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
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.
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.
Professor Konrad Schmuedgen
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
Saskatoon
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
University of Saskatchewan
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
Last update: May 5, 2024