Ergodic Theory with connections to Additive Combinatorics

 
Course higher education credits: 7,5
 
Course is normally given: Periods 3-4. Course first taught 2014.
 
Graduate school Mathematics
Department Mathematical Sciences
 
Course start 2014-01-20
Course end 2014-05-16
 
Contact information: Michael Björklund
 
COURSE DESCRIPTION:
 
Ergodic theory, which has its origin in celestial and statistical 
mechanics, was for a long time solely concerned with the
analysis of the long-time (average) behaviour of various physical 
systems. An elegant mathematical formalization was
initiated by von Neumann and others in the early 1930's. Unexpected 
connections to geometry, number theory and combinatorics, were 
discovered much later.

The geometric aspects were discussed by Mostow, Margulis and others; 
especially in connection to rigidity phenomenon
for locally symmetric spaces.

The utility of ergodic theory for certain problems in diophantine 
approximation was investigated by Furstenberg, Dani,
Margulis and many others, and remains until today a very potent area 
of mathematics (as Lindenstrauss' fields
medal shows).

The beautiful connections to combinatorics (Ramsey theory) were 
pointed out by Furstenberg in the 1970's in his ergodic-theoretic 
proof of Szemeredi's Theorem. His ideas have spurred a plethora of 
interesting new directions in both ergodic
theory and additive combinatorics (Gowers norms etc.).

This course will be mostly concentrated on the last part. We shall 
begin by discussing the rudiments of ergodic theory (recurrence, 
existence result, structure theory). After this we shall set up some 
topological/harmonic machinery (almost periodicity, stone-cech 
compactification), discuss how to apply this machinery in various 
settings, and towards the end of the course we shall prove Szemeredi's 
Theorem on arithmetic progressions. If time permits, we will also 
survey some more
modern developments.

Prerequisites:

Basic knowledge of functional analysis and measure theory (Banach 
spaces and their duals, weak*-topology, Riesz representation etc.).

Publicerad: ti 11 jun 2013. Ändrad: må 04 jun 2018