Short talks (by students and postdocs):
- Anna Blaszczok (Silesia Katowice): On immediate extensions of valued fields
A valued field extension is called immediate if the corresponding value group and residue field extensions are trivial. In such a case the value group and the residue field extensions carry minimal information about the extension of valued fields. This makes the structure of immediate extensions complicated. A better understanding of such extensions turned out to be important for questions in algebraic geometry,
real algebra and the model theory of valued fields.
We present results which describe the structure of maximal immediate extensions of valued fields. We give conditions for a valued field to admit maximal immediate extensions of infinite transcendence degree. We further answer the question when an infinite algebraic extension (L, v) of a maximal field (K, v) can be again maximal.
We also investigate the problem of uniqueness of maximal immediate extensions. Kaplansky proved that under a certain condition, which he called "hypothesis A", a valued field admits maximal immediate extensions which are unique up to isomorphism. We study a more general case, omitting one of the conditions of hypothesis A. We describe the structure of maximal immediate extensions of valued fields under such weaker assumptions. We consider also an opposite case and show that there is a class of valued fields which admit two nonisomorphic maximal immediate extensions of completely different structure.
- Erin Caulfield (UIUC): On expansions of the real field by complex subgroups
Abstract: We construct a class of finite rank multiplicative subgroups of the complex numbers such that the expansion of the real field by such a group is model-theoretically well-behaved. As an application we show that a classification of expansions of the real field by cyclic multiplicative subgroups of the complex numbers due to Hieronymi does not even extend to expansions by subgroups with two generators.
- Saskia Chambille (Leuven): Cell decomposition in P-minimal structures
Motivated by the theory of o-minimality for real closed fields, Haskell and Macpherson defined an analogue for p-adically closed fields, called P-minimality. These two concepts of minimality are however not as alike as one might have hoped for. I will discuss some of the similarities and differences between the o-minimal and the P-minimal setting, with an emphasis on the difficulties that one encounters in defining a notion of cells in P-minimal structures. I will also give an idea how we deal with these difficulties in the cell decomposition that we are developing.
(Joint work with Pablo Cubides Kovacsics and Eva Leenknegt.)
- Derya Çiray (Konstanz): Mild parametrization in o-minimal structures
The application of mild parametrization, which is a smooth parametrization with some control on the derivatives, was first introduced 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. Further he has shown that mild parametrization with sufficient uniformity in parameters would be sufficient to establish Wilkie's conjecture.
In this talk, I will discuss whether certain o-minimal expansions of the reals admit mild parametrization or not.
- Juan de Vicente (Madrid): Simply Connected Locally Nash Groups
We study locally Nash groups over the real field, which appear naturally when considering universal coverings of real algebraic groups and which are interesting in the Nash category, which in turn it is part of real algebraic and analytic geometry .
A classification of one-dimensional locally Nash groups was given by Madden and Stanton in . We give a description of two-dimensional
simply-connected abelian locally Nash-groups based on results of Painlevé on meromorphic functions admitting an Algebraic Addition
(Joint work with E. Baro and M. Otero.)
 Masahiro Shiota. Nash manifolds, volume 1269 of Lecture Notes in
Mathematics. Springer-Verlag, Berlin, 1987.
 James J. Madden and Charles M. Stanton. One-dimensional Nash
groups. Pacific J. Math., 154(2):331-344, 1992.
 E. Baro, J. de Vicente, M. Otero, Two-dimensional simply connected
locally Nash groups, http://arxiv.org/abs/1506.00405
(Leeds): The Jacobian property in T-convex fields
Abstract: I will discuss the interest in showing that an n-dimensional version of the Jacobian property used in motivic integration holds in T-convex fields. This interest includes mainly the existence of non-archimedean Whitney stratifications for definable sets. This n-dimensional Jacobian property has been proved for other classes of valued fields and I will discuss how similar techniques should provide it for T-convex fields whenever T is a polynomially bounded o-minimal theory.