Grassmannians, Flag Manifolds, and Schubert Calculus

Instructor: Harry Tamvakis

Lectures: Tuesday and Thursday 2:00 - 3:15, Math 0303

Office: Math 4419
Office hours: By appointment
Telephone: (301)-405-5120

Course guide:

Main text: W. Fulton, "Young tableaux, with applications to representation theory and geometry", Cambridge Univ. Press, Cambridge 1997.

Our goal in this course is to study Grassmannians and flag manifolds from many different points of view, all related to each other:

- Geometry : Homology and cohomology theories for these manifolds; Schubert varieties, Chern classes and Schubert calculus. Applications to classical enumerative geometry. Equivariant cohomology and degeneracy loci of vector bundles.

- Algebra : The structure of the cohomology rings of these spaces is described by the beautiful theory of Schur polynomials (for Grassmannians) and Schubert polynomials (for flag manifolds). We will make connections with the representation theory of the symmetric and general linear groups.

- Combinatorics : Partitions: Young diagrams and tableaux, calculus of tableaux. Permutations: the Bruhat order, divided difference operators, the combinatorics of Schur and Schubert polynomials.

Background necessary:
I will attempt to keep the discussion as self-contained as possible. This means I will NOT assume familiarity with algebraic geometry, cohomology theories, characteristic classes, etc. Rather than give a full exposition of all the background needed (as this is impossible to do in one semester), I will tell you enough about each topic so that you can understand what is going on. I hope that what people see in the course motivates them to learn more about each individual subject (it worked for me!). In the first part of the course we will be studying geometry, so it will help to have a general familiarity with geometric reasoning.

For the most part we'll concentrate on the SL(n) case. The aim of the course is to understand as much as we can about these examples; even here there is current research with many unsolved problems. There is room in the above description to cover some different topics (for example, quantum cohomology or K-theory), depending on the interests of the people attending.

I plan to distribute some notes and homework problems during the course. The homework will include some material that could not be part of the lectures for lack of time.

Some relevant references in book form:

- W. Fulton, "Intersection Theory", 2nd edition, Springer-Verlag 1998.

- W. Fulton and P. Pragacz, "Schubert Varieties and Degeneracy Loci", Springer-Verlag 1998.

- L. Manivel, "Symmetric Functions, Schubert Polynomials and Degeneracy Loci", A. M. S. 2001.

- I. Macdonald, "Symmetric Functions and Hall Polynomials", Clarendon Press, Oxford 1995.

- R. Stanley, "Enumerative Combinatorics", Vol. II, Cambridge Univ. Press, Cambridge 1999.