The TA for this course was Matthew Sunohara.
What was the course about?
The course centered about the question of classifying objects. We discussed examples on ways we can classify different types of mathematical objects (knots, POSETS, Groups, etc.) and how we can create tools to distinguish between them.
We covered the following topics (with different degrees of depth):
- What do we mean by classification and by invariants?
- Polynomial and combinatorial invariants of knots: tricolorability, Conway’s polynomial, Jones Polynomial.
- Process of tabulations and explored the question of “when is a classification complete”?
- Complete and Partial Invariants.
- POSETS and examples of families of them.
- The Wallace-Bolyai-Gerwien Theorem and Hilbert’s Third problem.
- The Dehn Invariant and examples of it.
- Groups, families of groups and the (very long) process of finding all the finite groups.
- The idea of “exceptional” objects in a classification.
What motivated me?
When I was designing this course, I wanted to show the students something that is not a classical subject to study. Instead of picking a topic (as for example, topology or group theory) I thought it would be interesting to discuss an idea that permeated all of mathematics. I asked myself what do we do as mathematicians in any field and after some time of thinking I came to the conclusion that classification is a big topic.
Exercises we worked on:
Note: The exercise section is not always complete. Many extra things were added on the spot as the course moved forward, depending on the questions of the public, and sometimes I do not have record of all of it. Sometimes the problems have mistakes and I might not have corrected all of them.
Malors 