MIT   spring 2008

18.319     combinatorics and geometry

PROBLEM SET 2 IS AVAILABLE

class meets:   Tuesday and Thursday, 2:30 - 4 pm, room 2-136

instructor:   Alexander Postnikov

office hour: Tuesday 4-5pm.

description:
Connections between combinatorics and geometry (and algebra). Discussion of combinatorial problems that arise in algebraic geometry, convex geometry, and algebraic topology. Topics include toric varieties, polytopes and fans, hyperplane arrangements, triangulations and tilings, matroids, topological combinatorics, Schubert calculus.

course level:   graduate

books: (the students are not required to buy these books)
-  Geometric Combinatorics, E. Miller, V. Reiner, B. Sturmfels, eds., IAS/Park City Mathematics Series, vol. 13, 2007,
    including R. Stanley's lecture notes on hyperplane arrangements.
-  Lectures on Polytopes, G. Ziegler, Springer, 1995.
-  Introduction to Toric Varieties, W. Fulton, Princeton University Press, 1993.
-  Discriminants, Resultants, and Multidimensional Determinants, I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Birkhauser, 1994.

problem sets:

• Problem Set 1 --- due April 1
• Problem Set 2 --- due May 15

lectures:

• T 02/05/08 Lecture 1. Introduction. Polytopes. Face numbers, volumes and lattice points, Pick's formula, Ehrhart polynomial.

• R 02/07/08 Lecture 2. Polar duality. Examples of polytopes: simplex, hypercube, crosspolytope, hypersimplices and Eulerian numbers, permutohedron.

• T 02/12/08 Lecture 3. Birkhoff polytope. Transportation polytopes. Chan-Robbins polytope and flow polytopes. Kostant partition function.

• R 02/14/08 Lecture 4. Counting lattice points and volumes.

• T 02/19/08 - no class - Monday schedule

• R 02/21/08 Lecture 5. Counting lattice points (cont'd). The algebra of rational polyhedra. Brion's formula.

  Additional reading for lectures 4-5: Barvinok's lectures in "Geometric Combinatorics."

• T 02/26/08 Lecture 6. Counting lattice points (cont'd). Generalized Ehrhart (quasi)-polynomials. Ehrhart-Macdonald reciprocity.

• R 02/28/08 Lecture 7. Matroids and matroid polytopes.

• T 03/04/08 Lecture 8. Grassmannian. Matroid strata. Schubert cells.

  Additional reading for lectures 7-8: Gelfand, Goresky, MacPherson, Servanova "Combinatorial geometries, convex polyhedra, and Schubert cells," Advances of Mathematics 63, no. 3 (1987), 301-316.

• R 03/06/08 Lecture 9. Hyperplane arrangements.

  Additional reading: Stanley's lectures in "Geometric Combinatorics."

• T 03/11/08 Lecture 10. Hyperplane arrangements (cont'd): Finite field method.

• R 03/13/08 Lecture 11. Catalan arrangement. Shi arrangement. Parking functions.

• T 03/18/08 Lecture 12. guest lecture by D. Kleitman.

• R 03/20/08 Lecture 13. guest lecture by I. Tyomkin.

• T 03/25/08 - spring break

• R 03/27/08 - spring break

• T 04/01/08 Lecture 14. Applications of the finite field method.

• R 04/03/08 Lecture 15. Parking functions and distance enumerator.

• T 04/08/08 Lecture 16. Root systems.

• R 04/10/08 Lecture 17. Coxeter groups. Reduced decompositions.

• T 04/15/08 Lecture 18. Coxeter elements. Root system Catalan combinatorics.

• R 04/17/08 Lecture 19. Associahedron and its realizations.

• T 04/22/08 - no class - partiots day

• R 04/24/08 Lecture 20. Momement graphs (aka GKM-graphs) and equivariant cohomology.

• T 04/29/08 Lecture 21. Moment graphs (cont'd).

• R 05/01/08 Lecture 22. Morse theory for moment graphs.

• T 05/06/08 Lecture 23. Schubert calculus. Schubert polynomials.

• R 05/08/08 Lecture 24. Schubert calculus (cont'd).