Schubert calculus, involutions and symmetric matrices

Abstract: In his “Kalkul der abzahlende Geometrie”, Schubert introduced a powerful non-rigorous method for solving enumerative geometry problems known as Schubert calculus. Making this method rigorous motivated a great deal of 20th century algebraic geometry. Since the combinatorics of permutations is central to this theory, combinatorialists have made major contributions to Schubert calculus broadly construed. We will overview some highlights from this field, develop a parallel theory based on the combinatorics of  involutions in the symmetric group and conclude by using this work to describe fundamental properties of the geometry of symmetric matrices. This work is joint with Eric Marberg and Brendan Pawlowski.