Math 669: Schubert Calculus

Winter 2012

Course meets: Tuesday and Thursday 1:10-2:30 in 3096 East Hall.

Instructor: Sergey Fomin, 2858 East Hall, 764-6297,

Course homepage:

Level: introductory graduate.

Prerequisites: none (for graduate students).

Student work expected: several problem sets.

This is an elementary introduction to the combinatorial aspects of Schubert Calculus, the part of enumerative geometry dealing with such classical varieties as Grassmannians and flag manifolds.

A prototypical Schubert Calculus problem asks how many subspaces in a given vector space are in a special position with respect to a given collection of subspaces. For example: given a generic configuration of four 2-dimensional subspaces in a complex 4-dimensional space, how many 2-dimensional subspaces intersect each of these four in a line?

In modern language, problems of this kind translate into calculations in the corresponding cohomology ring. These calculations involve the combinatorial machinery of Young tableaux, Bruhat orders, symmetric functions, and Schubert polynomials. Most of the course will be devoted to the development of this machinery.

More advanced topics may include (time-permitting): Schubert Calculus in other Lie types; quantum cohomology and calculation of Gromov-Witten invariants; and K-theoretic and equivariant analogues.

No special background in combinatorics, topology, algebraic geometry, commutative algebra, or Lie theory is required.

Course Outline
I. Schubert calculus on Grassmann manifolds.
II. Schubert calculus on flag manifolds. Schubert polynomials.
III. Variations on the theme of Schubert.

Texts. The course will not strictly follow a particular text. Principal sources:
[F] W.Fulton, Young tableaux, Cambridge University Press, 1997.
[M] L.Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, AMS, 2001.