(organized by Franz-Viktor Kuhlmann and Salma Kuhlmann)

at the

Department of Mathematics and Statistics

University of Saskatchewan

106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

Phone: (306) 966-6081 - Fax: (306) 966-6086

**Dear Friends and Colleagues of Murray Marshall!**

We celebrated Murray Marshall's 60th birthday on
Friday March 24th and Saturday March 25th, 2000,
at the Department of Mathematics and Statistics in Saskatoon.
We had two colloquium talks on Friday afternoon, and six talks
on Saturday.

We were very pleased and honoured to have the following two speakers deliver the colloquium talks on Friday:

4:00 p.m.

**Professor Albrecht Pfister
Universitaet Mainz, Germany**

gave a talk on

**Abstract:**

In the first part of my talk I introduce some preliminary statements
about quadratic forms, Galois cohomology and algebraic K-theory which
are necessary to formulate the Milnor Conjectures. Then there will be
some metamathematical remarks about the impact of these conjectures.
The second part will outline the various attempts (from 1970 till
now) to prove the conjectures, it also contains several applications.

5:00 p.m.

**Professor Konrad Schmuedgen
Universitaet Leipzig, Germany**

gave a talk on

**Abstract:**

Let K be a closed subset of **R**^{d}. The K-moment problem
asks under what conditions for a given multisequence s=(s_{n} ;
n \in **N**_{0}^{d}) there exists a positive Borel
measure \mu on **R**^{d} such that the support of s is
contained in K and s is the moment sequence of the measure \mu, that is,
s_n = \int t^{n} d\mu(t) for all n \in
**N**_{0}^{d}.

After a brief excurse to the historical roots two approaches to this problem are explained. Particular emphasis is placed on the case when K is a semialgebraic set. Then there is a close interrelation between the K-moment problem and the archimedean Positivstellensatz for K. For a compact semialgebraic set K, a solution of the K-moment problem can be given by using the Positivstellensatz of G. Stengle and conversely the archimedean Positivstellensatz can be proved by means of the K-moment problem. Two recent variants of the archimedean Positivstellensatz (due to M. Marshall and due to T. Jacobi and A. Prestel) are discussed. Some results for non-compact sets K and some open problems are mentioned.

Professor Schmuedgen visited our department for two weeks, from March
17th to 31st; and Professor Pfister for one week, from March 21st
to March 28th.

10:15

**Professor Ludwig Broecker
Universitaet Muenster, Germany**

gave a talk on

The talk describes the development from the study of quadratic forms over formally real fields in the seventies to some modern aspects of real algebraic geometry. In particiular it includes some remarks on Marshalls work and beyond.

11:15

**Professor Victoria Powers
Emory University, USA**

gave a talk on

This is joint work with Bruce Reznick.

Let

2:00 p.m.

**Professor Max Dickmann
Universite Paris VII, France**

gave a talk on

In this talk I will outline and compare the proofs, by F. Miraglia (Sao Paulo, Brazil) and myself, of:

(1) Marshall's signature conjecture for quadratic forms over Pythagorean fields (Inventiones Math., 1998).

(2) Lam's generalization of (1) to arbitrary formally real fields (proved in February 1999, unpublished).

I will point out, as well, a consequence of (1) concerning the representation of forms of a given degree by linear combinations of Pfister forms of a given degree.

3:00 p.m.

**Jonathon Funk
Saskatoon**

gave a talk on

The concept of a branched cover can be used to obtain orderings of a braid group. The orderings obtained in this way are precisely the ones of ``finite type'', as described by B. Wiest and H. Short, ``Orderings of mapping class groups after Thurston''.

4:00 p.m.

**Professor Alexander Lichtman
University of Wisconsin-Parkside**

gave a talk on

We construct a family of discrete valuations in group rings of residually torsion free nilpotent groups and extend these valuations to the Malcev-Neumann power series skew fields of these group rings. We apply our results and methods for study of the universal fields of fractions of free algebras and the universal fields of fractions of the Magnus power series ring; we give a description of the centralizer of a non-central element in this skew field. We obtain new methods for constructing the universal fields of fractions for free algebras.

5:00 p.m.

**Professor Franz-Viktor Kuhlmann
University of Saskatchewan**

gave a talk on

Hardy Fields encode in an algebraic way the asymptotic behaviour of real-valued functions. We consider the Hardy Fields obtained from germs of polynomially bounded and exponentially bounded functions. We describe their value groups and residue fields with respect to convex valuations. We apply the results to various problems concerning the asymptotic behaviour of definable functions in o-minimal expansions of the reals.

There was a *limerick contest* in Murray's honour. You can see
the results at the web page

On the following web page, you will find information about the University, Saskatoon, Saskatchewan, and much more:

*Last update: April 24, 2000*