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