Miscellaneous Math
For thirteen summers, I have been an active staff member at Canada/USA Mathcamp, a summer program for mathematically talented high school students. Here are a few of my favorite Mathcamp classes:
Note: These class notes are not thoroughly polished---they may still contain typos, or even errors, so exercise caution!
- How To Make Rings That Do Terrible Things: These are notes from a one-day class I taught in 2011 and 2012. The sequence of ideas comes from a lovely 1978 paper by George M. Bergman called "The Diamond Lemma for Ring Theory." The prerequisite for this class was Mathcamp's intro ring theory class, which went through some basic definitions and proofs, and then dove into quotient constructions, with an emphasis on constructing familiar rings using quotients of free algebras.
- Infinite Trees: This is a set theory class which I've taught several times. The major result in the class is the construction of a Suslin tree using the diamond axiom. The notes require a fair degree of mathematical sophistication, but should be accessible to anyone who's seen basic cardinality and the definition of a graph.
- Model Theory: This is an adaptation of a class on Skolem's Paradox that I have co-taught twice. At some point I wrote up notes figuring I might solo-teach it, but never quite got around to it. Prereqs are a little subtle---basic cardinality is a must, and familiarity with some mathematical structures like graphs or rings would be helpful but not necessary.
- Measure and Martin's Axiom: This class explores measure-theoretic ideas from the perspective of math logic. The countable union of measure zero sets is measure zero. But what about the union of uncountably many---but not continuum many---measure zero sets?
- The Continuum Hypothesis: This class walks through the basics of the proof of the independence of the continuum hypothesis through Cohen forcing. Significant overlap with both Model Theory and Measure and Martin's Axiom. Still a bit rough in the proof of the forcing theorem.
- Diamond, Continuum, and Forcing: This was a brief follow-up to the Continuum Hypothesis class. These notes give the definition of the Diamond Axiom, show that it implies the Continuum Hypothesis, and show how forcing can be used to prove the consistency of Diamond. The forcing proof follows Steven Jackson's notes.
- By request, I am posting my messy handwritten notes for the class I am currently teaching called Infinite-ness. Here is Day 1.
Note: These class notes are not thoroughly polished---they may still contain typos, or even errors, so exercise caution!
