of the

Mathematical Sciences Group

University of Saskatchewan

106 Wiggins Road

Saskatoon, SK, S7N 5E6, Canada

Phone: (306) 966-6081 - Fax: (306) 966-6086

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

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:

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

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

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
Universidad Autonoma de Madrid**

gave a talk in the Department Colloquium on

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

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

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

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

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

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

(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

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

**Professor Hans Schoutens
Wesleyan University, Middletown**

gave a talk in the Department Colloquium on

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

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

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

**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 Schmüdgen
Universität Leipzig, Germany**

gave a talk in the First Colloquiumfest on

**Abstract:**

Let K be a closed subset of **R**^{d}. The K-moment problem
asks under what conditions for a given multisequence s=(s_{n} ;
n \in **N**_{0}^{d}) there exists a positive Borel
measure \mu on **R**^{d} 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 t^{n} d\mu(t) for all n \in
**N**_{0}^{d}.

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

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

This is joint work with Bruce Reznick.

Let

2:00 p.m.

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

gave a talk in the First Colloquiumfest on

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
Saskatoon**

gave a talk in the First Colloquiumfest on

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

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

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

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

In September and October of 2000

**Herve Perdry
Besancon, France**

gave talks in the Algebra Seminar on

November 22, 2000

**Professor Danielle Gondard
Universite Paris VI, France**

gave a talk in the Algebra Seminar on

In January of 2001

**Dr. Jonathan Funk
Saskatoon**

gave talks in the Algebra Seminar on

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

**Matthias Aschenbrenner
Urbana, Illinois, USA**

gave a talk in the Algebra Seminar on

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

**Matthias Aschenbrenner
Urbana, Illinois, USA**

gave a talk in the Algebra Seminar on

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

**Matthias Aschenbrenner
Urbana, Illinois, USA**

gave a talk in the Algebra Seminar on

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

**Markus Schweighofer
Konstanz, Germany**

gave a talk in the Algebra Seminar on

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

**Markus Schweighofer
Konstanz, Germany**

gave a talk in the Algebra Seminar on

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

**Professor
Eberhard
Becker University of Dortmund, Germany**

gave a talk in the Department Colloquium on

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

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

**Professor
Claus
Scheiderer
University of Duisburg, Germany**

gave a talk in the Algebra Seminar on

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

**Professor
Claus
Scheiderer
University of Duisburg, Germany**

gave a talk in the Algebra Seminar on

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

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

gave a talk in the Algebra Seminar on

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

**Professor Alexander Prestel
Konstanz, Germany**

gave a talk in the Algebra Seminar on

11:30 a.m.

**Professor
Deirdre
Haskell
McMaster University**

gave a talk in the Algebra Seminar on

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

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

gave a talk in the Second Colloquiumfest 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 in the Second Colloquiumfest 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

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

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

gave a talk in the Second Colloquiumfest 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 in the Second Colloquiumfest 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 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

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

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

Given polynomials f

(1)

(2)

(3)

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:

In [Logic Colloquium '95, Lecture Notes of Logic

**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:

In this talk we describe a nice syntactic characterization of a structural property of general algebras (the

**Professor Toniann Pitassi
University of Toronto**

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

**Professor Bradd Hart
McMaster University and Fields Institute**

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

**Ziv Shami
McMaster University and Fields Institute**

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

Let

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:

Let

**Professor Jim Loveys
McGill University, Montreal**

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

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:

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:

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
University of Wisconsin-Madison**

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

For a formally real field

**Professor Hans Schoutens
Rutgers University, New Brunswick**

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

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
**A**_{K}^{n}, 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:

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 x

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
O_{P}\cap K. Assume that R is Nagata if \dim(R)=2, that the
group v_{P}(F) /v_{P}(K) is torsion-free, that the
extension FP|KP is separable, and that (K,v_{P}) 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

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

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

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

**Dr. Alexander Nenashev
University of Saskatchewan**

gave a talk in the Colloquium on

We define an infinite series e

**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

2000-2001 + 2001-2002 + 2002-2003 + 2003-2004 + 2004-2005 + 2005-2006

*Last update: February 5, 2008
--------- created and maintained by Franz-Viktor Kuhlmann*