
University of Michigan

Abstract


Szemer\'edi's theorem states that every set
of integers of positive upper density contains arbitrarily
long arithmetic progressions. Equivalently, for
any $\delta > 0$ and any $k$, every subset of $\{1,2,...,N\}$
of size at least $\delta N$ contains an arithmetic progression
of length $k$, provided only that $N$ is sufficiently large.
Until recently, the only known proofs of the theorem
(Szemer\'edi's and a different approach by Furstenberg
using ergodic theory) gave almost no information about
the dependence of $N$ on $k$ and $\delta$. The purpose
of these three lectures will be to explain a new, more
quantitative approach. I hope to organize the three lectures
roughly as follows.

Monday, October 29:  Lecture 1. Reception following the lecture, East Hall Atrium.

Tuesday, October 30:  Lecture 2. 
Wednesday, October 31:  Lecture 3. 
The Ziwet lectures were established at the bequest of Alexander Ziwet, a professor in the UM Mathematics Department during 18881925.