Logic, Set Theory and Topology Ical Atom

I will introduce the definitions of mutual and tight stationarity due to Foreman and Magidor. These notions generalize the property of stationarity from subsets of a regular cardinal to sequences of subsets of different regular cardinals (or, by some interpretations, to singular cardinals). Tight stationarity will then be related to pcf theory, and from a certain pcf-theoretic assumption we will define a ccc forcing which arranges a particularly nice structure in the tightly stationary sequences.

I will introduce the definitions of mutual and tight stationarity due to Foreman and Magidor. These notions generalize the property of stationarity from subsets of a regular cardinal to sequences of subsets of different regular cardinals (or, by some interpretations, to singular cardinals). Tight stationarity will then be related to pcf theory, and from a certain pcf-theoretic assumption we will define a ccc forcing which arranges a particularly nice structure in the tightly stationary sequences.

In this talk I will prove that any pseudo-finite group contains an infinite abelian subgroup. Additionally, I shall also discuss some other results concerning pseudo-finite groups and centralizers.

This is joint work with Nadja Hempel.

In the study of the connection between automorphism groups of models and the models themselves (or their theories, or their bi-interpretability class), the Small Index Property (SIP) has played a central role. The work of Hodges, Lascar, Shelah and Rubin among others has established in many cases when a model of a first order theory T has the Small Index Property.

With Ghadernezhad, we have studied this property for more general homogeneous classes. We have isolated properties of closure notions that allow to prove the SIP for some non-elementary cases, including Zilber’s pseudo-exponentiation and other examples.

I will present a panorama of these results, including our more recent generalizations of the Lascar-Shelah proof of SIP for uncountable structures. This last part is joint work with Zaniar Ghadernezhad.

We observe that several elementary definitions in point-set topology can be reformulated in terms of finite topological spaces and elementary category theory. This includes compactness of Hausdorff spaces, being connected, discrete, the separation axioms.

Though elementary, these observations raise a few open questions. For example, I was not able to prove that this reformulation of compactness gives the correct answer for non-Hausdorff spaces, or whether implications between various topological properties can also be proved entirely in terms of finite topological spaces, without any additional axioms.

In the framework of positive model theory I will give (recall) a definition of ``structural approximation’’ which is used in my paper on model-theoretic interpretation of quantum mechanics. I will then present some general theory as well as a few examples, if time permits.

For every finite ordered graph $H$ there is a natural number $k(H)>1$ such that whenever all copies of $H$ in the ordered inverse limit of all finite ordered graphs are partitions to finitely many Borel parts, then there is a (closed) copy of the inverse limit graph in itself whose copies of $H$ meet at most $k(H)$ many parts.

The probability that a random ordered graph on $n$ vertices satisfies $k(H)=1$ tends to 1 as $n$ grows.

Joint work with S. Geschke and S. Huber.

In my seminar talk on 29.12.2015, I introduced the notion of quasi-ordered fields, proved Fakhruddin’s dichotomy. In this talk, I will present a version of a classical theorem in real algebra (the Baer-Krull theorem) for quasi-ordered fields.

A linearly ordered structure is weakly o-minimal if every definable set is a finite boolean combination of convex sets. A weakly o-minimal expansion of an ordered group is non-valuational if it admits no non-trivial definable convex sub-groups. By a theorem of Baizalov-Poizat if M is an o-minimal expansion of a group and N is a dense elementary substructure then the structure induced on N by all M-definable sets is weakly o-minimal non-valuational.

It is natural to ask whether all non-valuational structures are obtained in this way. We will give examples showing that this is not the case. We will show, however, that if M is non-valuational then there exists M^, an o-minimal structure embedding M densely (as an ordered set) such that M (as a pure set) extended by all M^-definable sets is precisely the structrue M. We will give a complete axiomatisation of the theory of the pair (M^,M), show that it depends only on the theory of M, and that it shares many common features with the theory of dense o-minimal pairs. In particular (M^,M) has dense open core (i.e., the reduct consisting only of definable open sets is o-minimal).

Based on joint work with E. Bar-Yehuda and Y. Peterzil.