**
COLLOQUIUM **

of 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

Friday, July 21, 2000, 4:00 p.m.

**Dr. Mark V. Lawson
University of Wales, Bangor, UK**

gave a talk on

Motivated by questions in the physics of quasi-crystals, J. Kellendonk showed how to associate an inverse semigroup, called a tiling semigroup, with each tiling in R^n. In this talk, I shall explain how Kellendonk's construction can best be understood in terms of partial group actions. I shall also describe the first steps in the theory of those tiling semigroups coming from 1-dimensional tilings.

Thursday, July 27, 2000, 3:30 p.m.

**Mr. Troy McConaghy**

gave a talk on

Friday, September 15, 2000, Room 206 ARTS

4:00 p.m.

**Herve Perdry
Universite de Franche-Comte, Besancon, France**

gave a talk 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 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.

Friday, October 6, 2000, 4:00 p.m.

**Professor Artur
Sowa Yale University, New Haven, Connecticut**

gave a talk on

I will discuss a system of nonlinear equations that extends the Maxwell theory. In particular, I intend to cover the following topics.

1. Geometric motivation for the equations.

2. A sketch of how the existence of vortex-lattice solutions can be seen by means of a discrete theory custom-designed for these equations. In particular, it will be shown that emergence of nontrivial solutions has the characteristic of a phase transition.

3. An indication that the existence of static vortex-lattice solutions in three dimensions is tied to certain topological questions regarding foliations of 3-manifolds.

4. A sketch of how the generalized Maxwell theory is tied to certain topics in the Materials Science (e.g. the quantum Hall effects and high-Tc superconductivity) via the central theme of composite particles with fractional statistics.

Friday, October 20, 2000, 4:00 p.m.

**Dr. Sherry May
Director, Mathematics Learning Centre, Memorial University of
Newfoundland**

gave a talk on

Working memory is a pool of attentional resources that can be deployed for cognitive tasks. Deliberate assignment of these resources to a task allows that task to be completed. However, these resources are limited so the number of mental operations that can be performed at any given time is restricted. Competition between tasks for attentional resources lowers performance on all tasks. If a student is required to concentrate a great deal of working memory on the performance of a subskill, performance on a higher level skill will be affected.

*(Dr. May, herself a mathematician, has been doing research on the
causes and remedies of poor math skills in first-year students, a
problem with which we are all familiar. She is happy to share her
insights with our Department as we search for ways of addressing this
concern.)*

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

**Professor Masoud
Khalkhali University of Western Ontario**

gave a talk on

I will explain some of the basic ideas in noncommutative geometry starting with the fundamental algebra-geometry duality, leading to cyclic homology and K-theory. I will emphasize the theory of crossed product algebras for group actions and how it can be generalized to actions of Hopf algebras, leading to an equivariant K-theory and cyclic theory.

Friday, December 1, 2000, 4:00 p.m.

**Dr. Sarjinder Singh
University of Saskatchewan**

gave a talk on

In the present investigation, the problem of estimation of variance of the general linear regression estimator has been considered. It has been shown that the efficiency of the low level calibration approach adopted by S rndal [J. Amer. Statist. Assoc. 1992 ] is less than or equal to that of a class of estimators proposed by Deng and Wu [ J. Amer. Statist. Assoc. 1987 ]. A higher level calibration approach has also been suggested. The efficiency of higher level calibration approach is shown to improve on the original approach. Several estimators are shown to be the special cases of this proposed higher level calibration approach. An idea to find a non - negative estimate of variance of the GREG has been suggested. Results have been extended to a stratified random sampling design. The well known statistical package, GES, developed at Statistics Canada can further be improved to obtain better estimates of variance of GREG using the proposed higher level calibration approach under certain circumstances discussed in this paper.

Recent developments in survey sampling using higher order calibration approach suggested by Singh, Horn and Yu [ Survey Methodology 1998] are due to Singh et al. [Australian & NewZealand J. Statist. 1999], Singh and Horn [Biom. J. 1999], Tracy and Singh [ Metron 1999], Singh [Ann. Inst. Math. Statist. 2000], Singh [Calcutta Statist. Assoc. Bull. 2000] and Singh [South African Statist. J. 1999] will be discussed.

Friday, January 12, 2001, 4:00 p.m.

**Professor
Mik
Bickis
University of Saskatchewan**

gave a talk on

A ternary relation can be defined on the power set of vertices of a graph by the property of a subset separating two other subsets. Properties of this relation can be abstracted to a set of axioms for an algebraic structure called a semigraphoid. Relations satisfying the semigraphoid axioms can be defined on a number of mathematical structures. These axioms have been proposed by A.~P.~Dawid as capturing mathematically the essence of the concept of irrelevance.

In particular, conditional independence of two random variables given a third satisfies the semigraphoid axioms. Moreover, if the vertices of a graph are labelled by random variables which satisfy a Markov property relative to the graph, then the probabilistic semigraphoid defined by the random variables may be represented by the separation semigraphoid of the graph. Not all probabilistic semigraphoids can be represented by graphs, but the class which can be so represented can be enlarged by considering digraphs or even graphs with several edge types along with creative definitions of separation.

A digraph with vertices labelled by random variables (often called a Bayes' net) is a parsimonious representation of a joint probability distribution. The directions of the edges can be interpreted as causal arrows, although in general the directions are not uniquely determined from the probabilistic structure.

Friday, January 26, 2001, 4:00 p.m.

**Professor David Cowan
University of Saskatchewan**

gave a talk on

If A is a finite set, a subset of the free monoid A* is called a formal language over A. The class of rational languages over A is the least class containing the finite languages and closed under the boolean operations, concatenation, and the star operation, which, for a given language L, is just the free submonoid L* of A* generated by L. In the early seventies, a hierarchy of families of star-free languages was introduced by Brzozowski as a means by which such languages might be studied and classified (a star-free language is a rational language that can be obtained from the finite languages without resorting to the star operation). A fundamental question in this approach is one of decidability: given a star-free language L, is it decidable where in the hierarchy L first appears. This question for Brzozowski's and other related hierarchies will be discussed.

Friday, February 2, 2001, 4:00 p.m.

**Professor Keith
Taylor University of Saskatchewan**

gave a talk on

If G is a group and V is a vector space of scalar valued functions on G that is closed under translations, then one can ask when the set of translates of a given f in V are linearly independent. We will explore this question when V = l^2(G), the space of square summable functions from G into the complex numbers. There is an important conjecture that, when G is a torsion free group, if f is a nonzero element of l^2(G), then the translates of f are linearly independent (this would imply that the group ring of G over the complex numbers has no nontrivial zero divisors, for example).

Peter Linnell proved this conjecture for the case of G right orderable (that is, when G can be given a total order that is respected by right multiplication). I will present a short proof of Linnell's theorem by exploiting the geometric structure of the Hilbert space l^2(G).

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

**Professor Xikui
Wang
Department of Statistics, University of Manitoba**

gave a talk on

I will discuss some important issues and challenging problems in the theory of bandit processes and its applications, focusing on three key aspects: models, computations, and applications. The issues and problems are selected to address on concerns about theoretical or methodological limitations, practical or computational constraints, and other difficulties.

Friday, March 2, 2001, 4:00 p.m.

**Professor Salma
Kuhlmann
University of Saskatchewan**

gave a talk on

(joint work with Murray Marshall)

The K-moment problem originates in Functional Analysis: for a linear functional L on

Friday, March 9, 2001, 4:00 p.m.

**Professor Shaobin Tan
Xiamen University (China), presently visiting the
Fields Institute and York University**

gave a talk on

In this talk, we will give a unified view of some vertex operator constructions, in both the principal and homogenerous pictures, for some Lie algebras including the affine Kac-Moody algebra and some Lie algebras coordinatized by certain quantum tori.

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

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

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

gave a talk 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.

Friday, March 23, 2001, 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

Friday, April 6, 2001, 4:00 p.m.

**Professor Michael
Stephens Simon Fraser**

gave a talk on

The famous X-square (chisquare) test of fit was developed by Karl Pearson in 1900. The birthday will be celebrated by examining two problems, one old, one new, where the test is used or misused.

Problem 1: Testing normality given a histogram.

It is common in applied work for a statistician to be presented with a histogram derived from continuous observations by 'binning' them into cells. This may have occurred because it is easier to round off the observations to the nearest gram (or, in backward countries, to the nearest ounce) if, say, weighing newborn animals. Then the original observations are not available to the tester. It is also possible that the tester has binned the observations by choice, although all the original observations are available, because the histogram has the advantage that it gives an idea of the density.

Historically the test for normality would then be made using Pearson's X-squared, probably because the applied worker stopped at Stat 101, or more recently because it is the only test in the package. Furthermore, the X-square test would a.s. be done incorrectly, relying on folklore developed during 100 years. However, recent work by Richard Lockhart and two hangers-on has made available other statistics for discrete data; also, many other statistics which use all the original observations have been developed over the last century.

Problem 2:

Very recently, statisticians in Canada have been consulted on whether lotteries are fair, for example, the Lottery 6/49. It is easy to make an error in the use of chisquare to test, for example, whether each number, or each pair of numbers, (or triples, etc.) appears with equal probability. The celebration in this seminar will take the form of some comparisons of the tests old and new. (It is not necessary to understand what is meant by a.s. to follow the seminar - the emphasis will be on the application of the test.)

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

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

**Professor Winfried
Grassmann
Department of Computer Science, University of Saskatchewan**

gave a talk on

Many queueing problems lead to tridiagonal lambda-matrices containing polynomials that have, except for the diagonal, non-negative coefficients. This paper deals with the question, addressed in literature only for special cases, whether the eigenvalues corresponding to such lambda-matrices are real. In most cases, they are, as the theorems of this paper show, but sometimes, complex eigenvalues occur. Our results are derived by using Sturm sequences. In addition to simplifying the proofs of our theorems, Sturm sequences are helpful to actually determine the eigenvalues numerically.

Tuesday, May 29, 2001, 4:00 p.m.

**Dr. Jonathan Funk
**

gave a talk on

In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of a 2-sphere with finitely many marked points. Such a groupoid carries an order structure: it is a `circle' (in a sense that we make precise). The Artin group of braids of n-1 strands has an order-invariant action in the ramification groupoid of the 2-sphere with n marked points. Left-invariant linear orderings of the braid group may be retrieved. In particular, we show how to obtain the (inverse of) the linear ordering first obtained by Patrick Dehornoy. This approach uses topologically singular coverings in the sense of Ralph Fox, which we explain using the framework of cosheaf spaces.

Using the PYTHON programming language, we implement an algorithm derived from this method of ordering braids that compares two given braids with respect to the ordering inferred from a given element of the ramification groupoid. This computer program also decides the word problem for Artin braid groups (Dehornoy has also implemented these tasks on a computer, for his ordering).

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

Wednesday, May 30, 2001, 4:00 p.m.

**Professor Maria Golenishcheva-Kutuzova
Department of Mathematics, University of Florida at Gainesville**

gave a talk on

The notion of a Vertex Algebra was introduced 15 years ago by Borcherds in order to formalize a new algebraic structure which originally appeared as a basic building block of Conformal Field Theory. It has found since many applications in group theory, theory of automorphic functions, representation theory of infinite-dimensional Lie Algebras, Theory of integrable systems and Mathematical Physics.

I will explain foundations of the theory of Vertex Algebras with the variety of examples and will consider some applications to representation theory of Infinite-Dimensional Lie Algebras and Integrable Equations.

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

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

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

gave a talk:

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 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, August 3, 2001, 4:00 p.m.

**Professor
Horst
Lange University of Cologne, Germany**

gave a talk on

The nonlinear

For the linear case the method consists in three main steps:

(i) We show that the stationary linear Wigner equation is equivalent to
a singular **'evolutionary problem'** for specific initial functions
depending on the given inflow data;

(ii) by taking Fourier transforms it is shown that the evolutionary
system is equivalent to a characteristic initial value problem for a
linear **wave equation**;

(iii) the wave equation is solved first for **even initial data**
whereas the **odd case** is reduced to the even one by a multiplier
transformation.

(Joint work with A.Arnold (Saarbrücken) and L. Barletti (Florence))

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

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

**Dr. Alexander Nenashev
University of Saskatchewan**

gave a talk on

We define an infinite series e

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

*Last update: January 26, 2008
--------- created and maintained by Franz-Viktor Kuhlmann
*