Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

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:
1. Combinatorics of Macdonald polynomials, related operators, and associated modules, and links with rectangular Catalan combinatorics and their connections to the elliptic Hall algebra,
2. Combinatorial and representation theoretic analogs in several sets of variables,
3. Properties of plethysms of symmetric functions, and the Foulkes conjecture.
The second of these exploits ideas that I proposed a few years back concerning the expansion to multidiagonal versions of the questions that have been so fruitful in the bidiagonal case. About this, it may be worth underlining that certainly more than a hundred significant papers have appeared in top journals in relation to this case ( k =2) since the mid 1990s. An expansion to the multidiagonal case ( k >2) is bound to multiply this research impact, as well as give more fundamental reasons why all of this is so pregnant with significant new knowledge.
I have also come up with original new techniques to construct the algebraic counterparts (modules of polynomials) for the combinatorial constructions and symmetric function objects involved in the theory. This has been a long-standing important missing part in this research area. My new approach is bound to furnish an original satisfying explanation for a fundamental leitmotif in this context: the Schur positivity of the symmetric functions involved; and explain why this positivity phenomenon is so predominant.
This last aspect brings me to the last portion of my program regarding many new ways of understanding a conjecture of Foulkes that dates back almost 70 years. In particular, I propose a new and original q -analog.