University of Toronto's Symplectic Geometry Seminar

Monday January 17, 2005, 14:10--15:00
SS 5017A

Alexander Yong

Fields Institute

Enumerative formulas in Schubert calculus: problems, theorems and conjectures

Abstract: Schubert calculus studies classical questions of enumerative algebraic geometry (and their generalizations). For example: how many red lines touch four randomly placed blue lines in complex 3-space? The answer is 2. There has been substantial recent activity and progress over the past ten years on the following type of problem: give a combinatorial counting rule for computing (classical, K-theoretic, quantum, and/or equivariant) Schubert calculus problems on a (Grassmannian, flag variety, or G/P). The demand is that the rule be "positive": it manifestly outputs nonnegative integers. Work on these problems has relations to, and uses methods from, a diversity of mathematical disciplines, e.g.: algebraic geometry, combinatorics, representation theory, and symplectic geometry. A sample connection is this: mysteriously, Schubert calculus on the Grassmannian is computed by the tensor product multiplicity rule for complex GL(n) irreducible representations; no direct explanation of this fact is known yet. I will survey some of the developments in this subject, including joint projects with A. Buch, A. Knutson, A. Kresch, C. Lenart, E. Miller, M. Shimozono, F. Sottile and H. Tamvakis.