Research in Algebraic Combinatorics

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My program of research sits at the frontier of algebra and combinatorics, with specific emphasis on representation theory of finite groups and their invariants, and connections with algebraic geometry, topology, mathematical physics and statistical mechanics. One of my aims is to expand and bring together recent important developments at the frontier of four areas, namely: algebra (diagonal modules of polynomials), special functions (operators on symmetric functions), combinatorics (rectangular Dyck paths and parking functions), and knot theory (skein algebra of (m,n)-torus knot). For several years, I have been at the forefront of research in the first three of these areas, and my project sits at the very center of main open questions regarding their interactions. Moreover, there are exciting recent developments in each of these areas, and a large number of profound problems are arising. Together with my students, postdocs and collaborators, I am currently making significant progress in each of these areas. Indeed, our work has clarified many of the central questions that need to be solved in this interaction, and more recently I have explicitly outlined which main directions this endeavor should now go toward. More explicitly, my proposal is articulated around the four following main axes: