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Freshman seminar: the story of the alternating sign matrix conjecture -- Fall 2019

**Lectures:** Mondays 3-5:30pm, Science Center 530
**Professor: ** L. Williams
(Science Center 510, e-mail `williams@math.harvard.edu`)

**Office Hours:** Thursdays 10-11:30am, Science Center 510.

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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).

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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).

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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.)

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Lectures

Lecture 1 (Sept. 9): Introduction to the ASM conjecture (where it came from)
Lecture 2 (Sept. 16): Introduction to partitions and plane partitions
Lecture 3 (Sept. 23): Symmetric functions.
Lecture 4 (Sept. 30): Equivalence of three definitions of Schur polynomials.
Lecture 5 (Oct. 7): Applications to plane partitions. Plus ASM's, monotone triangles, square ice, fully-packed loops,
6-vertex model.
No lecture (Oct. 14): Columbus Day/ Indigenous Peoples' Day
Lecture 6 (Oct. 21): The Yang-Baxter equation.
Lecture 7 (Oct. 28): The Izergin-Korepin theorem.
Lecture 8 (Nov. 4): special guest: Richard Stanley.
Lecture 9 (Nov. 11): Kuperberg's proof of the ASM conjecture.
Lecture 10 (Nov. 18): Finishing the proof of the ASM conjecture.
Lecture 11 (Nov. 25): Final presentations:
Lecture 12 (Dec. 2): Final presentations:
12pm Monday December 16: Final paper due.