COLLOQUIUM
of the
Department of Mathematics and Statistics
Phone: (306) 966-6081
Fax: (306) 966-6086

## Colloquium Talks 1997/98

#### Colloquium chair: Salma Kuhlmann

Friday, October 3, 1997, 4:00 p.m.

Professor Mahmood Khoshkam

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

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

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

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

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

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.

### Centre for Algebra, Logic and Computation

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