ALGEBRA AND LOGIC GROUP
of the
Mathematical Sciences Group
Phone: (306) 966-6081 - Fax: (306) 966-6086

Past Talks of Our Guests

Friday, November 19, 1997, 4:00 p.m.

Professor Sibylla Priess-Crampe
Ludwig Maximillian Universität München, Germany

gave a talk in the Department Colloquium on

## Fixed Point and Coincidence Theorems for Ultrametric Spaces

Abstract:
An ultrametric space (X,d,G) is a set X with an ultrametric distance functions d from X to G , where G is a partially ordered set with a smallest element 0. d has the same properties as a metric but instead of the triangle inequality the following one:

For all g of G , if d(x,y) and d(y,z) are at most g then also d(x,z) is at most g .

A special role for ultrametric spaces play spherically complete ultrametric spaces. "Sperically complete" corresponds to the property "maximal valued" for valued fields.

For spherically complete ultrametric spaces there holds a fixed point theorem which looks like Banach's fixed point theorem for metric spaces. One has furthermore a generalization of this singlevalued fixed point theorem to multivalued mappings (again as it is the case in the metric situation). Some hints to applications of the theorems will be given.

Friday, February 13, 1998, 4:00 p.m.

Professor Hans Brungs
University of Alberta

gave a talk in the Department Colloquium on

## Extending Valuation Rings

Abstract:
For any field extension K/F every valuation ring of F can be extended to K. There are three competing definitions for noncommutative valuation rings and their extension properties will be discussed; in particular extensions of valuation rings in the center of a finite dimensional division algebra. These results can be used to show that every rooted tree can be realized as the graph of all valuation rings of a finite dimensional division algebra.

Friday, March 6, 1998, 4:00 p.m.

Professor Niels Schwartz
University of Passau, Germany

gave a talk in the Department Colloquium on

## From rings of continuous functions to real closed rings

Abstract:
Rings of continuous functions into the real numbers are obviously an important tool in topology. Therefore they have been studied intensely from many points of view. Their algebraic analysis is a difficult problem, partly because, as a category, rings of continuous functions have very poor properties. There are only very few ring theoretic constructions that produce rings of continuous functions when applied to such rings. A related fact is that there is no axiomatization in first order model theory. One major topic in real geometry is the investigation of semi-algebraic spaces. A semi-algebraic space is always defined with reference to a real closed field. Similar to rings of continuous functions on topological spaces, the functions that are most useful for the analysis of semi-algebraic spaces are the rings of continuous semi-algebraic functions into the real closed fields associated with the spaces. These rings have the same poor category theoretic and model theoretic properties as rings of continuous functions. But there is a larger class of rings, called real closed rings, which is the smallest axiomatizable class of rings containing the rings of continuous semi-algebraic functions. Thus, real closed rings can be studied by model theoretic methods. The category theoretic properties are also very favorable; the class is closed with respect to o large number of the usual ring theoretic constructions. All rings of continuous functions are real closed. In fact, currently the real closed rings are the smallest known class of rings that contains the rings of continuous functions, is axiomatizable and is closed under so many ring theoretic constructions. Thus, although they were first introduced as a tool for real geometry, the real closed rings also have a potential for applications in topology.

Friday, September 4, 1998, 4:00 p.m.

Professor M.R. Gonzalez Dorrego

gave a talk in the Department Colloquium on

## Curves on a Kummer Surface

Abstract:
Let K be an algebraically closed field of characteristic different from 2. A Kummer surface in P^3_K is a quartic surface with 16 ordinary double points. We shall discuss the geometry of a Kummer surface in P^3_K and (16,6) configurations. We shall describe curves on a general Kummer surface.

Wednesday, September 9, 1998, 4:00 p.m.

Professor Mark Spivakovsky
University of Toronto

gave a talk in the Department Colloquium on

## Resolution of Singularities

Abstract:
The problem of resolution of singularities and related questions account for a sizable portion of all the research in algebraic geometry in this century. The purpose of this talk is to introduce the audience to the subject of resolution of singularities. We will state the desingularization problem and outline the methods used in its solution. We will give a proof of resolution of singularities of curves, but with a view to the general case: as much as possible, we will point out how to extend all the concepts and results to higher dimensions.

Friday, October 2, 1998, 4:00 p.m.

Professor Alexander Prestel
Universität Konstanz, Germany

gave a talk in the Department Colloquium on

## Positive definite polynomials and the Moment Problem

Abstract:
The Moment Problem is asking for conditions on a linear form on the algebra of real polynomials in n variables to be induced by some positive Borel measure with compact support. This problem is closely related to certain representations for real polynomials being positive on a compact semi-algebraic set. We shall be talking about this conection.

5:00 p.m.

Professor Eberhard Becker
Universität Dortmund, Germany

gave a talk in the Department Colloquium on

## Symbolic solving of systems of polynomial equations with finitely many solutions

Abstract:
The talk will report on recent developments in symbolic computation which allows solution of systems of polynomial equations and inequalities in several variables provided the system has only finitely many solutions over the complex field C. The solutions in C^n are encoded as the roots of a univariante polynomial in T, and the coordinates of the solutions are given as rational functions in T. One of the main methods is the Buchberger Algorithm for computing Gröbner bases. The talk will also present some examples.

Monday, October 5, 1998, 4:00 p.m.

Professor Bernard Teissier
Ecole Normale Superieure, Paris

gave a talk in the Department Colloquium on

## Valuations and Binomial Ideals

Abstract:
The topic of the lecture is resolution of singularities; I will show how to resolve singularities in the very special case of irreducible algebraic varieties defined by binomial equations, and give some ideas of a strategy to reduce to this case the proof of local uniformization, which is a local form of resolution of singularities.

Friday, July 16, 1999, 4:00 p.m.

Professor Sudesh Kaur Khanduja
Punjab University, Chandigarh, India

gave a talk in the Department Colloquium on the

## Generalized Hensel's Lemma

Abstract:
We intend to talk about a generalization of the classical Hensel's Lemma and some of its applications using prolongations of a valuation v, defined on a field K to a simple transcendental extension K(x) of K.

Friday, July 23, 1999, 4:00 p.m.

Professor Peter Roquette
Universität Heidelberg

gave a talk in the Department Colloquium on

## The Grunwald and Wang Story

Abstract:
(1) Explain the statement of the famous Grunwald existence theorem.
(2) Report on its origin and its significance within Number Theory.
(3) Tell about the shock to the mathematical community when Wang found a counter example, and about the attempts to mend matters.
(4) Reinterpretation of the (corrected) theorem within valuation theory.

Friday, July 30, 1999, 7:30 p.m.

Professor Paulo Ribenboim
Queens University, Kingston

gave a public lecture on

## These marvelous prime numbers

### This lecture was supported by the Special Lectureship Fund of the University of Saskatchewan.

Friday, August 6, 1999, 4:00 p.m.

Professor Hans Schoutens
Wesleyan University, Middletown

gave a talk in the Department Colloquium on

## The image of an analytic map; the ultrametric case

Abstract:
Consider an analytic map f:Y->X (either complex, real or rigid analytic). We want to describe the image I=Im f of f viewed as a subset of X. Unfortunately, I is not necessarily a semi-analytic set. With a semi-analytic set we mean a subset given by (a Boolean combination of) norm-inequalities |p(x)|<|q(x)| between analytic functions p and q. We [S. + Gardener] show that in the rigid case, I can be described though by norm-inequalities if we allow for the functions p,q,... in this description to be (iterated) quotients of analytic functions. Moreover, after a well-choosen blowing up process \pi:\tilde X -> X, we can arrange for \pi^{-1}(I) to become semi-analytic. In fact, we show how to flatten the original map f by blow ups, after which we apply the Theorem of Raynaud which asserts that the image of a flat (affinoid) map is always semi-analytic. To give an idea of the proof, which is a rigid analytic adaptation of Hironaka's proof for the real case, I will work out the Osgood example, where f(s,t)=(s, st, s exp(t)).

Friday, September 10, 1999, 4:00 p.m.

Professor Michael Baake
University of Tübingen

gave a talk in the Department Colloquium on

## Which distributions of matter diffract

Abstract:
Mathematical diffraction theory is concerned with the spectral analysis of unbounded complex measures, and has important applications in crystallography and diffraction physics, and to the understanding of quasicrystals in particular.

After introducing the basic questions and methods, I will briefly survey the present state of affairs by means of characteristic examples from lattices, quasicrystals, and stochastic structures such as lattice gases. My emphasis will be on cases that can be treated explicitly, but without loss of rigour, and I'll also point out some open questions.

Friday, February 18, 2000, 4:00 p.m.

Professor Alfred Weiss
University of Alberta, Edmonton

gave a talk in the Department Colloquium on

## L-values and multiplicative Galois module structure

Abstract:
Zeta functions 'know' much about the arithmetic of their algebraic number field. Examples of this phenomenon are the analytic class number formula and the Main Conjecture of Iwasawa theory. This appears to persist also 'relatively', i.e., for the Galois action on an extension of number fields. An example here is the Frohlich-Taylor theory of Galois structure of the ring of integers when the extension is tame. The multiplicative analogue of this, the Galois structure of units, seems to continue the pattern in a suitable sense. Proofs of this seem to require strengthening aspects of all of the examples above. Perhaps this is not surprising as the Galois structure of units is at the core of class field theory.

Friday, March 24, 2000, 4:00 p.m.

Professor Albrecht Pfister
Universität Mainz, Germany

gave a talk in the First Colloquiumfest 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.

Universität Leipzig, Germany

gave a talk in the First Colloquiumfest 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.

Saturday, March 25, 2000, 10:15

Professor Ludwig Bröcker
Universität Münster, Germany

gave a talk in the First Colloquiumfest 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 in the First Colloquiumfest 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 in the First Colloquiumfest 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.

Dr. Jonathon Funk

gave a talk in the First Colloquiumfest 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 in the First Colloquiumfest 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.

Friday, September 15, 2000, Room 206 ARTS
4:00 p.m.

Herve Perdry
Universite de Franche-Comte, Besancon, France

gave a talk in the Department Colloquium on

## A constructive approach to Henselian fields

Abstract:
We briefly describe what constructive mathematics is. We outline the construction of the real closure of an ordered field. Then we give a classical background about valued fields and in particular, Henselian fields. After that, we give a construction for the Henselisation of a valued field, as well as constructive proofs of classical theorems about valued fields.

Friday, September 22, 2000, 4:00 p.m.

Professor Niels Schwartz
University of Passau, Germany

gave a talk in the Department Colloquium on

## The role of partially ordered rings in real algebraic geometry

Abstract:
Real algebraic geometry is an old topic, but has grown only recently into being a major special branch of algebraic geometry. Therefore many of its basic tools and techniques still need to be identified and developed. One algebraic structure that is going to be fundamental in real geometry are partially ordered rings. It will be shown in the lecture why this is the case. The whole range of future applications is not foreseeable at present, but some substantial uses that have already been recognized will be pointed out.

Thursday, September 28, 2000, 2:30 pm

Professor Alexander Prestel
University of Konstanz, Germany

gave a talk in the Algebra Seminar on

## Positive Polynomials over non-archimedean Fields

In September and October of 2000

Herve Perdry
Besancon, France

gave talks in the Algebra Seminar on

## Explicit Construction of the Henselization of a Valued Field

November 22, 2000

Professor Danielle Gondard
Universite Paris VI, France

gave a talk in the Algebra Seminar on

## From Hilbert's 17th problem to valuation fans

In January of 2001

Dr. Jonathan Funk

gave talks in the Algebra Seminar on

## Inverse Semigroups and Order Etendue

Monday, March 5, 2001, 10:30 a.m.

Matthias Aschenbrenner
Urbana, Illinois, USA

gave a talk in the Algebra Seminar on

## Asymptotic Couples, H-Fields and their Liouville Extensions, I

Thursday, March 8, 2001, 2:30 p.m.

Matthias Aschenbrenner
Urbana, Illinois, USA

gave a talk in the Algebra Seminar on

## Asymptotic Couples, H-Fields and their Liouville Extensions, II

Friday, March 9, 2001, 10:30 a.m.

Matthias Aschenbrenner
Urbana, Illinois, USA

gave a talk in the Algebra Seminar on

## Asymptotic Couples, H-Fields and their Liouville Extensions, III

Wednesday March 14, 2001, 10:30 a.m.

Markus Schweighofer
Konstanz, Germany

gave a talk in the Algebra Seminar on

## Extending Schmüdgen's Theorem to non-compact varieties, I

Friday March 16, 2001, 10:30 a.m.

Markus Schweighofer
Konstanz, Germany

gave a talk in the Algebra Seminar on

## Extending Schmüdgen's Theorem to non-compact varieties, II

Friday, March 16, 2001, 4:00 p.m.

Professor Eberhard Becker
University of Dortmund, Germany

gave a talk in the Department Colloquium on

## The cone of positive polynomials - from an optimization point of view

Abstract:
Researchers in optimization are interested in the cone of positive polynomials (up to a certain degree). In particular, the search for a "concrete" barrier function for the entire cone or for subcones is one of the main tasks. On the other hand, algebraists develop an interest in the concepts of modern optimization to study the above cone from the point of view of real algebra. The talk outlines some of the few results together with an application to finding minima of polynomials on compact basic closed semi-algebraic sets.

Monday, March 19, 2001, 10:30 a.m.

Professor Jaka Cimpric
Ljubljana, Slowenia

gave a talk in the Algebra Seminar on

## Artin-Schreier theory for semigroups

Tuesday, March 20, 2001, 1:00 p.m.

Professor Claus Scheiderer
University of Duisburg, Germany

gave a talk in the Algebra Seminar on

## The moment problem for non-compact semialgebraic sets, I

Wednesday, March 21, 2001, 10:30 a.m.

Professor Claus Scheiderer
University of Duisburg, Germany

gave a talk in the Algebra Seminar on

## The moment problem for non-compact semialgebraic sets, II

Thursday, March 22, 2001, 2:30 p.m.

Professor Victoria Powers
Emory University, Atlanta, USA

gave a talk in the Algebra Seminar on

## Constructive approaches to Hilbert's Theorem on ternary quartics

Friday, March 23, 2001, 10:30 a.m.

Professor Alexander Prestel
Konstanz, Germany

gave a talk in the Algebra Seminar on

## Representation theorems for archimedian' rings

11:30 a.m.

McMaster University

gave a talk in the Algebra Seminar on

## Elimination of imaginaries in Algebraically closed valued fields

Friday, March 23, 2001, 4:00 p.m.

Professor Victoria Powers
Emory University, Atlanta, USA

gave a talk in the Second Colloquiumfest 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 in the Second Colloquiumfest 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 Schmüdgen. I will present these facts, and in the end try to discuss a few recent results for non-compact K.

Saturday, March 24, 2001, 10:15 a.m.

Professor Max Dickmann
Universite Paris 7, France

gave a talk in the Second Colloquiumfest 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 in the Second Colloquiumfest on

## Extension of Schmüdgen'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 Schmüdgen'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 Schmüdgen's result to the non-compact case.

2:30 p.m. --- This talk was supported by the University of Saskatchewan Role Model Speaker Fund

Professor Isabelle Bonnard
Angers, France

gave a talk in the Second Colloquiumfest 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 in the Second Colloquiumfest on

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 in the Second Colloquiumfest 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 Gröbner 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.

Saturday, June 2, 2001

Professor Michael Makkai
McGill University, Montreal

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Logic and D. G. Quillen's homotopical algebra

Abstract:
In [Logic Colloquium '95, Lecture Notes of Logic 11, Springer, 1998; 153-190] the author introduced First Order Logic with Dependent Sorts (FOLDS) for certain foundational purposes. The main semantical feature of FOLDS is the presence, associated with any given FOLDS signature L, of a concept called L-equivalence, taking the place of the usual notion of isomorphism of L-structures. In this talk, modeltheoretical results for FOLDS, of both of a general and of an applied nature, will be presented. In particular, connections with Quillen's classical approach to homotopy theory via the so-called model categories will be developed. An example of the connections is the fact that L-equivalence, with a suitable L, for simplicial Kan-complexes is the same as homotopy equivalence.

Professor Ross Willard
Department of Pure Mathematics, Waterloo

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Palyutin's h-formulas and a problem from universal algebra

Abstract:
In this talk we describe a nice syntactic characterization of a structural property of general algebras (the strict refinement property) in terms of certain formulas (h-formulas) defined by E. A. Palyutin in Categorical Horn classes, I. Algebra and Logic 19(1980), 377-400.

Professor Toniann Pitassi
University of Toronto

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Topics in propositional proof complexity

McMaster University and Fields Institute

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## The stable forking conjecture

Ziv Shami
McMaster University and Fields Institute

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Towards a binding group theorem for simple theories

Abstract:
Let T be simple. Let p Î S(Æ) be internal in Q Î L and suppose the global algebraic closure is weakly transitive on QC, that is DCL(aclR(Q)) Í ACL(QCÈA) for every one-to-finite definable relation R defined over A (where ACL(S) (DCL(S)) for a set S denotes those elements (in Ceq) which have finite orbit (an orbit of size 1 resp.) with respect to the action of Aut(Ceq/QC). aclR(Q)={b | | R(b,[(c)]) for some tuple [`(c)] from QC}). Then Aut(p/Q)={s| pC | s Î Aut(C/QC) } with its action on pC is type-definable in Ceq over Æ. If p is Q-internal via p-regular generic parameter (i.e. every forking extension of the type of the generic parameter is orthogonal to p) then Aut(p/Q) is direct-limit-definable in Ceq over Æ and its action on pC is definable. This is a joint work with B. Hart.

Sunday, June 3, 2001

Professor Robert E. Woodrow
University of Calgary

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Absorbing sets in arc-coloured tournaments

Abstract:
Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of T absorbing if from each vertex of T no in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches based on notions from the Theory of Relations. Most of the work deals with finite tournaments, but extensions to the infinite are also discussed.

Professor Jim Loveys
McGill University, Montreal

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Tiny models of strongly minimal theories

Monday, June 4, 2001

Professor Zoe Chatzidakis
CNRS, Universit‚ Paris 7, France

gave a plenary talk and a session talk in the CMS Summer 2001 Meeting:

## Model theory of generic difference fields

Abstract:
A difference field is a field with a distinguished automorphism. A generic difference field is a difference field such that every system of difference equations which has a solution in a difference field extension, has a solution in the field.

In the first part of the talk I will state the main model-theoretic results obtained on these fields and explain their significance and importance. In the second part of the talk, I will mention some applications obtained by Hrushovski to the solution of diophantine problems (e.g., the Manin-Mumford conjecture and the Jacobi conjecture for difference fields). I will also mention some intriguing questions, which lie at the boundary of model theory and diophantine geometry.

Professor Chris Miller
Ohio State University, Columbus, Ohio

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Hausdorff dimension, analytic sets and transcendence

Abstract:
Every analytic (in the sense of descriptive set theory) set of real numbers having positive Hausdorff dimension contains a transcendence base. Equivalently, every analytic proper real-closed subfield of the reals has Hausdorff dimension zero. (Joint work with G. A. Edgar.)

Reed Solomon

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## The effective content of spaces of orders on groups and fields

Abstract:
For a formally real field F, the space of orders X(F) has a natural topology which makes it a Boolean space (compact, Hausdorff, and totally disconnected). Craven proved that given any Boolean space B, there is a formally real field F such that B is homeomorphic to X(F). Metakides and Nerode examined the computable content of this correspondence by considering effectively closed subspaces of the Cantor space (called P01 classes) in place of Boolean spaces. They showed that for every P01 class C, there is a computable field F such that C is homeomorphic to X(F) by a Turing degree preserving homeomorphism. Downey and Kurtz asked whether a similar correspondence holds for computable orderable groups. In this talk, we will discuss these results as well as give a negative answer to the Downey and Kurtz question for the classes of torsion-free abelian and nilpotent groups.

Professor Hans Schoutens
Rutgers University, New Brunswick

gave a talk in the Special Session on Model Theoretic Algebra at the CMS Summer 2001 Meeting:

## Determining the number of equations of an affine curve

Abstract:
It is in general a very hard problem to find the minimal number of generators of an ideal in a finitely generated algebra over a field. In fact, there can be no computer algebra system that calculates this number exactly (I will briefly discuss a counterexample due to SCHMIDT). The obstruction lies in the possible unboundedness of the degrees of a minimal set of generators and is also related to the presence of non-trivial line bundles.

I will show that for the defining ideal of an affine curve C Ì AKn, such a uniform bound does exist, except possibly when C is locally generated by the least possible minimal number of generators (namely n-1). This answers a question raised by VAN DEN DRIES for affine curves. Consequently, there exists an algorithm calculating the exact number of generators of the ideal of an affine reduced curve C (barring the exceptional case) provided we take the arithmetic of the field as an oracle. In the exceptional case (which includes the smooth case!), we know at least that the ideal of C requires either n-1 or n generators.

The proof uses a non-standard argument together with the Forster-Swan Theorem and (the positive solution of) the EE-Conjecture.

Tuesday, June 5, 2001, 4:00 p.m.

Dr. Hagen Knaf
Institute for Industrial Mathematics, Kaiserslautern, Germany

gave a talk in the Department Colloquium:

## On uniformization of Abhyankar places over base rings of small dimension

Abstract:
In the year 1995 A.J. de Jong proved that given a geometrically integral variety X over the field K there exists a finite extension L|K and a regula} alteration Y\rightarrow X xK L. In particular, every K-trivial place P of F=K(X) posesses a prolongation Q to a finite extension E=K(Y) of FL such that the valuation ring OQ dominates a regular local ring O\subset E essentially of finite type over L (local uniformization of P after finite extension of F). The latter was independently proved by means of purely valuation-theoretic methods in 1998 by F.-V. Kuhlmann. More recently it became clear that for a so-called Abhyankar place P local uniformization is possible without extending F. The method used to verify this result can also be applied when working over a base ring R instead of the field K, but does not directly yield local uniformization in this case.

The aim of the talk is to show how one can combine this method with Abhyankar's results on uniformization in dimension \leq 2 to prove:

(A) Let P be an Abhyankar place of the function field F|K and R\subseteq K a regular local ring of dimension dim(R)\leq 2 that is dominated by OP\cap K. Assume that R is Nagata if \dim(R)=2, that the group vP(F) /vP(K) is torsion-free, that the extension FP|KP is separable, and that (K,vP) is defectless. Then there exists an R-scheme X of finite type such that P is centered in a regular point of X, and K(X)=F holds.

For a discrete valuation ring R de Jong has proved similiar results as in the case of a base field, so that (A) for dim(R)=1 can be deduced from his results at least up to finite extensions.

Some higher dimensional cases in which (A) is true as well as a more general version of this statement will also be discussed in the talk.

This talk was supported by the Colloquium Fund of the Department of Mathematics and Statistics.

Friday, July 20, 2001, 4:00 p.m.

Professor Edward Mosteig
Tulane University, New Orleans

gave a talk in the Department Colloquium on

## Applications of Gröbner Bases

Abstract:
Gröbner bases are a computational tool used in solving systems of polynomial equations by exact means. Currently, they are employed in many fields of mathematics including commutative algebra, algebraic geometry, algebraic combinatorics, statistics, linear programming, numerical analysis, and differential equations. Although they were developed in the 1960's, they have only recently appeared at the forefront of computational mathematics. The advent of the personal computer has permitted computations that were previously impossible to perform by hand.

Some immediate applications that have elegant expositions via Gröbner bases include the following.

• Solving the Three Color Problem
• Determining Dependency within Networks
• Computing Hilbert series
• Constructing Minimal Polynomials of Algebraic Numbers
• Geometric Theorem Proving
• Multi-dimensional Reed-Solomon Codes
• Robotics
My goal is to introduce Gröbner bases from an elementary standpoint and examine their development as given by Bruno Buchberger. From there, I will highlight a few key results and demonstrate their importance. Along the way, I will demonstrate how certain examples from a few different fields of mathematics can be solved using Gröbner bases.

This talk was supported by the University of Saskatchewan Visiting Lecturers' Fund.

Friday, September 7, 2001, 4:00 p.m.

Dr. Alexander Nenashev

gave a talk in the Colloquium on

## Invariants of quadratic forms over exact categories with duality

Abstract:
We define an infinite series e0, e1, ... of invariants of the Witt classes of symmetric bilinear forms over an exact category with duality. These invariants take their values in suitable subquotients of Quillen K-groups of the category in question. In the lower degrees, we recover the generalized dimension index and generalized discriminant. In the classical case of quadratic forms over a field of characteristic different from 2 we expect to recover the invariants related to the Milnor conjecture. Our definition of these invariants is explicit and is based on the use of a self-dual model for the K-theory space of the category.

This talk was supported by the Colloquium Fund of the Department of Mathematics and Statistics.

Tuesday, September 25, 2001, 10:00 am

Professor Niels Schwartz
University of Passau, Germany

gave a talk in the Algebra and Logic Seminar on