Contents

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

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Elementary Number Theory

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

• Homework (22 Oct 2011).
• 3-part series on very basic modular arithmetic. No background required. It includes some exercises.
  • Number Theory Notes (22 Oct 2011) – Part I
  • Number Theory Notes (22 Oct 2011) – Part II
  • Number Theory Notes (22 Oct 2011) – Part III
• Number Theory and Calculus/Analysis : gives a very brief idea of p-adic analysis.
• Homework (29 Oct 2011)
• A short two-parter: motivated by concepts in group theory.
  • Order of an Element Modulo m and Applications – Part I
  • Order of an Element Modulo m and Applications – Part II
• Number Theory Homework (2 Weeks)
• 4-part series on quadratic residues, including quadratic reciprocity.
  • Quadratic Residues – Part I
  • Quadratic Residues – Part II
  • Quadratic Residues – Part III
  • Quadratic Residues – Part IV (Applications)
• Sample Problem Solving + Homework Hints : more examples of problem solving in number theory.
• Modular Arithmetic Deluxe Edition : on means of extending modular arithmetic to rational numbers in some instances.

Thoughts on a Problem

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

• Thoughts on a Problem
• Thoughts on a Problem II
• Thoughts on a Problem III

Algebra in Olympiad Problems

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

• On symmetric polynomials and their applications to solving Olympiad-type problems.
  • Symmetric Polynomials (I)
  • Symmetric Polynomials (II)
  • Symmetric Polynomials (III)
• Power series and generating functions, with applications to solving combinatorial problems.
  • Power Series and Generating Functions (I): Basics
  • Power Series and Generating Functions (II): Formal Power Series
  • Power Series and Generating Functions (III) – Partitions
  • Power Series and Generating Functions (IV) – Exponential Generating Functions

Linear Algebra (Misc)

Various notes on linear algebra concepts.

• Matrices and Linear Algebra
• Linear Algebra: Inner Products
• Why Do We Need Eigenvalues and Eigenvectors?

Miscellaneous Topics

• Estimating Sums Via Integration
• Pick’s Theorem and Some Interesting Applications
• Polynomial Multiplication, Karatsuba and Fast Fourier Transform
• Random Walk and Heat Equation
  • Random Walk and Differential Equations (I)
  • Random Walk and Differential Equations (II)

Calculus

• Some motivation for the definition of curvature.
  • What is Curvature? (I)
  • What is Curvature? (II)
• Series on multivariate calculus, which places emphasis on the idea that no particular set of parameters or coordinates should be given undue focus.
  • Thinking Infinitesimally – Multivariate Calculus (I)
  • Thinking Infinitesimally – Multivariate Calculus (II)

Physics: Thermodynamics

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

• Thermodynamics for Mathematicians (I)
• Thermodynamics for Mathematicians (II)
• Thermodynamics for Mathematicians (III)
• Thermodynamics for Mathematicians (IV)
• Kinetic Theory, Entropy and Information Theory

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. ]

• Combinatorial Game Theory I : distinguishing between a winning & losing position.
• Combinatorial Game Theory II : solving Nim.
• Combinatorial Game Theory III : solving a “sum” of impartial games.
• Combinatorial Game Theory IV : solving take-and-break games.
• Combinatorial Game Theory Quiz 1
• Combinatorial Game Theory V : loopy games and games with draw.
• Combinatorial Game Theory VI : basic CGT concepts; game comparisons & Nim values.
• Combinatorial Game Theory VII : numbers in CGT; simplicity rule.
• Combinatorial Game Theory VIII : infinitesimals in CGT; toads-and-frogs game.
• Combinatorial Game Theory IX : number avoidance.
• Combinatorial Game Theory Quiz 2
• Combinatorial Game Theory X : canonical form of games.
• Combinatorial Game Theory XI : more examples of game computations.
• Combinatorial Game Theory XII : tinies and minies.
• Combinatorial Game Theory Quiz 3

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.

• Casual Introduction to Group Theory (1) : permutation group.
• Casual Introduction to Group Theory (2) : axioms of a group.
• Casual Introduction to Group Theory (3) : subgroups.
• Casual Introduction to Group Theory (4) : cosets + Lagrange’s theorem.
• Casual Introduction to Group Theory (5) : normal subgroups + group quotients.
• Casual Introduction to Group Theory (6) : group homomorphisms + isomorphism theorems.
• Intermediate Group Theory (0) : motivations + road map.
• Intermediate Group Theory (1) : group actions.
• Intermediate Group Theory (2) : Sylow’s theorems.
• Intermediate Group Theory (3) : automorphisms + semidirect products.
• Intermediate Group Theory (4) : determining structure of finite groups.
• Intermediate Group Theory (5) : free groups + relations.
• Intermediate Group Theory (6) : direct sums + direct products + universal properties.

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.

• Introduction to Ring Theory (1) : definition and motivation.
• Introduction to Ring Theory (2) : subrings.
• Introduction to Ring Theory (3) : ideals and ring quotients.
• Introduction to Ring Theory (4) : ring homomorphisms and isomorphism theorems.
• Introduction to Ring Theory (5) : correspondence between ideals/subrings of R and those of R/I, Chinese Remainder Theorem.
• Introduction to Ring Theory (6) : table of summary.
• Introduction to Ring Theory (7) : polynomial rings.
• Introduction to Ring Theory (8) : matrix rings.

On Unique Factorisation Domains

A good start in algebraic number theory.

• Topics in Commutative Rings: Unique Factorisation (1) : finite factorisation.
• Topics in Commutative Rings: Unique Factorisation (2) : unique factorisation in rings.
• Topics in Commutative Rings: Unique Factorisation (3) : examples and sample computations.

Basic Analysis

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

• Sequence Convergence
  
  • Basic Analysis: Sequence Convergence (1) : definition and basic properties of convergent sequences.
  • Basic Analysis: Sequence Convergence (2) : more advanced properties, e.g. monotone convergence theorem, squeeze theorem.
  • Basic Analysis: Sequence Convergence (3) : limits at ∞ or -∞; limits inferior and superior.
  • Basic Analysis: Sequence Convergence (4) : convergent sums, absolute convergence, alternating sign series.
• Limits and Continuity
  • Basic Analysis: Limits and Continuity (1) : definition and basic properties of limit of a function.
  • Basic Analysis: Limits and Continuity (2) : advanced properties and definition of continuity.
  • Basic Analysis: Limits and Continuity (3) : continuity in higher dimensions and greater generality, open subsets.
• Differentiation
  • Basic Analysis: Differentiation (1) : definition and properties, chain rule, higher derivatives.
  • Basic Analysis: Differentiation (2) : local extrema, Rolle’s and mean value theorem, l’Hopital’s rule.
• Basic Analysis: Uniform Convergence : pointwise convergence and uniform convergence of a sequence of functions.
• Basic Analysis: Closed Subsets and Uniform Continuity : closed subsets of real line, uniform continuity of a function.

Advanced Combinatorics

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.
  • Burnside’s Lemma and Polya Enumeration Theorem (1)
  • Burnside’s Lemma and Polya Enumeration Theorem (2)

Point Set Topology

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

• Basic Definitions
• Bases and Subbases : the equivalence of “generating sets” for topology.
• Subspaces : topology for a subset; ’nuff said.
• Product Spaces (I) : only covers product of finitely many spaces.
• Disjoint Unions
• Continuous Maps : maps which respect the underlying topologies of the spaces.
• Limits and Convergence : re-looks at limits from the point-of-view of topology; gives a nice “holistic” view of all limits.
• Nets and Points of Accumulation : nets are a useful generalisation of sequences, for topological spaces.
• Cauchy Sequences and Uniform Continuity
• Closure : the smallest closed subset containing a set S.
• Interior : the largest open subset contained in a set S.
• Product Spaces (II) : covers infinite products; uses universal properties quite a lot.
• Hausdorff Spaces and Dense Subsets : truth be told, not quite related concepts.
• Complete Metric Spaces : includes completion of a metric space.
• Sequentially Compact Spaces and Compact Spaces : attempts to motivate the concept of compact spaces.
• More on Compact Spaces
• Finite Intersection Property : a fascinating application of the compactness concept.
• One-Point Compactification and Locally Compact Spaces : locally compact criterion is essential for the one-point compactification to be Hausdorff.
• Connected Spaces
• Path-Connected Spaces
• Locally Connected and Locally Path-Connected Spaces
• Separation Axioms : (T1) to (T4); roughly speaking, these describe how close a topological space is to being metrisable.
• Topological Groups : groups whose underlying actions are continuous.
• Quotient Topology and Gluing : useful for constructing new topological spaces; again, the universal property is important.
• Quotients of Topological Groups : not quite as obvious as one might think; with topology involved, there’re always traps for the unwary.
• More on Algebra and Topology : a general article to cover various concepts in algebra and whether they still make sense when topology is added.

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:

• Quick Guide to Character Theory (I): Foundation
• Quick Guide to Character Theory (II): Main Theory
• Quick Guide to Character Theory (III): Examples and Further Topics

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.

• Elementary Module Theory (I)
• Elementary Module Theory (II)
• Elementary Module Theory (III): Approaching Linear Algebra
• Elementary Module Theory (IV): Linear Algebra

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.

• From Euler Characteristics to Cohomology (I)
• From Euler Characteristics to Cohomology (II)

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.

• Simple Modules
• Semisimple Rings and Modules
• Structure of Semisimple Rings
• Application to representation theory:
  • The Group Algebra (I)
  • The Group Algebra (II)
  • The Group Algebra (III)
• Noetherian and Artinian Rings and Modules
• Radical of Module
• Composition Series
• Jacabson Radical
• Local Rings
• Krull-Schmidt Theorem
• Exact Sequences and the Grothendieck Group
• Hom Functor
• On tensor products
  • Tensor Product and Linear Algebra
  • Tensor Product over Noncommutative Rings
• Projective Modules and Artinian Rings
• Projective Modules and the Grothendieck Group
• Modular Representation Theory (I)
• Modular Representation Theory (II)
• Modular Representation Theory (III)
• Idempotents and Decomposition
• Modular Representation Theory (IV)

Polynomials and Representations

This is a series of the relationship between symmetric polynomials, representations of the symmetric group , and representations of the general linear group . This gives an interesting interplay between combinatorics and representations theory.

• Polynomials and Representations I : elementary symmetric polynomials.
• Polynomials and Representations II : partitions.
• Polynomials and Representations III : complete symmetric polynomials.
• Polynomials and Representations IV : power sum polynomials.
• Polynomials and Representations V : symmetric polynomials in infinitely many variables.
• Polynomials and Representations VI : Young tableaux.
• Polynomials and Representations VII : Kostka numbers.
• Polynomials and Representations VIII : matrix balls.
• Polynomials and Representations IX : Schur polynomials.
• Polynomials and Representations X : Cauchy’s identity.
• Polynomials and Representations XI : Pieri’s formula.
• Polynomials and Representations XII : Lindström–Gessel–Viennot lemma.
• Polynomials and Representations XIII : skew tableaux and Schur polynomials.
• Polynomials and Representations XIV : sliding algorithm on tableaux.
• Polynomials and Representations XV : tableaux and words.
• Polynomials and Representations XVI : word problem and solution.
• Polynomials and Representations XVII : two theorems on tableaux product.
• Polynomials and Representations XVIII

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

• Polynomials and Representations XIX : representations of .
• Polynomials and Representations XX : irreducible representations of .
• Polynomials and Representations XXI : sign twist of representations.
• Polynomials and Representations XXII : correspondence with symmetric polynomials.
• Polynomials and Representations XXIII : power sum polynomials and the character table.
• Polynomials and Representations XXIV : Specht modules.
• Polynomials and Representations XXV : Young symmetrizer.
• Polynomials and Representations XXVI : row tableau and column tableau.
• Polynomials and Representations XXVII : (continuation).

Now we shift gears and look at the general linear group .

• Polynomials and Representations XXVIII : continuous representations of topological groups.
• Polynomials and Representations XXIX : characters and orthogonality.
• Polynomials and Representations XXX : the general linear group and the unitary group.
• Polynomials and Representations XXXI : comparison between the two groups’ representations.
• Polynomials and Representations XXXII : weak Peter-Weyl theorem.
• Polynomials and Representations XXXIII : symmetric polynomials and representations of
• Polynomials and Representations XXXIV : weight space decomposition.
• Polynomials and Representations XXXV : Schur-Weyl duality.
• Polynomials and Representations XXXVI : the Schur functor.
• Polynomials and Representations XXXVII.
• Polynomials and Representations XXXVIII : determinant modules.
• Polynomials and Representations XXXIX : a little invariant theory.

Commutative Algebra

• Commutative Algebra 0 : introduction, summary of things you should already know.
• Commutative Algebra 1 : some operations on ideals of a ring.
• Commutative Algebra 2 : radical of ideal and ideal division.
• Commutative Algebra 3 : a look at basic algebraic geometry.
• Commutative Algebra 4 : some consequences of Hilbert’s Nullstellensatz.
• Commutative Algebra 5 : maps in algebraic geometry.
• Commutative Algebra 6 : a look at two non-trivial examples.
• Commutative Algebra 7 : modules over a ring.
• Commutative Algebra 8 : more about modules.
• Commutative Algebra 9 : direct sums and products; first look at universal properties.
• Commutative Algebra 10 : algebras over a ring.
• Commutative Algebra 11 : a look at affine k-varieties and k-schemes.
• Commutative Algebra 12 : some basic stuff on noetherian posets and Zorn’s lemma.
• Commutative Algebra 13 : Zariski topology on a ring.
• Commutative Algebra 14 : more on Zariski topology.
• Commutative Algebra 15 : unique factorization domains.
• Commutative Algebra 16 : principal ideal domains and Euclidean domains.
• Commutative Algebra 17 : Gauss’s lemma.
• Commutative Algebra 18 : basic category theory and functors.
• Commutative Algebra 19 : natural transformations and representable functors.
• Commutative Algebra 20 : Yoneda lemma, products and coproducts.
• Commutative Algebra 21 : exact sequences and exact functors.
• Commutative Algebra 22 : localization of rings.
• Commutative Algebra 23 : more on localization.
• Commutative Algebra 24 : localization of modules.
• Commutative Algebra 25 : localization and local properties.
• Commutative Algebra 26 : left-exact and right-exact functors (on modules).
• Commutative Algebra 27 : free modules and projective modules.
• Commutative Algebra 28 : tensor products of modules.
• Commutative Algebra 29 : more on tensor product, induced modules.
• Commutative Algebra 30 : tensor product of A-algebras.
• Commutative Algebra 31 : flat modules.
• Commutative Algebra 32 : more on flat modules, examples of flat algebras.
• Commutative Algebra 33 : f.p. modules, projectivity is local for such modules.
• Commutative Algebra 34 : Nakayama’s lemma.
• Commutative Algebra 35 : noetherian modules and rings.
• Commutative Algebra 36 : on noetherian topological spaces.
• Commutative Algebra 37 : composition series and length of modules.
• Commutative Algebra 38 : artinian rings and zero-dimensional rings.
• Commutative Algebra 39 : integral extensions of rings.
• Commutative Algebra 40 : spectra of integral extensions of rings.
• Commutative Algebra 41 : transcendence bases of field extensions.
• Commutative Algebra 42 : Noether normalization theorem.
• Commutative Algebra 43 : prime chains and catenary rings.
• Commutative Algebra 44 : fractional ideals and invertible fractional ideals.
• Commutative Algebra 45 : Dedekind domains and discrete valuation rings.
• Commutative Algebra 46 : properties of Dedekind domains.
• Commutative Algebra 47 : computing Picard groups (in number theory & geometry).
• Commutative Algebra 48 : category theory – pushouts, fibre products, colimits.
• Commutative Algebra 49 : category of diagrams, limits.
• Commutative Algebra 50 : adjoint functors and properties.
• Commutative Algebra 51 : limits and exactness, direct limits.
• Commutative Algebra 52 : direct limits of rings, duality of constructions.
• Commutative Algebra 53 : graded rings and graded modules.
• Commutative Algebra 54 : filtrations and completions.
• Commutative Algebra 55 : more properties of completions, Artin-Rees lemma.
• Commutative Algebra 56 : consequences of Artin-Rees lemma; 𝔞-adic completions.
• Commutative Algebra 57 : more properties of 𝔞-adic completions; Hensel’s lemma.
• Commutative Algebra 58 : annihilators, supports and associated primes of modules.