Polynomials are the basic building blocks of algebra. Polytopes are sets in high-dimensional space with flat sides.
Karola Mészáros, Mathematics, is using 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?
Despite the beautiful formulas developed over the past three decades, the coefficients of Schubert and Grothendieck polynomials remain mysterious. It is not even evident how to tell if a given coefficient is nonzero. Mészáros is investigating convexity properties of the coefficients of these polynomials. She is dedicated to understanding the coefficients of the polynomials, with an eye for exposing polytopal reasons for their properties.