Lecture Notes
These are my (handwritten) lecture notes that arose during discussions with many juniors and colleagues. Most of them are incomplete and will be updated from time to time. These may contain errors. If any viewer of these notes find such errors, kindly send me a mail mentioning that.
Representation Theory of Finite Groups: A Detailed Exposition
These notes are based on the book Linear Representations of Finite Groups by J. P. Serre. These notes also contain the solutions to the exercises.
1. Definitions and Basics of Character Theory
2. Decomposition into Irreducibles
3. Constructions in Representation Theory, Induced Representation
4. Basic Examples to Appear Later
Short Course in Representation Theory of Finite Groups
These are the notes that arose out of my own study of semisimple rings and the representation theory of finite groups. It is a brief overview of the vast topic of representation theory. I have mainly followed TIFR Notes on Semisimple Rings, Central Simple Algebras, etc. Apart from that, I have sometimes referred to Lectures on Forms in Many Variables by M. J. Greenberg, Algebra by S. Lang, Linear Representations of Finite Groups by J. P. Serre, Representation Theory of Finite Groups by B. Steinberg, etc.
Semisimple Modules (2-14)
Isotypical Components (15-23)
Semisimple Rings (23-30)
Simple Rings (30-40)
Radical and Semisimplicity (40-54)
Central Simple Algebras and The Brauer Group
Central Simple Algebras (3-19)
The Brauer Group (19-23)
The Skolem-Noether Theorem (24-37)
Splitting Fields (37-43)
Existence of Galois Splitting Field (43-49)
Reduced Norm (49-63)
Real Numbers (2-9)
Finite Fields (9-17)
Tsen's Theorem (17-43)
Representation Theory of Finite Groups Overview
The Group Algebra (2-8)
Representations (9-22)
Irreducible Representations, i.e., Simple K[G]-modules (23-31)
Characters and The Orthogonality Relations (32-45)
Integrality of Characters (45-51)
Group Theory and Solvability (51-65)
Burnside's Theorem (65-73)
A little elaborated discussion is the following.
Introduction and Connection with Noncommutative Ring Theory
Here are two additional lecture notes that arose out of an introductory discussion with some students.
Characteristic Classes
These notes arose out of my discussion on characteristic classes. The main references followed are Characteristic Classes by Milnor and Stasheff, Smooth Manifolds by John M. Lee, and Differential Topology by Guillemin and Pollack.
Lecture 1: Abstract Smooth Manifolds
Lecture 2: Smooth Maps and Tangent Spaces
Algebraic Topology
These lectures consist of basic algebraic topology meant for a standard graduate course. I closely followed books such as A Concise Course in Algebraic Topology by J. P. May, Algebraic Topology by A. Hatcher, An Introduction to Algebraic Topology by J. J. Rotman, Algebraic Topology by Tammo tom Dieck and Algebraic Topology from a Homotopical Viewpoint by M. Aguilar, S. Gitler and C. Prieto. The lecture notes are given below.
Applications of Fundamental Group of Circle and Categorical Prerequisites
Metric Spaces
In these lectures we covered mainly compactness, connectedness and completeness in the context of metric spaces. Theory of basic properties of real numbers, sequences and introductory point set topology of metric spaces was assumed and is written briefly. The lectures are made following the books: A Basic Course in Real Analysis by Ajit Kumar and S. Kumaresan; Topology of Metric Spaces by S. Kumaresan; Mathematical Analysis by T. M. Apostol and Principles of Mathematical Analysis by W. Rudin. For the parts we have assumed one can look into the first two books for an excellent reference. The link to the lectures are given below.
The materials below are result of some doubt discussion sessions. They contain various exercises and some theorems in Real Analysis.
Here are some problems on basic point set topology on real numbers.
Analysis of Several Variable Real Valued Functions
These lectures were given to some Master's students to introduce them to Multivariable Differential and Integral Calculus. I am grateful to Prof. Kingshook Biswas who taught me a course of Multivariable Calculus. Many of the materials covered here are taken from his lectures. Besides I followed the books Calculus on Manifolds by M. Spivak, Analysis on Manifolds by J. R. Munkres, Mathematical Analysis by T. M. Apostol, A Course in Differential Geometry and Lie Groups by S. Kumaresan. The links to the lecture notes are given below.
Additionally,
Metric Spaces and Introduction to Topology
These notes on metric spaces have arisen from discussions with undergraduate students in order to drive them through the transition of analysis to topology with the help of metric spaces. The references are A Basic Course in Real Analysis by Ajit Kumar and S. Kumaresan; Topology of Metric Spaces by S. Kumaresan; Mathematical Analysis by T. M. Apostol and Principles of Mathematical Analysis by W. Rudin. These notes are to be referred alongside the notes of the first few lectures of the course on Multivariable Calculus and Introduction to Differential Geometry.
Additionally,
Introductory Algebraic Number Theory
These are lectures on introductory Algebraic Number Theory. I am grateful to Prof. Mrinal Kanti Das who taught me a course on Algebraic Number Theory, which help me to learn the subject. Also most of the material here is made following his lectures. The link to the lecture notes are given below.
Subgroup of a free abelian group of finite rank
Introduction to Multivariable Calculus and Differential Geometry
These lectures were given to some undergraduate students to introduce them to Multivariable Differential and Integral Calculus and to the basics of differential geometry. I am grateful to Prof. Kingshook Biswas who taught me a course of Multivariable Calculus. Many of the materials covered here are taken from his lectures. Besides I followed the books Calculus on Manifolds by M. Spivak, Analysis on Manifolds by J. R. Munkres, Mathematical Analysis by T. M. Apostol, A Course in Differential Geometry and Lie Groups by S. Kumaresan. The links to the lecture notes are given below.
Lecture 1: Introduction and Preliminaries
Lecture 2: Continuity in Metric Spaces and Topological Spaces
Lecture 3: Examples and Non-examples of Continuous Functions on Euclidean Spaces
Lecture 4: Limits and Attempts to Find a Proper Definition of Differentiability in Higher Dimensions
Lecture 5: Differentiability in Higher Dimensions and Basic Properties
Lecture 6: Examples of Computation of Derivative
Lecture 7: Examples of Partial and Directional Derivatives
Lecture 8: Equivalent criterion for a function to be C^1, Chain Rule
Lecture 9: Formulations of the Chain Rule and Mean Value Theorem
Lecture 10: Contraction Mapping Principle and the Background for Inverse Function Theorem
General Topology
This notes focuses on general topology. Here I have briefly discussed point set topology with more focus to quotient spaces. All the references are listed in the pdf below. Here are the links to the lecture notes.
The link to the (incomplete) pdf created by assembling the lecture notes is given below. I have a plan to completely prepare it so as to contain more involved materials. However due to lack of time this is not done yet. I will post the updated pdf time to time replacing this link and will denote the update no. within bracket adjacent to the title of the link.
Topology: An Algebraic Viewpoint (1)
The next section contains some more lectures on General Topology. It was discussed with a different set of students. It is sometimes a better-structured presentation of the materials covered in the lectures above.
An Introduction to The Notion of Subbasis
Construction of Various Topologies
Characteristic or Universal Properties of Various Constructions
Fundamental Theorem of Algebra Using Fundamental Group
The next few lectures are notes of some discussions on Introductory Algebraic Topology. The first lecture is on Category Theory and for motivation to introduce Algebraic Topology. The next few lectures cover some basic topics in Algebraic Topology ending with a brief introduction to Singular Homology Theory. I have mostly followed Algebraic Topology by Allen Hatcher and An Introduction to Algebraic Topology by Joseph J. Rotman in these lectures.
The next two notes are on Affine Spaces and on properties of Finite and Countable Sets. These are sometimes useful in studying topology. The references are Linear Algebra Done Right by S. Axler, An Expedition to Geometry by S. Kumaresan and G. Santhanam and An Introduction to Algebraic Topology by J. J. Rotman. For the countability part the references are A Course on Borel Sets by S. M. Srivastava and Topology by J. Munkres.
The next few lectures are basic introduction to point set topology both from metric space and general topology viewpoint.
Field and Galois Theory
These lectures consist of some topics in Field and Galois Theory. The main references used are Field and Galois Theory by P. Morandi and Algebra: A Graduate Course by I. M. Issacs. Besides some good references that we have used sometimes are Algebra by T. W. Hungerford and Algebra by S. Lang. Here are the link to the lectures. The first set of lectures are made from Morandi's excellent reference.
The next few lectures are made following Issac's reference with some different bunch of students.