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