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