**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

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:**

- September 5: Introduction, Grassmannians and Schubert varieties

- September 10: General statements about cohomology and intersection theory

- September 12: Duality and Pieri rules

- September 17: Symmetric functions; Schur functions, Jacobi-Trudi, bialternant formula

- September 19: Pieri rule and Littlewood-Richardson rule for Schur function

- September 24: Equivariant cohomology (Borel construction, properties)

- September 26: Goresky-Kottwitz-Macpherson and Chang-Skjelbred theory

- October 1: Equivariant Schubert calculus for Grassmannians

- October 3: Algebraic construction of equivariant Schubert classes

- October 8: Equivariant Pieri rule

- October 10: Factorial Schur polynomials, Jacobi-Trudi formula, tableaux formula

- October 15: no class (Holiday)

- October 17: Flag varieties, Schubert cells, Schubert varieties

- October 22: Equivariant cohomology of flag varieties, localization formulae

- October 24: Equivariant Monk's formula, Borel homomorphism

- October 29: Schubert polynomials

- October 31: Double Schubert polynomials, nilCoxeter algebra

- November 5: Pipe dreams, back stable Schubert polynomials

- November 7: no class (Thomas is out of town)

- November 12: Stanley symmetric functions, Edelman-Greene insertion

- November 14: coproduct formula, back stable double Schubert polynomials

- November 19: bumpless pipe dreams I

- November 21: bumpless pipe dreams II

- November 26: infinite and affine Grassmannians and flag varieties

- November 28: no class (Thanksgiving)

- December 3: no class (cancelled due to snow)

- December 5: affine Schubert calculus I

- December 10: affine Schubert calculus II

A previous version of this class is here.