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 will be held in March 2001. We will have 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

will 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

will 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

will 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

will 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

will 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

will 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

will 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 will visit our Department for the whole month of March. Our seminar will meet (at least) three times a week. Daily seminars will be held March 19-22. For the program of the seminars, see the web page

Algebra Seminar

The following guests from abroad also have announced their visits in March:

Eberhard Becker (Dortmund, Germany),
Jaka Cimpric (Slowenia),
Deirdre Haskell (McMaster),
Alexander Prestel (Konstanz, Germany),
Niels Schwartz (Passau, Germany)

Here, you will find the dates of visits of our guests in March 2001:

Dates of Visits


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

Useful information


Dates of Visits

The Colloquiumfest Series

The Valuation Theory Home Page

Centre for Algebra, Logic and Computation

The Algebra and Logic Seminar


Last update: January 22, 2004 --------- created and maintained by Franz-Viktor Kuhlmann