Donau-Rhein Modelltheorie und

Anwendungen

Anwendungen

3rd meeting

Basel - July 13th 2018

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.

Here is a link to the webpage of the program.

The event will take place in Hörsaal -101, Alte Universität, Rheinsprung 9, 4051 Basel.

**Organisers:**

Universität Basel: Philipp Habegger;

Albert-Ludwigs-Universität Freiburg: Amador Martin-Pizarro;

Universität Konstanz: Salma Kuhlmann, Margaret Thomas;

Universität Passau: Tobias Kaiser.

**Local organisers:**

**Speakers:**

Fabrizio Barroero (Basel)

Pantelis Eleftheriou (Konstanz)

Arno Fehm (Dresden)

Gabriel Lehéricy (Konstanz)

**Schedule**

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

11:30-12:15 Talk: Fabrizio Barroero

12:15-13:30 Lunch

13:30-14:15 Talk 2: Arno Fehm

14:25-15:10 Talk 3: Pantelis Eleftheriou

15:10-16:00 Break (coffee/tea)

16:00-16:45 Talk 4: Gabriel Lehéricy

18:00 Dinner

**Titles and abstracts**

Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible.

After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.

Pantelis Eleftheriou (Konstanz): Counting rational points in tame expansions of o-minimal structures

The Pila-Wilkie theorem states that if a set X is definable in an o-minimal structure M over the real field R and contains "many" rational points, then it contains an infinite semialgebraic set. We extend this theorem to an expansion (M, P) of M by a dense set P, which is either an elementary substructure of M, or it is independent, as follows. If X is definable in (M, P) and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is 0-definable in (R, P).

Arno Fehm (Dresden): Pseudo-algebraic fields are dense in their real and p-adic closures

In joint work with Sylvy Anscombe and Philip Dittmann we show that every model of the theory of fields that are algebraic over the rationals is dense in all its real closures and all its p-adic closures. In this talk I will explain the interplay of various ingredients from model theory, number theory and arithmetic, and I will present some applications of this result.

Gabriel Lehéricy (Konstanz): The differential rank of a differential-valued field

The rank is an important invariant of a valued field and has several characterizations. Recently, several notions of ranks for valued fields endowed with an operator have appeared, for example the exponential rank of an exponential ordered field and the difference rank of a difference valued field. In this talk, we will introduce a notion of differential rank for differential-valued fields. We will see how far we can push the analogy between the differential rank and the previous notions of ranks. We will also see how one can realize any totally ordered set as the differential rank of some field of generalized power series endowed with a derivation. This is joint work with Salma Kuhlmann.

**Where**

**Accommodation**

Here you can find a list of suggested Hotels with prices.

**Travel information**

The talks are located at the Alte Universität building in Basel. To get there from

- the train station Basel SBB: take Tram 11 towards St. Louis Grenze and exit at the stop Schifflände (travel time is approximately 10 minutes)
- the train station Basel Badischer Bahnhof: take Tram 6 towards Allschwil, Dorf and exit at the stop Schifflände (travel time is approximately 10 minutes)
- the EuroAirport Basel Mulhouse Freiburg: take Bus 50 towards Basel SBB, change at Kannenfeldplatz to Bus 31 towards Riehen, Friedhof am Hörnli or Bus 38 towards Wyhlen, Siedlung and exit at the stop Schifflände (travel time is approximately 15 minutes)

Please note that the power plugs and sockets from Switzerland may differ from the ones of your home country.