Lauren K. Williams (Department of Mathematics)
First-Year Seminar 51E (Fall Term) Enrollment: Limited to 12
Thursday, 3-5:15pm CANVAS SITE
Prerequisites: This seminar is recommended for students with a strong interest in mathematics, including some familiarity with proofs. It would be helpful to have some exposure to combinatorics (permutations, binomial coefficients) and linear algebra (matrix multiplication and determinants of
n by n matrices).
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 story of 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 x 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 proofs of the conjecture. In this course we will survey the story of the alternating sign matrix conjecture, building up to Kuperberg’s proof. If time permits, we will also get a glimpse of very recent activity in the field, for example the Razumov-Stroganov conjecture.