Combinatorics in Geometry and Physics

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This project lies at the interface of algebraic combinatorics and particle physics. Combinatorics is the study of discrete structures (such as permutations) and their classification and enumeration. Algebra is the study of equations and their solutions. Algebraic combinatorics seeks to combine the two subjects, solving algebraic problems using combinatorics, and vice versa. In particle physics, one tries to model and predict outcomes of experiments involving collisions of elementary particles. In recent years, such calculations, called "scattering amplitudes", have been related to combinatorial structures, called "positive geometries", appearing in algebraic combinatorics. Roughly speaking, the combinatorics of positive geometries controls special positions for elementary particles. This project aims to advance the field by further improving our combinatorial understanding of positive geometries, and establishing more robust relations with physics. The project provides research training opportunities for graduate students.The project is in algebraic combinatorics. The PI aims to further study the recently defined positive geometries and integral functions on these spaces. These functions have applications to scattering amplitudes and to mirror symmetry, and include special functions such as Beta functions and Bessel functions. One of the main examples of positive geometries are totally positive spaces and the PI will study the combinatorics and topology of these spaces. The PI will further develop a program relating Langlands reciprocity to mirror symmetry for Fano varieties. Finally, the PI will study the cohomology of cluster varieties, which has applications to combinatorics (rational Catalan numbers) and representation theory (higher extensions of Verma modules).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.