This project investigates combinatorial structures arising in algebra and geometry, including further development of the theory and applications of cluster algebras. Combinatorics deals with discrete objects such as finite sets, graphs, and permutations; many continuous phenomena allow for a discrete representation, lending themselves amenable to combinatorial methods of study as well. Cluster algebras and the combinatorics underlying them were discovered only relatively recently, but their importance for application in other areas of mathematics and physics is becoming increasingly apparent. It is often the case that similar combinatorial structures underlie seemingly unrelated mathematical entities, revealing hidden connections between them and allowing the transfer of insights and techniques from one discipline to another. This project aims to extend and deepen these connections.
This research is motivated by several classical areas of mathematics. The main tools come from combinatorics, including combinatorial topology, the machinery of quiver mutations, symmetric functions and Young tableaux, and combinatorial ring theory. Cluster algebras, and the combinatorics of quiver mutations underlying them, have found applications in several mathematical disciplines including representation theory, Teichmüller theory, mathematical physics, and enumerative and geometric combinatorics. A new application of cluster theory to the study of isolated singularities of plane algebraic curves will be studied. Another research direction concerns the theory of noncommutative Schur functions and related questions of combinatorial ring theory. The project will also investigate applications of algebraic combinatorics to planar projective geometry and algebraic complexity theory.
