Primers

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

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
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
Kolmogorov Complexity
Information Distance
Parameterized Complexity of Vertex Cover and Kernelizations
Communication Complexity

Probability and Statistics
Finite Probability Theory
Conditional Probability
Probabilistic Bounds (Markov, Chebyshev, Chernoff-Hoeffding)
Martingales and the Optional Stopping Theorem

Learning Theory
Probably Approximately Correct – A Formal Theory of Learning
A problem that’s not properly PAC-learnable
Occam’s Razor and PAC-learning

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

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17 thoughts on “Primers

  1. Hi,

    I’m impressed. Good introductions to difficult topics. Especially I liked your article about psychodelic pictures or about search engines (Once I developed a search engine myself).

    Wish you the Best for the New Year and a lot more of your articles
    Thomas Nitsche

    Like

  2. Awesome!

    2 comments:
    What I learned as discrete math in cs classes is what you have here as miscellaneous, and what you have under discrete math I learned under topologies

    Second, is it possible to get all the topics in one big pdf? what is your licensing on the content?

    Like

    • Well number theory and set theory are not particularly discrete in nature, at least as they occur in mathematics.

      I don’t have all the topics in one big pdf. The licensing on my blog is Creative Commons non-commercial, so you’re welcome to assemble one and distribute it, as long as you give attribution and don’t sell it.

      Like

    • This looks like a very great and in-depth tutorial. I am coincidentally just finishing up my own (less detailed) Bezier curve tutorial! I’ll be sure to link to this one in that post.

      Like

  3. This is a wonderful blog! You have a fantastic writing style and I especially liked the graph theory post. I remember how much I struggled to explain the Seven Bridges of Konigsberg problem to my sister, and after reading your post about it I feel like exclaiming to the world the wonders of graph theory.

    Like

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