**Fall 2008**

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

**Instructor:**
Sergey Fomin, 2858 East Hall, 764-6297,
fomin@math.lsa.umich.edu

**Course homepage:** http://www.math.lsa.umich.edu/~fomin/665f08.html

**Level:** introductory graduate.

**Prerequisites:** none (for graduate students).

**Student work expected:** several problem sets.

Hermann Schubert (1848-1911)

** Synopsis**

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.