Problem Sets from Mathcamp
Mathcamp is a most wonderful place to be, and probably the most wonderful place I've ever taught. Among the many activities competing for students' attention are the many problem sets handed out by teachers. While I'm quite certain that all of my students did their problem sets promptly, I will still archive them here for those students or any other interested viewers (and, indeed, for myself).
Proof Techniques 2004
Quite possibly my most well-received class ever. The class covered proof by contradiction, the pigeon hole principle, proof by induction, infinite descent, invariants, and whatever else could be thrown into that wonderful stew. It served as an essential toolbox for students to draw on throughout the rest of the camp. It was also a lot of fun.
- Problem set 1 (PS, PDF)
- Covered proof by contradiction, and a refresher on direct proofs.
- Problem set 2 (PS, PDF)
- More proof by contradiction, and the pigeon hole principle.
- Solutions to problem set 2 (PS, PDF)
- Problem set 3 (PS, PDF)
- Covered infinite descent. Also the first of the induction problems (two easy ones, at the end).
- Problem set 4 (PS, PDF)
- Proof by induction, invariants.
What is Distance? 2004
This class was an introduction to topology via metric spaces. The goal was to introduce topological ideas and thinking through metric spaces; in particular, a number of pathological metric spaces were introduced. In addition, the class aimed to make students think about equivalence; we emphasized the contrast between an isometry and a homeomorphism, and considered what properties of spaces were metric properties and what properties were topological in nature.
The class should have probably moved a bit faster, had one more problem set, and gone on for a bit longer. A new and improved version is hopefully on the way for 2005!
- Problem set 1 (PS, PDF).
- Covered ultrametrics, other variations on metric spaces. The primary goal was to get comfortable with the abstractions of the subject.
- Problem set 2 (PS, PDF).
- Cauchy sequences, completions.
Construction of the Real Numbers
A class in progress. I'm debating where to stick it --- it could go before a class on metric spaces, or in a class on numbers in general. The class constructs the real numbers from the rationals by Dedekind cuts.
- Problem set 1 (PS, PDF).
- This problem set is still a work in progress. There's a lot still to be considered for it, including cementing out the contents of the class itself!
- This over-long problem set allows the students to fill in the proof of the uniqueness of the Dedekind cut closure of the rationals. It introduces the notion of an ordered field properly, and gets students comfortable with algebraic abstractions. It gives a non-constructive categorization of the real numbers ("the unique ordered field with the least upper bound property"), which is a kind of definition students should get used to! Sadly, it is also far too long.