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.

4:00 p.m.

**Professor
Victoria
Powers
Emory University, Atlanta, USA**

gave a talk on

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

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

Coffee and cookies will be available in the lounge between 3:30 and 4:00 p.m.

7:00 p.m.

Marquis Hall

10:15 a.m.

**Professor Max Dickmann
Universite Paris 7, France**

gave a talk on

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 cm

11:15 a.m.

**Markus Schweighofer
Konstanz, Germany**

gave a talk on

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.

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

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 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

Given polynomials f

(1)

(2)

(3)

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*