# Mini Courses

## Homogeneous structures, omega-categoricity and amalgamation constructions

Dates: September 17th, 19th and 24th, 10:00 am - 12:00 am

Lecturer: David Evans

Abstract: The aim of these lectures is to provide an introduction to homogeneous structures, omega-categorical structures and their automorphism groups.

The plan of the lectures is:

- Homogeneous structures, Fraisse's theorem and examples; omega-categoricity, the Ryll-Nardzewski Theorem, more examples.
- Automorphism groups as topological groups; imaginaries and biinterpretability for omega-categorical structures.
- Generalisations of the Fraisse construction. Hrushovski's predimension construction and amalgamation. Using the Hrushovski construction to produce omega-categorical structures.

The talks are expository and aimed at non-specialists, though some familiarity with basic model theory will be assumed.

## Continuous first-order model theory for metric structures

Dates: October 17th, 21th and 23th, 10:00 am - 12:00 am

Lecturer: Ward Henson

Abstract: A metric structure is based on a complete metric space (M,d) of finite diameter. The rest of the structure consists of operations, which are distinguished M-valued functions on M; predicates, which are distinguished [0,1]-valued functions on M; and constants, which are distinguished elements of M. Each operation and predicate must be uniformly continuous. Metric structures arise in all areas of mathematics, especially in analysis, probability, and geometry. Examples include: measure algebras; balls in Banach spaces, -lattices, -algebras, etc; in C*-algebras and in asymptotic cones of finitely generated groups; and metric spaces themselves.

Continuous first-order logic is a [0,1]-valued generalization of the usual first-order logic. Its propositional fragment corresponds to the logic of Lukasiewicz, which was introduced in about 1930. With quantifiers corresponding to the operations of sup and inf on the interval [0,1], this logic was studied in the 1960s, and then dropped without being substantially developed and without model theoretic applications. Recently it has re-emerged as the appropriate logic for the model theory of metric structures. Its theoretical features have been developed rapidly and its applications are being actively pursued.

Continuous logic is a beautiful and natural extension of classical logic with suitable analogues of essentially all the key properties: logical compactness, existence of rich and highly homogeneous models, characterizations of quantifier elimination and categoricity, fundamental tools of model-theoretic stability, etc. It resonates well with the ultraproduct construction for metric structures, a tool that has played a significant role in research on structures in functional analysis, including Banach spaces, C*-algebras, and tracial von Neumann algebras, in geometric group theory, and elsewhere. Many important classes of metric structures are axiomatizable in continuous logic, and thus are susceptible to being treated model-theoretically Indeed, this logic provides model theorists and analysts/geometers with a common language; this is due to its being closely parallel to first-order logic while also using familiar constructs from analysis (e.g., sup and inf in place of the universal and existential quantifiers, and continuous functions on [0,1] in place of propositional connectives).

The purpose of this short course is to present the syntax and semantics of continuous logic for metric structures, to indicate some of its key theoretical features, and to show a few of its recent application areas. Throughout, there will be an emphasis on specific metric structures of general mathematical interest and the methods used by model theorists to study them. Included will be a discussion of such structures as Urysohn's metric space and Gurarij's Banach space, which fit into the *universality and homogeneity*theme of this HIM program.

## Topological dynamics for automorphism groups of homogeneous structures

Dates: November 11th, 12th and 13th, 10:00 am - 11:00 am

Lecturer: Todor Tsankov

Abstract: A surprising connection between Ramsey theory for Fraïssé classes and the structure of the universal minimal flow for the automorphism groups of their Fraïssé limits was discovered a few years ago by Kechris, Pestov, and Todorcevic, thus stimulating a lot of new research on the topological dynamics of those groups as well as reviving interest in structural Ramsey theory. More precisely, the universal minimal flow of the group encodes the most general obstruction to a Ramsey theorem holding for the Fraïssé class and in all known situations (under reasonable assumptions), both the obstructions and the flow have turned out to be rather manageable objects, in sharp contrast with the situation for locally compact groups. These ideas have also given hope of proving fairly general structural Ramsey theorems using methods from topological dynamics, as opposed to combinatorial techniques. In the mini-course, I will explain the connection between Ramsey theory and topological dynamics and discuss a conjectural picture for a wide class of groups as well as some partial progress towards it (joint with Lionel Nguyen Van Thé). Another opportunity given by the manageability of the universal minimal flows is the classification of all minimal flows of certain groups; I will also discuss some results in this direction.

## Continuous Fraïssé limits

Dates: November 18th and 25th, 11:00 am - 13:00 pm

Lecturer: Wiesław Kubiś

Abstract: I will present a recently developed theory of continuous Fraïssé limits, in the framework of categories enriched over metric spaces. The theory explains fundamental properties of almost homogeneous objects like the Gurarii space (in Banach space theory), the pseudo-arc and the pseudo-circle (in geometric topology). As a new application, I will present a construction of a universal projection on the Gurarii space, which again can be considered as a continuous Fraïssé limit in the appropriate category. If time permits, some other examples will be shown.

The lectures are based on the following preprint:

W. Kubiś, Metric-enriched categories and approximate Fraïssé limits (arXiv:1210.6506)