Itai Ben Yaacov (Lyon)
Title: From the model theory of Banach spaces to the model theory of non
standard function and number fields
Abstract: The sum formula for a function field says that given a non zero rational function f , the sum of zeros and poles of f with the appropriate multiplicities is zero, or equivalently, that the degree of the divisor (f) is zero. Very much similarly, given a non zero element x of a number field K , the sum of all valuations of x (including the Archimedean ones) with appropriate weights is zero.
Both situations can be formalised as a pair (K,E) where K is a field and E an L^1 lattice of "divisors", along with a lattice-valued valuation v: K -> E such that the integral of v(x) is always zero.
The class of such pairs is elementary and inductive in the appropriate (continuous, unbounded) first order logic, and therefore natural to study from a model-theorist's point of view. The model theory of L^1 Banach lattices, which is already well understood, is a crucial ingredient of this study.
This is work in progress, joint with E. Hrushovski.
Gregory Cherlin (Rutgers)
Title: Around homogeneous graphs
Abstract: We will discuss some developments related to Henson's work on homogeneous graphs.
Julien Melleray (Lyon)
Title: Continuous logic and the theory of Polish groups
Abstract: The language of continuous logic turns out to be useful to the study of general Polish goups. I'll discuss some examples of that phenomenon: a reformulation of an oscillation theorem due to Hjorth in the framework of continuous logic; a weaker version of the notion of ample generics that has interesting applications, especially when it comes to automatic continuity theorems; and a characterization of extreme amenability in terms of metric Frass classes.
Anand Pillay (Leeds)
Title: Topological dynamics and connected components
Abstract: I will discuss Newelski’s recent work which tries to interpret /recover various model-theoretic objects such as G/G^00 in/from Ellis’ theory (in topological dynamics). I will give some positive results along these lines, generalizing the situation for stable groups.
Christian Rosendal (UIC)
Title: On isometric representations and maximal symmetry
Abstract: A classical result going back to Weyl and Auerbach states that any infinite-dimensional Banach space admits an isometry-invariant inner product, which thus provides an optimal, namely Euclidean, norm on the space. This was extended by Sz. Nagy and Dixmier around 1950, who showed that if G is a bounded amenable group of invertible operators on a Hilbert space, there is a G-invariant inner product, and thus G can be seen as a group of unitary operators for another Euclidean norm. However, the question of whether every Banach space admits an optimal norm, realising maximal symmetry, has remained open for a number of years. We shall answer this and other related questions and indicate how this is related to Dixmier's unitarisability problem.
This is joint work with V. Ferenczi.
David Sherman (Virginia)
Title: Model theory of operator algebras
Abstract: Classical ultraproducts facilitate several beautiful theorems in (first-order) model theory. In functional analysis, the related notion of metric ultraproduct has been used consciously for about forty years, but it does not satisfy the same statements. Over time the analysis community largely forgot about the logical meaning of ultraproducts, although Ward Henson and others constructed variations of first-order model theory for which metric ultraproducts are suitable.
The past few years have brought increased interest in problems involving metric ultraproducts, and, serendipitously, also a compatible model theory based on continuous logic and the syntax of analysis. Ilijas Farah, Bradd Hart, and I have been studying the model theory of operator algebras, in particular $II_1$ factors (a certain basic type of von Neumann algebra that I will review). I will explain some of our ideas and results.
Henry Towsner (UCLA)
Title: Extracting complexity bounds from proofs
Abstract: One of the central themes of proof theory is the extraction of numeric bounds from seemingly non-effective proofs. Even when there are no quantities in a theorem for which numeric bounds make sense, we can sometimes identify an appropriate notion of complexity, usually indexed by ordinals, and extract bounds with respect to this notion of complexity. We'll discuss some simple examples of ordinal bounds in combinatorics, as well as a potential application to the nonlinear theory of Banach spaces. (No knowledge of either proof theory or Banach spaces is assumed!)