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Complex Analysis and Approximation
A conference in honour of Anthony G. O'Farrell
NUI Maynooth17–19 June 2013
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Titles and abstracts
Approximate identities in Banach function algebras
Let $(A, \|\cdot\|)$ be a natural Banach function algebra on a compact
space $K$; thus every maximal ideal in $A$ has the form
$M_x= \{f \in A: f(x)=0\}$ for some $x\in K$. A maximal ideal $M_x$
has a contractive approximate identity (CAI) if there is a net
$(f_\alpha)$ in $M_x$ such that $\| f_\alpha\| \leq
1$ for each $\alpha$ and $\lim_\alpha ff_\alpha =f$ for each $f \in
M_x$. The algebra $A$ is contractive if each maximal ideal
$M_x$ has a CAI. For example, set $A =C(K)$. Then $A$ is contractive.
We shall discuss how many other Banach function algebras have this
property.
Towards this we shall introduce a 'BSE-norm' on a Banach function algebra.
This talk is based on joint work with Ali Ülger (Istanbul).
Holomorphic functions over infinite dimensional domains
I will discuss various types of expansions of holomorphic functions over
different kinds of locally convex spaces and give an example of a Banach
space where the monomials form a Schauder basis for the space of all
holomorphic functions. (Joint work with Jorge Mujica (Campinas))
Classicalisation of Swiss cheeses
Three of my students (PhD/MSc) have made significant contributions to this
(ongoing) work: Matthew Heath, Jonathan Mason, and Hongfei Yang.
In this talk we will discuss various types of Swiss cheese set, and their
applications. Here, a Swiss cheese set is a compact plane set obtained by
deleting the union of some suitable sequence of open discs from some initial
closed disc. Of course, without some additional conditions on the discs,
this would mean that every compact plane set was a Swiss cheese set. In
practice we place requirements on the positions and/or the radii of the
deleted discs to ensure that the resulting set has desirable
properties. Such Swiss cheese sets are a very useful source of examples in
rational approximation theory and in the theory of uniform algebras.
We discuss some of the standard applications of Swiss cheese sets from the
literature, including an example of O'Farrell (1979) of a regular uniform
algebra with a continuous point derivation of infinite order. (This example
is $R(X)$ for a suitable Swiss cheese set $X$.) We then describe a process
which we call the classicalisation of Swiss cheeses, which enables us to
modify a Swiss cheese set $X$ in order to improve its topological
properties, while attempting to retain desired properties of $R(X)$. One
direct application of this classicalisation procedure is to produce examples
of essential, regular uniform algebras on locally connected topological
spaces. However, rather more care is required in order to classicalise the
example of O'Farrell. We discuss some of the issues, and how they can be
overcome.
The growth of bounded and related functions in the unit disk
In this talk, an answer is supplied to the following question of Rozenblum:
"How fast can a bounded analytic function tend to zero outside an
appropriate exceptional set"? The answer is obtained first for Blaschke
products and is then extended to the Nevanlinna class of functions of
bounded characteristic, i.e. meromorphic functions which are the ratio of
two bounded functions, by the representation of such functions as the ratio
of two Blaschke products multiplied by an exponential term. For Blaschke
products $B(z)$ it is shown that
$(1-|z|)\log|B(z)|\to 0$
as $|z|\to 1$ for $z$ outside an exceptional F-set. This is defined in the
introduction as the union of a suitable set of disks lying in the unit disk.
It is also shown that these results are essentially sharp.
Function algebras invariant under group actions
Motivated by his work on a conjecture of William Arveson in operator theory,
Ronald Douglas raised a question regarding function algebras on the unit
sphere in complex $n$-space invariant under the torus action. Surprisingly,
the answer depends on the dimension.
The speaker's work on Douglas' question led him to formulate a conjecture
regarding function algebras that are invariant under a transitive group
action. Results related to this conjecture will be presented. Among these
results is a complete description of all function algebras on the circle
invariant under a transitive group action. This description shows, in
particular, that the conjecture referred to above holds for function
algebras on the circle.
Algebraic aspects of the Dirichlet problem
It is well known that a solution of the Dirichlet problem in the ball in
$R^n$ with a polynomial data is a harmonic polynomial. It is less well known
but nevertheless true that the same holds for ellipsoids. In $R^2$ one can
easily show that the rational data on the circle implies a solution to be a
rational harmonic function of $x$ and $y$. Moreover, this property
characterizes disks ( S. Bell, P. Ebenfelt, DK and H. S. Shapiro, 2006). The
latter result and its generalizations lead to a more general task of
locating the singularities of solutions to the Dirichlet problem posed on
algeraic closed curves. Remarkably, the latter problem in two dimensions is
closely related to complexified version of the "$R^n$ closed lightning bolts"
introduced by Arnold and Kolmogorov in the 1950s to study Hilbert's 13th
problem.
Recent results on finely monogenic functions
In my talk, I recall briefly the theory of finely holomorphic
functions which has been developed by B. Fuglede, T.J. Lyons,
A.G. O'Farrell and others since 1970s. Then I review results about
finely monogenic functions which are a generalization of finely holomorphic
functions to higher dimensions in the context of hypercomplex analysis.
In particular, I shall talk about fine differentiability.
Conformal Welding
We will discuss some recent applications of conformal welding,
including the uniformization of triangulations and quadrangulations of
the sphere, conformal maps to the complements of trees and dendrites,
and their connection to the combinatorics of maps (embedded graphs)
and Grothendieck's dessins d'enfants. In particular this gives a
"hands-free" method for computing Shabat polynomials and Belyi
functions.
Swiss cheese & champagne bubbles
The first part of the talk reveals why and where Swiss Cheese Sets are
important in mathematical analysis, in particular, in Banach function
algebras and approximation theory. The second part presents new results
obtained jointly with Wolfhard Hansen on champagne subdomains. These
subdomains which have to do with Brownian motion and harmonic measure have
recently been studied by several authors. We were able to find optimal
results on the size of bubbles for unavoidable champagne subregions.
Embeddings, Carleson measures, and weights
In the sixties, Lennart Carleson characterised those positive Borel measures
$\mu$ on the unit disc $\mathbb D$ such that the Hardy space inclusion
$H^p(\mathbb D) \subseteq L^p(\mathbb D, \mu)$ is continuous. Nowadays,
such measures are called classical Carleson measures. In this talk we will
discuss how this embedding problem has been solved for other spaces of
analytic functions such as Dirichlet and Bloch spaces. Special attention
will be paid to Carleson-type measures for weighted Bergman spaces
$A^p_{\omega}$ when $\omega$ is a Bekollé-Bonami weight, and when
$\omega$ is a rapidly increasing weight as recently introduced by
Peláez and Rättyä.
Vector valued multivariate spectral multipliers, Littlewood-Paley
functions, and Sobolev spaces in the Hermite setting
In this talk we find new equivalent norms in
$L^p(\mathbb{R}^n,\mathbb{B})$ by using multivariate Littlewood-Paley
functions associated with Poisson semigroup for the Hermite operator,
provided that $\mathbb{B}$ is a UMD Banach space with the property
($\alpha$). We make use of $\gamma$-radonifying operators to get new
equivalent norms that allow us to obtain
$L^p(\mathbb{R}^n,\mathbb{B})$-boundedness properties for (vector valued)
multivariate spectral multipliers for Hermite operators. As application of
this Hermite multiplier theorem we prove that the Banach valued Hermite
Sobolev and potential spaces coincide.
On a conjecture of Lorentzen and Ruscheweyh
We consider the dynamics of compositions of analytic self-maps of a
hyperbolic domain. Twenty years ago, Lorentzen and Ruscheweyh made a
conjecture about the behaviour of such compositions for a particular class
of analytic maps related to continued fractions. We outline a programme for
proving their conjecture when the domain is a polygon.
Universal reversibility of JC*-triples
Jordan triple systems of operators on Hilbert space
are those closed subspaces $E\subseteq B(H)$ with the property that
$ab^*c + cb^*a \in E$ whenever $a,b,c\in E$ and their internal
structure has been studied extensively. In particular the open
unit ball of $E$ has a transitive biholomorphic automorphism
group, like the Möbius group of the disk. If two JC*-triples
are the same in the sense of being internally indistinguishable,
they may not be embedded in $B(H)$ in the same way. However, the
number of distinct operator space structures is often limited
and we describe an approach to this aspect (joint work with L. J.
Bunce). Universally reversible triples occur rather frequently among
JC*-triples and we show that they are especially well-behaved
from our point of view.
Personal Reminiscences
I will discuss various ways in which Tony O'Farrell had a major impact on
NUI Maynooth, both in the Department of Mathematics and elsewhere. I will
also discuss some aspects of Tony's contribution to the state of mathematics
in Ireland.
Picard Theorems, 1879–2013
A bird's-eye view of the mathematics surrounding the Picard Theorems (Great
and Small) and their close relative, Montel's Theorem, over the past 1 1/3
centuries, with an emphasis on recent progress.