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 1997/98
Friday, October 3, 1997, 4:00 p.m.
Professor Mahmood Khoshkam
University of Saskatchewan
gave a talk on
Toeplitz algebras in K-theory
Abstract:
Toeplitz algebras have played a significant role in K-theory of C*
algebras. The purpose of my talk is to give an account of various
notions of Toeplitz algebras which have been developed in connection
with K-theoritical computations. In particular, I will talk about the
Toeplitz algebras and extensions associated with automorphisms and with
endomorphisms of C* algebras. Recall that the simplest Toeplitz operator
is the unilateral shift on a separable infinite dimensional Hilbert
space H and the corresponding extension is
0 -> K(H) -> T -> C(S1) -> 0 .
The extensions that I am going to discuss are in one way or another
related to the above extension.
Friday, October 17, 1997, 4:00 p.m.
Dr. Franz-Viktor Kuhlmann
University of Saskatchewan
gave a talk on
On resolution of singularities in arbitrary
characteristic
Abstract:
The resolution of singularities of algebraic surfaces is probably one of
the most prominent problems in mathematics and one of the main subjects
of algebraic geometry. It has applications in many branches of
mathematics, such as real algebraic geometry, algebraic number theory
and differential equations. While the desingularization of curves was
already done at the beginning of this century, the modern developments
were shaped by the work of Oskar Zariski and promoted by his students J.
Lipman, S. Abhyankar and H. Hironaka. Zariski used a valuation
theoretical approach. His idea was to first resolve singularities
locally; this is called local uniformization. He proved it for varieties
over fields of characteristic 0. He used this result for the
desingularization of algebraic surfaces (= dimension 2) in char. 0. In
1965, Hironaka proved resolution of singularities for all dimensions in
char. 0. Abhyankar solved the case of surfaces in arbitrary
characteristic. Since then, it is an open question whether res. of sing.
is possible for all dimensions in arbitrary characteristic. In 1995, de
Jong proved his famous result that it can be done if one takes into the
bargain a finite extension of the function field of the variety (which
is not really what we want). Using again Zariski's valuation theoretical
approach, Mark Spivakovsky is working on resolution of singularities via
local uniformization. Recently, I was able to give a purely valuation
theoretical proof for local uniformization after finite extension of the
function field. This result follows from the more general result of de
Jong. But in contrast to his proof, mine does not use the very hard
theory of moduli spaces. More important, it gives a valuation
theoretical description of the extension of the function field (which is
important for applications). In certain cases, I can show that no
extension is needed (this is a result towards what Spivakovsky is trying
to prove). Very recently, I have succeeded to get the extension to be
Galois, which for the case of global desingularization is still an open
problem (that de Jong was not able to solve). As my methods show
similarities to those used in the theory of moduli spaces, there is some
hope that my result could help to solve this important open problem and
to improve de Jong's results.
Friday, November 19, 1997, 4:00 p.m.
Professor Sibylla Priess-Crampe
Ludwig Maximillian Universitaet Muenchen, Germany
gave a talk on
Fixed Point and Coincidence Theorems for Ultrametric
Spaces
Abstract:
An ultrametric space (X,d,G) is a set X with an ultrametric distance
functions d from X to G , where G is a partially ordered set with a
smallest element 0. d has the same properties as a metric but instead of
the triangle inequality the following one:
For all g of G , if d(x,y) and d(y,z) are at most g then also d(x,z) is
at most g .
A special role for ultrametric spaces play spherically complete
ultrametric spaces. "Sperically complete" corresponds to the property
"maximal valued" for valued fields.
For spherically complete ultrametric spaces there holds a fixed point
theorem which looks like Banach's fixed point theorem for metric spaces.
One has furthermore a generalization of this singlevalued fixed point
theorem to multivalued mappings (again as it is the case in the metric
situation). Some hints to applications of the theorems will be given.
Friday, November 28, 1997, 4:00 p.m.
Professor Salma Kuhlmann
University of Saskatchewan
gave a talk on
Formally Exponential Fields
Abstract:
The class of Formally Real Fields (i.e. fields which admit a total
ordering) is well understood both algebraically (Artin-Schreier Theory)
and model-theoretically (Tarski-Seidenberg Theorem on projections of
semi-algebraic sets). Moreover, Valuation Theory of Ordered Fields is an
important tool there: for example, it provides elegant construction
methods of non-archimedean Real Closed Fields.
Analogously, we define a Formally Exponential Field to be an ordered
field which admits an exponential function (i.e. an order preserving
isomorphism from its ordered additive group onto its multiplicative
group of positive elements). An example is the real exponential field,
i.e. the field of the reals with the real exponential function. But the
Algebra and Model Theory of this structure is far from being well
understood and presents deep open problems.
In this talk we develop Valuation Theory of Ordered Exponential Fields
and apply it to obtain structure theorems for these fields. We use this
characterization to construct non-archimedean models of real
exponentiation, that is, non-archimedean Exponential Fields that
nevertheless have the same elementary properties as the real exponential
field. these models can be used to solve open conjectures about the
asymptotic behaviour of real-valued functions.
Friday, January 16, 1998, 4:00 p.m.
Professor Salma Kuhlmann
University of Saskatchewan
gave a talk on
Exponentiation in power series fields
Abstract:
Power series fields with exponents in an arbitrary ordered abelian
group, and coefficients in the reals are a natural domain for the study
of analysis in non-archimedean extensions of the reals. They were
studied by several mathematicians (such as Levi-Civita, Neder, Laugwitz,
A. Robinson, Gonshor, Conway, and others) in an attempt to develop the
foundations of such an analysis.
One of the first concerns was to define on these fields the elementary
functions (such as the exponential and logarithm) that are known to us
from real analysis. It was known that the exponential (and more
generally, all C-infinity functions) are definable for the
INFINITESIMALS of the field, through its Taylor expansion. But the
problem of defining a total exponential (for the infinitely large
elements of the field) remained open.
We answered this open problem to the negative. Indeed, we showed that NO
power series field admits a total exponential function. In this talk, we
present a proof of this fact, by showing that certain functional
equations in lexicographic products of chains are not solvable. On the
other hand, we show that EVERY power series fields admits a
non-surjective logarithm. This last fact can be used to construct
exponential fields as countable union of power series fields.
(Joint work with F.-V. Kuhlmann and S. Shelah.)
Friday, January 23, 1998, 4:00 p.m.
Dr. Franz-Viktor Kuhlmann
University of Saskatchewan
gave a talk on
Maps on ultrametric spaces: an alternative to fixed
point theorems
Abstract:
As a continuation to Sibylla Priess' Colloquium talk, we
will study maps on sperically complete ultrametric spaces.
We give a criterion for such a map to be onto and for the
image to be spherically complete again. This criterion
has found interesting applications:
1) It can be used to give a short proof of Hensel's Lemma.
(This lemma is equivalent to the valuation-theoretic
version of the Implicit Function Theorem.)
2) It can be used to show that spherically complete
differential fields admit integration. This has an important
application to the construction of universal domains for Hardy
fields.
3) It can be used to deduce a property of power series fields
over fields of positive characteristic which has not been known
before. This adds quite unexpected new information to our
knowledge about the model theory of these fields.
Friday, February 6, 1998, 4:00 p.m.
Professor Yuly Billig
University of New Brunswick
gave a talk on the
Lie group of pseudodifferential operators
Abstract:
In a recent paper, Khesin and Zakharevich introduced a new important
construction in the theory of integrable systems. Using the duality
between the Lie algebras of differential operators and integral
operators, one can introduce the Poisson structure on the Lie group of
pseudodifferential operators. This procedure can be also carried out
for the central extensions of these algebras. While the formula for the
central extension of the Lie algebra of differential operators is
well-known from work of Kac and Peterson, the dual central extension is
introduced using the logarithm of the operator of differentiation. The
space of Hamiltonians is the center of the Poisson algebra of invariant
functions on this Lie group. Reductions to various Poisson submanifolds
give Hamiltonian flows corresponding to many important integrable
systems.
Friday, February 13, 1998, 4:00 p.m.
Professor Hans Brungs
University of Alberta
gave a talk on
Extending Valuation Rings
Abstract:
For any field extension K/F every valuation ring of F can be extended to
K. There are three competing definitions for noncommutative valuation
rings and their extension properties will be discussed; in particular
extensions of valuation rings in the center of a finite dimensional
division algebra. These results can be used to show that every rooted
tree can be realized as the graph of all valuation rings of a finite
dimensional division algebra.
Thursday, February 26, 1998, 12:30 p.m. (Analysis Seminar)
Professor Karim Seddighi
University of Calgary & Shiraz University, Shiraz, Iran
gave a talk
On the Commutant of 2 \times 2 Operator Matrices
Abstract:
Let ${\cal B}$ be a direct sum of spaces of functions on each of which
the operator $M_z$ of multiplication by $z$ $(f\longrightarrow zf)$ is
bounded. We determine the commutant of the direct sum of the operators
of multiplication by $z$ on certain Hilbert spaces of functions (Banach
spaces of functions). Also we characterize the commutant of $M_z$ and
multipliers of Lipschitz algebras. Let $\mu$ be a compactly supported
measure on ${\bf C}$ and $t\ge 1$.We determine the commutant of the
operator $M_z$ on $P^t(\mu)$, the closure of polynomials in $L^t(\mu)$,
thus extending a result of M. Raphael for the case $t=2$.
Friday, February 27, 1998, 4:00 p.m.
Professor Karim Seddighi
University of Calgary & Shiraz University, Shiraz, Iran
gave a talk on
Operator Theory and Complex Analysis
Abstract:
Let $G$ be a bounded open subset of the complex plane {\bf C}. Given a
function $\phi:G\longrightarrow {\bf C}$ there are various ways that
$\phi$ can induce an operator $T$ defined on a Hilbert space ${\cal H}$
of functions defined on $G$. We would like to discuss the properties of
$T$ in terms of those of its symbol $\phi$.
The interaction between operator theory and function theory is quite
interesting. This area has been developed for the last two or three
decades and this activity shows no sign of waning. Many of the people
working in operator theory know the connections. It is nice that many on
the other side do too.
The basic theory of Hilbert spaces of analytic functions on a bounded
open set will be presented first. Next we define the concept of
multipliers and explore their properties. Some examples of Hilbert
spaces of analytic functions, such as Hardy spaces, Bergman spaces,
Dirichlet spaces, etc. will be presented. The works of Axler, Cowen,
Shields, Shapiro, Stroethoff, etc. will be touched on.
A multiplication operator defined on a Hilbert space ${\cal H}$ has the
form $f\rightarrow hf$. We are interested in determining the commutant,
cyclicity, and other important operator theoretic properties of such
operators. A composition operator $C_{\phi}$ is of the form
$f\rightarrow f\circ\phi$. It is good to know when these operators are
normal, hyponormal, or subnormal.
Let $L^2=L^2({\bf D},dA)$ be the space of measurable functions on
{\bf D} that are square integrable with respect to the area measure $A$.
Let $L_a^2$ be the closed subspace of $L^2$ consisting of analytic
functions and let $P:L^2\rightarrow L^2_a$ be the projection onto
$L^2_a$. For $\phi\in L^{\infty}$ the Toeplitz operator $T_{\phi}$ on
$L^2_a$ is defined by $T_{\phi}f=P(\phi f)$. The Hankel operator is
defined by projecting onto $(L^2_a)^{\bot}$. The study of these
operators plays a crucial role both in operator theory and function
theory.
Friday, March 6, 1998, 4:00 p.m.
Professor Niels Schwartz
University of Passau, Germany
gave a talk on
From rings of continuous functions to real closed rings
Abstract:
Rings of continuous functions into the real numbers are obviously an
important tool in topology. Therefore they have been studied intensely
from many points of view. Their algebraic analysis is a difficult
problem, partly because, as a category, rings of continuous functions
have very poor properties. There are only very few ring theoretic
constructions that produce rings of continuous functions when applied to
such rings. A related fact is that there is no axiomatization in first
order model theory. One major topic in real geometry is the
investigation of semi-algebraic spaces. A semi-algebraic space is always
defined with reference to a real closed field. Similar to rings of
continuous functions on topological spaces, the functions that are most
useful for the analysis of semi-algebraic spaces are the rings of
continuous semi-algebraic functions into the real closed fields
associated with the spaces. These rings have the same poor category
theoretic and model theoretic properties as rings of continuous
functions. But there is a larger class of rings, called real closed
rings, which is the smallest axiomatizable class of rings containing the
rings of continuous semi-algebraic functions. Thus, real closed rings
can be studied by model theoretic methods. The category theoretic
properties are also very favorable; the class is closed with respect to
o large number of the usual ring theoretic constructions. All rings of
continuous functions are real closed. In fact, currently the real closed
rings are the smallest known class of rings that contains the rings of
continuous functions, is axiomatizable and is closed under so many ring
theoretic constructions. Thus, although they were first introduced as a
tool for real geometry, the real closed rings also have a potential for
applications in topology.
Thursday, April 23, 1998, 10:30 p.m. (General Seminar)
Professor Keith Taylor
University of Saskatchewan
gave a talk on
Spectral Theory Problems from Chemistry
Abstract:
For a few years, Dr. Shigeru Arimoto and I have been working
together on some questions which arise from a simplified
model for large hydrocarbon atoms. In the end, one is concerned
with the fine detail of the spectra of large matrices associated
with these molecules. We apply functions to the matrices via
the functional calculus and consider the asymptotic behaviour
as the size of the matrix goes to infinity. I will introduce
the mathematical formulation, give some of our results and
pose some questions.
Last update: January 26, 2008
--------- created and maintained by Franz-Viktor Kuhlmann