Professor: L. Williams (Science Center 510, e-mail williams@math.harvard.edu)
Office Hours: Wednesdays 1:05pm-2:30pm.
Course assistant: Charles Wang, e-mail m.charles.wang@gmail.com.
Charles' office hours: Fridays 10am-12pm, near 421b.
This course will provide an elementary introduction to the combinatorial aspects of Schubert calculus, the part of enumerative geometry dealing with classical varieties such as Grassmanians, flag varieties, and their Schubert varieties. A classical example of a Schubert calculus question is the following: 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? To be able to answer this and related questions, one needs to concretely understand the structure of the cohomology ring of the Grassmanian. In this course we will develop the necessary combinatorial machinery to answer such enumerative questions, including Young tableaux, the Bruhat order, symmetric functions, and Schubert polynomials. We may also discuss related topics including singularities of Schubert varieties, quantum cohomology rings, real Schubert calculus, and Macdonald polynomials.
Prerequisites: Although this class will be focusing on combinatorial aspects of the theory, one of the main topics will be the cohomology ring of the Grassmannian, so familiarity with cohomology (from e.g. Math 231a) and the notion of a projective variety (from Math 137 or 232A) will be helpful. If you have not taken these classes but are very motivated to take my class, then reading about these notions might suffice: I'd recommend the appendix to Manivel's book; Cox-Little-O'Shea's Ideals, Varieties, and Algorithms; Hatcher's Algebraic Topology book; and Harris' Algebraic Geometry (a first course). Familiarity with some representation theory may also be helpful.
Grades will be based on problem sets (50%), and a final paper (50%).
A main reference for the course will be Laurent Manivel's book ``Symmetric functions, Schubert polynomials and degeneracy loci," which you can rent or buy here. I will also draw from Fulton and Harris' ``Representation theory," Fulton's ``Young tableaux," Stanley's ``Enumerative Combinatorics 2," as well as some research articles.