CAREER: Integer Point Transforms of Polytopes

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Polynomials are the basic building blocks of algebra. Polytopes are sets in high-dimensional space with flat sides. The aim of this project is to use polytopes to answer questions about the coefficients of special families of polynomials in many variables. For example, which coefficients are nonzero? How do the coefficients compare to each other in size? If we plot the exponent vectors of the monomials with nonzero coefficients, do they form the integer points of a polytope? If we plot a histogram of the coefficients, do the ratios of consecutive heights form a decreasing sequence? This project also has an educational component, with the core goal of enabling young women to succeed in STEM careers by introducing them to discrete mathematics, an area of growing importance for computer science and data science.Schubert and Grothendieck polynomials are multivariate polynomials representing cohomology and K-theory classes on the flag manifold, respectively. Despite the beautiful formulas developed for them over the past three decades, the coefficients of these polynomials remain mysterious. It is not even evident how to tell if a given coefficient is nonzero! The PI will investigate convexity properties of the coefficients of Schubert and Grothendieck polynomials. For instance, are the Newton polytopes of these polynomials saturated? Are their coefficients log-concave along lines? Is there a polytope whose integer point transform specializes to Schubert and Grothendieck polynomials? This project is dedicated to understanding the coefficients of these polynomials, with an eye for exposing polytopal reasons for their properties.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.