THE SECOND ANNUAL COLLOQUIUMFEST
in honour of the 60th Birthday of Murray Marshall (but wait,
this was already last year! Never mind, let's celebrate it
again.)
(organized by Franz-Viktor and Salma Kuhlmann and
Murray Marshall)
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
The Second Annual Colloquiumfest was held in March 2001.
We had seminar talks daily from Monday 19th until Thursday
22nd, two colloquium talks on Friday 23rd, and five talks on Saturday
24th.
Friday, March 23, 2001, Room 206 ARTS
4:00 p.m.
Professor
Victoria
Powers
Emory University, Atlanta, USA
gave a talk on
Real algebraic geometry and convex optimization
Abstract:
Semidefinite programming is an important tool for solving many problems
in applied math and engineering, for example in systems and control
theory. In this talk we will give an overview of the interaction of
concepts in real algebraic geometry and semidefinite programming. In
particular, we will talk about applications to convex optimization
problems. Much of the talk will be based on recent work of Pablo
Parrilo, who has developed practical methods for studying semidefinite
programming using ideas from real algebraic geometry. No prior knowledge
of semidefinite programming or convex optimization will be assumed.
5:00 p.m.
Professor
Claus
Scheiderer
University of Duisburg, Germany
gave a talk on
Sums of squares and the moment problem
Abstract:
The question whether a non-negative polynomial is always a sum of
squares of polynomials was raised in the 1880s by Minkowski and
answered by Hilbert. I'll first discuss the generalization of this
question to polynomial functions on affine real algebraic sets.
The hardest case is that of compact curves and surfaces. These
questions are directly related to the (multi-dimensional) moment
problem from analysis. The latter asks for a characterization of
the possible moment (multi-) sequences of positive Borel measures
with support in a given closed subset K of Rn.
The case when K is compact is solved completely by a theorem of
Schmuedgen. I will present these facts, and in the end try to
discuss a few recent results for non-compact K.
Coffee and cookies will be available in the lounge between 3:30 and
4:00 p.m.
7:00 p.m.
BANQUET
Marquis Hall
Saturday, March 24, 2001, Room 206 ARTS
(tentative schedule)
10:15 a.m.
Professor Max Dickmann
Universite Paris 7, France
gave a talk on
Bounds for the representation of quadratic forms
Abstract:
The (affirmative) solution to Marshall's signature conjecture for
Pythagorean fields implies that, for fixed integers n,m >= 1,
there is a uniform bound on the number of Pfister forms of degree n
over any Pythagorean field F necessary to represent (in the Witt ring
of F) any form of dimension m as a linear combination of such forms
with non-zero coefficients in F. "Uniform" means that the bound does
not depend either on the form nor on the field F; it is given by a
recursive function f of n and m. We single out a large class of
Pythagorean fields and, more generally, of reduced special groups for
which f has a simply exponential bound of the form cmn-1
(c a constant). Such a class is closed under certain - possibly
infinitary - operations which preserve Marshall's signature conjecture.
In the case of groups of finite stability index s, we obtain an upper
bound for f which is quadratic on [m/2n], where the
coefficient c depends on s.
11:15 a.m.
Markus Schweighofer
Konstanz, Germany
gave a talk on
Extension of Schmuedgen's Positivstellensatz to algebras of finite
transcendence degree
Abstract:
We investigate the iterated real holomorphy ring of rings as introduced
by Becker and Powers. First we give a new and simple proof for their
stationarity result. Then we prove the conjecture of Monnier saying that
Schmuedgen's Positivstellensatz holds true not only for affine algebras
but also for algebras of finite transcendence degree. From this it
follows that the stationary object of Becker and Powers is exactly the
archimedean hull of the subsemiring of sums of squares. As a corollary
we obtain a new proof of Marshall's generalization of Schmuedgen's
result to the non-compact case.
LUNCH BREAK
2:30 p.m. --- This talk is supported by the University of
Saskatchewan Role Model Speaker Fund
Professor Isabelle Bonnard
Angers, France
gave a talk on
Nash constructible functions
Abstract:
A Nash constructible function on a real algebraic set is defined as a
linear combination (with integer coefficients) of Euler caracteristic of
fibres of regular proper morphisms intersected with connected components
of algebraic sets. The aim of the talk is to prove that Nash
constructible functions on a compact set coincide with sums of signs of
semialgebraic arc-analytic functions.
3:30 p.m.
Raf Cluckers
Kathlieke Universiteit Leuven, Belgium
gave a talk on
Semi-algebraic p-adic geometry
Abstract:
Semi-algebraic p-adic geometry is the p-adic counterpart of real
semi-algebraic geometry. In both cases semi-algebraic sets have a
well-defined dimension which is invariant under semi-algebraic
isomorphisms and which corresponds to the algebro-geometric dimension of
the Zariski-closure. In the real case there is also an Euler
characteristic to the integers; this Euler characteristic together with
the dimension leads to a classification of the real semi-algebraic sets
up to semi-algebraic isomorphism. In the p-adic case, D. Haskell and R.
Cluckers proved that every (abtstract) Euler characteristic on the
p-adic semi-algebraic sets is trivial. Nevertheless, it was possible to
give a classification of p-adic semi-algebraic sets up to semi-algebraic
isomorphism.
4:30 p.m.
Matthias Aschenbrenner
Urbana, Illinois, USA
gave a talk on
Ideal membership in polynomial rings over the integers:
Kronecker's Problem
Abstract:
Given polynomials
f0(X), f1(X),..., fn(X) in Z[X],
X = (X1,..., XN), are there g1(X),...,
gn(X) in Z[X] such that f0 =
g1f1 +...+ gnfn? This is the
ideal membership problem for polynomial rings over the integers. It
constitutes a key problem in Kronecker's "finite type" mathematics. A
decision procedure has been known for about 40 years. More recently, the
method of Groebner bases has led to a procedure whose number of steps
could be explicitly bounded in terms of the size of the coefficients,
degrees of the fj's, and the number N of variables. While
for fixed N this upper bound is primitive recursive, as a function of
N it involves the notorious Ackermann function (and thus is not
primitive recursive).
In this talk, we will present a novel approach to this problem. We will
discuss the following three aspects:
(1) existence of bounds for the degrees and coefficients of
g1,..., gn (in terms of the degrees and
coefficients of f0,..., fn);
(2) decidability of ideal membership
by a primitive recursive algorithm;
(3) definability:
dependence on parameters, from an arithmetic-logical viewpoint. In
particular, our method yields bounds which drastically improve the
previously known ones.
Matthias Aschenbrenner and Markus Schweighofer visited our Department
for the whole month of March.
Last update: May 2, 2024
--------- created and maintained by Franz-Viktor Kuhlmann