Program

Algebraic combinatorics is an area of mathematics that employs methods in abstract algebra in combinatorial contexts, and that uses combinatorial methods to approach problems in algebra. Some important topics are symmetric functions, Young tableaux, matroids, Coxeter combinatorics. There are links to computer algebra (sage-combinat), number theory (L-functions), representation theory, and mathematical physics through Macdonald processes and integrability. The work on the totally positive Grassmannian also gave rise to beautiful results in mathematical physics: for example KP solitons and Scattering Amplitudes. Schubert calculus is an important part of algebraic combinatorics and is now at the frontier with k-Schur functions, which first came up in the theory of Macdonald polynomials. This semester will have a special focus on • Integrable systems and dynamical combinatorics • Combinatorial representation theory • Geometry of polynomials • Combinatorial varieties and connections to symmetric function theory • Cluster algebras We will also have a transversal theme on computation with Sage Combinat and the use of machine learning for research in Algebraic Combinatorics. Most of the participants will be experts on several of these focuses. The semester will offer a unique opportunity for interaction between integrable/statistical physics, algebraic geometry, representation theory, theoretical computer science and combinatorics communities, and will foster exchanges on a wide range of problems and methods, via tutorials, discussions and eventually collaborations.