Programme

Welcome 10:30

10:45 - 11:30      Lou van den Dries: Hardy fields, transseries, and surreal numbers
This is about joint work with Matthias Aschenbrenner and Joris van der Hoeven. I will discuss how the three topics in the title are related.
Partly this is still conjectural. Valuation theory as in Paulo's Theorie des Valuations plays an important role in dealing with these structures.
Slides

11:45 - 12:30     Franziska Jahnke: Definable Henselian Valuations
Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years. We survey a number of these results which address the definability of concrete henselian valuations, the existence of definable henselian valuations on a given field, and questions of uniformity and quantifier complexity.

Lunch 12:30 -14:30

14:30 - 15:15     Sibylla Priess-Crampe: Asymptotic approximations - a short insight into my joint work with Paulo for our book on Ultrametric Spaces
A strictly contracting mapping from a spherically complete ultrametric space into itself has a unique fixed point. This fixed point can be approximated by a family of elements which are determined by the mapping. We will show how this general method can be applied to the case of ordinary differential equations of polynomial type over Hardy fields.

15:30 - 16:15     Paulo Ribenboim: Roots of Polynomials in Ultranormed Rings

Coffee and tea break 16:15 - 17:00

17:00 - 17:45     Daniel Bertrand: Pell equations over polynomial rings
Interest in these equations goes back to Abel. I will describe the new development brought by Masser and Zannier in the case of square free discriminants, in relation with their work on unlikely intersections in abelian schemes. The case of non separable discriminants leads to similar problems on generalized jacobians, where unexpected solutions sometimes appear. I will also discuss generalized Pell equations and their link with relative versions of the Mordell-Lang problem.
Slides

Birthday cocktail 18:00 - 20:00

This event is supported by IMJ-PRG, Projets Logique and Théorie des nombres, and by the GDR-STN.

   

Organizers: Zoé Chatzidakis, Françoise Delon, Max Dickmann, Danielle Gondard and Tamara Servi.