Mathcamp: Course Topics

Course Topics

Course offerings vary from year to year, depending on the interests of the students and faculty. The topics are chosen so as to be both interesting and accessible: many of these subjects are typically not covered until graduate school, although they have few formal prerequisites other than a capacity for abstract thought. It is not uncommon for many students to have never even heard of many of our classes before coming to camp.

Spotlight on a Class: Combinatorial Game Theory

We have three berry baskets: one with four strawberries, one with three blueberries, and one with two blackberries. We take turns eating them. In your turn, you may eat as many berries as you want, as long as they are all in the same basket. Then it is my turn. The player who eats the last berry wins. Will you beat me? What if we start with different amounts of berries?

In Combinatorial Game Theory, we analyze games of no chance (no drawing cards, no throwing dice). Or goal is a strategy that allows us to win from any position. Come to this class to learn the main techniques to analyze games, and to discover why Nim (the berry game above) is the model that we can use to solve a very large family of them.

Take a look at the lecture notes [PDF] and handout [PDF] from Alfonso's Combinatorial Game Theory class.

Some of the topics taught in previous years have included:

  • Discrete Mathematics
    • Combinatorics (enumerative, algebraic, geometric)
    • Generating functions and partitions
    • Graph theory
    • Ramsey theory
    • Probability
    • Finite geometries
    • Polytopes and polyhedra
    • Combinatorial Game Theory
  • Algebra and Number Theory
    • Linear algebra
    • Groups, rings, and fields
    • Primes and factorization algorithms
    • Congruences and quadratic reciprocity
    • Galois theory
    • Algebraic number theory
    • Analytic number theory
    • Fermat's Last Theorem for polynomials
    • p-adic numbers
    • Geometry of numbers
  • Geometry and Topology
    • Euclidean and non-Euclidean (hyperbolic, spherical, projective, inversive) geometries
    • Geometric transformations
    • Algebraic geometry
    • Point-set topology
    • Combinatorial topology
    • Knot theory
    • The Brouwer Fixed-Point Theorem
  • Calculus and Analysis
    • Topics in calculus
    • Fourier analysis
    • Complex analysis
    • Real analysis
    • Dynamical systems
  • Set Theory, Logic, and Foundations
    • Cardinals and ordinals
    • Gödel's Incompleteness Theorem
    • The Banach-Tarski Paradox
    • Model Theory
    • Category Theory
  • Computer Science
    • Theoretical CS
    • Complexity theory
    • Information Theory
    • Cryptography
    • Algorithms
  • Connections to Other Fields
    • Relativity and quantum mechanics
    • Neural networks
    • Mathematical biology
    • Game theory
    • Voting theory
    • Bayesian statistics
  • Discussions
    • Philosophy of Mathematics
    • Math Education
    • How to Give a Math Talk
    • College And Beyond
  • Problem Solving
    • Proof methods
    • Elementary and advanced techniques
    • Contest problems of various levels of difficulty
    • Relays and team competitions