Algebraic Combinatorics Graduate course offered in the spring semester, 2009. Instructor: Lior Pachter Course number: Math 249. Course control number: 54734. Lectures: Tuesdays and Thursdays, 9:30--11:00am in 101 Wheeler. Office hours: Monday 1-3pm.

About the course

In 1963 Gian Carlo Rota taught his first course on combinatorics at Harvard University. With his course, he began a tradition of algebraic combinatorics that draws on, and continues to inspire, a wide range of mathematics. At the the same time, his style combined an intuitive approach to the subject with a delightful sense of humor. He also emphasized applications ranging from algebraic geometry to probability theory.

The last time his course was offered was in the Fall of 1998 at MIT. In the 35 years that had elapsed since the course had been developed the field and its scope had changed enormously. There had also been a proliferation of new textbooks, such as the excellent "Enumerative Combinatorics" by Richard Stanley (who took the course in 1967). Starting with his favorite companion to combinatorics, lattice theory, Rota proceeded to highlight many of the classic (often forgotten) results that are routinely omitted from textbooks and courses. Fortunately, a student in the class, John N. Guidi, preserved his lectures and wit in a carefully compiled series of lecture notes. Our own course will be based on Stanley's book and Rota's notes, supplemented by other texts and recent research papers.

We will begin by reviewing sets and relations, and basic enumeration. We will then study lattices and projective space, followed by matching and matroid theory. Time permitting, we will discuss other topics such as geometric probability, tropical mathematics, and applications of algebraic combinatorics in biology.

Prerequisites

Basic knowledge of abstract algebra and linear algebra is essential. An undergraduate course in combinatorics would be helpful, but is not required.

Homework

Practice with homework problems is essential for learning the material. As Professor Rota was fond of saying "The Germans have aptly called Sitzfleisch the ability to spend endless hours at a desk doing grueling work. Sitzfleisch is considered by mathematicians to be a better guage of success than any of the attractive definitions of talent with which psychologists regale us from time to time."

Exercises will be assigned during the lectures, and will be due a week or two after being assigned. The final grade will be based solely on the homework. There will be no GSI due to the current budget crisis.

Reading material

Richard P. Stanley, Enumerative Combinatorics (1986,1997,1999).
Gian-Carlo Rota, Combinatorial Theory (1999).
Marie-Louise Dubreil-Jacotin
Reading project suggestions

Combinatorics in Berkeley

Course participants are encouraged to attend other combinatorics related activities around Berkeley such as the

Discrete Math Seminar.
Special seminar: Lauren Williams (Harvard) on Friday January 23rd, 4-5pm in 740 Evans Hall: The Grassmanian, hopping particles and orthogonal polynomials.
Bay Area Discrete Math Day, April 4.