Course overview -- Sommersemester 2017 (Archive)

The fundamental theorem of ultraproducts (Łos' Theorem) provides a transference principle between the finite structures and their limits. Roughy speaking, it states that a formula is true in the ultraproduct M of an infinite class of structures if and only if it is true for "almost every" structure in the class.

When applied to ultraproducts of finite structures, Łos' theorem presents an interesting duality between finite structures and their infinite ultraproducts. This kind of finite/infinite connection can sometimes be used to prove qualitative properties of large finite structures using the powerful known methods and results coming from infinite model theory, and in the other direction, quantitative properties in the finite structures often induce desirable model-theoretic properties in their ultraproducts.

These ideas were used by Hrushovski to apply ideas from geometric model theory to additive combinatorics, locally compact groups and linear approximate subgroups. More examples of this fruitful interaction were given by Goldbring and Towsner to provide proofs of the Szemerédi's regularity lemma and Szemerédi's theorem: every subset of the integers with positive density contains arbitrarily large arithmetic progressions.

The purpose of the talk will be to present these ideas and outline some of the applications to asymptotic combinatorics. If time permits, I will give a brief overview of the Erdos-Hajnal conjecture and present a proof (due to A. Chernikov and S. Starchenko) of the Erdos-Hajnal property for graphs without the order property using ultraproducts, pseudofinite dimensions and basic properties of stable formulas.

Zame and Lauwers recently showed connections between set theory and theoretical economics. In particular they showed that the existence of social welfare relations satisfying intergenerational equity imply the existence of non-constructible objects, such as non-Ramsey and non-measurable sets. In this talk I prove some connection also with another popular regularity property, i.e., the Baire property, and if there is any time left I propose to use Shelah's amalgamation in order to show that the two above implications does not reverse.

Gao and Kechris proposed in 2003 two somewhat related problems concerning ultrametric spaces, namely:

1) Determine the complexity of the isometry relation on locally compact Polish ultrametric spaces.

2) Characterize the Polish groups that are isomorphic (as topological groups) to the isometry group of some Polish ultrametric space.

We will present a construction strictly relating ultrametric spaces and a special kind of trees which helps in tackling these two problems. This technique applies to both separable and non-separable complete ultrametric spaces, and allows us to e.g. show that they are unclassifyiable up to isometry even when considering only discrete spaces. (Joint work with R. Camerlo and A. Marcone.)

In a paper by J. Moore following the seminal work of Kechris-Pestov-Todorsevic a correspondence between a certain combinatorial property of a Fraisse class, called convex Ramsey property, and amenability of the automorphism group of the Fraisse limit has been found. In this paper we review similar results for the automorphism groups of generic structures and especially show that automorphism groups of certain generic structures are not amenable by showing that a certain point-line geometries are realized in the generic structure.

The talk will be about the fine structure of (very) well-behaved complete first order theories. Totally categorical structures of disintegrated type (i.e. the underlying strongly minimal set is trivial) can analysed by a chain of finite covers. A finite cover of some structure is an extension by a new sort and new relations such that the old structure is stably embedded (i.e. every automorphism of the old structure extends to the cover) and there is some definable finite-to-one function from the new sort to the old sorts. Now we have that non-trivial phenomena in this chain of finite covers are connected to something called higher amalgamation, that is the ability to amalgamate certain systems of types. We will investigate higher amalgamation over parameters in a more general setting, i.e. in theories with a good notion of independence (e.g. strongly minimal, stable, simple). We give a general finite cover construction to force failure of higher amalgamation and apply it to the totally categorical structure (Z/4Z)^\omega such that higher amalgamation over some parameter fails while it holds over the empty set. This tells us that the analysis of general totally categorical structure via covers has another complication. But on the other hand as we can, after adding a sequence of finite covers, force every omega-categorical theory to have higher amalgamation over any parameter set, we could potentially have a starting point for some sort of classification of general totally categorical theories via covers.

A countably infinite first order structure is homogeneous if every isomorphism between finitely generated substructures extends to a total automorphism. By Fraisse Theorem, homogeneous structures arise as the Fraisse limits of amalgamation classes. Moreover, a free homogeneous structure is a homogeneous relational structure whose age has the free amalgamation property. In a joint work with Solecki, we show that free amalgamation classes has a 'coherent' form of the extension property for partial automorphisms (EPPA). We further discuss some group-theoretic consequences of this result on the automorphism group of any free homogeneous structure such as the existence of ample generics and a dense locally finite subgroup.

Abstract: D. Marker and A. Pillay proved that in a reduct of an algebraically closed field F, which is non-locally modular and expanding the additive structure, an infinite field is interpretable and then the multiplication on F is definable in this reduct. In their work, they use a result of B. Poizat, which states an infinite field K which is definable in the pure algebraically closed field F is definably isomorphic to F. I will present this result and its proof.

The terminology "Wesentlich verschiedene Abbildungen" (which means "essentially different functions") is taken from Hausdorff's work  "Über zwei Sätze von Fichtenholz und Kantorovich'' (1935).

We will follow Hausdorff's proof of the existence of  continuum many essentially different functions: i.e. there is some $H \subseteq {^\omega \omega}$ of size continuum such that for every finitely many $f_0, \dots, f_i \in F$ there is a level $x \in \omega$ such that $f_l(x) \neq f_j(x)$ for $l<j \leq i$.

We will then see how to generalize the result to find a family of size continuum of "independent functions"  using a construction with trees. If the audience is interested, we could also compare it with some other well known  (but less pictorial) proofs. If time remains, we will show how the existence of continuum many independent functions applies to prove that a finite support iteration of σ-centred forcing notions is again σ-centred (this is a question asked by Goldstern and answered by Blass in Mathoverflow).

A formula φ(x;y) is an equation in x, if its instances have the DIC: that is, the collection of finite intersections of instances has the descending chain condition. A theory is then equational if every formula is equivalent to a boolean combination of equations.

The colored pseudospace is one of only two known examples of stable, non-equational theories. It was introduced by Hrushovski and Srour in an unpublished paper. We will see an alternative axiomatization of the colored pseudospace as a colored lattice. This simplifies the proofs to some extent. Furthermore, we will see a criterion for non-equationality that requires no knowledge of Morley rank.