THE FOURTH ANNUAL COLLOQUIUMFEST


Titles and Abstracts of Talks



Dr. Lou van den Dries
Urbana

will talk on

Solving linear differential equations in H-fields

Abstract:
I will report on joint work in progress with Matthias Aschenbrenner and Joris van der Hoeven. An H-field is a field with an ordering and a derivation subject to some compatibilities. Hardy fields and fields of transseries are H-fields. An optimistic conjecture is that each H-field can be embedded, as ordered differential field, into a field of transseries, analogous to the fact that each valued field of equal characteristic zero can be embedded as valued field into a field of generalized power series. As a step towards this conjecture we wish to understand which algebraic differential equations over a given H-field can be solved in suitable H-field extensions. This understanding can be achieved for linear differential equations.


Dr. Philip Ehrlich
Ohio

will talk on

The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small

Abstract:
This talk will provide a brief introduction to the theory of surreal numbers, an overview of some of my mathematical work on the subject, and insight into the historical and philosophical motivation of that work.


Dr. Tobias Kaiser
Urbana

will talk on

Energy and capacity in o-minimal structures

Abstract:
We want to show that capacity resp. energy is a tame concept in o-minimal structures: We proof that the capacity of a definable set equals the capacity of its closure. Moreover we show that the capacity-density exists in o-minimal structures and we give some connections to the volume-density.


Dr. Margarita Otero

will talk on

Examples of transfer for o-minimal fields

Abstract:
It is well known that many results of real semialgebraic geometry generalize to any real closed field. In o-minimal geometry, based on an o-minimal expansion of an ordered field, we do not have an analogue of the Tarski-Seidenberg theorem, and even in semialgebraic geometry sometimes the concepts involved do not allow to apply this transfer principle. We illustrate the ``transfer approach'' in o-minimal geometry: once few basic transfer principles are proved, many other results admit an easy transfer from the corresponding result over the reals.

(Joint work with Alessandro Berarducci.)


Dr. Daniel Pitteloud
Lausanne

will talk on

Some Algebraic Properties in Rings of Generalized Power Series

Abstract (postscript file)


Dr. Thomas Scanlon
Berkeley

will talk on

Rosy pseudo-real closed fields and closed ordered differential fields

Abstract:
We discuss some recent work of A. Onshuus on the model theory of pseudo-real closed fields and M. Singer's theory of closed ordered differential fields. Specifically, we show that these fields fit into the context of rosy theories, and are, thus, equipped with a good independence theory and dimension theory.


Dr. Patrick Simonetta
Paris 7

will talk on

Non-abelian C-minimal groups

Abstract:
The notion of C-minimality, introduced by D. Macpherson and C. Steinhorn, provides a natural setting for the study of algebraically closed valued fields and some valued groups. This notion can be seen as a variant of o-minimality where the order is replaced by a ternary relation satisfying some specific axioms. However, C-minimal groups are far less understood than the o-minimal ones. In this talk we will study the structure of non-abelian C-minimal valued groups. After giving an example wich is not virtually abelian we will prove that C-minimal valued groups are virtually nilpotent. We will also give further information about C-minimal valued groups for which the class of nilpotency is not 2.


Dr. Patrick Speissegger
Madison

will talk on

Hausdorff limits in o-minimal expansions of the real field

Abstract:
I will outline a geometric proof of the following well-known theorem for o-minimal expansions of the real field: the Hausdorff limits of a compact, definable family of sets are definable.

(Joint work with Jean-Marie Lion.)


Dr. Charles Steinhorn
Vassar College

will talk on

On tame expansions of densely ordered structures

Abstract:
This is joint work with Chris Miller, and builds on work in [Miller-Speissegger 1998, Fund. Math.]. Let $R$ be an ordered structure and let the open core $R^{\circ}$ of $R$ be the reduct of this structure whose basic relations are the open definable sets in $R$ in any number of variables. We consider (model-theoretic) properties that imply that $R^{\circ}$ is o-minimal and several related issues. Examples also will be discussed.


Dr. Marcus Tressl
Regensburg

will talk on

Completion in Stages of o-minimal Structures

Abstract:
For an o-minimal expansion R of a real closed field and a set V of T-convex valuation rings, we construct a ``pseudo completion'' with respect to V. This is an elementary extension S of R generated by all completions of all the residue fields of valuation rings from V, when these completions are embedded into a big elementary extension of R. We'll show that S does not depend on the various embeddings up to an R-isomorphism. For polynomially bounded R we can iterate the construction of the pseudo completion in order to get a completion in stages S of R with respect to V. S is the ``smallest" extension of R such that all residue fields of the unique extensions of all valuations from V to S are complete.


The Fourth Annual Colloquiumfest


Last update: February 24, 2003 --------- created and maintained by Franz-Viktor Kuhlmann