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Algebraic CombinatoricsGraduate course offered in the fall semester, 2005.Instructor: Lior PachterCourse number: Math 249. Course control number: 55173. Lectures: Tuesdays and Thursdays, 3:30--5:00pm in 85 Evans Hall. Office hours: Wednesdays 1--3pm in 1081 Evans Hall. |
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 this material, 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.
Lectures
This section will contain summaries of the lectures as the semester progresses.
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.
Reading material
Gian-Carlo Rota, Combinatorial Theory (1999). 14 copies are available for purchase directly from me.
Richard P. Stanley, Enumerative Combinatorics (1986,1997,1999)
Combinatorics in Berkeley
Course participants are encouraged to attend other combinatorics related activities around Berkeley such as the