
Ahlfors Lecture series
September 1718, 2015, Organizers: HT Yau and ST Yau
Harvard University, Science Center Hall A (Thu) and D (Fri)
Speaker and Program
Jacob Fox (Stanford University)

September 17, 2015:
Lecture I 4:155:15 PM in SC Hall A

The Graph Regularity Method
It is a fundamental problem to understand the structure of
large graphs, as it can yield critical insights into topics ranging
from the spread of diseases to how the brain works. Szemerédi's
regularity lemma gives a rough structural result for all graphs. It is
one of the most powerful tools in graph theory, with many applications in
combinatorics, number theory, discrete geometry, and computer science.
Roughly speaking, it says that every graph can be partitioned into
a small number of parts such that between almost all pairs of parts
the graph is randomlike. Variants of the regularity lemma have since
been established with many further applications. In this talk, I will
introduce the regularity method and its applications, and survey recent
progress in understanding the quantitative aspects of these results.

September 18, 2015:
Lecture II 4:155:15 PM in SC Hall D
 Sparse Regularity and Prime Patterns
The celebrated GreenTao theorem states that there are
arbitrarily long arithmetic progressions in the primes. In this talk,
I will explain the ideas of the proof and recent joint work with David
Conlon and Yufei Zhao simplifying the proof. A key ingredient in the
proof of the GreenTao theorem is a relative Szemerédi theorem,
which says that any subset of a pseudorandom set of integers of positive
relative density contains long arithmetic progressions. Our main advance
is a simple proof of a strengthening of the relative Szemerédi
theorem, showing that a much weaker pseudorandomness condition is
sufficient. The key component in our proof is an extension of the
regularity method to sparse hypergraphs.

