Math 665: Schubert Calculus

Fall 2008

Course meets: Tuesday and Thursday 1:10-2:30 in 3866 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.

Hermann Schubert (1848-1911)

This course will provide 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 example of a Schubert calculus question is the following: Given mp subspaces of dimension p in general position in a complex vector space of dimension m+p, how many subspaces of dimension m intersect all these mp subspaces nontrivially?

To be able to answer such questions, one needs to gain a concrete understanding of the structure of the cohomology ring of the corresponding variety (in the example above, it will be the Grassmannian Gr(m,m+p)). Computing intersection numbers like the one defined above requires development of extensive combinatorial machinery involving Young tableaux, Bruhat orders, symmetric functions, and Schubert polynomials.

More advanced topics may include (time-permitting): intersection-theoretic computations in (partial) flag manifolds related to classical semisimple Lie groups; quantum cohomology rings and calculation of Gromov-Witten invariants; and K-theoretic and equivariant analogues.

The presentation will be essentially self-contained and elementary, and will require no special background in combinatorics, topology, algebraic geometry, commutative algebra, or Lie theory.

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.