18.218 MIT Spring 2020
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 Sets:
-
Problem Set 1 (due Friday, March 6), several problems
added on 02/03/2020.
Lectures:
(with suggested additional reading)
-
M 02/03/2020. Introduction.
[Ziegler, Section 0]
-
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]
- 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]
- M 02/10/2020.
Normal fans of polytopes. Minkowski sums.
[Ziegler, Sections 7.1, 7.2]
- 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]
- 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.
- Tuesday (!) 02/18/2020. Regular triangulations and
regular zonotopal tilings.
[Ziegler, Sections 7.5, 9.1]
- 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]
- F 02/21/2020. Generic affine arrangements. Unimodular
vector configurations and zonotopes. Graphical arrangements
are unimodular.
- M 02/24/2020. Intersection poset and
characteristic polynomial of an arrangement. Zaslavsky's theorem.
[Stanley,
Lectures 1 and 2]
- 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]
- F 02/28/2020.
Crosscut theorem for lattices. Möbius algebra. Proof of
Whitney's and Zaslavsky's theorems.
[Stanley,
Lecture 2]
- M 03/02/2020. Finite field method.
Example: Shi arrangement.
[Stanley,
Lecture 5]
- W 03/04/2020. Properties of the characteristic polynomial of an
arrangement: alternating coefficients, constant term, etc.
- F 03/06/2020.
Problem Set 1 due:
pdf file
Cohomology of complements of complex hyperplane arrangements.
Geometric lattices.
[Stanley,
Lecture 3]
- M 03/09/2020. Discussion of Problem Set 1:
students' presentations of solutions.
- 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.
- M 03/30/2020.
- W 04/01/2020.
- F 04/03/2020.
- M 04/06/2020.
- W 04/08/2020.
- F 04/10/2020.
- M 04/13/2020.
- W 04/15/2020.
- F 04/17/2020.
M 04/20/2020.
Patriots' day - vacation.
- W 04/22/2020.
- F 04/24/2020.
- M 04/27/2020.
- W 04/29/2020.
- F 05/01/2020.
- M 05/04/2020.
- W 05/06/2020.
- F 05/08/2020.
- 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:
- [P] A. Postnikov,
Permutohedra, associahedra, and beyond,
International Mathematics Research Notices 2009, no. 6, 1026-1106.
Related courses from past years:
last updated: March 26, 2020