18.315   MIT   Fall 2014

Combinatorial Theory: Schubert Calculus and Beyond

Lectures: Wednesday and Friday, 1-2:30 pm, room E17-139

Instructor: Alexander Postnikov

Synopsis: This is a graduate-level course in combinatorial theory. The content varies year to year.

This year the course is about combinatorics related to the Grassmannian and flag varieties. The main players are Young tableaux, Schur polynomials and Schubert polynomials, Bruhat order, Littlewood-Richardson rule and its variations, wiring diagrams and reduced decompositions, matroids, the positive Grassmannian and positroids, quantum cohomology of the Grassmannian, quantum Schubert polynomials... We start with classical topics and (as time allows) move to topics of more recent research.

The class will be accessible to first year graduate students.

Course Level: Graduate

Recommended textbooks:
*  W. Fulton: Young Tableaux, Cambridge University Press, 1997.
*  R. P. Stanley: Enumerative Combinatorics, Volumes 1 and 2, Cambridge University Press, 1996 and 1999.
*  L. Manivel: Symmetric Functions, Schubert Polynomials and Degeneracy Loci, SMF/AMS Texts and Monographs, Vol 6 and Cours Specialises Numero 3, 1998.

Grading: Based on several problem sets

Problem Sets:

Lecture Notes (by Carl Lian) http://web.mit.edu/clian/www/315_notes.pdf

Additional Reading:

last updated: November 21, 2014