# Foundations of Combinatorics

### Fall, 2017 + Winter, 2018 Prof. Sara Billey

Monday, Wednesday, Friday 12:30-1:20

### Textbooks

• Networks:An Introduction by Mark Newman. Published 2010 by Oxford Scholarship Online. Accessible via UW Library.
• Handbook of Graph Theory edited by Jonathan L. Gross, Ping Zhang, Jay Yellen. Published 2018 by Taylor and Francis. (The "Download" button worked for me now!).

### Syllabus

Summary:This three quarter topics course on Combinatorics includes Enumeration, Graph Theory, and aspects of Algebraic and Geometric Combinatorics. Combinatorics has connections to all areas of mathematics, industry and many other sciences including biology, physics, computer science, and chemistry. We have chosen core areas of study which should be relevant to a wide audience. The main distinction between this course and its undergraduate counterpart will be the pace and depth of coverage. In addition we will assume students have a substantial knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research. New students are welcome to join in Winter or Spring with approval from the instructor depending on prior background.

The outline for the full year is the following. Dr. Vasu Tewari will teach the third quarter.

• Fall = Enumeration: Every discrete process leads to questions of existence, enumeration and optimization. This is the foundation of Combinatorics. In this quarter we will present the basic combinatorial objects and methods for counting various arrangements of these objects.
• Basic counting methods.
• Sets, multisets, permutations, and graphs.
• Inclusion-exclusion.
• Recurrence relations and integer sequences.
• Generating functions.
• Partially ordered sets.
• Complexity Theory

• Winter = Graph Theory: Graphs are among the most important structures in combinatorics. They are universally applicable for modeling discrete processes. We will introduce the fundamental concepts and some of the major theorems. The existence questions of combinatorics are prevalent in graph theory.
• Basic graph structures
• Matroids
• Matchings
• Planar graphs
• Colorings
• Ramsey Theory
• Random graphs and probabilistic methods
• Graph minor theorem

• Spring = Algebraic and Geometric Combinatorics: This quarter will focus on the topic of hyperplane arrangements. A hyperplane arrangement is a finite collection of codimension one subspaces in a finite-dimensional vector space over some field. The theory of hyperplane arrangements is a thriving field of research and its richness is demonstrated by the fact that it shows up in areas as diverse as topology, combinatorics and geometry. Surprisingly, many questions of geometric or topological flavor regarding hyperplane arrangement are essentially combinatorial questions. For the most part, this course aims at making the preceding statement precise before discussing some recent developments.
• Hyperplane arrangements and the associated intersection lattice
• The characteristic polynomial of a hyperplane arrangement
• The finite field method following Crapo-Rota and Athanasiadis
• Graphic arrangements and connections with chromatic polynomials
• Coxeter groups and reflection arrangements
• The topology of the complement of arrangements, and the Orlik-Solomon algebra
• Connections to zonotopes and oriented matroids
• Connections to diagonal harmonics

Textbooks:

• Fall: "Enumerative Combinatorics; Volumes 1" by Richard Stanley, Cambridge Studies in Advanced Mathematics, 2011.
• Winter: (Tentative) "Graph Theory" by Reinhard Diestel, Springer-Verlag, GTM vol. 173 Second Edition, 2000. Electronic version.
• Spring: (Tentative) "Arrangement of Hyperplanes" by Orlik and Terao, Springer-Verlag, 1992.

Exercises: The single most important thing a student can do to learn combinatorics is to work out problems. This is more true in this subject than almost any other area of mathematics. Exercises will be assigned each week but many more good problems are to be found, with solutions, in your textbooks. Do as many of them as you can.

Problem sets: Grading will be based mostly on weekly problem sets due on Wednesdays. The problems will come from the text or from additional reading. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. We will have weekly problem sessions to discuss the harder problems. The time will be determined in the first week of class.

Grading Problem Sets: Students will be asked to grade another students problem set. More instructions to come with the first assignment.

No final: There will not be a final for this class.

Optional presentations: Several recently published research articles will be presented in class by students. You will have the option of doing a presentation sometime this year. If you do a presentation, your lowest homework grade will be dropped.

Computing: Use of computers to verify solutions, produce examples, and prove theorems is highly valuable in this subject. Please turn in documented code if your proof relies on it. If you don't already know a computer language, then try SAGE, Maple, Mathematica or GAP. All three are available for free on our math department servers.

### Tentative Schedule:

* Lecture 1: Introduction to graphs and the Graph Isomorphism Problem. Read Chapter 1. PS# handed out.

* Lecture 2: Connectivity and Trees.

* Lecture 3: Spanning trees and the Matrix Tree Theorem.

* Lecture 4: Hamiltonian and Eulerian graphs. PS#2 handed out.

* Lecture 5: More on Hamiltonian and Eulerian graphs with applications to de Bruijn sequences.

* Lecture 6: Matchings

Sara Billey