This project aims to address several questions at the interface of combinatorics and physics. More specifically, it will apply algebraic and combinatorial tools to questions involving the exclusion process, Macdonald polynomials, cluster structures in Schubert varieties, mirror symmetry in cluster varieties, and the amplituhedron.
More concretely, this project concerns several interrelated questions surrounding the multispecies asymmetric simple exclusion process (ASEP) on both a lattice with open boundaries, and on a ring. This combinatorial model is intimately connected to both Koornwinder polynomials and Macdonald polynomials; and a better understanding of the ASEP should lead to combinatorial formulas for Koornwinder and Macdonald polynomials. A second direction is to understand the cluster structure on Schubert varieties in the Grassmannian. This should have applications to toric degenerations and mirror symmetry on the Grassmannian and its Schubert varieties. A final direction concerns the amplituhedron, which was introduced by physicists in the context of scattering amplitudes in N=4 super Yang Mills theory. Conjecturally, the volume of the amplituhedron computes certain scattering amplitudes, which measure how massless particles interact. Understanding this conjecture should lead to a new insights concerning the positive Grassmannian.
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.
