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Titles and Abstracts of Talks**

**Dr. Lou van den Dries
Urbana**

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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**

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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**

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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
**

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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**

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**Dr. Thomas Scanlon
Berkeley**

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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**

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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**

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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**

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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**

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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.

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