Math 665 Fall 2016
Lectures: Tuesday and Thursday 11:30-1
Instructor: Thomas Lam, 2834 East Hall,
Office Hours: Tuesday and Thursday 2-3:30pm
No office hours on Oct 4 or Oct 6.
Familiarity with linear algebra is essential. Some experience with algebraic geometry and algebraic topology is helpful.
There will be problem sets roughly every two weeks.
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:
- Sep 6: Translation to Grassmannians
- Sep 8: Schubert varieties
- Sep 13: Plucker embedding, Plucker relations
- Sep 15: Class of a subvariety, Schubert classes
- Sep 20: Pieri rule
- Sep 22: Schur functions
- Sep 24: Jacobi Trudi formula, Bialternant formula
- Sep 27: Pieri rule for Schur functions, Relation between symmetric functions and cohomology
- Sep 29: Stembridge's "concise" Littlewood-Richardson rule
- Oct 4: Equivariant cohomology
- Oct 6 (Guest lecture by Zach Hamaker): RSK correspondence
- Oct 11: GKMCS graphs and conditions
- Oct 13: Equivariant cohomology of Grassmannians
- Oct 18: Fall Break
- Oct 20: Equivariant Schubert classes
- Oct 25: Equivariant Pieri rule
- Oct 27: Factorial Schur functions
- Nov 1: Tableaux and Jacobi Trudi formulae for factorial Schur functions
- Nov 3: Vanishing property of Factorial Schur functions
- Nov 8: Flag variety
- Nov 10: Equivariant cohomology of flag variety
- Nov 15: Divided different operators
- Nov 17: Equivariant Monk's rule
- Nov 22: Schubert polynomials
- Nov 29: Pipe dreams, nilCoxeter algebra
- Dec 1: Double Schubert polynomials
- Dec 6: Schubert classes for equivariant cohomology of flag variety
- Dec 8: Quantum or affine Schubert calculus
- Dec 13: Affine or quantum Schubert calculus