18.218    Topics in Combinatorics: Polytopes

Class meets: MWF 2 pm   Room 4-145

Instructor: Alexander Postnikov   apost at math   room 2-367

Course webpage: http://math.mit.edu/~apost/courses/18.218/

Synopsis:

The course will focus on convex polytopes and their connections with algebraic and enumerative combinatorics.

We'll start with a discussion of classical notions, such as, f-vectors and h-vectors of polytopes, volumes and Ehrhart polynomials, triangulations, etc. However, the main focus of the course will not be on the general theory of convex polytopes, but rather on special classes of polytopes that appear in combinatorics, algebra, Lie theory, algebraic geometry, etc.

We'll show how various algebraic and combinatorial structures (such as matroids, Schur polynomials, Littlewood-Richardson coefficients, RSK, etc.) can be interpreted in terms of convex polytopes, and how this "polytopal point of view" helps to understand these stuctures.

The list of possible topics (that will be covered in the course as time permits) is:

• Two combinatorial aspects of convex polytopes: face enumeration (f-vectors, h-vectors, etc.) and valuations (volumes, number of lattice points, etc.)

• Ehrhart theory: Ehrhart polynomials and reciprocity.

• Two Stanley's poset polytopes: the order polytope and the chain polytope.

• Permutohedra and zonotopes. Connections with hyperplane arrangements.

• Brion's theorem. Khovanskii-Pukhlikov's theorem. The algebra of polyhedra.

• Matroids and matroid polytopes. Gelfand-Goresky-MacPherson-Serganova's theorem. Connections with geometry of the Grassmannian and Schubert calculus.

• Positive Grassmannian and positroid polytopes.

• Generalized permutohedra.

• Associahedra, cyclohedra, graph-associahedra, and nestohedra. Gamma-vector and Gal's conjecture.

• Triangulations of products of simplices. Ardila-Develin's tropical oriented matroids. Root polytopes.

• Polytopes in Lie theory. Root systems.

• Chapoton-Fomin-Zelevinsky's generalized associahedra. From triangulations of n-gons to cluster algebras.

• Schur polynomials and Littlewood-Richardson coefficients. Gelfand-Tsetlin polytopes. Berenstein-Zelevinsky's polytopes and Knutson-Tao's honeycombs.

• Polytopes in algebraic geometry. Newton polytopes, fans, and toric varieties. Mixed volumes. Bernstein-Khovanskii-Kushnirenko's theorem.

• Gelfand-Kapranov-Zelevinsky's secondary polytopes. Discriminants and triangulations.

• Birkhoff polytope. Transportation polytopes.

• Kostant's partition function. Flow polytopes. Chan-Robbins-Yuen polytope.

• Box-spline theory. Zonotopal algebra.

• Affine Weyl group. Affine Coxeter arrangement and alcoved polytopes. Polypositroids.

• Polytopes beyond convex geometry: the positive Grassmannian, the amplituhedron of Arkani-Hamed and Trnka, Lam's Grassmann polytopes, etc.

• ...

Course Level: Graduate

The course should be accessible to first year graduate students.

Grading: The grade will be based on problem sets.

1. Problem Set 1 plus Additional Problems for Problem Set 1 (due Wednesday, March 2, 2016)

2. Problem Set 2 plus Additional Problems for Problem Set 2 (due Wednesday, May 4, 2016)

Lectures:

1. W 02/03/2016. Introduction. What is this course about? Two combinatorial aspects of polytopes: face enumeration and valuations.

2. F 02/05/2016. Basic definitions. Polytopes and polyhedra. Supporting faces. Simple polytopes.
   [Ziegler, Sections 1.1, 2.1, 2.2, 2.5, 3.1, 3.2]

3. M 02/08/2016. f- and h-vectors. f(q) = h(q+1). Dehn-Sommerville equations h_i = h_{d-i}. Example: permutohedron - Stirling and Eulerian numbers. Euler triangle. Minkowki sums and zonotopes.
   [Ziegler, Sections 8.3, 7.3]

4. W 02/10/2016. Faces of zonotopes and central hyperlane arrangements. Newton polytopes. Vandermonde determinant --> Permutohedron is a zonotope.
   [Ziegler, Section 7.3], [P1, Section 2]

5. F 02/12/2016. Graphical zonotopes. Spanning trees and forests. Unimodular zonotopes. Bases and idependent sets. Zonotopal tilings.
   [P1, Section 2], [Ziegler, Section 7.5]

6. Tuesday! 02/16/2016. Normal fan of a Minkowski sum. Regular and non-regular subdivisions of polytopes. Regular zonotopal tilings and affine hyperplane arrangements. Pseudoline arrangements.
   [Ziegler, Sections 7.1, 7.5]

7. W 02/17/2016. Pappus's theorem and example of non-regular tiling of 2n-gon. Zaslavsky's formula for the number of (all/bounded) regions in a hyperplane arrrangement. The intersection semi-lattice and Mobius function. The poset of independent subsets.
   [Ziegler, Example 7.28], [Stanley-arrangements, Lectures 1 and 2]

8. F 02/19/2016. Valuations (volumes, the number of lattice points) of polytopes which are not zonotopes. Examples: the permutohedron P(a_1,...,a_n), the hypersimplices. The Ehrhart polynomial of an integer polytope. The number of lattice points vs the volume. An introduction to Brion's formula.
   [P1, Section 2], [Barvinok, Lecture 1]

9. M 02/22/2016. Brion/Khovanski-Pukhlikov theory. The algebra of polyhedra. Heaviside functions. Local cones of polyhedra.
   [Barvinok, Lecture 2], [P1, Section 19 "Appendix"]

10. W 02/24/2016. Brion/Khovanski-Pukhlikov theory (cont'd). Example: formula for volume of the permutohedron P(a_1,...,a_n). Proof of the 1st version of Brion's formula.
    [P1, Sections 3 and 19]

11. F 02/26/2016. Brion/Khovanski-Pukhlikov theory (cont'd). The algebra of rational polyhedra.
    [Barvinok, Lectures 3 and 4], [P1, Section 19]

12. M 02/29/2016. Brion/Khovanski-Pukhlikov theory (cont'd). Formulas for numbers of lattice points and volumes of rational polytopes. The series q/(1-e^{-q}) and Bernoulli numbers.
    [Barvinok, Lectures 3 and 4], [P1, Section 19]

13. W 03/02/2016. Problems Set 1 discussion.

14. F 03/04/2016. Brion's formula vs Weyl's character formula (determinant formula for Schur polynomials). Combinatorial formula for Vol(P(\lambda)) in terms of permutations with given descent sets. Mixed Eulerian numbers.
    [P1, Sections 3, 16]

15. M 03/07/2016. Deformation cone of a simple polytope P. The volume polynomial V_P(z) and the generalized Ehrhart polynomials I_P(z).
    [PRW, Section 15 "Appendix"], [P1, Section 19 "Appendix"]

16. W 03/09/2016. Classical Euler-Maclaurin formula. Bernoulli formula. Todd operator. Khovanskii-Pukhlikhov's Euler-Maclaurin formula for polytopes.
    [P1, Section 19 "Appendix"]

17. F 03/11/2016. Proof of Euler-Maclaurin formula. Rado theorem and permutohedra. Generalized permutohedra.
    [P1, Sections 2, 6]

18. W 03/14/2016. Generalized permutohedra (cont'd). Submodular functions. Matroids and polymatroids.
    [P1, Section 6]

19. W 03/16/2016. Hypergraphs and hypergraph-permutohedra P_H (Minkowski sums of simplices). Hypergraphical generalizations of the chromatic polynomial, acylic orinetations, and Stanley's theorem.
    [P1, Section 6]

20. F 03/18/2016. Dragon marriage theorem. Hypertrees. Formulas for the volume and the number of lattic points of hypergraph-permutohedra in terms of hypertrees.
    [P1, Sections 5, 9, 11].

    03/21/2016 - 03/25/2016. no classes - Spring vacation

21. M 03/28/2016. Example of hypergraph-permutohedra: Stanley-Pitman polytope and parking functions. Mixed volumes of polytopes.
    [P1, Section 8.5, Example 9.7]

22. W 03/30/2016. Bernstein's theorem for the number of solutions of a system of algebraic equations in terms of mixed volume.
    [P1, Section 9]

23. F 04/01/2016. Proof of the formula for volume of hypergraph-permutohedra via Bernstein's theorem.
    [P1, Section 9]

24. M 04/04/2016. Polyhedral subdivisions. Mixed subdivisions of Minkowski sums. Fan arrangements.
    [P1, Section 14]

25. W 04/06/2016. Cayley trick. Examples of triangulations and mixed subdivisions.
    [P1, Section 14]

26. F 04/08/2016. Triangulations of the product of two simplices and spanning trees of the complete bipartite graph K_{m,n}.
    [P1, Section 12]

27. M 04/11/2016. Root polytopes and their triangulations. Example: The triangulation given by non-crossing alternating trees.
    [P1, Section 12]

28. W 04/13/2016. Trimmed hypergraph-permutohedra. Volume(root polytope) = number of lattice points of trimmed hypergraph-permutohedron.
    [P1, Sections 12, 14]

29. F 04/15/2016. Duality of hypergraph-permutohedra. Left- and right-degree vectors of bipartite trees. Example: Graphical zonotopes are dual to graphical matroids.
    [P1, Sections 11, 12]

    M 04/18/2016. no class - Patriots Day

30. W 04/20/2016.

31. F 04/22/2016.

32. M 04/25/2016.

33. W 04/27/2016.

34. F 04/29/2016.

35. M 05/02/2016.

36. W 05/04/2016.

37. F 05/06/2016.

38. M 05/09/2016.

39. W 05/11/2016.

Texts (books and papers):

• [Ziegler] Gunter M. Ziegler, Lectures on Polytopes, Springer, 1998. (A good textbook on the general theory of polytopes. We'll cover only a small portion of this book.)

• [Stanley-arrangements] Richard P. Stanley, An Introduction to Hyperplane Arrangements, IAS/Park City Mathematics Series, Vol. 13.

• [Stanley-posets] Richard P. Stanley, Two poset polytopes, Discrete Comput. Geom. 1:9-23 (1989).

• [GGMS] I. M. Gelfand, R. M. Goresky, R. D. MacPherson, V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Advances in Mathematics 63 (1987), no. 3, 301-316.

• [Fomin-Reading] Sergey Fomin, Nathan Reading, Root Systems and Generalized Associahedra, IAS/Park City Mathematics Series, Vol. 14, 2004.

• [GKZ] I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhauser, 1994.

• [Barvinok] Alexander Barvinok, Lattice Points, Polyhedra, and Complexity, IAS/Park City Mathematics Series, 2004.

• [DeConcini-Procesi] C. De Concini, C. Procesi, Topics in Hyperplane Arrangements, Polytopes and Box-Splines, Springer, 2010.

• [P1] A. Postnikov, Permutohedra, associahedra, and beyond, International Mathematics Research Notices 2009, no. 6, 1026-1106.

• [PRW] A. Postnikov, Victor Reiner, Lauren Williams, Faces of generalized permutohedra, Documenta Mathematica 13 (2008), 207-273.

• [LP] Thomas Lam, A. Postnikov, Alcoved Polytopes I, Discrete & Computational Geometry 38 no. 3 (2007) 453-478.

• [P2] A. Postnikov, Total positivity, Grassmannians, and networks.

last updated: February 5, 2016