Ergodic Ramsey Theory

Master Lecture: Department of Mathematics, SNSB (Scoala Normala Superioara Bucuresti), Winter Semester 2010/2011

Lecturer: Laurentiu Leustean

Time and location: Tuesday, 10:00-14:00, Lecture Hall "Grigore Moisil" (412), IMAR

Ergodic Ramsey theory was initiated in 1977 when Hillel Furstenberg proved a far reaching extension of the classical Poincare recurrence theorem and derived from it the celebrated Szemeredi's theorem, which states that any subset of integers of positive upper density must necessarily contain arbitrarily long arithmetic progressions.

Since then, Furstenberg's ergodic approach was used to establish many more types of recurrence theorems, which (via the Furstenberg's correspondence principle) yield a number of highly non-trivial combinatorial theorems. Many of the results obtained by these ergodic techniques are not known, even today, to have any "elementary" proof, thus testifying to the power of this method.

Lectures :

Lecture 1: a general presentation of the course.

Lecture 2: Topological Dynamical Systems: definitions, examples. Basic constructions: homomorphisms, (strongly) invariant sets.

Lecture 3: Basic constructions continued: subsystems, direct products, disjoint unions. Transitivity.

Lecture 4: Minimality. Recurrence. Application to a result of Hilbert, presumably the first result of Ramsey Theory.

Lecture 5: Multiple Recurrence Theorem.

Lecture 6: Ramsey Theory: van der Waerden Theorem.

Lecture 7: Ultrafilter approach to Ramsey Theory.

Lecture 8: Hales-Jewett Theorem. Measure-preserving systems.

Lecture 9: Ergodic Theory: measure-preserving systems, induced operator, Bernoulli shift.

Lecture 10: Different notions of density. Furstenberg correspondence principle.

Lecture 11: Ergodicity. Maximal ergodic theorems. Birkhoff ergodic theorem.

Lecture 12: Mean ergodic theorem in uniformly convex Banach spaces. Finitary version.

Lecture 13: Recurrence. Szemeredi theorem - finitary and ergodic versions.

Lecture 14: Mixing. Szemeredi property for compact and weak mixing systems.

Lecture 15: Proof of Roth Theorem.

Useful links:


Lecture notes, surveys, essays: