18.218 M.I.T. Spring 2016
18.218 Topics in Combinatorics: Polytopes
Class meets:
MWF 2 pm
Room 4145

Instructor:
Alexander Postnikov
apost at math
room 2367
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, fvectors and hvectors 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,
LittlewoodRichardson 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 (fvectors,
hvectors, 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. KhovanskiiPukhlikov's theorem. The algebra of polyhedra.
 Matroids and matroid polytopes.
GelfandGoreskyMacPhersonSerganova's theorem.
Connections with geometry of the Grassmannian
and Schubert calculus.
 Positive Grassmannian and positroid polytopes.
 Generalized permutohedra.
 Associahedra, cyclohedra, graphassociahedra, and nestohedra.
Gammavector and Gal's conjecture.

Triangulations of products of simplices. ArdilaDevelin's tropical oriented matroids.
Root polytopes.
 Polytopes in Lie theory. Root systems.
 ChapotonFominZelevinsky's generalized associahedra.
From triangulations of ngons to cluster algebras.

Schur polynomials and LittlewoodRichardson coefficients.
GelfandTsetlin polytopes. BerensteinZelevinsky's polytopes
and KnutsonTao's honeycombs.
 Polytopes in algebraic geometry.
Newton polytopes, fans, and toric varieties.
Mixed volumes. BernsteinKhovanskiiKushnirenko's theorem.
 GelfandKapranovZelevinsky's secondary polytopes. Discriminants and triangulations.
 Birkhoff polytope. Transportation polytopes.
 Kostant's partition function. Flow polytopes.
ChanRobbinsYuen polytope.
 Boxspline theory. Zonotopal algebra.
 Affine Weyl group. Affine Coxeter arrangement and alcoved polytopes.
Polypositroids.
 Polytopes beyond convex geometry: the positive Grassmannian,
the amplituhedron of ArkaniHamed 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.
Problem Sets:

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

Problem Set 2
plus
Additional Problems for Problem Set 2
(due Wednesday, May 4, 2016)
Lectures:
 W 02/03/2016. Introduction. What is this course about? Two combinatorial aspects of polytopes: face enumeration and valuations.
 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]
 M 02/08/2016.
f and hvectors. f(q) = h(q+1).
DehnSommerville equations h_i = h_{di}.
Example: permutohedron  Stirling and Eulerian numbers.
Euler triangle.
Minkowki sums and zonotopes.
[Ziegler, Sections 8.3, 7.3]
 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]
 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]
 Tuesday! 02/16/2016.
Normal fan of a Minkowski sum. Regular and nonregular subdivisions of polytopes.
Regular zonotopal tilings and affine hyperplane arrangements.
Pseudoline arrangements.
[Ziegler, Sections 7.1, 7.5]
 W 02/17/2016.
Pappus's theorem and example of nonregular tiling of 2ngon.
Zaslavsky's formula for the number of (all/bounded) regions
in a hyperplane arrrangement. The intersection semilattice and Mobius
function. The poset of independent subsets.
[Ziegler, Example 7.28],
[Stanleyarrangements, Lectures 1 and 2]
 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]
 M 02/22/2016.
Brion/KhovanskiPukhlikov theory.
The algebra of polyhedra. Heaviside functions.
Local cones of polyhedra.
[Barvinok, Lecture 2],
[P1, Section 19 "Appendix"]
 W 02/24/2016.
Brion/KhovanskiPukhlikov 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]
 F 02/26/2016.
Brion/KhovanskiPukhlikov theory (cont'd).
The algebra of rational polyhedra.
[Barvinok, Lectures 3 and 4], [P1, Section 19]
 M 02/29/2016.
Brion/KhovanskiPukhlikov theory (cont'd).
Formulas for numbers of lattice points and volumes of rational
polytopes. The series q/(1e^{q}) and Bernoulli numbers.
[Barvinok, Lectures 3 and 4], [P1, Section 19]
 W 03/02/2016.
Problems Set 1 discussion.
 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]
 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"]
 W 03/09/2016.
Classical EulerMaclaurin formula.
Bernoulli formula.
Todd operator. KhovanskiiPukhlikhov's EulerMaclaurin formula
for polytopes.
[P1, Section 19 "Appendix"]
 F 03/11/2016.
Proof of EulerMaclaurin formula.
Rado theorem and permutohedra. Generalized permutohedra.
[P1, Sections 2, 6]
 W 03/14/2016.
Generalized permutohedra (cont'd).
Submodular functions. Matroids and polymatroids.
[P1, Section 6]
 W 03/16/2016.
Hypergraphs and hypergraphpermutohedra P_H (Minkowski sums of simplices).
Hypergraphical generalizations of the chromatic polynomial, acylic orinetations,
and Stanley's theorem.
[P1, Section 6]
 F 03/18/2016.
Dragon marriage theorem. Hypertrees. Formulas for the volume and
the number of lattic points of hypergraphpermutohedra in terms of hypertrees.
[P1, Sections 5, 9, 11].
03/21/2016  03/25/2016. no classes  Spring vacation
 M 03/28/2016.
Example of hypergraphpermutohedra: StanleyPitman polytope and parking
functions. Mixed volumes of polytopes.
[P1, Section 8.5, Example 9.7]
 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]
 F 04/01/2016.
Proof of the formula for volume of hypergraphpermutohedra via Bernstein's
theorem.
[P1, Section 9]
 M 04/04/2016.
Polyhedral subdivisions. Mixed subdivisions of Minkowski sums.
Fan arrangements.
[P1, Section 14]
 W 04/06/2016.
Cayley trick. Examples of triangulations and mixed subdivisions.
[P1, Section 14]
 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]
 M 04/11/2016.
Root polytopes and their triangulations.
Example: The triangulation given by noncrossing alternating trees.
[P1, Section 12]
 W 04/13/2016.
Trimmed hypergraphpermutohedra. Volume(root polytope) = number of lattice points of trimmed hypergraphpermutohedron.
[P1, Sections 12, 14]
 F 04/15/2016.
Duality of hypergraphpermutohedra. Left and rightdegree vectors
of bipartite trees. Example: Graphical zonotopes are dual to graphical matroids.
[P1, Sections 11, 12]
M 04/18/2016. no class  Patriots Day
 W 04/20/2016.
 F 04/22/2016.
 M 04/25/2016.
 W 04/27/2016.
 F 04/29/2016.
 M 05/02/2016.
 W 05/04/2016.
 F 05/06/2016.
 M 05/09/2016.
 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.)
 [Stanleyarrangements] Richard P. Stanley,
An
Introduction to Hyperplane Arrangements,
IAS/Park City Mathematics Series, Vol. 13.
 [Stanleyposets] Richard P. Stanley,
Two poset polytopes, Discrete Comput. Geom. 1:923 (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, 301316.
 [FominReading] 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.
 [DeConciniProcesi] C. De Concini, C. Procesi,
Topics
in Hyperplane Arrangements, Polytopes and BoxSplines, Springer, 2010.
 [P1] A. Postnikov,
Permutohedra, associahedra, and beyond,
International Mathematics Research Notices 2009, no. 6, 10261106.
 [PRW] A. Postnikov, Victor Reiner, Lauren Williams,
Faces of generalized permutohedra,
Documenta Mathematica 13 (2008), 207273.
 [LP] Thomas Lam, A. Postnikov,
Alcoved Polytopes I,
Discrete & Computational Geometry 38 no. 3 (2007) 453478.
 [P2]
A. Postnikov,
Total positivity,
Grassmannians, and networks.
last updated: February 5, 2016