Program
Schedule
10:30-11:15 Arrival & Coffee
11:20-12:10 Talk 1: Daniel Palacín
12:15-13:30 Lunch
13:30-14:20 Talk 2: Andre Opris
14:30-15:20 Talk 3: Elías Baro
15:20-16:00 Coffee Break
16:00-16:50 Talk 4: 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" (arXiv:1801.08744), 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: TBA
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.
We will give a rough sketch of the following theorem: Let L be the language of fields and P be a predicate. Let R be an RCF and (R,P) a d-minimal structure. Under some model theoretic assumptions, we have that every definable C1 function on an open connected domain is the restriction of some L-definable function to this domain.