Campuses:

Seminorms and Inverse Theorems in Additive Combinatorics and Ergodic Theory

Monday, September 29, 2014 - 10:15am - 11:05am
Keller 3-180
Tim Austin (New York University)
This course will be a gentle introduction to some of the technical tools
that underlie recent approaches to Szemeredi's Theorem in additive
combinatorics and associated questions about multiple averages in ergodic
theory.

We will begin with a synopsis of Furstenberg's work relating Szemeredi's
Theorem to the ergodic theoretic phenomenon of multiple recurrence, and
with some of the technical background needed for this connection to bear
fruit.

We will then quickly sketch Gowers' approach to Szemeredi's Theorem, which
yielded the best-known bounds and pointed towards various extensions, such
as the Green-Tao result for the primes. A key tool introduced by Gowers is
a family of seminorms for functions on Z/n, together with an `inverse
theorem' describing the structure of those functions for which these
seminorms take large values. These seminorms have counterparts in ergodic
theory, introduced by Host and Kra for their work on convergence of
multiple averages, also along with an inverse theorem (closely related to
another analysis by Ziegler, not in terms of seminorms). These ideas will
be summarized with an emphasis on their commonalities. In both settings,
the known inverse theorems give not only proofs of new or tightened
results, but a much improved understanding of the structures responsible
for them.

Having done this, the course will go into more detail about the
ergodic-theoretic inverse theorems. The classical case of triple averages
will be treated carefully, and its generalization covered more tersely.

Finally, we will sketch the difficulties of extending the above work,
either additive combinatorial or ergodic theoretic, to results for
higher-rank Abelian groups. Here the desired inverse theorems remain quite
mysterious. Depending on time, we will finish with a sketch of some recent
work on a simple `extremal' version of the inverse problem in higher
dimensions, where one already finds considerable new structure.
MSC Code: 
41A27
Keywords: