Overview

Every summer, I teach at Canada/USA Mathcamp, a summer camp for high school students talented in mathematics. The level of the material is accessible to students without an extensive math background while still often covering graduate school material. Teachers and students are both excited to be there and a wonderful synergy occurs that creates one of the best learning environments I've ever seen.

Each year I teach several classes, at various levels of difficulty and with various levels of homework. During the summer of 2006, I taught:

• Hyperbolic Geometry, a one-week class
• Theoretical Computer Science, a three-week class discussing abstract models of computation and various complexity classes
• Understanding Infinity, a one-week class on notions of infinity: countability, Cantor's diagonalization argument, ordinals and cardinals
• The Reeb Foliation of the 3-Sphere, a one-day class that I've given many times, describing ways of studying objects native to four-dimensional space, in particular a generalization of the "sphere" we're familiar with in our own space
• A Topological Smorgasbord, a one-week class giving an introduction to various topological spaces (tori and projective space), quaternions, and culminating in the Hopf fibration of the 3-sphere
• Quantum Computation, a one-week class describing quantum circuits and Shor's algorithm

The only class of these to generate significant quantities of problem sets was Theoretical Computer Science, which had many handouts, largely revisions of the same class from Mathcamp 2005. The problem sets were much more refined this year, and well-received by students. If you're interested in the material, although you'll probably recognize some of the problems, you'll also enjoy doing others.

In prior years I've had other classes that also generated significant material. You can see these archives below.

• Mathcamp 2005, which also has material on metric spaces and proof techniques.
• Mathcamp 2004, which has early incarnations of some of this material; I recommend looking primarily at 2005 and 2006.

Theoretical Computer Science

This class ran for three weeks. It covered Turing machines, the Halting problem, decidability/undecidability, P vs. NP, PSPACE, the polynomial hierarchy/alternating hierarchy, and a teensy bit of random computation (BPP).

The class was a lot of fun, and was very well received. It will probably run again at a future Mathcamp.

Class Handouts

• Sorts
• A Turing machine handout, hopefully with a coherent description of their operation
• Turing machine worksheet

Homework Assignments

• Homework 0
• Homework 1 and solutions.
• Homework 2 and solutions.
• Homework 3