Complex Function Theory
Org: Ian Graham (Toronto) and Eric Schippers (Manitoba) [PDF]
 ROGER BARNARD, Texas Tech University, Lubbock, Texas
Icebergtype problems in two dimensions
[PDF] 
We consider the complex plane C as a space filled with two
different media, separated by the real axis R. Let H denote the
upper halfplane. For a planar body E, the icebergtype problem is
to estimate characteristics of the invisible part E \H from
the characteristics of the whole body E and its visible part E ÇH.
In this talk, we outline the methods we use to determine the maximal
draft of E as an explicit function of the logarthmic capacity of E
and the area of E ÇH.
 TOM BLOOM, University of Toronto, 40 St. George St., Toronto
Random polynomials and pluripotential theory
[PDF] 
I will report on results giving the expected distribution of zeros
of certain ensembles of random polynomials in one and several
complex variables as the equilibrium measure of compact sets.
This is joint work with B. Shiffman.
 ANDRÉ BOIVIN, University of Western Ontario
Weighted Hardy spaces for the unit disc: Approximation
properties
[PDF] 
We will state some basic properties of the weighted Hardy space for
the unit disc with the weight function satisfying Muckenhoupt's
(A^{p}) condition (1 < p < ¥). Approximation properties in that
space of the system of rational functions e_{k}(z) = \dfrac1(2pi)(1[`(a)]_{k}z), where {a_{k}} is a sequence satisfying the
Blaschke condition å_{k=1}^{¥} (1a_{k}) < ¥, will then
be discussed.
 MARITZA BRANKER, Niagara University, Lewiston, NY
Weighted pluripotential theory on Kahler manifolds
[PDF] 
This talk will describe an extension of weighted pluripotential
theory, based on quasiplurisubharmonic functions on compact Kahler
manifolds.
The work described is a collaborative effort with M. Stawiska.
 RICHARD FOURNIER, Dawson CollegeCRM, 3040 Sherbrooke West, Montreal
Asymptotics of the Bohr radius for polynomials of fixed
degree
[PDF] 
We obtain a characterization and precise asymptotics of the Bohr
radius for the class of complex polynomials in one variable. Our work
is based on the notion of boundpreserving operators.
 DAVID HERRON, University of Cincinnati
Euclidean Quasiconvexity
[PDF] 
A metric space is quasiconvex provided it is bilipschitz equivalent to
a length space: each pair of points can be joined by a rectifiable
path whose length is comparable to the distance between its
endpoints.
We consider a closed set in Euclidean nspace and ask when is its
complement quasiconvex. In dimension n=2, a complete description is
available, at least for closed sets with finitely many components. In
general, there are sufficient conditions that such a complement be
quasiconvex; one such condition is that the set have zero
(n1)dimensional Hausdorff measure.
We exhibit, for each dimension d in [n1,n], a compact totally
disconnected set with positive finite dmeasure whose complement is
quasiconvex. On the other hand, we also construct a compact totally
disconnected set with nonzero (n1)measure whose complement fails
to be quasiconvex.
Joint work with Hrant Hakobyan.
 DAN JUPITER, Department of Systems Biology and Translational Medicine,
College of Medicine, Texas A&M Health Science Center
Weak Stein Neighbourhood Bases
[PDF] 
It is of interest to understand whether the closure of a pseudoconvex
domain in C^{n} has a neighbourhood basis of pseudoconvex
domains. Another question of interest is to understand when
holomorphic functions on a pseudoconvex domain can be approximated by
holomorphic functions on a larger set. We discuss some aspects of the
relationship between these two types of approximation properties.
 GABRIELA KOHR, BabesBolyai, Romania

 DANIELA KRAUS, University of Würzburg
Critical points of inner functions, nonlinear partial
differential equations, and an extension of Liouville's
theorem
[PDF] 
Theorem 1
Let {z_{j}} Í D be a Blaschke sequence. Then there
exists a Blaschke product with critical points {z_{j}} (counted with
multiplicity) and no others.
The case of finitely many critical points has been proved earlier by
Heins 1962, Wang & Peng 1979, Bousch 1992 and Zakeri 1996
(topological proofs) and by Stephenson 2005 (discrete methods). It
has found applications in complex dynamics by Milnor.
The proof of Theorem 1 is based on an extension of Liouville's
classical representation theorem for solutions of the partial
differential equation Du = 4e^{2u} combined with methods from
nonlinear elliptic PDE. Our work is closely related to the
BergerNirenberg problem in differential geometry.
Joint work with Oliver Roth.
 JAVAD MASHREGHI, Laval
Integral representations of the derivatives of functions in
H(b)
[PDF] 
Let H^{p}(C_{+}) stand for the Hardy space of the upper half
plane C_{+}, and for j Î L^{¥} (R), let
T_{j} stand for the Toeplitz operator defined on
H^{2}(C_{+}) by
T_{j} (f) : = P_{+} (jf), 
æ è

f Î H^{2}(C_{+}) 
ö ø

, 

where P_{+} denotes the orthogonal projection of L^{2}(R)
onto H^{2}(C_{+}). Then, for j Î L^{¥}(R), j_{¥} £ 1, the de BrangesRovnyak
space H (j), associated to j, consists of
those H^{2}(C_{+}) functions which are in the range of the
operator (IdT_{j} T_{[`(j)]})^{1/2}. It is a Hilbert
space when equipped with the inner product
á(IdT_{j} T_{[`(j)]})^{1/2}f, (IdT_{j}T_{[`(j)]})^{1/2} g ñ_{j} = áf,gñ_{2}, 

where f,g Î H^{2}(C_{+}) \ominus ker(IdT_{j}T_{[`(j)]})^{1/2}. In particular, if b is an inner
function, then (IdT_{b} T_{[`(b)]}) is an orthogonal projection
and H(b) is a closed (ordinary) subspace of
H^{2}(C_{+}) which coincides with the socalled model spaces
K_{b} = H^{2}(C_{+}) \ominus b H^{2}(C_{+}).
We give some integral representations for the boundary values of
derivatives of functions of the de BrangesRovnyak spaces
H(b), where b is an extreme point of the unit ball of
H^{¥} (C_{+}).
 DAVID MINDA, University of Cincinnati, Cincinnati, Ohio
Geometric variations of Schwarz's Lemma
[PDF] 
The classical version of Schwarz's Lemma deals with holomorphic
selfmaps of the unit disk D that fix the origin; the
extremal functions for Schwarz's Lemma are rotations about the origin.
We consider holomorphic maps of D into a region W
that satisfies some geometric property that holds for the unit disk.
For example, W has diameter at most 2. There are regions of
diameter 2 that are not contained in a disk of radius 1, so this
case properly contains the classical framework. Landau and Toeplitz
in considered this situation in 1907. Other geometric conditions on
W involve the area, length of the boundary and higherorder
diameters, including the transfinite diameter. In all cases we obtain
sharp analogs of the classical Schwarz Lemma and identify the extremal
functions.
 JERRY MUIR JR., University of Scranton, Scranton, PA
Preservation of geometric properties by a class of extension
operators
[PDF] 
We consider operators that extend locally univalent mappings of the
unit disk D of C to locally biholomorphic mappings
of the Euclidean unit ball B of C^{n}. For such an
operator F, we ask whether F(f) is a convex (resp. starlike) mapping of B whenever f is a convex (resp. starlike)
mapping of D or whether e^{t} F( e^{t} f(·,t)), t ³ 0, is a Loewner chain on B whenever f(·, t),
t ³ 0, is a Loewner chain on D. Answers will be provided
for a class of operators that are perturbations of the well known
RoperSuffridge extension operator.
 RAJESH PEREIRA, University of Saskatchewan
Inequalities in the Analytic Theory of Polynomials
[PDF] 
We show how techniques in matrix theory and majorization can be used
to derive inequalities relating the zeros and critical points of a
polynomial. These inequalities stengthen known results such as the
GaussLucas Theorem and Mahler's inequality. We will also present
some partial results and conjectures on a KrooPritsker type
inequality between the Bombieri norm of a polynomial and the product
of the Bombieri norms of its linear factors.
 JOHN PFALTZGRAFF, University of North Carolina, Chapel Hill, NC 275993250,
USA
Conformal mapping of multiply connected slit domains
[PDF] 
The first general formula for a SchwarzChristoffel mapping of a
canonical domain of connectivity m > 2 onto a conformaly equivalent
polygonal domain appears in work with DeLillo and Elcrat [DEP,04].
Construction of the mapping and its formula uses infinite sequences of
iterated reflections in circles, repeated use of the reflection
principle and invariance of the preSchwarzian to obtain an infinite
product representation of the derivative of the map and an integral
formula for the mapping function. The method can be interpreted as a
form of the "method of images" in electrostatics.
The problem of implementing the formula numerically and graphically is
pursued in [DDEP,06]. Developing a robust code and a complete, easy
to apply procedure remains a challenging problem. In current work
with DeLillo, Driscoll and Elcrat, interesting special features have
appeared when the target domains are certain canonical slit domains.
For example, the direct construction of a formula for the mapping
function that produces f(z) explicitly without requiring an
integration of the derivative.
Remark
The results in [DEP,04] were presented by Elcrat and Pfaltzgraff in
2003 at international meetings ICIAM in Australia and AMSRSME,
Seville, Spain.
References
 [DEP,04]

T. DeLillo, A. Elcrat and J. Pfaltzgraff,
SchwarzChristoffel mapping of multiply connected domains.
J. Anal. Math. 94(2004), 1747.
 [DDEP,06]

T. DeLillo, T. Driscoll, A. Elcrat and J. Pfaltzgraff,
Computaton of multiply connected SchwarzChristoffel maps for
exterior domains.
Comput. Methods Funct. Theory 6(2006) 301315.
 DAVID RADNELL, American University of Sharjah, United Arab Emirates
Interactions between conformal field theory and Teichmueller
spaces
[PDF] 
Conformal Field Theory (CFT) arose in physics as a special class of
twodimensional quantum field theories. The mathematics of CFT
requires the study of Riemann surface whose boundary components are
parameterized. The moduli space of these rigged Riemann surfaces
arises naturally in the mathematical description. The sewing of two
Riemann surfaces by identifying the boundary components is a
fundamental operation.
We have recently applied results from Teichmueller theory, such as
conformal welding, to CFT. In particular, the sewing operation was
shown to be holomorphic.
In ongoing work, these results and ideas from CFT are used to give a
new structure to the infinitedimensional Teichmueller space of
bordered Riemann surfaces. We show that this Teichmueller space is a
complex fiber space over the finitedimensional Teichmueller space of
punctured surfaces. The fibers are spaces of conformal maps with
quasiconformal extensions that are closely related to the universal
Teichmueller space.
This introductory talk will overview the new and rewarding interplay
between these fields.
This is joint work with E. Schippers.
 THOMAS RANSFORD, Université Laval, Québec (QC), G1K 7P4
Computation of capacity
[PDF] 
I shall describe a method for computing the logarithmic capacity of a
compact plane set. The method yields upper and lower bounds for the
capacity. If the set has the Hölder continuity property, then these
bounds converge to the value of the capacity. I shall discuss several
examples, including the Cantor middlethird set, for which we estimate
c(E) » 0.220949102189507.
Joint work with Jérémie Rostand.
 OLIVER ROTH, University of Würzburg
On the isolated singularitites of a class of subharmonic
functions
[PDF] 
We extend a classical result proved by Nitsche in 1957 about the
isolated singularities of the solutions of the Liouville equation
Du = 4e^{2u} to solutions of the Gaussian curvature equation
Du = 4k(z) e^{2u} where k is a strictly negative
Hölder continuous function. This yields growth and regularity
theorems for strictly negatively curved conformal Riemannian metrics
close to their singular points which complement the corresponding
existenceanduniqueness results due to Heins, Troyanov, McOwen and
others.
Joint work with Daniela Kraus.
 STEPHAN RUSCHEWEYH, Department of Mathematics, Wuerzburg University, 97074
Wuerzburg, Germany
Universally convex univalent functions
[PDF] 
A function f analytic in the slit domain C[1,¥] is
called universally convex if it maps every circular domain containing
the origin but not the point 1 univalently onto a convex domain. We
give a complete characterization of those functions in terms of
Hausdorff moment sequences, and show that this set is closed under
convolutions (Hadamard product). Some generalisations are also
mentioned.
Joint work with L. Salinas, Valparaíso, and T. Sugawa, Hiroshima.
 ALEXANDER SOLYNIN, Texas Tech University, Department of Mathematics and
Statistics, Lubbock, TX 79409, USA
Hyperbolic convexity and the analytic fixed point function
[PDF] 
We will discuss properties of the analytic fixed point function
introduced recently by D. Mejia and Ch. Pommerenke. In particular, we
solve one of the problems raised by D. Mejía and Ch. Pommerenke by
showing that the analytic fixed point function is hyperbolically
convex in the unit disc. We also prove some extremal properties of
such functions related to mappings from the unit disk onto symmetric
Riemann surfaces.
 TED SUFFRIDGE, Department of Mathematics, University of Kentucky, Lexington,
KY 40506
Invariant mappings on the ball and extremal problems
[PDF] 
The concept of "linear invariant family" was introduced by
Pommerenke in his 1964 paper in Mathematische Annalen. A family
F of functions that are analytic on the unit disk and
normalized by f(0) = 0, f¢(0) = 1 with f¢(z) ¹ 0 when z < 1 is linear invariant provided that the function K_{j} f Î F whenever f Î F. Here, j is a
holomorphic automorphism of the unit disk, and K_{j} f is
obtained by forming the composition f°j and normalizing
the result. The functions that have the property K_{j} f = f
for certain automorphisms j are of particular interest and in
fact the solution of many extremal problems on a family F
is one of these "invariant" functions. We discuss the extension of
these ideas to mappings f : B® C^{n}, where
B is the Euclidean ball in C^{n}, and in fact characterize
the invariant mappings for given linear invariant familes of mappings,
in a theorem that gives a procedure for constructing all such
mappings.
This is joint work with J. A. Pfaltzgraff.
 DROR VAROLIN, Stony Brook University
Nonnegative Hermitian polynomials and quotients of squared
norms
[PDF] 
Hilbert's 17th problem asks whether any nonnegative polynomial can be
written as a sum of squares of rational functions. While a positive
answer was established by Artin, there is no known way to construct
the rational functions. In this talk we describe our solution of a
Hermitian analog of Hilbert's 17th problem posed by John D'Angelo
about 14 years ago.
 BROCK WILLIAMS, Texas Tech University, Lubbock, Texas
Circle Packings, Welding Operations, and Applications
[PDF] 
We will discuss various welding operations in the plane and on Riemann
surfaces and their circle packing analogues. For example, circle
packings can be used to approximate classical quasisymmetric welding,
such as arises in the work of Radnell and Schippers in string theory.
We will also consider other weldingtype deformations included
Thurston's earthquakes and McMullen's complex earthquakes.
This is joint work with Roger Barnard, J'Lee Bumpus, and Eric Murphy.
