Decomposing morphisms in abelian categories

In this short post we state the basic decomposition result that any morphism in an abelian category decomposes canonically into an epimorphism followed by a monomorphism, and derive some very useful consequences.

Abelian categories

In this short post we define abelian categories (which takes some work), which were introduced as abstractions of some core properties of categories like abelian groups and modules over a ring; so for example most of homological algebra can be carried over any abelian category, which is neat.

Representation theory of the symmetric group: basics

In this post we lay out some introductory material on the representation theory of $ S_n$ over $ \mathbb{C}$.

Talagrand's concentration inequality

In this post we basically go over a proof of Talagrand's concentration inequality, following material from Bela Bollobas' part III course on probabilistic combinatorics. Things can get rather messy, so hopefully writing this will force me to properly understand it.

The finite-dimensional spectral theorem

A spectral theorem is a statement of the form `any sufficiently nice operator admits an (orthonormal) basis of eigenvectors'. I can't put it in words how important those are, so I won't try. In this post we give first-principles proofs of spectral theorems for operators ranging from Hermitian to normal. The difficulty gradually builds up as we use the same basic tricks with increasing skill.

Representation theory: Reducibility and Maschke's theorem

Reductionism being one of our most useful ways of thinking about math, in this post we discuss assumptions under which any representation can be decomposed into a number of `atomic' representations.

The Hausdorff metric inherits compactness

In this post we prove that for a compact metric space $ X$, the set of closed subsets $ F(X)$ can be given a metric that captures `closeness' in a way that turns $ F(X)$ in a compact metric space in its own right. This has applications in various optimization problems, such as the isoperimetric inequality, where we want to argue an optimum exists by approximating it with a sequence of sets.

Representation theory: Introduction

In this post, we develop the basics of representation theory, loosely following some notes from Dennis Gaitsgory's Math 122 at Harvard. For the most part, we motivate definitions and give different ways of thinking about them; what we say applies to representations of any group over any field.

Roth's theorem via Fourier analysis

Roth's classical theorem says that a subset of $ \{1,\ldots,n\}$ of density $ \Omega \left(\frac{1}{\log\log n}\right)$ contains a length-3 arithmetic progression. In this post we give an exposition of a proof from Tim Gowers' part III course on additive combinatorics using Fourier analytic methods on $ \mathbb{Z}_n$.

The Erdos-Szekeres theorem

This nice classical theorem says simply that in any sequence of $ ab+1$ numbers there is either an increasing subsequence of length $ a+1$ or a decreasing subsequence of length $ b+1$. In this post, we'll prove it, give some applications, and show a high-dimensional generalization.