*This conference is organized by Salma
Kuhlmann and dedicated to
Professor Danielle Gondard, on the occasion of her 60th
Birthday*

**Confirmed Speakers: **

- Doris Augustin (Regensburg)

- Ron Brown (Hawai)

- Andreas Fischer (Saskatchewan)

- Pawel Gladki (Saskatchewan)

- Murray Marshall (Saskatchewan)

- Tim Netzer (Konstanz)

- Rajesh Pereira (Saskatchewan)

- Daniel Plaumann (Konstanz)

- Marcus Tressl (Regensburg)

** SCHEDULE**

**Thursday October 5 morning:**

**10:00--11:30: Marcus Tressl, Algebra and Logic Seminar
Room will be announced.
**

** **

**
ROOM ARTS 104:
Thursday October 5 afternoon:
4:00 to 4:10:
Opening and Greetings to Madame Gondard by Murray Marshall.
**

**
4:10--5:00 Ron Brown
5:05--5:35 Andreas Fischer
5:45--6:15 Pawel Gladki
6:20-- 7:10 Daniel Plaumann
**

**
Friday October 6 afternoon:
**

**
4:00--4:50 Tim Netzer
5:00--5:30 Doris Augustin
5:40--6:10 Rajesh Pereira
6:15--7:00 Murray Marshall
Closing Greetings
**

**
**

Speaker:
**Doris Augustin**

Title: The membership problem for preorders - the one-dimensional case

Abstract:

The membership problem for a preorder in the polynomial ring over a
real
closed field is solvable if the set of coefficients of the polynomials
which
belong to the preorder is weakly semialgebraic; this means: the
intersection of this set of coefficients with every finite dimensional
subspace of the polynomial ring is semialgebraic.
In my talk I will concentrate on finitely generated preorders in the
polynomial ring over the field of real numbers in one variable. By
describing the structure of preorders generated by one polynomial in
the
local power series ring at a point we will derive conditions which are
necessary and sufficient for a polynomial f to lie in the preorder
generated by another polynomial g if the basic closed semialgebraic set
generated by g is compact. These conditions imply that the membership
problem is solvable for preorders generated by one single polynomial.
For finitely generated preorders we are also able to prove that the
membership problem is solvable - again by looking at the structure of
these preorders in the local power series ring at a point.

Speaker: **Ron Brown**

Title: Lifting generalized orders on fields to field extensions

Abstract:

We will argue for the utility of Andre Weil's notion of an
absolute value on a field which can take the value infinity (ie.,
an "extended" absolute value) as a central concept of valuation theory.
In particular the properties a field Henselian with respect to an
extended absolute value and generalized orders and order closures
with respect to an extended absolute value will be discussed. (The
Henselizations at real extended absolute values are the HRRC fields
of Becker, Berr and Gondard, and the closures with respect to p-adic
and real extended absolute values are the p-adically closed and
real closed fields, respectively.) These notions allow a unification
of parts of the theories of formally p-adic and formally real fields,
including, for example, generalizations of place existence
theorems of Lang and of Prestel and Roquette. Two more recent
results wii be presented: an extension existence theorem for
generalized orders and a count of the number of extensions of an
order on a formally p-adic field to a formally p-adic algebraic
extension of that field.

Speaker: **Andreas Fischer
**

Title: John functions in o-minimal structures

Abstract

A John function is a continuously differentiable function $f:U-->R$
such that the product of the norm $\nabla f(x)$
and the distance of $x$ to the complement of $U$ is bounded.
We consider the following problem. Given an open subset $U$ of $R^n$.
Are there finitely many John functions $f_1,...,f_r$ such that
$dist(x,R^n\setminus U)\sum_{i=1}^r\norm{\nabla f_i(x)}$
is also bounded from below by a positive number?
If $U$ is definable in some o-minimal structure we can give an
affirmative answer, and in this case
the $f_i$ are logarithms of definable functions.

Speaker:** Pawel Gladki**

Title: The pp conjecture for the space of orderings of the field R(x,y).

Abstract:

In this talk we consider the space of orderings $(X_{\R(x,y)},
G_{\R(x,y)})$ of the field of rational functions over $\R$ in two
variables. It is shown that the pp conjecture fails to hold for such a
space; an example of a simple positive primitive formula which is not
product-free and one-related is investigated and it is proven, that
although the formula holds true for every finite subspace of
$(X_{\R(x,y)}, G_{\R(x,y)})$, it is false in general. This provides a
negative answer to one of the questions raised in: M. Marshall, {\em
Open questions in the theory of spaces of orderings}, J. Symbolic Logic
67 (2002), 341-352. In a certain sense this work is a continuation of
previous results presented in: P. Gladki, M. Marshall, {\em The pp
conjecture for spaces of orderings of rational conics}, to appear in J.
Algebra Appl., however here new, 'valuation theory free' methods are
developed and used.

Speaker: **Murray Marshall**

Title: New proofs of old results

Abstract:

A new short proof is given for Jacobi's extension of the
Kadison-Dubois Theorem. This yields a short uniform proof of
bothresults. The Kadison-Dubois Theorem and Jacobi's extension of it are
then
applied to give short proofs of results of Reznick and of Prestel and
Jacobi.

Speaker: **Tim Netzer **

Title: The Moment Property and Representations of Polynomials.

Abstract:

A finitely generated preordering in the real polynomial ring is said to
have the Strong Moment Property, if its closure with respect to the
finest
locally convex topology equals its saturation. In the literature, there
has been introduced a property which implies the Strong Moment
Property,
known as the "Double-Dagger Property". We discuss an example which
shows
that this property is strictly stronger than the Strong Moment
Property.
And unlike the Strong Moment Property, a preordering does not
necessarily
have it if all preorderings corresponding to the fibers of some bounded
polynomials have it.

Speaker:** Rajesh Pereira **

Title: Matrices over semirings and their applications.

Abstract:

A semiring is a set (R,+,x) with a multiplication and addition
which satisfies all of the axioms of a ring with identity except the
existence of additive inverses. We give many examples of semirings and
then discuss the properties of matrices over semirings including some
results and conjectures on diagonally dominant matrices over semirings.
(The original portion of this material in this talk is joint work with
M.
A. Vali).

Speaker: **Daniel Plaumann**

Title: Sums of squares on reducible real curves

Abstract:

Scheiderer has classified all irreducible real affine curves for which
every non-negative regular function is a sum of squares in the
coordinate ring. We show how to extend some of these results to
reducible curves. We also discuss the moment problem for reducible
curves and applications to the moment problem in dimension 2.

Speaker:** Marcus Tressl **

Title: Weakly semi algebraic orderings and generic extensions
of orderings on polynomial rings".

Abstract:

A weakly semi-algebraic ordering of the polynomial ring
over a real closed field in several indeterminates is
an ordering for which the membership problem is solvable, i.e.
for each given degree, the set of coefficient vectors of polynomials
contained in the ordering of that degree, is a semi-algebraic set.
The Marker-Steinhorn Theorem implies that in the case
where R is the real field, every ordering is weakly semi-algebraic.
This can be proved by an induction on the dimension, based on the
analysis of
generic extensions
of orderings (so called heirs and coheirs), where the
one-dimensional
case is trivial.
In the talk I will explain this strategy, characterize
generic extensions of certain orderings and discuss possible
applications to the mebership problem for orderings in the case
where R is an arbitrary real closed field.

*updated January 21, 2007*