List of things you should know about if you want to do ComputerScience.
1. Foundations and logic
Why: This is the assembly language of mathematics—the stuff at the bottom that everything else complies to.
- Propositional logic.
- Predicate logic.
- Axioms, theories, and models.
- Proofs.
- Induction and recursion.
2. Fundamental mathematical objects
Why: These are the mathematical equivalent of data structures, the way that more complex objects are represented.
- Naive set theory.
- Predicates vs sets.
- Set operations.
- Set comprehension.
- Russell's paradox and axiomatic set theory.
- Functions.
- Functions as sets.
- Injections, surjections, and bijections.
- Cardinality.
- Finite vs infinite sets.
- Sequences.
- Relations.
- Equivalence relations, equivalence classes, and quotients.
- Orders.
- The basic number tower.
- Countable universes: ℕ, ℤ, ℚ. (Can be represented in a computer.)
- Uncountable universes: ℝ, ℂ. (Can only be approximated in a computer.)
- Other algebras.
- The string monoid.
- ℤ/m and ℤ/p.
- Polynomials over various rings and fields.
3. Modular arithmetic and polynomials
Why: Basis of modern cryptography.
- Arithmetic in ℤ/m.
- Primes and divisibility.
- Euclid's algorithm and inverses.
- The Chinese Remainder Theorem.
- Fermat's Little Theorem and Euler's Theorem.
- RSA encryption.
- Galois fields and applications.
4. Linear algebra
Why: Shows up everywhere.
- Vectors and matrices.
- Matrix operations and matrix algebra.
- Geometric interpretations.
- Inverse matrices and Gaussian elimination.
5. Graphs
Why: Good for modeling interactions. Basic tool for algorithm design.
- Definitions: graphs, digraphs, multigraphs, etc.
- Paths, connected components, and strongly-connected components.
- Special kinds of graphs: paths, cycles, trees, cliques, bipartite graphs.
- Subgraphs, induced subgraphs, minors.
6. Counting
Why: Basic tool for knowing how much resources your program is going to consume.
- Basic combinatorial counting: sums, products, exponents, differences, and quotients.
- Combinatorial functions.
- Factorials.
- Binomial coefficients.
- The 12-fold way.
- Advanced counting techniques.
- Inclusion-exclusion.
- Recurrences.
- Generating functions.
7. Probability
Why: Can't understand randomized algorithms or average-case analysis without it. Handy if you go to Vegas.
- Discrete probability spaces.
- Events.
- Independence.
- Random variables.
- Expectation and variance.
- Probabilistic inequalities.
- Markov's inequality.
- Chebyshev's inequality.
- Chernoff bounds.
- Stochastic processes.
- Markov chains.
- Martingales.
- Branching processes.
8. Tools
Why: Basic computational stuff that comes up, but doesn't fit in any of the broad categories above. These topics will probably end up being mixed in with the topics above.
These you will have seen before:
- How to differentiate and integrate simple functions.
- Things you may have forgotten about exponents and logarithms.
These may be somewhat new:
- Inequalities and approximations.
- ∑ and ∏ notation.
- Computing or approximating the value of a sum.
- Asymptotics.