COLLOQUIUMFEST
in honour of the 60th Birthday of
Murray Marshall
(organized by Franz-Viktor Kuhlmann and Salma Kuhlmann)
at the
Department of Mathematics and Statistics
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.

Friday, March 24, 2000

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

## On the Milnor Conjectures: History, Influence, Applications

(in particular, among the applications, Marshall's signature conjecture was emphasized)

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.

Universitaet Leipzig, Germany

gave a talk on

## The Classical Multidimensional Moment Problem

(and its relations and analogies to semialgebraic geometry, in particular to the Positivstellensatz, and Marshall's recent generalizations)

Abstract:
Let K be a closed subset of Rd. The K-moment problem asks under what conditions for a given multisequence s=(sn ; n \in N0d) there exists a positive Borel measure \mu on Rd 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 tn d\mu(t) for all n \in N0d.

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.

Saturday, March 25, 2000

10:15

Professor Ludwig Broecker
Universitaet Muenster, Germany

gave a talk on

## From Murrays miraculous lemma to real algebraic geometry

Abstract:
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

## A new bound for Polya's Theorem with applications to polynomials positive on polyhedra

Abstract:
This is joint work with Bruce Reznick.
Let R[X] := R[x1,...,xn]. Polya's Theorem says that if f \in R[X] is homogeneous and positive on the simplex
{(x1,..., xn) | xi \geq 0, \sumi xi = 1},
then for sufficiently large N \in N all the coefficients of
(x1 +...+ xn)N f(x1,...,xn)
are positive. We give an explicit bound for N, improving a previous bound by de Loera and Santos, and give an application to some special representations of polynomials positive on polyhedra.

2:00 p.m.

Professor Max Dickmann
Universite Paris VII, France

gave a talk on

## Proof of Murray's signature conjecture and generalizations

Abstract:
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

gave a talk on

## Branched covers and orderings of braid groups

Abstract:
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

## Valuation methods in group rings and skew fields

Abstract:
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

gave a talk on

## Valuation Theory of Exponential Hardy Fields

This is joint work with Salma Kuhlmann.
Abstract:
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

### Limericks for Murray Marshall

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

### Useful information

Last update: April 24, 2000