Schubert polynomials are multivariate polynomials representing cohomology
classes on the flag manifold. Despite the beautiful formulas developed for them
over the past three decades, the coefficients of these polynomials remained
mysterious. I will explain Schubert polynomials from a polytopal point of
view, answering, at least partially, the questions: Which coefficients are
nonzero? How do the coefficients compare to each other in size? 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 polynomials? As the questions themselves suggest, we
will find that polytopes play an outsized role in our understanding.
The talk is based on joint works with Alex Fink, June Huh, Ricky Liu, Jacob
Matherne, Linus Setiabrata and Avery St. Dizier.