Topics in Combinatorics: Polytopes and Hyperplane Arrangements

This is the old 18.218 course webpage! For the information about (virtual) lectures starting March 30, 2020, see     the new course webpage

Class meets: MWF 2pm   Room 4-237

Instructor: Alexander Postnikov

Office hours: Mondays 1-2pm or by appointment (in Room 2-367)

Grader: Yibo Gao

Course webpage: math.mit.edu/18.218

Announcements:

Starting Monday, March 30, all lectures will be given virtually via zoom until further notice. The lectures will be given during the regular class time (MWF 2-3pm). I'll start each zoom session 10-15 min before the class starts, so that you can join the meeting a little bit before the lecture and make sure that your connection works.

The lectures will be recorded and posted on this webpage. So, if you cannot attend virtural lectures in real time, you can watch a recording later. But I encourage everbody to try to join the lectures in real time, if you can. Your participation in class and questions are always welcomed!

Office hours will also be held virtually over zoom.

Solutions for forthcoming problem sets will be handed in and returned electronically.

I will send announcements with instructions on how to connect to zoom by email.

If you have not received an email from me recently, please write me and I'll add your email address to the mailing list.

More technical details to follow...

Synopsis:

We'll focus on convex polytopes and hyperplane arrangements, their combinatorial, enumerative, and geometric properties.

We'll discuss classical notions such as f-vectors and h-vectors of polytopes, Ehrhart polynomials, fans, triangulations, intersection lattices and characteristic polynomials of hyperplane arrangements, etc.

We'll show how many constructions from algebraic combinatorics (e.g., Young tableaux, hook length formula, Schur polynomials, Littlewood-Richardson rule, Robinson-Schensted-Knuth correspondence, etc.) can be interpreted and understood in terms of volumes and integer lattice points of polytopes.

We'll explain how to count regions of hyperplane arrangements using Möbius inversion and the finite field method.

We'll pay special attention to links between polytopes and arrangements and other areas:

• algebraic geometry: Newton polytopes, toric varieties, discriminants, triangulations and Gelfand-Kapranov-Zelevinsky's secondary polytopes, Brion's formula and Khovanskii-Pukhlikov's Euler-MacLaurin formula, tropical geometry.
• Lie theory: permutohedra, Coxeter arrangements, Shi and Catalan arrangements, root polytopes, Kostant's partition function and flow polytopes, Gelfand-Tsetlin bases and polytopes, crystals and Berenstein-Zelevinsky's polytopes.
• graph and matroid theories, optimization: graphical arrangements, Tutte polynomial, matroid polytopes, polymatroids, submodular functions, Birkhoff polytope and transportation polytopes.
• topology: braid groups and arrangements, cohomology of hyperplane arrangements and Orlik-Solomon algebras, knot invariants via polytopes.
• cluster algebras: associahedra and Chapoton-Fomin-Zelevinsky's generalized associahedra.
• approximation theory: Dahmen-Michelli box-splines, vector partition functions, zonotopal algebra.
• physics: positroids, cosmological polytopes and Feynman diagrams, amplituhedra...

Course Level: Graduate

The course should be accessible to first year graduate students, and even some advanced undergraduate students.

The main prerequisite for the course is linear algebra. We'll discuss some combinatorial constructions related to algebraic geometry, representation theory, and topology. So some familiarity with these areas of math would be helpful, but not required.

Grading: The grade will be based on several problem sets.

• Problem Set 1 (due Friday, March 6), several problems added on 02/03/2020.

Lectures: (with suggested additional reading)

1. M 02/03/2020. Introduction.
   [Ziegler, Section 0]

2. W 02/05/2020. Basic definitions: Convex sets, polytopes, polyhedra, supporting faces. Example: Faces of the permutohedron and ordered set partitions.
   [Ziegler, Sections 1.1, 1.2, 2.1, 2.2]

3. F 02/07/2020. Simple polytopes. f-vectors and h-vectors. Dehn-Sommerville equations. Example: The permutohedron, Stirling numbers of 2nd kind, and Eulerian numbers.
   [Ziegler, Sections 2.5, 8.3]

4. M 02/10/2020. Normal fans of polytopes. Minkowski sums.
   [Ziegler, Sections 7.1, 7.2]

5. W 02/12/2020. Zonotopes. Central hyperplane arrangements. Newton polytopes. Kushnirenko's theorem. The permutohedron and the Vandermonde determinant.
   [Ziegler, Section 7.3], [Stanley, Section 1.1], [P, Section 2, Theorem 9.9]

6. F 02/14/2020. Graphical zonotopes. Spanning trees and forests. Zonotopal tilings. Bases and independent sets. Affine hyperplane arrangements.
   [P, Section 2], [Ziegler, Section 7.5]

   M 02/17/2020. President's day - no classes.

7. Tuesday (!) 02/18/2020. Regular triangulations and regular zonotopal tilings.
   [Ziegler, Sections 7.5, 9.1]

8. W 02/19/2020. Regular zonotopal tilings are dual to affine hyperplane arrangements. Pseudoline arrangements. Pappus theorem and example of non-regular tiling.
   [Ziegler, Section 7.4, Example 7.28]

9. F 02/21/2020. Generic affine arrangements. Unimodular vector configurations and zonotopes. Graphical arrangements are unimodular.

10. M 02/24/2020. Intersection poset and characteristic polynomial of an arrangement. Zaslavsky's theorem.
    [Stanley, Lectures 1 and 2]

11. W 02/26/2020. Graphical arrangements: characterictic polynomial = chromatic polynomial. Deletion-contraction for graphs and deletion-restriction for hyperplane arrangements. Whitney's theorem.
    [Stanley, Lecture 2]

12. F 02/28/2020. Crosscut theorem for lattices. Möbius algebra. Proof of Whitney's and Zaslavsky's theorems.
    [Stanley, Lecture 2]

13. M 03/02/2020. Finite field method. Example: Shi arrangement.
    [Stanley, Lecture 5]

14. W 03/04/2020. Properties of the characteristic polynomial of an arrangement: alternating coefficients, constant term, etc.

15. F 03/06/2020. Problem Set 1 due: pdf file
    Cohomology of complements of complex hyperplane arrangements. Geometric lattices.
    [Stanley, Lecture 3]

16. M 03/09/2020. Discussion of Problem Set 1: students' presentations of solutions.

17. W 03/11/2020. Matroids and geometric lattices.

    03/13/2020--03/27/2020. --- no classes ---

    After the break, the lectures will resume virtually via zoom.

18. M 03/30/2020.

19. W 04/01/2020.

20. F 04/03/2020.

21. M 04/06/2020.

22. W 04/08/2020.

23. F 04/10/2020.

24. M 04/13/2020.

25. W 04/15/2020.

26. F 04/17/2020.

    M 04/20/2020. Patriots' day - vacation.

27. W 04/22/2020.

28. F 04/24/2020.

29. M 04/27/2020.

30. W 04/29/2020.

31. F 05/01/2020.

32. M 05/04/2020.

33. W 05/06/2020.

34. F 05/08/2020.

35. M 05/11/2020.

Texts:

• [Ziegler] Günter M. Ziegler, Lectures on Polytopes, Springer, 1995: book pdf.

• Geometric Combinatorics (E. Miller, V. Reiner, and B. Sturmfels, eds.), IAS/Park City Mathematics Series, vol. 13, AMS, 2007; including the following chapters:
  • [Barvinok] Alexander Barvinok, Lattice Points, Polyhedra, and Complexity.

  • [Stanley] Richard P. Stanley, An Introduction to Hyperplane Arrangements.

  • [Fomin-Reading] Sergey Fomin, Nathan Reading, Root Systems and Generalized Associahedra.

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

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

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

last updated: March 26, 2020