Schubert polynomials are polynomials indexed by permutations with nonnegative integer coefficients; they reflect Coxeter combinatorics of the permutation as well as geometric information about certain algebraic varieties. Any invertible n x n matrix can be written in the form LwU where w is a uniquely determined permutation matrix and L, U are lower and upper triangular respectively. For a fixed w, the closure of the set of matrices LwU is called a matrix Schubert variety, and the corresponding Schubert polynomial is its torus-equivariant cohomology class.

I will explain the approach of Knutson and Miller which relates this geometry to the combinatorics in a natural way, via Gröbner bases and Stanley-Reisner rings of simplicial complexes. No knowledge of algebraic geometry will be assumed beyond the correspondence of ideals and varieties. Speaker(s): Brendan Pawlowski (University of Michigan)