**
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 16, 1999, 4:00 p.m.

**Professor Sudesh Kaur Khanduja
Punjab University, Chandigarh, India**

gave a talk 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
Universitaet Heidelberg**

gave a talk 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 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)).

Thursday, August 19, 1999, 2:00 p.m.

**Professor D. H. Sattinger
Utah State University**

gave a talk on

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

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

A

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

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

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 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

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

This talk introduces the concept of

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

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

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

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

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

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

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

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

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
*