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


Colloquium Talks 2000/2001

Colloquium chair: Salma Kuhlmann


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

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

gave a talk on

Tiling semigroups

Abstract:
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

Spacecraft trajectories and analog circuit synthesis: some recent adventures in applied mathematics

Troy is a former U of S student who is now pursuing graduate work at Purdue University. The talk was directed towards undergraduate and graduate students, however, EVERYONE WAS WELCOME to attend.


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

Herve Perdry
Universite de Franche-Comte, Besancon, France

gave a talk 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 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.


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

Professor Artur Sowa
Yale University, New Haven, Connecticut

gave a talk on

Vortices in a generalized Maxwell theory-analysis and interpretation

Abstract:
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

Automaticity in learning mathematics

Abstract:
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

Noncommutative geometry and Hopf algebras

Abstract:
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

Calibration of the estimators of variance

Abstract:
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

Graphs and semi-graphoids: Conditional independence and causal inference

Abstract:
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

Concatenation Hierarchies of Rational Languages

Abstract:
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

Linear Independence of Translates

Abstract:
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

Bandit processes and applications: some issues and problems

Abstract:
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

Positivity, Sums of Squares, and the Multi-Dimensional Moment Problem
(joint work with Murray Marshall)

Abstract:
The K-moment problem originates in Functional Analysis: for a linear functional L on R[X1,...,Xn] (the polynomial ring over the reals), one asks whether there exists a positive Borel measure \mu on Euclidean space Rn, supported by some given closed subset K of Rn, such that for every f in R[X1,...,Xn] we have L(f) = \int f d\mu . We consider the case where K is a basic closed semi-algebraic set defined by a finite set of polynomials g1,...,gs. Via Haviland's Theorem, the K-moment problem is closely related to the problem of representation of positive semidefinite polynomials on K in the preordering generated by g1,...,gs. For K compact, Konrad Schmuedgen has provided positive solutions to both the moment and the representation problems. We study the case where K is not compact, and extend Schmuedgen's results in some cases (e.g., for cylinders with compact cross-sections). We present several open problems.


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

General Construction of Quantum Tori Lie Algebras

Abstract:
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

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.


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

Professor Victoria Powers
Emory University, Atlanta, USA

gave a talk on

Real algebraic geometry and convex optimization

Abstract:
Semidefinite programming is an important tool for solving many problems in applied math and engineering, for example in systems and control theory. In this talk we will give an overview of the interaction of concepts in real algebraic geometry and semidefinite programming. In particular, we will talk about applications to convex optimization problems. Much of the talk will be based on recent work of Pablo Parrilo, who has developed practical methods for studying semidefinite programming using ideas from real algebraic geometry. No prior knowledge of semidefinite programming or convex optimization will be assumed.

5:00 p.m.

Professor Claus Scheiderer
University of Duisburg, Germany

gave a talk on

Sums of squares and the moment problem

Abstract:
The question whether a non-negative polynomial is always a sum of squares of polynomials was raised in the 1880s by Minkowski and answered by Hilbert. I'll first discuss the generalization of this question to polynomial functions on affine real algebraic sets. The hardest case is that of compact curves and surfaces. These questions are directly related to the (multi-dimensional) moment problem from analysis. The latter asks for a characterization of the possible moment (multi-) sequences of positive Borel measures with support in a given closed subset K of Rn. The case when K is compact is solved completely by a theorem of Schmuedgen. I will present these facts, and in the end try to discuss a few recent results for non-compact K.


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

Professor Michael Stephens
Simon Fraser

gave a talk on

One hundred years of chi-squared

Abstract:
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

Real Eigenvalues of Certain Tri-diagonal Matrix Polynomials, with Queueing Applications

Abstract:
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

The Hurwitz action and braid group orderings

Abstract:
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

Introduction to Vertex Algebras

Abstract:
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:

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 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.

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

Professor Horst Lange
University of Cologne, Germany

gave a talk on

The stationary Wigner equation with inflow boundary conditions

Abstract:
The nonlinear Wigner equation models the evolution of the Wigner function w=w(x,v,t), a quasi-density distribution of particles in a medium, and plays a fundamental role in quantum semi-conductor theory. It couples an evolution equation for the Wigner function (which contains the potential of the system in a nonlocal form) self-consistently with a Poisson equation for this potential (with the particle density as right hand side); the particle density is the integral of the Wigner funtion over velocity space. We consider the stationary Wigner equation for dimension 1 with inflow boundary conditions on a slab of length 1. Without using a cut-off at v=0 (which often is needed) we can show the existence of solutions the for the 'linear' and 'nonlinear' potential case satisfying the inflow boundary conditions.

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

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.


Colloquium Talks 2002-2005

Colloquium Talks 2002/2003

Colloquium Talks 2001/2002

Colloquium Talks 1999/2000

Colloquium Talks 1998/99

Colloquium Talks 1997/98

Algebra and Logic Seminar

Cryptography & Coding Theory Student Seminar

Centre for Algebra, Logic and Computation


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