| Math 194c |
|
| Combinatorial Number Theory | |
| Spring 2006 | |
|
WF 3:00 - 4:30 PM, 123 Lauritsen |
\
|
Instructor:
Peter
Keevash, 280 Sloan, 395-4369, keevash@caltech.edu
|
|
|
|
|
|
|
|
In recent years, there has been much progress in understanding the combinatorial structures arising from arithmetic operations, using techniques from many branches of mathematics, including probability, Fourier analysis, and ergodic theory. Highlights are proofs of Freiman's structure theorem and Szemeredi's theorem on arithmetic progressions that are considerably shorter and easier to understand than the originals. In this class we will work through some of the current research in this field, culminating with the recent result of Green and Tao showing that the primes contain arbitrarily long arithmetic progressions. |
|
|
| Grades will be
calculated according to the following scheme: 50%: A presentation in class by the student. I will suggest some potential topics. If there is something you are particularly interested in presenting, then that will be even better. 25%: Participation in class. "The mind is not a vessel to be filled but a fire to be kindled." (Plutarch) 25%: Final examination. I'll assemble a list of problems at some point for you to think about during the term. The final examination will be oral: I'll make appointments for students to come and tell me their solutions or partial progress on the questions set. |
|
|
|
There is no textbook currently available on the material for this course,
but there are many well-written sets of lecture notes and research
papers that should provide ample material for us to work with. Lecture notes Terence Tao:
Some highlights of
arithmetic combinatorics Papers Green and Tao:
The primes
contain arbitrarily long arithmetic progressions Tentative course plan 1. Freiman's Theorem (following notes by Ben Green) |