Notes
Algebra
Group Theory [Click] These notes cover the theory of fintie group action and Sylow theorems.
Perron-Frobenius Theorem [Click] We discuss geometric proofs of the Perron-Frobeninus theorems.
Representation Theory of Finite Groups. [Click] These notes contains the basic material on representation theory of finite groups that would usually be covered in a first course on the subject.
Unique Factorization Domains. [Click] We review the basic theory of Euclidean Domains, Principle Ideal Domains, and Unique Factorization Domains, along with the Gauss Lemma.
Linear Algebra
Linear Transformations [Click] We discuss some basic objects of linear algebra: vector spaces, linear maps, isomorphism theorems, dual spaces, transpose of a linear map, annihilators etc.
The Minimal Polynomial [Click] We discuss the notion of the minimal polynomial of a linear endomorphism on a finite diemsnional vector space and discuss how it connects with the notion of eigenvalues, upper triangulability, and diagonalizability.
Bilinear Forms [Click] We discuss the notion of a bilinear form on a vector space. In particular we discuss orthogonality, symmetric and skew-symmetric bilinear foms.
Inner Product Spaces [Click] We discuss the standard theorems about inner product spaces.
Analysis
Banach Alaoglu Thoerem. [Click] We recall the notion of weak* topology and prove the Banach-Alaoglu Thoerem.
Gelfand-Naimark Thoerem. [Click] We develop the thoery of Banach algebras and give a proof of the Gelfand-Naimark Theorem. We then apply it to prove the various avatars of the Spectral Theorem.
Peter-Weyl Thoerem. [Click] We discuss the Peter-Weyl Theorem for compact groups.
Combinatorics
Warning's Theorem. [Click] We prove warning's theorem and give combinatorial applications.
Entropy Arguments. [Click] We discuss the notion of entropy of a random variable and see several combinatorial applications. We breifly present an information theoretic viepoint of entropy and discuss the source coding theorem.
Dynamics
Killing Isects using Toplogical Dynamics. [Click] Suppose an insect is walking on the unit circle such that at the $n$-th second the insect is at the point whose angle is $n^2\alpha (mod 1)$, where $\alpha$ is an irrational number. We show that any amount of glue put anywhere on the unit circle will eventually trap the insect.
An Application of Jonings. [Click] We prove the weak ergodic thoerem by using the notion of joinings.
Periodic Tiling Conjecture. [Click] The periodic tiling conjecture states that every finite subset of $\mathbb Z^n$ which can tessellate $\mathbb Z^n$ by translations always admits a periodic tessellation. This was proved by S. Bhattacharya for the case $n=2$ in 2016 using ergodic theoretic methods. This article is an exposition of Bhattacharya's proof.
General Topology
Cantor Set. [Click] A short exposition on the Cantor set.
Metric Spaces. [Click] Covering the basic theory of metric spaces with some applications.
Metrization. [Click] A short exposition on metrization of topological spaces culminating in the proof of the Urysohn's metrization theorem.
Theorem of Stadje. [Click] A beautiful property of compact connected metric spaces due to Wolfgang Stadje.
Number Theory
Geometry of Numbers. [Click] We prove the Miskowski's convex body theorem and apply it to Diophantine approximation.
Miscellaneous
Automata and Monoids. [Click] We see the equivalence of an NFA with a DFA via an algebraic approach using monoid theory..