Reading seminar on solvable lattice models in Fall 2023

Tuesdays, 11:00am - 12:10pm, Kerchof 111

Solvable lattice models originated in mathematical physics (Ising model, square ice). They found numerous applications to combinatorics (enumerative and algebraic - keywords are nonsymmetric Macdonald polynomials and Hecke algebras), representation theory (keyword - quantum groups), and more recently, probability. We will start with classical results, such as the Izergin-Korepin determinant and the entropy of the square ice. We’ll select further topics based on the audience’s interests.

Schedule of meetings

1. August 29, 2023

   Leo Petrov - Introduction

2. September 5, 2023

   Nick Sweeney - The Alternating Sign Matrix conjecture
   after Chapter 1 of the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud

3. September 12, 2023

   Petch Chueluecha - Dodgson condensation
   after Section 3.5 of the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud

4. September 19, 2023

   Mikhail Tikhonov - The Yang-Baxter equation and the Izergin-Korepin determinant
   after Wheeler-Zinn-Justin 2015 (Section 4.2 with $u=1$) and Petrov 2020 (Yang-Baxter equation is Proposition 2.1; and the determinant is proven in Section 3.4 with $\gamma=1$, $s_0=0$)

5. September 26, 2023

   Leo Petrov - From the Izergin-Korepin determinant to the number of alternating sign matrices

6. October 10, 2023

   Jacob Campbell - R-matrices from quantum groups (after Kassel “Quantum Groups”, 1995, Section 8.1); see also this part on the definition of the quantum $sl_2$

7. October 17, 2023

   Suren Kyurumyan - The stochastic six-vertex model (after Gorin-Nicoletti’s lecture notes, Section 4.1)

8. October 24, 2023

   Break

9. October 31, 2023

   Leo Petrov - From the stochastic six-vertex model to asymptotics (after Gorin-Nicoletti’s lecture notes, Section 5; and also papers on Schur measures)