Ergodic Theory towards Szemerdi’s Theorem

This course serves as an introductory course to ergodic theory. The objective of the course is to present Furstenberg’s proof of Szemeredi’s theorem, which states that every positive density subset of integers contains arbitrarily long arithmetic progressions. In order to provide a dynamical proof of this result, Furstenberg developed a “Correspondence Principle” and Structure Theorems for measure-preserving transformations. These results are now considered fundamental concepts in ergodic theory.

Topics:

• Some Topological dynamics and van der Waerden’s Theorem
• Measure preserving transformations and Poincare recurrence
• Ergodicity, weak mixing, factors and conditional expectations
• Furstenberg’s Correspondence Principle
• Ergodic proof for Szemeredi’s Theorem