Math 580A
Matroids and Hyperplane Arrangements
Fall 2019 – Spring 2020
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Index
Teacher
Thomas Zaslavsky
Office: WH-216
Email: zaslav@math.binghamton.edu
Office hours
by appointment.
Course goal
Develop theoretical understanding of advanced combinatorial linear algebra. (I adapted this from my 304 linear algebra class, and it fits.)
Syllabus
We will cover the following, Chronos willing:
- 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.
Sources:
- 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.
Written homework assignments
- 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.
Exam policy
There will be no written exams. There will be periodic oral exams.
That is, besides those in Stanley's E&A.
- 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: An−1 is not Bn. The former is a set of vectors. The latter is the dual hyperplane arrangement, for which better notation is An*. 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.
In Orlik's booklet you will find the definition of the algebra A at the beginning of Ch. 3. I will develop it for a simple matroid, which is more general than Orlik's treatment for an arrangement, but the properties and proofs are the same.
Some missing proofs are provided in my notes on A(M), M a simple matroid.
We'll omit the algebra B, which follows A in Ch. 3. (B is isomorphic to A but more complicated.)
The connection to the complex complement is in Theorem 6.3.
Exercises for OS
- 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.
- ?
Dates are due dates. Sections §s.n mean readings. Boldface problem numbers (s.n) are to be handed in. Problem numbers (s.n) are particularly (but not exclusively) to work for yourself.
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.
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- Late Sept.: Finish reading Lecture 2.
- W 10/2: §3.1.
- F 10/4: §3.2.
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- F 1/24: Solve the characteristic polynomial of the hollow Catalan arrangement, Cn° = A[{±1}Kn].
Optional: Solve that of the extended Catalan arrangement, Cn,k = A[{0,±1,...,±k}Kn].
- (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).