The Story of the Alternating Sign Matrix Conjecture
FYSEMR 51E
Subject & Catalog Number
Course Information
Description
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 seminar 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.
NOTE: This seminar will meet Mondays, 3pm-5:30pm.
NOTE - PREREQUISITE: 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).
Class Notes
First-Year Seminars are available only to first-year students. You may apply to both Fall 2026 and Spring 2027 First-Year Seminars via the FYS lottery between July 6 and August 6, 2026 at 11:59PM-midnight.
You may apply to as many seminars each term as you would like, but we recommend you apply to at least six in fall and three in spring.
As part of your application, you must provide a brief statement on why you are interested in each seminar. You will be notified of lottery results for both fall and spring seminars at 5 pm on August 11. If you are unsuccessful in the lottery, you may still join any seminar with open seats. A list of open seminars and instructions on next steps will be available on the First-Year Seminar Program website August 11 at 12 Noon.
Not Available for Cross Registration
