In one direction we have seen many applications of tools and emergence of objects from algebraic combinatorics in integrable models in statistical mechanics. The major applications concern recent advances in [colored] vertex models and interacting particle systems using symmetric functions, specifically Schur and Macdonald polynomials and their generalizations. Another example is the study of various exclusion processes (e.g. TASEP and its relatives) and their steady states where polynomials of algebro-combinatorial significance emerge (e.g. Schubert polynomials). The third example is the emergence of dimer models and electrical networks from the algebra of the positive Grassmannian.
In the opposite direction, tools and ideas from probability and statistical physics have seen application in problems from Algebraic Combinatorics. Two examples are the asymptotics of various structure constants of representation theoretic significance (e.g. Kostka, Littlewood-Richardson, and Kronecker coefficients) and symmetries of polynomials and rational functions arising from Yang-Baxter equations.
We have seen how probability motivates new research directions in algebraic combinatorics and how algebraic combinatorics leads to new discoveries in probability. The aim of the workshop is to further stimulate the cross-infiltration of the ideas between two fields.
(University of California, Berkeley (UC Berkeley))
Alejandro Morales (University of Quebec Montréal)
Greta Panova (University of Southern California (USC))