# Universality and Homogeneity

Hausdorff Trimester Program

September 2 - December 20, 2013

Organizers: Alexander Kechris, Katrin Tent, Anatoly Vershik

The goal of the program was to study and link universality phenomena in different areas of pure mathematics. These areas included model theory, combinatorics, descriptive set theory, group theory, dynamical systems, especially ergodic theory and topological dynamics, random matrices and representation theory. In this way we promoted interactions in an unprecedented way between these different areas.

Universality is a pervasive notion in mathematics, and is suggestive of useful analogies between different areas. Examples are Fraïssé theory in logic, universal graphs in combinatorics, the universal Urysohn space in topology, universality in algebraic geometry, and so on. We considered it important to initiate a discussion of universality as a global phenomenon in mathematics, and brought together some of the main contributors.

The main reason for studying universal objects is that they may act as a frame of reference for what kind of results to expect, what kind of objects might exist and how homogeneous certain structures are. If a class of structures allows universal objects, studying these objects yields important insights into the class of structures and on individual structures of this class. Universal objects often serve both as examples for and counterexamples to theorems and conjectures. They are typically very homogeneous and reflect all properties that may or may not be realized. Model theory provides techniques for constructing and studying such objects. Also constructing a structure by a 'random' process often leads to the universal object of a certain class. Therefore, we obtained feedback in both directions from the universal object to random processes and conversely.