Which problems can be solved by systematic algorithms, and which cannot? One of the marvelous qualities of mathematics is that it can analyze its own limitations. In this class, we'll study problems from various domains—geometry, algebra, analysis, number theory, computer science, and logic itself—and we'll explore which ones can or cannot be solved. The problems will all be relatively elementary (no specialized background will be required beyond a clear understanding of proofs and abstraction, obtained from 18.100 and one further course), but they will be fundamental to how we think about mathematics.
In this class, as in all the undergraduate seminars, the students will do almost all of the lecturing and will write papers. We'll begin by covering some of the basics. Once we have a firm foundation, each student will choose a favorite topic to investigate in more detail. This means the syllabus will be heavily based on the preferences of the students, although of course I'll offer suggestions and assistance and I'll provide guidance regarding feasibility.
Whenever possible, we'll examine the original research papers. This is not always so easy, and we'll certainly fall back on modern expositions whenever the original papers are overly obscure or difficult to read, but it's important to understand where the ideas actually came from. Mathematics is not just a collection of facts, but also a story, so we need to understand both what and why.