# Contents

Here’s a list of topics currently available. Updated on 12 Jan 2018.

[ Tip: if the text is too small, pressing Ctrl+= on most browsers will expand the page, and the formulae will be re-rendered at a higher resolution. ]

## Elementary Number Theory

Notes from a short series of lectures on elementary number theory.

## Thoughts on a Problem

A series of posts on the thought processes behind problem-solving.

By algebra, we mean elementary algebra, not abstract algebra.

• On symmetric polynomials and their applications to solving Olympiad-type problems.
• Power series and generating functions, with applications to solving combinatorial problems.

## Linear Algebra (Misc)

Various notes on linear algebra concepts.

## Calculus

• Some motivation for the definition of curvature.
• Series on multivariate calculus, which places emphasis on the idea that no particular set of parameters or coordinates should be given undue focus.

## Physics: Thermodynamics

An attempt to explain thermodynamics in a (somewhat) mathematical approach.

## Combinatorial Game Theory (Lecture Series)

This is a short course on CGT conducted in the first half of 2008, in NUS High School. Lectures 1-4 cover a simplified theory of impartial games, without going into full-fledged CGT. Lecture 5 is a slight diversion on loopy games. Lectures 6 onwards cover CGT in greater generality, which allows for the possibility of partial games. As a consequence, the materials in lectures 1-4 become a special case of this general theory.

[ The original course had an extra lesson on Dots-and-Boxes, but after reviewing the notes, I found some concepts poorly described, so it’s put on hold for now. ]

## Abstract Algebra: Group Theory

Two sets of notes on group theory. The term “casual introduction” may be a misnomer, since most of it is rather technical. That being said, I do attempt to motivate every definition and new concept. Whether or not the attempt has fulfilled its goal, that’s up to the reader to judge.

## Introduction to Ring Theory

A list of notes on ring theory. This tends to be a little fast-paced since there’re lots of parallels between basic group theory and basic ring theory.

## On Unique Factorisation Domains

A good start in algebraic number theory.

## Basic Analysis

Some notes on undergraduate level analysis (involving lots of ε-δ stuff).

Typically includes algebraic combinatorics, e.g. group theory, representation theory.

• Burnside’s Lemma and Polya Enumeration Theorem : theory and lots of computational examples, useful in finding the number of objects up to symmetry.

## Point Set Topology

Some notes on point-set topology. In this series, I attempt to provide motivation for every definition or concept.

## Representation Theory of Finite Groups

The first part talks about the “semisimple case”, where the base field has characteristic zero. The second part discusses modular representation theory and requires a significant amount of non-commutative algebra.

Semisimple case:

For modular representation theory, see later.

## Elementary Module Theory

An attempt to cover modules in the general context of non-commutative rings, including a discussion of linear algebra over division rings. Surprisingly, non-commutativity can actually clarify certain concepts. E.g. the dual of a left module is a right module. The series is not quite complete.

## Cohomology: Starting from Basics

A series of articles to talk about cohomology, beginning from the basic concept of Euler characteristics. Definitely unfinished, and I’m not quite sure how to proceed.

## Non-commutative Algebra

On non-commutative rings: the selection of topics is far from complete, but it’s primarily meant as a foundation to modular representation theory. I’m reasonably happy with the outcome. Please feel free to email me if you find any mistakes.

## Polynomials and Representations

This is a series of the relationship between symmetric polynomials, representations of the symmetric group $S_n$, and representations of the general linear group $GL_m(\mathbb{C})$. This gives an interesting interplay between combinatorics and representations theory.

This concludes the combinatorial segment of the notes. The subsequent parts are on representation theory.

Now we shift gears and look at the general linear group $GL_m(\mathbb{C})$.

### 7 Responses to Contents

1. anonymous says:

Hi! Mind me ask for the references to your notes. Though most of the stuff you presented are standard, I find some parts quite illuminating, and wonder where you have learnt them from. Thanks!

2. limsup says:

Hi there. Generally I don’t use a single source of reference. I’m glad you found some of the articles useful.

3. Ana says:

Thank you for making your course on combinatorial games available for use. I am impressed you taught the course in a high school.
You write very well and explain well, so… any progress in Dots and Boxes?

• limsup says:

Hi sorry for taking so long to reply. I’ve not touched Dots and Boxes for a long time, but thanks for the reminder. One of these days, I gotta get back to them 🙂

4. JoseBrox says:

Dear limsup, where do your more recent posts about polynomials and representations fit? Why are they numbered as they are? (They seem to be pretty high-numbered and with some jumps).

• limsup says:

Hi JoseBrox. The series on “polynomials and representations” is an independent one. It starts from article I (https://mathstrek.blog/2016/05/03/polynomials-and-representations-i/), and ends at XXXIX only because I wasn’t sure how to continue the series. There’re no gaps in between.

The first part (I to XVIII) talks about symmetric polynomials, Young tableaux and related combinatorial topics. It does not require any background. The second part talks about representations of the symmetric group and GL(n) so it requires quite a bit of knowledge of representation theory.

It’s missing from the Contents page only because I was lazy in updating. 😛

5. a says:

Hey, just leaving a post of appreciation, I randomly stumbled into your blog while finally looking to see noncommutative tensor is what I imagined, and ended up reading a huge number of your notes :).