Math 8680: Topics in combinatorics
Arrangements of hyperplanes and subspaces

Spring 2000

Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256, Telephone (with voice mail): 625-6682, E-mail: 
Classes: Mon-Wed-Fri 2:30-3:20pm, Vincent Hall 1. 
Office hours: Mondays and Wednesdays 11:15-12:05;
one can always arrange to meet me by appointment 
Course content: An arrangement of hyperplanes is a finite collection of codimension one subspaces in a vector space over some field. Their study lies at the intersection of combinatorics, topology, and geometry, as most enumerative, geometric or topological questions about them reduce to combinatorics. The answers to many of these questions have been worked out, mostly in the last three decades. Similarly, questions about arrangements of subspaces (not necessarily of codimension one) are often controlled by combinatorics, although here many more questions remain unanswered. In this course, we hope to explain the beautiful combinatorial answers to many of these questions. Here are some of the topics we'll think about covering:
  • The characteristic polynomial of a hyperplane arrangement A, and its relation to:
    - counting proper colorings of a graph
    - the number of chambers cut out by the hyperplanes over R
    - enumerating points off the hyperplanes over a finite field F_q
    - the Tutte polynomial and broken circuits of the matroid of A
    - the Poincar\'e polynomial of the complexified complement of A in C
    - the Smith normal form of Varchenko's matrix associated to A
    - Terao's theory of free arrangements
  • Computation of topological invariants of the complexified complement A, including:
    - Randell and Arvola's presentation for the fundamental group
    - Orlik and Solomon's presentation for the cohomology ring
    - Goresky and MacPherson's formula for the homology groups of a subspace arrangement
  • The connection between the topology of certain subspace arrangements and computational complexity theory, due to Bj\"orner, Lovasz and Yao.
  • The determinant of Varchenko's matrix associated to A
  • Connections to zonotopes and oriented matroids.
Prerequisites: No absolute requirements. Of course a little bit of combinatorics would be useful. We will discuss some algebraic and topological things, so experience with algebra (like rings, modules) and algebraic topology (like fundamental groups, homology and cohomology) might be useful. But we'll try to review the basic facts about them anyway, often without proof.  
Source materials, course requirements, and grading: We'll borrow heavily from the text
Arrangements of hyperplanes, P. Orlik and H. Terao, Springer-Verlag 1992
available from the bookstore, but is not absolutely required.

There is also a source list of papers, containing both useful surveys and other smaller papers in the subject.

There will be no homeworks or exams, but registered students are expected to attend, and must give one or two talks during the semester on papers chosen from the above source list, or any others proposed by the student that I approve. Students will be expected to meet with me at least once after I have approved their choice of paper to speak on, but before they give a talk, to discuss which aspects of the paper they will cover during the talk.
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