Research in Algebraic Combinatorics

我的研究项目位于代数和组合学的前沿，特别强调有限群及其不变量的表示理论，以及与代数几何、拓扑学、数学物理和统计力学的联系。我的目标之一是扩展和汇集四个领域的最新重要发展，即：代数(多项式的对角模)、特殊函数(对称函数上的算子)、组合学(矩形Dyck路和停车函数)和纽结理论((m，n)环面纽结的skein代数)。几年来，我一直站在这些领域中前三个领域的研究前沿，我的项目位于关于它们相互作用的主要公开问题的中心。此外，在这些领域中的每一个领域都有令人振奋的最新事态发展，并出现了大量深刻的问题。与我的学生、博士后和合作者一起，我目前在这些领域都取得了重大进展。事实上，我们的工作已经澄清了在这种互动中需要解决的许多核心问题，最近我明确提出了这项努力现在应该朝着哪些主要方向发展。更明确地说，我的建议围绕以下四个主要轴心提出：

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: