18.217   Combinatorial Theory:
Combinatorics and Geometry

Fall 2024, MIT

Instructor: Alex Postnikov
Grader: Sveta Gavrilova
Time: MWF 1-2 pm
Place: Room 2-190

Description:

We will discuss relations between combinatorics and geometry. Here "geometry" stands for "convex geometry", "algebraic geometry", "tropical geometry", "topology", and "metric geometry".

The list of topics includes:

• Convex polytopes, polyhedral fans, and hyperplane arrangements
• Graphs, hypergraphs, matroids and polymatroids
• Triangulations and polyhedral tilings
• F-vectors, h-vectors, and gamma-vectors
• Volumes and lattice points of polytopes, Ehrhart theory, Brion's formula
• Toric varieties and Newton polytopes
• Mixed volumes and Bernstein-Khovanskii-Kushnirenko theorem
• Gelfand-Kapranov-Zelevinsky's secondary polytopes and Billera-Sturmfels' fiber polytopes
• Zaslavsky's formula for hyperplane arrangements and Mobius function
• Braid arrangements, Coxeter arrangements, Arnold-Orlik-Solomon algebras
• Grassmannians and flag manifolds, Schur and Schubert polynomials
• Kostant's partition function and flow polytopes
• Gelfand-Tsetlin bases, Berenstein-Zelevinsky's polytopes
• Tutte polynomials and knot invariants
• Hypersimplices, permutoheda, associahedra, and Birkhoff polytopes
• Positroids, polypositroids, and cluster algebras
• Dahmen-Michelli's box-splines and zonotopal algebra
• etc.

Course Level: The course should be accessible to first year graduate students.

Grading: Based on several problem sets.

Problem Set 1  due Friday, October 18, 2024

Problem Set 2  due Friday, December 6, 2024

Lectures: with suggestions for additional reading and lecture notes by Ilani Axelrod-Freed

1. W 09/04/24. Introduction. Combinatorics and geometry. Main players: graphs, polytopes, hyperplane arrangements, matroids, hypergraphs, polymatroids, etc. Polytopes: f-vectors and h-vectors, volumes, and the numbers of lattice points. Example: The permutohedron and the Stirling numbers, the Eulerian numbers, the numbers of labelled trees and labelled forests.
   [Ziegler, Section 0]
   Lecture Notes

2. F 09/06/24. Basic definitions for polytopes: convex sets, the convex hull, polytopes, and polyhedra. The fundamental theorem of convex polytopes: polytopes and exactly bounded polyhedra. Polar duality. Supporting hyperplanes and supporting faces. Example: Faces of the permutohedron are products of smaller permutohedra.
   [Ziegler, Sections 1.1, 1.2, 2.1, 2.2]
   Lecture Notes

3. M 09/09/24. Simple polytopes vs simplicial polytopes. Examples: the tetrahedron, the cube, and the octahedron. Is ∅ a face? Face numbers of the polar dual polytopes: f_i(P) = f_d-i-1(P^*). A combinatorial interpretation of the h-vector in terms of in-degrees of vertices in 1-skeleton of P. Corollary: Dehn-Sommerville equations: h_i(P) = h_d-i(P).
   [Ziegler, Sections 2.5, 8.3]
   Lecture Notes

4. W 09/11/24. Minkowski sums and differences. Zonotopes. Graphical zonotopes. Newton polytopes. Vandermonde determinant and the permutohedron.
   [Ziegler, Section 7.3]
   Lecture Notes

5. F 09/13/2024. Acyclic orientations of graphs and graphical zonotopes. Hypergraphical polytopes. Example: the associahedron. Binary trees and binary search labelling.
   [Post, Section 8.2]
   Lecture Notes

6. M 09/16/2024. Examples of h-vectors of simple polytopes: simplices, cubes, associahera, and permutohedra. Narayana numbers and Eulerian numbers. Normal fans of polytopes. Normal fans of Minkowski sums. Hyperplane arrangements. Graphical hyperplane arrangements.
   [Ziegler, Section 7.1]
   Lecture Notes

7. W 09/18/2024. Zonotopal tilings. Bases and independent sets of vector configurations. Zonotopes as projections of cubes; and zonotopal tilings as projections of collections of faces of cubes.
   [Ziegler, Section 7.5]
   Lecture Notes

8. M 09/23/2024. Volumes of Minkowski sums of dilated polytopes. Volumes of graphical zonotopes. Unimodularity of graphical vector configurations. Regular zonotopal tilings.
   Lecture Notes

9. W 09/25/2024. Mixed subdivisions of Minkowski sums and fan arrangements.
   Lecture Notes

10. F 09/27/2024. Polyhedral subdivisons and triangulations. Examples: a non-regular triangulation, non-regular zonotopal tilings from Pappus' and Desargues' theorems.
    [Ziegler, Sections 5.1, 7.4, Example 7.28, Section 9.1]
    Lecture Notes

11. M 09/30/2024. Mixed volumes, Bernstein-(Kushnirenko-Khovanskii) theorem.
    Lecture Notes

12. W 10/02/2024. Ehrhart polynomial. Ehrhart reciprocity. Chromatic polynomial. Stanley's theorem on acyclic orientations.
    Lecture Notes

13. F 10/04/2024. Chromatic and Ehrhart polynomials (cont'd). Intersection poset of a hyperplane arrangement.
    Lecture Notes

14. M 10/07/2024. Counting regions of hyperplane arrangements. Motivating examples: braid arrangements, Shi arrangement, Catalan arrangement. Mobius function of a poset. Characteristic polynomial. Zaslavsky's theorem.
    [Stanley-arrangements, Lectures 1 and 2]
    Lecture Notes

15. W 10/09/2024. Hyperplane arrangements (cont'd). Deletion-restriction recurrence. Whitney's theorem. Rota's Crosscut theorem.
    [Stanley-arrangements, Lecture 2]
    Lecture Notes

16. F 10/11/2024. Hyperplane arrangements (cont'd). Proof of Crosscut theorem. Mobius algebra of a lattice. Affine arrangements with generically translated hyperplanes.
    Lecture Notes

17. W 10/16/2024. Finite field method. Example: Catalan arrangement.
    [Stanley-arrangements, Lecture 5, part 5.4]
    Lecture Notes

18. F 10/18/2024. Shi arrangement. Linial arrangement. Alternating trees and local-binary-search trees.
    [Stanley-arrangements, Lecture 5, part 5.2]
    Lecture Notes

19. M 10/21/2024. Braid, Catalan, Shi, Linial arrangements and 4 kinds of labelled binary trees. Extended Catalan and extended Shi arrangements.
    Lecture Notes

20. W 10/23/2024. Discussion of problem set.

21. F 10/25/2024. Orlik-Solomon algebra.
    Lecture Notes
    Lecture Notes from 2020 (second half)

22. M 10/28/2024. Orlik-Solomon algebra (cont'd). No-broken-circuit (NBC) basis.
    [Stanley-arrangements, Lecture 4, part 4.1]
    Lecture Notes

23. W 10/30/2024. Arnold-Orlik-Solomon algebra and two games on graphs. Volumes of root polytopes.
    Lecture Notes

24. F 11/01/2024. Brion's formula.
    Lecture Notes from 2020 (second half)

25. M 11/04/2024. Brion's formula (cont'd). Space of rational polyhedra. Ruled polyhedra.
    [Post, Section 19: Appendix]
    Lecture Notes
    Lecture Notes from 2020

26. W 11/06/2024. Brion's formula (cont'd).
    Lecture Notes
    Lecture Notes from 2020

27. F 11/08/2024. Volume of permutohedra. G-Schur polynomials.
    Lecture Notes

28. W 11/13/2024. Second formula for volume of permutohedra.
    [Post, Section 3: Descents and divided symmetrization]
    Lecture Notes
    Lecture Notes from 2020

29. F 11/15/2024. Hypersimplices and the Eulerian numbers. Calculating volumes of hypersimplices using inclusion-exclusion and by constracting their triangulations. Triangulation of the 2nd hypersimplex and thrackles.
    Lecture Notes
    Lecture Notes from 2020 (the second half)

30. M 11/18/2024. Short presentations by Ilani Alxelrod-Freed, Elisabeth Bullock, Ryota Inagaki, and Dora Woodruff
    on binary tree bijections, mixed volumes of hypersimplices, (higher) Bruhat orders and zonotopal tilings.
    Lecture Notes

31. W 11/20/2024. Guest lecture by Colin Defant: Posets, Coxeter Groups, Root Systems, etc.
    Slides

32. F 11/22/2024. Guest lecture by Colin Defant: Random Combinatorial Billiards.
    Slides

Additonal reading materials:

• [Ziegler] Günter M. Ziegler, Lectures on Polytopes (Graduate Texts in Mathematics, 152), Springer.

• [Post] A. Postnikov, Permutohedra, associahedra, and beyond, IMRN 2009, no. 6, 1026-1106; arXiv:math/0507163

• [Stanley-arrangements] Richard P. Stanley, An Introduction to Hyperplane Arrangements, webpage with pdf file.

Related courses taught in the past:

• Lauren Williams: Math 287Z: Geometric and Topological Combinatorics, Harvard, Spring 2024.

• Alex Postnikov: 18.218 Topics in Combinatorics: Polytopes and Hyperplane Arrangements, MIT, Spring 2020.

• Alex Postnikov: 18.218 Topics in Combinatorics: Polytopes, MIT, Spring 2016.

• Richard P. Stanley: 18.315 Combinatorial Theory: Hyperplane Arrangements, MIT, Fall 2004.

This webpage will be updated periodically. All information related to the course, including problem sets, will be posted here.

last updated: November 26, 2024