space structure. This gives rise to the beautiful Lie Theory. The target of this course is to give a glimpse of it and see how beautifully various different aspects of mathematics like Analysis, Linear Algebra, Abstract Algebra, Topology and Geometry join hands to create a wonderful theory of these groups. Time permitting, we shall also have a look at the Representation Theory of such groups and their corresponding Lie algebras.

  • Pre-Requisites: A working knowledge of Groups and Vector Spaces, multivariable calculus, topological aspects of metric spaces.
  • Primary Reference:
    • Matrix Groups For Undergraduates – Kristopher Tapp
  • Group Leaders:
    • Shahbaz A. Khan (B3)
    • Siddharth Acharya (B3)
  • Topics Covered:
    • Matrices and their properties
    • Some properties of Matrix Groups and factorisations
    • Orthogonal and Symmetry Groups
    • Topology of Matrix Groups
    • Elements of Lie algebras
    • Exponentiation of Matrices and Lie Correspondence
    • Some idea of Matrix Groups as Manifolds
    • Basics of representation theory of groups and lie algebras.

The following people, other than the leaders, have completed the reading project by going through, in details, all the assigned parts and participating in the discussions and classes which occurred. They have also presented in these classes time to time.

  • Aprameya Girish Hebbar (B2)
  • Harshul Khanna (B3)
  • Srigyan Nandi (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.