# Stevo Todorcevic (University of Toronto)

**CONFÉRENCE DU PRIX CRM-FIELDS-PIMS 2012**

Le vendredi 16 mars 2012 / Friday, March 16, 2012, 4:00 pm

Centre de recherches mathématiques

Pavillon André-Aisenstadt, Université de Montréal

Salle / Room 6214

### Ramsey degrees of mathematical structures and some of their applications

The original motivation behind the discovery of Ramsey's theorem in the late 1920's is an algorithm that would test the validity of universal relational sentences. On the combinatorial level this reduces to classifying relations on finite symmetric cubes of the integers not in terms of their global structure on the set **N** of all integers but rather their restrictinctions on arbitrarily thin infinite subsets of **N** This initiated the corresponding Ramsey-classification theory in the realm of other mathematical structures and we shall list some of the main contributions to this area. The second part of our lecture will concentrate to some of the applications. For example, we shall point out a close relationship of this theory and some area of topological dynamics. More precisely we shall explain how Ramsey classification results and the numerical invariants ("Ramsey degrees") they give to objects of a given class *K* of mathematical structures naturally lead to the description of the universal minimal flow of the corresponding group of automorphisms. Conversely, we shall explain which progress on the level of topological dynamics would lead to new advances in the structural Ramsey theory which at this point still stays reachable only to the typically complex combinatorial methods.

### Une réception suivra la conférence au Salon Maurice-L'Abbé, Pavillon André-Aisenstadt (Salle 6245).

### A reception will follow at the Salon Maurice-L'Abbé, Pavillon André-Aisenstadt (Room 6245).