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 1999/2000
Friday, July 16, 1999, 4:00 p.m.
Professor Sudesh Kaur Khanduja
Punjab University, Chandigarh, India
gave a talk on the
Generalized Hensel's Lemma
Abstract:
We intend to talk about a generalization of the classical Hensel's Lemma
and some of its applications using prolongations of a valuation v,
defined on a field K to a simple transcendental extension K(x) of K.
Friday, July 23, 1999, 4:00 p.m.
Professor Peter Roquette
Universitaet Heidelberg
gave a talk on
The Grunwald and Wang Story
Abstract:
(1) Explain the statement of the famous
Grunwald existence theorem.
(2) Report on its origin and its significance within Number Theory.
(3) Tell about the shock to the mathematical community when Wang found a
counter example, and about the attempts to mend matters.
(4) Reinterpretation of the (corrected) theorem within valuation theory.
Friday, July 30, 1999, 7:30 p.m.
Professor Paulo Ribenboim
Queens University, Kingston
gave a public lecture on
These marvelous prime numbers
This lecture was supported by the
Special
Lectureship Fund of the University of Saskatchewan.
Friday, August 6, 1999, 4:00 p.m.
Professor Hans Schoutens
Wesleyan University, Middletown
gave a talk on
The image of an analytic map; the ultrametric case
Abstract:
Consider an analytic map f:Y->X (either complex, real or
rigid analytic). We want to describe the image I=Im f of f
viewed as a subset of X. Unfortunately, I is not necessarily a
semi-analytic set. With a semi-analytic set we mean a subset given by (a
Boolean combination of) norm-inequalities |p(x)|<|q(x)| between
analytic functions p and q. We [S. + Gardener] show that in the rigid
case, I can be described though by norm-inequalities if we allow for
the functions p,q,... in this description to be (iterated) quotients
of analytic functions. Moreover, after a well-choosen blowing up
process \pi:\tilde X -> X, we can arrange for \pi^{-1}(I) to become
semi-analytic. In fact, we show how to flatten the original map f by
blow ups, after which we apply the Theorem of Raynaud which asserts that
the image of a flat (affinoid) map is always semi-analytic. To give an
idea of the proof, which is a rigid analytic adaptation of Hironaka's
proof for the real case, I will work out the Osgood example, where
f(s,t)=(s, st, s exp(t)).
Thursday, August 19, 1999, 2:00 p.m.
Professor D. H. Sattinger
Utah State University
gave a talk on
Soliton Interactions at High Mach Numbers
Abstract:
A numerical study of soliton interactions in the ion acoustic plasma
equations was carried out at various Mach numbers. The numerical results
are compared with exact two soliton solutions of the Korteweg-deVries
equation. The interactions are strongly elastic, even for high Mach
numbers, while analomies in the scattering shifts are observed. A
description of the computation of higher order corrections is given.
Friday, September 10, 1999, 4:00 p.m.
Professor Michael Baake
University of Tuebingen
gave a talk on
Which distributions of matter diffract
Abstract:
Mathematical diffraction theory is concerned with the spectral analysis
of unbounded complex measures, and has important applications in
crystallography and diffraction physics, and to the understanding of
quasicrystals in particular.
After introducing the basic questions and methods, I will briefly survey
the present state of affairs by means of characteristic examples from
lattices, quasicrystals, and stochastic structures such as lattice
gases. My emphasis will be on cases that can be treated explicitly, but
without loss of rigour, and I'll also point out some open questions.
Friday, September 17, 1999, 4:00 p.m.
Professor Barry Monson
University of New Brunswick
gave a talk on
Realizations of Regular Toroidal Maps
Abstract:
A regular abstract polytope P is a poset having the essential
structural features of the face lattice of a regular convex polytope,
including transitivity of Aut P on flags. Other examples include
the regular star polyhedra, maps on compact surfaces, and tessellations
of spherical, Euclidean or hyperbolic space of any dimension. (In
general, however, P need not be a lattice or have a particularly
nice geometric realization.)
Indeed, McMullen (1989) has developed the basic theory of
realizations for P, basically by using geometric methods
to describe certain real representations for Aut P.
Asia Ivic Weiss and I have recently examined the case
that P is a finite, regular toroidal map of type
{4,4}, {3,6} or {6,3}. In a pretty and unexpected way, the
data for the realizations (and group representations)
are encoded in a simple picture of the map.
There are very many other examples of finite, regular
abstract polytopes, but our understanding of their
realizations is quite limited.
Friday, September 24, 1999, Room 206 ARTS
4:00 p.m.
Professor Mahmood Khoshkam
University of Saskatchewan
gave a talk on
Categorical Constructions in C*-algebras
Abstract:
Recently, Gert Pedersen has initiated the study of pullback and pushout
constructions in the theory of C*-algebras. These constructions are
intrinsically related to the theory of extensions and amalgamated free
products in C*-algebras.
In my talk I will investigate the notions of limits and colimits in the
category of C*-algebras. Both pullbacks and pushouts constructions arise
as special cases of our constructions. Our main result shows that limits
and colimits diagrams of C*-algebras are stable under tensoring by a
fixed C*-algebra, and under crossed product with a fixed group.
Friday, October 29, 1999, Room 206 ARTS
4:00 p.m.
Professor Franz-Viktor Kuhlmann
University of Saskatchewan
tried to answer the question
What do we know about power series fields in positive
characteristic?
Abstract:
Early on in the development of Algebraic Number Theory, people noticed
the similarities between number fields (the finite extensions of the
field of rational numbers) and function fields in one variable over
finite fields. That is why both types of fields are subsumed under the
name "global field". The similarities also carry over to the
completions: the fields of p-adic numbers on the one hand, and the
fields of formal Laurent series over finite fields on the other hand.
For this reason, it had been suspected that Artin's conjecture would
hold for the fields of p-adic numbers, after the analogous assertion had
been proved by Serge Lang for the Laurent series fields. Ax and Kochen
used model theory to show that the assertion, in a somewhat weaker form,
carries over from the Laurent series fields to the fields of p-adic
numbers. In this way, a corrected, weaker version of Artin's conjecture
is proved (in fact, it was shown by Terjanian that the original
conjecture is false). After this pioneering work, Ax and Kochen studied
successfully the elementary properties of the fields of p-adic numbers
(and also of Laurent series fields over fields of characteristic 0).
However, the problem of determining the elementary properties of the
Laurent series fields over finite fields remains unsolved till the
present day.
In my talk, I will describe the differences between the two types of
fields and show why they make the problem for the Laurent series fields
over finite fields so much harder than for the fields of p-adic numbers.
In particular, I will sketch the role of the "defect" (also called
"ramification deficiency") of valued field extensions, and the role of
additive polynomials. Finally, I will state some of my own results which
are connected with this open problem.
Friday, November 5, 1999, 4:00 p.m.
Professor Murray Marshall
University of Saskatchewan
gave a talk on
Hilbert's 17th Problem
Abstract:
Hilbert's 17th Problem asks if a positive semi-definite polynomial is
necessarily a sum of squares of rational functions. The talk will look
at the history of Hilbert's 17th Problem, how it arose, how it was
solved, and its role in twentieth century developments in real
algebraic geometry, quadratic form theory, and in model theory.
Friday, November 12, 1999, 4:00 p.m.
Professor Keith Taylor
University of Saskatchewan
gave a talk on
Self-similar sets and wavelets
Abstract:
The first part of this talk will be a general introduction to
self-similar sets in Euclidean space following the ideas of Stricartz,
Grochenig and Madych and others. The role of self-similar sets in
constructing examples of wavelets will be explained. Q. Yang's
formulation in scalable locally compact groups will lead to an explicit
construction of wavelets based on the so-called Heisenberg group.
Friday, January 14, 2000, 4:00 p.m.
Dr. Dean Slonowsky
Fields Institute
gave a talk on
Set-Indexed Martingales
Abstract:
This talk introduces the concept of set-indexed martingales, a
special type of set-indexed stochastic process which serves as a
generalization of continuous parameter martingales. The study of
set-indexed martingales is a relatively new area in probability with
many statistical applications now emerging, such as: improved
multivariate tests-of-fit, modeling the spread of epidemics, testing the
Poisson hypothesis in R^d and sequential clinical trials.
In general, a set-indexed process is any family X = { X_A : A \in \cal
A } of random variables where \cal A is a collection of subsets
of a fixed set T. When T = R^d, and \cal A is a suitable
collection of Borel subsets of R^d, the prototypical example of
a set-indexed process is the empirical process, X(A) = 1/n
\sum_{i=1}^{n} {\rm{1{\!}I}}[ Y_i \in A ] where Y_1, Y_2,..., Y_n
are independent samples from a d-dimensional distribution F. By
multiplying the indicators in X(A) by i.i.d. zero-mean random
weights, X becomes a set-indexed martingale.
Friday, January 28, 2000, 4:00 p.m.
Professor Murray Bremner
University of Saskatchewan
gave a talk on
Identities for the associator in alternative algebras
Abstract:
The associator (a,b,c) := (ab)c - a(bc) is an alternating trilinear
product for any alternative algebra. We study this trilinear product in
three related algebras: the associator in a free alternative algebra,
the associator in the Cayley algebra, and the ternary cross product on
4-dimensional space. The last example is isomorphic to the ternary
subalgebra of the Cayley algebra which is spanned by the non-quaternion
basis elements. We determine the identities of degrees up to 7 satisfied
by these three triple systems. We discover two new identities in degree
7 satisfied by the associator in every alternative algebra and five new
identities in degree 7 satisfied by the associator in the Cayley
algebra. For the ternary cross product we recover the ternary derivation
identity in degree 5 introduced by Filippov. (This is joint work with
Irvin Hentzel of Iowa State University).
Friday, February 4, 2000, 4:00 p.m.
Professor D. Farenick
University of Regina
gave a talk on
The Krein-Milman Theorem in Noncommuative Convexity
Abstract:
A cornerstone of 20th century analysis is a 1943
theorem of M.G. Krein and D. Milman: if a set K is compact
and convex, then K has an extreme point and, moreover, the
smallest closed convex subset of K that contains all of
these extreme points is K itself.
Motivated by the needs of a number of subject areas, the
notion of a matrix-convex set was introduced several years
ago. Whereas classical convexity theory is carried out using
convex coefficients that are numbers, the coefficients in
matrix convexity are complex matrices satisfying certain
positivity conditions. In this lecture I will introduce the
notions of convex set and extreme point in the noncommutative
context, and I will discuss the analogue of the Krein-Milman
theorem for these convex sets.
Friday, February 11, 2000, 4:00 p.m.
Professor E. Kaniuth
University of Paderborn (Germany)
gave a talk on
Spectral Synthesis
Abstract:
Starting from L^1-algebras of locally compact abelian groups, the
purpose of the talk is to report on progress in spectral synthesis of
L^1-algebras and of Fourier algebras of non-abelian locally compact
groups. Though this concerns the past twenty years, some of the results
are very recent.
Friday, February 18, 2000, 4:00 p.m.
Professor Alfred Weiss
University of Alberta, Edmonton
gave a talk on
L-values and multiplicative Galois module structure
Abstract:
Zeta functions 'know' much about the arithmetic of their algebraic
number field. Examples of this phenomenon are the analytic class
number formula and the Main Conjecture of Iwasawa theory. This appears
to persist also 'relatively', i.e., for the Galois action on an
extension of number fields. An example here is the Frohlich-Taylor
theory of Galois structure of the ring of integers when the
extension is tame. The multiplicative analogue of this, the Galois
structure of units, seems to continue the pattern in a suitable sense.
Proofs of this seem to require strengthening aspects of all of the
examples above. Perhaps this is not surprising as the Galois structure
of units is at the core of class field theory.
Tuesday, April 25, 2000, 3:30 p.m.
Professor
John
Baker
Emeritus at the University of Waterloo
gave a talk on
The Dirichlet Problem for Ellipsoids
Abstract:
This talk is mainly concerned with two elementary (and perhaps somewhat
novel) solutions of the Dirichlet problem for ellipsoids in Euclidean
space (of arbitrary, finite dimension). It will be lightly spiced with
historical remarks.
After the talk, we had a
Retirement Reception in honour of Yvonne Cuttle at the Faculty Club.
Tuesday, May 9, 2000, 2:30 p.m.
Dr. Igor Fulman
University of Calgary
gave a talk on
Operator-algebraic approach to dynamical systems
Abstract:
I will shortly tell about operator algebras and discuss
how they are applied to study dynamical systems. An important problem
arising in this context is the following: describe all ideals in the
algebras associated to dynamical systems. I will talk about groupoid
algebras and bimodules -- they happen to be a useful tool for solving
this problem. I will also discuss algebras associated with graphs,
and finally I will tell about a construction generalizing both the
graph algebras and the algebras of dynamical systems. The talk is based
on joint work with Professor M. Lamoureux from the University of
Calgary.
Friday, May 12, 2000, 4:00 p.m.
Dr. David Richter
McGill University, Montreal
gave a talk on
Coloured brackets and 2-manifolds
Abstract:
Among algebras of matrix differential operators there is a class known
as Lie G-graded algebras, a generalisation of Lie superalgebras wherein
Z/2Z is replaced by a finite abelian group G. I will exhibit and discuss
a correspondence between these generalised "coloured" Lie brackets and
connected compact 2-dimensional manifolds when G is a 2-group.
Friday, May 19, 2000, 4:00 p.m.
Dr.
Holger
Teismann
North Dakota, Fargo
gave a talk on
Some analytical methods for the solution of Schroedinger
equations with singularities
Abstract:
The aim of the talk is to describe some analytical techniques which have
been used to study "difficult" Schroedinger equations. These "difficult"
equations are linear and nonlinear evolution equations of
Schroedinger-type arising in different branches of mathematical physics,
which contain certain singularities. For instance, the linear potential
may have a singular time dependence, or the unknown function may appear
in the denominator of a rational expression. The analytical techniques
include harmonic analysis, special function spaces, Nash-Moser methods,
etc.
Last update: January 26, 2008
--------- created and maintained by Franz-Viktor Kuhlmann