DRMTA - SS2017

# Donau–Rhein Modelltheorie und Anwendungen (DRMTA)

## Inaugural Meeting, 12 May 2017

Fachbereich Mathematik und Statistik, Universität Konstanz

Speakers: Derya Çıray (Universität Konstanz), Deirdre Haskell (McMaster University), Chris Miller (The Ohio State University), Daoud Siniora (Albert-Ludwigs-Universität Freiburg/University of Leeds), Patrick Speissegger (McMaster University/Universität Konstanz)

Local Organisers: Salma Kuhlmann, Margaret Thomas, Lothar Sebastian Krapp.

### Registration

There is no registration necessary to participate and no registration fee. However, if you are interested in participating in this meeting, but are not associated with one of the four organising universities, please contact one of the local organisers to communicate your interest so that we have a better idea of expected participation.

### Programme

All talks will take place in room F426. For titles and abstracts please see below.

10:45-11:30 Arrival (coffee/tea etc.)

11:30-12:15 Chris Miller

12:15-13:30 Lunch
-- will not be specially arranged but visiting participants are encouraged to join locals in obtaining lunch at the Mensa (cafetereia) on campus.

13:30-14:15 Patrick Speissegger (in conjunction with the Oberseminar Reelle Geometrie und Algebra)
14:20-15:05 Daoud Siniora

15:05-15:35 Break (coffee/tea etc.)

16:25-17:10 Derya Çıray

18:00 Dinner
-- we have a reservation at the restaurant at Hafen Halle, located on the Konstanz waterfront conveniently close to the Konstanz Hauptbahnhof, for those who need to catch the train afterwards. For those who are not driving to Konstanz, we will take the bus into town together following the last talk. For those driving, there are many possibilities to find parking in the town centre, the closest being the Parkhaus at the nearby shopping centre LAGO.

### Titles and Abstracts

Derya Çıray (Universität Konstanz)
Title: Expansions of the Real Field with Denjoy-Carleman Classes and their Mild Parametrization
Abstract: Mild parametrizations are smooth parametrizations of sets by means of functions with some special control on the derivatives. They were first applied in the realm of diophantine geometry by J. Pila, to obtain results about the density of rational points on the graphs of non-algebraic pfaffian functions. Furthermore he has shown that obtaining mild parametrization results with some uniformity in parameters would be sufficient to establish Wilkie's conjecture.
Quasianalytic Denjoy-Carleman classes are special classes of infinitely differentiable functions which are important in harmonic analysis and in other areas. Rolin, Speissegger and Wilkie (2003) proved that the expansions of the real field with quasianalytic Denjoy-Carleman classes are o-minimal, polynomially bounded and model complete. In this talk I will talk about mild parametrization of the sets definable in these structures.

Title: Residue field domination in theories of valued fields
Abstract: The fundamental intuition in the model theory of valued fields is that a valued field with appropriate closure properties is controlled in some sense by its value group and residue field. The classical theorem of Ax-Kochen and Ersov states a version of this at the level of the theory of a henselian valued field of characteristic $(0,0)$. In this talk, I will discuss some theorems which give a stronger statement: for certain henselian valued fields there are conditions on the value group and residue field for the isomorphism type of the field itself to be fixed.

Chris Miller (The Ohio State University)
Title: Component-closed expansions of the real line
Abstract: We consider expansions $M$ of the real line $(\mathbb{R},<)$ having the property that, for all sets $E$ definable in $M$, each connected component of $E$ is definable in $M$; we then say that $M$ is "component closed". Some notable examples are: (a) all o-minimal $M$; (b) $M =$ the expansion of the real field by a predicate for the set of integers; and (c) the "component closure" of $M$ (defined in an obvious way). I will demonstrate that, in contrast to cases (a) and (b), the question "Is $M$ component closed?" can be difficult to answer even if the theory of $M$ is well understood. This is very preliminary joint work with Athipat Thamrongthanyalak.

Daoud Siniora (Albert-Ludwigs-Universität Freiburg/University of Leeds)
Title: Free homogeneous structures
Abstract: 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.

Patrick Speissegger (McMaster University/Universität Konstanz)
Title: Holomorphic extensions of functions definable in $\mathbb{R}_{\textrm{an},\exp}$
Abstract: I describe a general extension theorem for the germs of functions definable in the o-minimal expansion of the field of reals by all restricted analytic functions and the exponential function. The extension theorem can be used to bound the complexity of, for instance, the compositional inverse of a definable germ in terms of the complexity of that germ.