DRMTA is a cooperation between the Universities of Basel, Freiburg, Konstanz and Passau, and brings together researchers interested in model theory and its applications from the Donau-Rhein region.

The 4th meeting will take place at the Institute of Mathematics of the Albert-Ludwigs-Universität Freiburg. You can find the directions here.

4th Floor, Room 404

Speakers

Elías Baro (Universidad Complutense de Madrid )

Andre Opris (Universität Passau)

Daniel Palacín (Albert-Ludwigs-Universität Freiburg)

Alex Savatovsky (Universität Konstanz)

Schedule

10:30-11:15 Arrival & Coffee (in Sozialraum 331, 3rd Floor)

11:20-12:10 Talk by Daniel Palacín

12:15-13:30 Lunch

13:30-14:20 Talk by Andre Opris

14:30-15:20 Talk by Elías Baro

15:20-16:00 Coffee Break

16:00-16:50 Talk by Alex Savatovsky

18:00 Dinner

Titles and abstracts

Elías Baro: Small groups definable in dense pairs of real closed fields .

A. Robinson proved the completeness of the theory of real closed fields (rcf) with a predicate for a proper dense real closed subfield. L. van den Dries later gave a characterization of definable sets and definable functions in dense pairs of rcf (in fact, of o-minimal expansions of ordered groups). A notion of "small" set played a crucial role. Roughly, a small set is the image of the predicate by a semialgebraic map. Recently, E. Pantelis has proved that any definable small set is, up to interdefinability, a subset of the predicate.

In the paper Small groups in dense pairs, we show that every small group definable in a dense pair of rcf has locally, up to interdefinability, an algebraic group law. In this talk I will present this result, which is a joint work with A. Martín-Pizarro.

Andre Opris: Tamm's Theorem for log-analytic functions.

In the paper Extending Tamm’s Theorem by L. van den Dries and C. Miller, a parametric result of Tamm’s theorem is given. This talk is about a generalisation of this result: Let X be a subset of R^{n+m} and f : X → R be a log-analytic function. This means that f is a global subanalytic function augmented by logarithmic terms. Then there exists a natural number N such that the function mapping y to f(x_0, y) is real analytic in a neighbourhood of y_0, whenever the function is C^N.

Daniel Palacín: On a question of Babai and Sós, a model theoretic approach.

In 1985, Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free set of size at least cn, where a product-free set of a group is a subset that does not contain three elements x, y and z satisfying xy=z. Gowers showed that the answer is no in the early 2000s, by linking the existence of product-free sets of large density to the existence of low dimensional unitary representations.

In this talk, I will provide an answer to the aforementioned question by model theoretic means. Furthermore, I will relate some of Gowers' results to definable compactifications of nonstandard finite groups.

Alex Savatovsky: Expansions of the real field which introduce no new C^\infty functions.

We will give a rough sketch of the following theorem: Let L be the language of rings and P be a predicate. Let R be a real closed field and (R,P) a d-minimal structure. Under some model theoretic assumptions, we have that every definable C^\infty function on an open connected domain is the restriction of some L-definable function to this domain.