18.315 | Fall 2004 | Graduate
Combinatorial Theory: Hyperplane Arrangements

Lecture Notes

These lecture notes on hyperplane arrangements are based on a lecture series at the Park City Mathematics Institute, July 12-19, 2004. They provide an introduction to hyperplane arrangements, focusing on connections with combinatorics, at the beginning graduate student level. Background material on posets and matroids is included, as well as numerous exercises. After going through these notes a student should be ready to study the deeper algebraic and topological aspects of the theory of hyperplane arrangements. Perhaps someday these notes will be expanded into a textbook on arrangements.

Chapter 1: Basic Definitions, the Intersection Poset and the Characteristic Polynomial (PDF)

1.1. Basic Definitions
1.2. The Intersection Poset
1.3. The Characteristic Polynomial

Chapter 2: Properties of the Intersection Poset and Graphical Arrangements (PDF)

2.1. Properties of the Intersection Poset
2.2. The Number of Regions
2.3. Graphical Arrangements

Chapter 3: Matroids and Geometric Lattices (PDF)

3.1. Matroids
3.2. The Lattice of Flats and Geometric Lattices

Chapter 4: Broken Circuits, Modular Elements, and Supersolvability (PDF)

4.1. Broken Circuits
4.2. Modular Elements
4.3. Supersolvable Lattices

Chapter 5: Finite Fields (PDF)

5.1. The Finite Field Method
5.2. The Shi Arrangement
5.3. Exponential Sequences of Arrangements
5.4. The Catalan Arrangement
5.5. Interval Orders
5.6. Intervals with Generic Lengths
5.7. Other Examples

Chapter 6: Separating Hyperplanes (Preliminary Version) (PDF)

6.1. The Distance Enumerator
6.2. Parking Functions and Tree Inversions
6.3. The Distance Enumerator of the Shi Arrangement
6.4. The Distance Enumerator of a Supersolvable Arrangement
6.5. The Varchenko Matrix

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Fall 2004
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