Math 580A
Matroids and Hyperplane Arrangements

Fall 2019 – Spring 2020

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Office hours by appointment.

Arrangements of hyperplanes, mainly in real and complex space.
Dually, arrangements of points and positive span.
The matroid theory behind hyperplane arrangements.
Matroid invariants for arrangements.
Arrangements described by graphs, signed graphs, and gain graphs.
The commutative algebra of arrangements (the Orlik-Solomon algebra) and cohomology of the complex complement.

Richard Stanley's Park City lectures, "An Introduction to Hyperplane Arrangements", in Ezra Miller, Victor Reiner, and Bernd Sturmfels, eds., Geometric Combinatorics (Park City, 2004), IAS/Park City Math. Ser., Vol. 13, pp. 389–496. American Mathematical Society, Providence, R.I., 2007. MR 2383123. Zbl 1136.52009.
You can download it from this Web site. We will omit Lecture 6 and pass lightly over much of Lecture 5. Do not overlook Stanley's errata and addenda and mine.
Elementary matroid theory: Stanley's lecture notes are enough.
Polynomial invariants of matroids and arrangements: Stanley's lecture notes are enough.
Graphs: Stanley's lecture notes.
The Orlik-Solomon algebra: Selections from Peter Orlik, Introduction to Arrangements, CBMS Regional Conf. Ser. Math., 72. American Math. Soc., Providence, R.I., 1989. MR 90i:32018. Zbl 722.51003.
Our lecture notes on gain graphs and arrangements.

I will assign and grade some written homework. [This turned out to be taking notes on gain graphs and their arrangements. No paper was required.]
Staple multiple sheets (paper clips and paper folds don't work).
Write your solutions by yourself in your own words. You may discuss the problems before writing up the solution, but you must write in your own words and formulas.

There will be no written exams. There will be periodic oral exams.

Example 1.2: This is not the "sweep hyperplane" method. The sweep line method (in general dimension, the "sweep hyperplane" method) is an entirely different method.
Prop. 1.1, proof: "By linear algebra, ..., then $H \cap x = x$ or $H \cap x = \emptyset$ or $\dim(x) - \dim(H \cap x) = 1$, ..."
Geometry of relative vertices: proof that they are translates of each other.
Lecture 5, page 3 [p. 63], middle: A_n−1 is not B_n. The former is a set of vectors. The latter is the dual hyperplane arrangement, for which better notation is A_n*. When A is a set of vectors, I will write A* for the dual hyperplane arrangement.

Here are the fully revised (up to now) combined lecture notes in PDF. The LaTeX and the separate sections are linked from a separate page, and they are not kept up to date.

We have a simple matroid M of rank l. Find the algebra A(M) when
- l = 1 (easy).
- l = 2 (cute; it's in Orlik and we do it in class).
- l = 3 (a serious exercise, begun in class).
l = 4 is far too complicated for a mere exercise.
?

Warning: This list is not up to date. See the emails for full assignments.

F 8/23: §§1.1-1.2.
M 8/26: Reread §§1.1-1.2 carefully. Start §1.3.
Late Sept.: Finish reading Lecture 2.
W 10/2: §3.1.
F 10/4: §3.2.
F 1/24: Solve the characteristic polynomial of the hollow Catalan arrangement, C_n° = A[{±1}K_n].
Optional: Solve that of the extended Catalan arrangement, C_n,k = A[{0,±1,...,±k}K_n].
(M 1/27, i.e.) W 1/29: §§5.1-5.2.
M 2/3: §§5.4, 5.7, and 5.3.
W 2/5: §§5.3 and parts of 5.5 that are about arrangements (starting at the bottom of page 461 in the book, page 77 in the separate file), not interval orders. You may omit the proof of Theorem 5.19.
M 2/10: §5.6 (but I will omit this from the lectures).