# 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.