The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics.
The applications chosen for this incarnation of the course will have little overlap (primarily braid theory) with the Fall 2003 course taught by Prof. Cohen although there will be a significant overlap in tools -- particularly the free differential calculus and the lower central series.
This course provides a basic introduction to finite dimensional, continuous time, deterministic control systems at the graduate level. The course is intended for PhD students in applied mathematics and engineering graduate students with a solid background in graduate real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for research at the interface of applied mathematics and control engineering. This will be a rigorous, proof-oriented systems theory course emphasizing controllability and stabilization.
I've always been a fan of the coin problem. It's basically a generalization of the question, \"what is the largest score in American football that cannot be achieved (say, ignoring safeties)\" It's got a built-in hook with the story of the \"McNugget numbers\". It very naturally motivates $\\gcd(a_1, \\ldots, a_n)$ and you can discuss existence of the Frobenius number. As I recall there's a nice graphical proof of the formula for two coins. The topic typically uses some basic number theory like Bezout's lemma, but if anything that's a feature and not a bug. 1e1e36bf2d