Instructor: Harry Tamvakis
Lectures: Tuesday and Thursday 2:00 - 3:15, Math 0403
Office: Math 4419
Office hours: By appointment
- Schubert calculus began with the work of
Schubert, Pieri, and Giambelli in the 19th century, which
studied the geometry of the Grassmannian manifold
parametrizing k-dimensional linear subspaces of a complex vector
space. This research was a precursor to the cohomology rings
of more general manifolds and the Chern classes of vector bundles
over them developed in the 20th century. Around the same time,
important connections between Schubert calculus and the much
older theory of symmetric functions and Schur polynomials were
- Thanks to the pioneering papers of Borel and Chevalley in
the 1950s, the subject today may be defined as the study of
the cohomology ring of the compact homogeneous spaces G/P of
complex Lie groups. Despite its long history, for the basic
examples of symplectic and orthogonal Grassmannians, the
analogues of the Schur polynomials were found only during the
last 15 years. Moreover, a new, intrinsic point of view in
Schubert calculus has emerged, which includes all of the
classical groups, and employs algebro-combinatorial tools such
as partitions, raising operators, Young tableaux, divided
differences, nilCoxeter algebras, and transition trees. This
has solved open problems in the field and revealed links with
other areas such as the theory of Hall-Littlewood functions
and Macdonald polynomials.
- This course will be an introduction to Schubert calculus,
with emphasis on the modern intrinsic perspective. I will
strive to keep the discussion as self-contained as possible,
so will NOT assume familiarity with algebraic geometry,
intersection theory, characteristic classes, etc. However, a
basic knowledge of Lie groups and reflection groups should
be helpful. Although the geometry will remain mostly in the
background, it is the motivation for much of the rich algebraic
combinatorics that is encountered in the
class. Understanding how these two areas interact with the
Lie theory is one of the end goals of this subject.
- I plan to distribute some homework problems
during the course. The homework will include material that could
not be part of the lectures for lack of time.
- The most relevant references are my two research-expository papers below. They give an overview
of the theory with examples, but include very few proofs.
- Giambelli and degeneracy locus formulas for classical G/P spaces,
Mosc. Math. J. 16 (2016), 125-177.
- Theta and eta polynomials in geometry, Lie theory, and combinatorics,
"First Congress of Greek Mathematicians", 243-284,
De Gruyter Proc. Math., De Gruyter, Berlin, 2020.
- Some older references in book form:
- H. Hiller, "Geometry of Coxeter Groups", Research Notes in Mathematics 54,
Pitman, Boston 1982.
- W. Fulton, "Young Tableaux, with Applications to Representation Theory and Geometry",
Cambridge Univ. Press, Cambridge 1997.
- L. Manivel, "Symmetric Functions, Schubert Polynomials and Degeneracy
Loci", Amer. Math. Soc., 2001.
- S. Kumar, "Kac-Moody groups, their Flag Varieties and Representation Theory",
Birkhaüser, Boston 2002.