Algebraic Combinatorics

Combinatorics studies discrete structures such as finite sets, graphs, and permutations. Many continuous phenomena allow for discrete representation, lending themselves amenable to combinatorial approaches. In algebraic combinatorics, it is often the case that similar combinatorial structures turn out to underlie seemingly unrelated mathematical phenomena. As a result, hidden connections are revealed, allowing the transfer of insights and techniques from one discipline to another. This project aims to extend and deepen such connections. It investigates combinatorial structures arising in algebra and geometry and is motivated by several classical areas of mathematics. On the geometric side, it aims to develop new combinatorial techniques in classical incidence geometry. This is a classical subject that has its roots in antiquity. It studies configurations of geometric objects, such as points, curves, and surfaces, focusing exclusively on their relative position. The algebraic side of the project concerns further development of the theory and applications of cluster algebras and their underlying combinatorial structures that have found applications in many areas of mathematics and theoretical physics. The project will involve students at various levels. Incidence geometry is famous for a panoply of beautiful theorems discovered over the course of centuries. A recently proposed combinatorial approach utilizes tilings of oriented surfaces to place all these results under one roof. This approach has already been used to discover new incidence theorems and to generalize several known ones. It suggests multiple directions of further research, including those involving non-Euclidean geometries and/or varieties of higher degree, as well as new connections with discrete integrable systems and low-dimensional topology. Another part of the project is dedicated to further development of the theory of cluster algebras. These algebras, and the underlying combinatorics of quiver mutations, have found applications in many mathematical disciplines including representation theory, Teichmüller theory, mathematical physics, and symplectic geometry. One research direction concerns the structural theory of cluster algebras, more specifically the study of quiver mutations and associated invariants. Another direction aims to deepen our understanding of real plane algebraic curves and related concepts of singularity theory and low-dimensional topology, by revealing and investigating associated cluster-algebraic structures. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.