Professor: L. Williams (Science Center 510, e-mail williams@math.harvard.edu)
Office Hours: Tuesdays 3-4pm and Thursdays 10:30am to 11:30am.
This seminar is intended to illustrate how research in mathematics actually progresses, using recent examples from the field of algebraic combinatorics. We will learn about the search for and discovery of a proof of a formula conjectured by Mills-Robbins-Rumsey in the early 1980's: the number of n by n alternating sign matrices. Alternating sign matrices are a curious family of mathematical objects, generalizing permutation matrices, which arise from an algorithm for evaluating determinants discovered by Charles Dodgson (better known as Lewis Carroll). They also have an interpretation as two-dimensional arrangements of water molecules, and are known in statistical physics as square ice. Although it was soon widely believed that the Mills-Robbins-Rumsey conjecture was true, the proof was elusive. Researchers working on this problem made connections to invariant theory, partitions, symmetric functions, and the six-vertex model of statistical mechanics. Finally in 1995 all these ingredients were brought together when Zeilberger and subsequently Kuperberg gave two different proofs of the conjecture. In this seminar we will survey these developments. If time permits, we will also get a glimpse of very recent activity in the field, for example the Razumov-Stroganov conjecture (now Cantini-Sportiello theorem).
A main reference for the course will be this book by David Bressoud. We will supplement this book with various articles including Kuperberg's paper.