Freshman seminar: the story of the alternating sign matrix conjecture -- Fall 2025

Lectures: Mondays 3-5:30pm, Science Center 530

Professor: L. Williams (Science Center 510, e-mail williams@math.harvard.edu)

Office Hours: Fridays 2:30-3:30, Science Center 510.

Course 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 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).

Prerequisites

The course will be fairly fast-paced; familiarity with proofs, and some basic notions from linear algebra will be helpful (for example the notion of the determinant of an n by n matrix).

References

A main reference for the course will be this book by David Bressoud, which you can also access here. We will supplement this book with various articles including Kuperberg's paper.

There is various software for playing with ASM's, including this code written by Dan Romik. (There's a Mathematica notebook and a Mac app.)

Here are some resources for writing a math paper, and here are some resources for writing in Latex.

Lectures

  • Lecture 1 (Sept. 8): Introduction to the ASM conjecture (where it came from)
  • Lecture 2 (Sept. 15): Introduction to partitions and plane partitions
  • Lecture 3 (Sept. 22): Symmetric functions.
  • Lecture 4 (Sept. 29): Equivalence of three definitions of Schur polynomials.
  • Lecture 5 (Oct. 6): Applications to plane partitions. Plus ASM's, monotone triangles, square ice, fully-packed loops, 6-vertex model.
  • No lecture (Oct. 13): Columbus Day/ Indigenous Peoples' Day
  • Lecture 6 (Oct. 20): The Yang-Baxter equation.
  • Lecture 7 (Oct. 27): The Izergin-Korepin theorem.
  • Lecture 8 (Nov. 3): Kuperberg's proof of the ASM conjecture.
  • Lecture 9 (Nov. 10): Finishing the proof of the ASM conjecture.
  • Lecture 10 (Nov. 17): Recent results and open problems on ASMs; Doron Zeilberger lecture!
  • Lecture 11 (Nov. 24): Final presentations: Grant and Tian (cyclic sieving), Quang (hook length), Kira (poset perspective)
  • Lecture 12 (Dec. 1): Final presentations: Austin and Tony (FPL and gyration), Mia (ASMs and Bruhat), Emma (Cantini-Sportiello theorem).
  • 12pm Tuesday December 9 (last day of reading period): Final paper due.