




Analysis / Analyse (T. Ransford, Organizer)
 L. BARIBEAU, Université Laval, Québec G1K 7P4
Iterated function systems and the llemma

To any finite family of contractions f_{i}: z®c_{i}(l)z+b_{i}(l) can be associated its socalled
attractor, which is the unique nonempty compact subset of the
zplane which is invariant under all the f_{i}'s. For instance, for
the pair of maps z® z/3 and z® z/3+2/3, the attractor
is the celebrated Cantor set. We will talk about the analytic
properties of the attractor when the c_{i}'s and b_{i}'s are
holomorphic, and discuss this in the context of the llemma of
Mañé, Sad and Sullivan.
 ALAIN BOURGET, Johns Hopkins, 3400 North Charles Street,
Baltimore, Maryland 21218, USA
Nodal statistics for the Lamé ensemble

Recently, there has been great interest in both mathematics and physics
circles in the asymptotic statistics of both eigenvalues and zeroes of
eigenfunctions of quantum Hamiltonian. Despite the abundance of
conjectures, we present some rigorous results concerning the asymptotic
statistics of the zeroes of the joint eigenfunctions (Lameé
harmonics) of an important class of quantum completely integrable
dynamical system: the Gaudin spin chains with real marked points.
 A. BRUDNYI, University of Calgary, Calgary, Alberta T2N 1N4
Bounded analytic functions on special Riemann surfaces

Let M be a noncompact connected Riemann surface of finite type, and
R Ì Ì M be a relatively compact domain such that
H_{1}(M,Z) = H_{1}(R,Z). Let [(R)\tilde]® R
be a covering and i: U\hookrightarrow[(R)\tilde] be a subdomain
such that the induced homomorphism i_{*} of fundamental groups is
injective. We study the algebra H^{¥}(U) of bounded holomorphic
functions on U. We will show that many properties of H^{¥}(U)
are essentially the same as those of H^{¥}(D), where
D Ì C is the open unit disk. In particular, we prove for
H^{¥}(U): a Forelli type theorem on projections, some theorems
on the characterization of interpolating sequences, an extension
theorem for matrices with entries in H^{¥}(U) (= a nonlinear
version of the corona theorem). Our methods are of a geometric nature
and have never been previously used in the theory of H^{¥}.
 Y. CHEN, Lakehead University, Thunder Bay, Ontario P7B 5E1
An improved bound for spectral variation near algebraic elements

Let A be a Banach algebra and a, b two algebraic elements of
degree at most 2 of A. Our earlier work showed that the Hausdorff
distance between the spectra s(a) and s(b) satisfies
the following inequality:
D^{2} 
æ è

s(a),s(b) 
ö ø

£ c 
max
 (a, b)ba, 

where c £ 3. In this talk, we will improve the bound by showing
that c £ Ö3+1.
 B. COLE, Brown University
Polynomial formulas for the disk and the bidisk

We produce polynomial formulas for the disk and the bidisk related to
the Pick and Agler interpolation problems. These formulas can be used
to deduce the von Neumann inequalities for polynomials in one and two
variables applied to linear operators on Hilbert space. This is joint
work with John Wermer.
 G. DAFNI, Concordia University, Montreal, Quebec H3G 1M8
A duality theorem for some function spaces related to H^{1} and
BMO

We consider the function spaces Q_{a}, 0 < a < 1, on n
dimensional Euclidean space, and identify them with the duals of
certain spaces of distributions which contain the Hardy space H^{1}.
The proof uses generalized Carleson measures, square functions, and
Hausdorff capacity. An atomic decomposition for the predual is also
obtained.
 R. FOURNIER, Department of Mathematics, Dawson College,
Montreal, Quebec H3Z 1A4
On certain spaces of metrics of hyperbolic type

Let D:={ z  z < 1} be the open unit disc of the
complex plane, H(D) the set of functions analytic in
D endowed with the usual topology and B :={ w Î H(D)  w(D) Í D}. For each w Î B we define w^{*} (z) = w¢(z)/(1w(z)^{2}) and W :={ w Î B  sup_{z Î D} w^{*}(z) < ¥}. Given an
outer function h Î B let W_{h} : = { w Î W  w^{*}(z) £ h(z), z Î D}. In this talk we shall study a natural
homeomorphism T : W_{h} ® B and elementary properties
of the extreme points of W_{h}.
 ANDREA FRASER, Dalhousie University
A characterization of some singular integral operators on the
Heisenberg group

The Heisenberg group H^{n} is the simplest example after
R^{n} of a homogeneous group; that is, a group which admits a
family of automorphic dilations. I consider the class of standard
CalderónZygmund singular integral operators on H^{n}. In
particular, this class includes the CauchySzegö projection. The
standard singular integral operators with radial convolution kernels
form a subclass of the Marcinkiewicztype spectral multipliers,
characterized by Müller, Ricci, and Stein in 1995. I establish a
stricter condition on the multipliers which characterizes this
subclass.
 P. GAUTHIER, Universite de Montreal, CP 6128 Succ. CentreVille, Montreal,
Quebec H3C 3J7
The existence of universal inner functions on the unit ball of
C^{n}

Together with XIAO Jie, we have shown that there exists an inner
function I on the unit ball B^{n} of C^{n} such
that each function holomorphic on B^{n} and bounded by 1
can be approximated by ``nonEuclidean translates'' of I.
 F. LARUSSON, Department of Mathematics, University of Western Ontario, London,
Ontario N6A 5B7
Recent developments in the theory of disc functionals

We describe recent work with Ragnar Sigurdsson in which we show that a
disc functional on a complex manifold has a plurisubharmonic envelope
if all its pullbacks by holomorphic submersions from domains of
holomorphy in affine space do and it is locally bounded above and upper
semicontinuous in a certain weak sense. For naturally defined classes
of disc functionals on manifolds, this result reduces a property
somewhat stronger than having a plurisubharmonic envelope to the affine
case. The proof uses a recent Stein neighbourhood construction of
Rosay, who proved the plurisubharmonicity of the Poisson envelope on
all manifolds. As a consequence, the Riesz envelope and the Lelong
envelope are plurisubharmonic on all manifolds; for the former, we make
use of new work of Edigarian. The basic theory of the three main
classes of disc functionals is thereby extended to all manifolds.
 N. LEVENBERG, Auckland University/Syracuse University
Distribution of nodes on algebraic curves in C^{N}

Given an irreducible algebraic curve A in C^{N}, let m_{d}
be the dimension of the complex vector space of all holomorphic
polynomials of degree at most d restricted to A. Let K be a
nonpolar compact subset of A, and for each d=1,2,..., choose
m_{d} points {A_{dj}}_{j=1,...,md} in K. Finally, let
L_{d} be the dth Lebesgue constant of the array {A_{dj}};
i.e., L_{d} is the operator norm of the Lagrange
interpolation operator L_{d} acting on C(K), where L_{d}(f) is the
Lagrange interpolating polynomial for f of degree d at the points
{A_{dj}}_{j=1,...,md}. Using techniques of pluripotential
theory, we show that there is a probability measure m_{K} supported
on ¶K such that for any array in K satisfying
limsup_{d® ¥}L_{d}^{1/d} £ 1, the discrete measures
m_{d}:= 
1 m_{d}


m_{d} å
j=1

d_{Adj}, d=1,2,..., 

converge weak* to m_{K}.
 J. MASHREGHI, Universite Laval, Quebec, G1K 7P4
Argument of outer functions

While a complete description of the modulus of an outer function is
known, a satisfactory characterization of the argument is not
available. Being interested in characterizing real function y
such that exp(iy(x)) = O(x)/O(x), for almost all x Î R, and for some outer function O, we show that if
y is a sawtoothLipschitz function, then it is the argument of an
outer function on the real line.
 PETR PARAMONOV, Moscow State University, Moscow 119899, Russia
Uniform approximation of (stationary) gravitational fields.
Extension and related capacity

Capacitative criteria for the approximation of (stationary)
gravitational fields on compact subsets of R^{3} will be
given. Subadditivity and density properties of the related capacity
will be established. Extension from balls to all of R^{3}
will also be discussed.
 E. POLETSKY, Syracuse University
Analytic multifunctions and Jensen measures

In the talk we will give a characterization of analytic multifunctions
through Jensen measures and discuss consequences. This approach allows
us to use methods of pluripotential theory to study the geometry of
multifunctions.
 D. ROCHON, Université du Québec à TroisRivières, Département de
Mathématiques et d'informatique, TroisRivières,
Québec G9A 5H7
A Cantor set for bicomplex numbers

This talk will be about a commutative generalization of the
complex numbers called bicomplex numbers. First, we will focus our
attention on the bicomplex dynamics. In particular, we will give a
generalization of the Mandelbrot set and the Julia sets in dimensions
three and four and we will establish some relationships between them.
Furthermore, from the properties of the Julia sets we will propose
a bicomplex Cantor set in dimension four.
 J. ROSTAND, Centre de recherches pour la defense, Ottawa, Ontario
Sur les automorphismes de la boule unité spectrale / On
the automorphisms of the spectral unit ball

Soit W_{n} l'ensemble des matrices n×n dont le rayon
spectral est inférieur à 1. Le but ultime est de répertorier
l'ensemble des automorphismes de W_{n}. Dans un premier temps,
nous discuterons des résultats connus dans le cas général. Puis,
nous porterons notre attention sur le cas particulier n=2 et verrons
qu'il est possible d'en savoir davantage.
Let W_{n} be the set of n×n matrices with spectral radius
less than 1. The ultimate goal is to make a list of all automorphisms
of W_{n}. We will first present what has been done in the general
case. Then we will focus on the special case n=2 and see what more
can be obtained.
(Transparencies will be in english et l'exposé en français.)
 ANATOLY SEDLETSKII, Moscow State University, Moscow 119899, Russia
MuntzSzasz type approximation

The problem of the description for complete systems of powers ( with
complex exponents) in the Lebesgue spaces on a finite interval will be
discussed.
 ZBIGNIEW SLODKOWSKI, Department of Mathematics, Statistics and Computer Science,
University of Illinois at Chicago, Chicago, Illinois 60607,
USA
Evolution of compact subsets of C^{2} by Levi curvature

(Text of abstract:)
The talk will report on the joint work of the speaker with Giuseppe
Tomassini(Pisa). Informally speaking, we consider a type of evolution
in which a domain bounded by a smooth hypersurface expands at a
strictly pseudoconcave point in the direction of the outside normal
vector at a rate proportional to the negative of its Levi curvature at
the point, but its rate of change is zero at the weakly pseudoconvex
points. Formal definition of such evolution is given in terms of weak
(viscosity) solutions of a certain nonlinear, degenerate parabolic
equation and global existence and uniqueness is obtained for every
compact subset K of C^{2}. We show that this evolution is continuous
in time, nondecreasing and that the sets added in time to the initial
one have local maximum property with respect to plurisubharmonic
functions. Thus the limit of the evolution is a kind of a hull; we
examine its relation the Stein hull. It is shown that the closures of
nonpseudoconvex (bounded) domains have nontrivial (nonconstant)
evolution, while smoothly bounded pseudoconvex domains are ``stable'',
i.e. preserved by evolution. To characterize stable sets, as
well as the evolution family in general, we introduce a maximal
function of a compact set defined in terms of subsolutions of the
elliptic part of the parabolic operator mentioned above. (This maximal
function has some resemblance to Siciak's maximal function.)
 P. VITSE, Departement de Mathematiques et Statistiques, Pavilon Vachon,
Universite Laval, Quebec, Quebec G1K 7P4
Functional calculus under Kreiss and Ritt condditions

A bounded operator T on a Banach space X is said to satisfy the
Kreiss condition (K) if  (TlI)^{1}  £ [(C)/(l 1)] for every l, l > 1. If
moreover X is of finite dimension N, it is known that T is power
bounded, namely sup_{n ³ 0} T^{n} £ e C (N+1). Here we
generalize this type of inequality to the polynomial calculus of
algebraic operators. For example, if T satisfies (K) and q(T) = 0, where q is a polynomial of degree N, we have  j(T)  £ aC (N+1)^{2}  sup_{z < 1}  j(z)  for every
polynomial j, where a does not depend on T.
Under the stronger Ritt condition (R),  (TlI)^{1}  £ [(C)/(l1)], l > 1, it is known that T is always
power bounded, more precisely sup_{n ³ 0} T^{n} £ C^{2}. For
the polynomial calculus, we get a bound in terms of the operator norm
in the space of multipliers of the CauchyStieltjes integrals, namely
 j(T)  £ C j_{mult}.
 J. WERMER, Brown University, Providence, Rhode Island 02912, USA
Approximation on the unit sphere in C2

Let X be a compact set on the unit sphere in C2. Problem: When
does R(X) = C(X), where R(X) is the uniform closure on X of
rational functions analytic on X. A key example was given by Basener
in 1972 and various authors have made contributions to the problem
since then. We shall discuss some recent results and conjectures
connected with the problem. (this is joint work by the author,
J. Anderson and A. Izzo).

