Interactions of Combinatorics and Physics

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In recent years there have been a number of extremely fruitful interactions between combinatorics and physics. This project aims to use combinatorial techniques to address questions from various fields of physics, including statistical physics (particle processes including the asymmetric simple exclusion process), quantum field theory (scattering amplitudes in N=4 super Yang Mills theory), and mirror symmetry. The mathematical topics and tools involved include special functions, the amplituhedron, cluster algebras, and cluster varieties. Graduate students will be trained as part of this project.More concretely, this project concerns the relationship between the asymmetric simple exclusion process (ASEP) and important families of polynomials in algebraic combinatorics, including Schubert polynomials, Macdonald polynomials, Askey-Wilson polynomials, and Koornwinder polynomials. It seeks to use the structure of the ASEP to obtain insight and new formulas for these polynomials. A second direction is to study the combinatorics of the amplituhedron, a generalization of the positive Grassmannian, and in particular the cluster adjacency conjecture, and an enumerative conjecture linking its tilings to plane partitions. A final direction is to use tools from cluster algebras to understand mirror symmetry for flag varieties and related spaces.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.