Algebraic Combinatorics

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. This project investigates combinatorial structures arising in algebra and geometry, focusing on further development of the theory and applications of cluster algebras.

Cluster algebras and the associated combinatorial and algebraic constructions were discovered at the turn of the century. Their importance for applications in other areas of mathematics and physics is becoming increasingly apparent. In algebraic combinatorics, it is often the case that similar combinatorial structures 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 these connections. The project will also involve graduate students.

This research is motivated by several classical areas of mathematics. The main tools come from combinatorics, including combinatorial topology, the machinery of quiver mutations, and combinatorics of bicolored planar graphs. Cluster algebras, and the combinatorics of quiver mutations underlying them, have found applications in many mathematical disciplines including representation theory, Teichmüller theory, mathematical physics, and enumerative and geometric combinatorics.

The project suggests new applications of cluster theory to the study of plane algebraic curves and related problems of singularity theory and low-dimensional topology. Other research directions involve applications of algebraic combinatorics to discrete and computational geometry, the study of planar projective configurations, and discrete integrable systems.

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.