18.S996 Fall 2019
Schubert Calculus

Lectures: Tuesday and Thursday 9:30-11, room 2-132

Instructor: Thomas Lam, room 2-169, tfylam@mit.edu

Course description: This is an introductory course in Schubert calculus, a branch of enumerative geometry with deep connections to combinatorics and representation theory. A typical problem in Schubert calculus is: fix four lines in three-dimensional space; how many other lines intersect all four? Some of the topics we may study include: cohomology of Grassmannians and flag varieties, Schur functions, Schubert polynomials, Littlewood-Richardson rules, equivariant Schubert calculus, K-theory Schubert calculus, affine Schubert calculus, quantum Schubert calculus, back stable Schubert calculus

Prerequisites:
Familiarity with linear algebra is essential. Some experience with algebraic geometry and algebraic topology is helpful.

Exams: There will be no exams.

Homework: There will be three to four problem sets. Homework solutions should be typed in LaTeX and submitted at the beginning of class. Late homework grades are penalized 20% per late day.
At the front of your homework solution, please acknowledge any books, online sources, etc. consulted, and indicate other students you worked with on the homework.

Term paper: Students seeking a grade will write a term paper. This can be purely expository or it can involve exploring an open problem. A list of possible topics will be made available half way through the semester.

References:
Cohomology of Grassmannians and flag varieties:
W. Fulton, Young tableaux, Cambridge University Press, 1997.
L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, AMS, 2001.
Equivariant cohomology of Grassmannians:
Knutson and Tao, Puzzles and (equivariant) cohomology of Grassmannians.
Goresky, Kottwitz, and Macpherson, Equivariant cohomology, Koszul duality, and the localization theorem.
Tymoczko, An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and Macpherson
Anderson, Introduction to Equivariant Cohomology in Algebraic Geometry

List of lectures:

A previous version of this class is here.