Algebraic Combinatorics

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This project is motivated by several classical areas of mathematics. The main tools come from combinatorics, including combinatorial topology, algebraic combinatorics, and the machinery of quiver mutations. Combinatorics deals with discrete objects such as finite sets, graphs, permutations, partial orders, etc. Many continuous phenomena allow for a discrete representation, lending themselves amenable to combinatorial methods of study. It is often the case that identical or similar combinatorial structures underlie seemingly unrelated mathematical entities, revealing hidden connections between these and allowing the transfer of insights and techniques from one discipline to another. A case in point is the theory of cluster algebras, which are the main focus of this project.This award supports the PI's research on combinatorial structures arising in algebra and geometry, with an emphasis on the theory and applications of cluster algebras. Cluster algebras, discovered by the PI in collaboration with A. Zelevinsky, have found applications in several mathematical disciplines including representation theory, Teichmueller theory, mathematical physics, and enumerative and geometric combinatorics. The investigator intends to further explore the connections between cluster algebras, on one hand, and classical invariant theory and projective geometry, on the other. Another research direction concerns algebraic complexity for computational models employing a restricted set of arithmetic operations.