As a fair warning to the reader, these primers are a bit more terse than what you’d find in your average textbook. I only introduce the bare minimum required to understand the main content posts, so there are carefully chosen gaps whose exclusion is necessary for time’s sake. If the reader is confused about something, or wants a deeper explanation of a concept we deliberately leave out, feel free to leave a comment asking about it and we will do our best to fill in the gaps.

Methods of Proof

Direct Implication

Contrapositive

Contradiction

Induction

Diagonalization

Abstract Algebra

Linear Algebra

Inner Product Spaces

Groups (motivations, basic definitions, homomorphisms, quotient groups)

Groups (first isomorphism theorem, presentations, classification theorem, free products)

Rings (basic definitions, zero-divisors, units, examples)

Rings (homomorphisms, ideals, quotients)

Tensor Products

Outer Products

Fields (mostly finite fields)

Fourier Analysis

The Fourier Series

The Fourier Transform

Generalized Functions and Tempered Distributions

The Discrete Fourier Transform

Discrete Math

Graph Theory (for the math-phobic)

Graph Coloring

Trees and Tree Traversal

Computing Theory

Determinism and Finite Automata

Turing Machines

Big-O Notation

Busy Beaver Numbers

P vs. NP (And a Proof Written in Racket)

Other Complexity Classes

NP-hard does not mean hard

Kolmogorov Complexity

Information Distance

Parameterized Complexity of Vertex Cover and Kernelizations

Communication Complexity

A Zero-Knowledge Proof for Graph Isomorphism

Coding Theory

A Proofless Introduction to Information Theory

Hamming’s Code

The Codes of Solomon, Reed, and Muller

Probability and Statistics

Finite Probability Theory

Conditional Probability

Probabilistic Bounds (Markov, Chebyshev, Chernoff-Hoeffding)

Martingales and the Optional Stopping Theorem

Markov Chain Monte Carlo

Learning Theory

Probably Approximately Correct – A Formal Theory of Learning

A problem that’s not properly PAC-learnable

Occam’s Razor and PAC-learning

The Boosting Margin, or Why Boosting Doesn’t Overfit

Topology

Metric Spaces

Topological Spaces (motivations, basic definitions, and examples)

Constructing Topological Spaces (subspaces, quotients, and gluing)

The Fundamental Group

Homology (definitions and examples)

Programming

A Dash of Python

A Pinch of Python (Random Psychedelic Art)

A Spoonful of Python (and Dynamic Programming)

A Taste of Racket, or How I Learned to Love Functional Programming

A Sample of Standard ML (the TreeSort algorithm, and Monoids)

Miscellaneous

Set Theory, Countability

Number Theory

Lagrangians for the Amnesiac