


Complex Variables
Org: Thomas Bloom (Toronto) and Paul Gauthier (Montréal) [PDF]
 GAUTAM BHARALI, University of Michigan, Department of Mathematics, East Hall,
Ann Arbor, MI 48109, USA
Polynomial approximation, local polynomial convexity, and
degenerate CR singularities
[PDF] 
This talk will present a few advances on some earlier work on the
following question: when is a smooth real surface S Ì C^{2} locally polynomially convex at a point p Î S? This
question is complicated by the presence of points in the surface S
that have complex tangents. Such points are called CR singularities.
Let p Î S be a CR singularity at which the order of contact of the
tangent plane with S is greater than 2; i.e., a
degenerate CR singularity. We will discuss a sufficient condition for
S to be locally polynomially convex at a degenerate CR singularity.
In demonstrating this sufficient condition, we will need a new result
about the uniform algebra on a closed disc in C generated
by z and a complexvalued continuous function F. This result may
be of independent interest because the function F here is allowed to
be nonsmooth; this result may thus be viewed as a Mergelyantype
theorem for complexvalued F.
 LEN BOS, University of Calgary, Calgary, Alberta
Metrics Associated to Polynomial Inequalities
[PDF] 
Suppose that K is a compact subset of R^{n}. We discuss a number of
metrics on K that are closely related to MarkovBernstein type
inequalities on the derivatives of polynomials, as well as to the
SiciakZahariuta extremal function for K as a subset of C^{n}. We
also discuss some conjectured relations between these metrics and the
distribution of extremal point sets such as Fekete points.
 ALEXANDER BRUDNYI, University of Calgary
Holomorphic Functions of Slow Growth on Coverings of
Pseudoconvex Domains in Stein Manifolds
[PDF] 
We study holomorphic functions of slow growth on coverings of
pseudoconvex domains in Stein manifolds. In particular, we extend and
strengthen certain results of Gromov, Henkin and Shubin on holomorphic
L^{2} functions on coverings of pseudoconvex manifolds in the case of
coverings of Stein manifolds.
 DAN BURNS, University of Michigan, Ann Arbor, MI 48109, USA
Exterior MongeAmpère solutions for real convex bodies
[PDF] 
We study solutions of the homogeneous complex MongeAmpère equation
(¶[`(¶)] u)^{n} = 0 in C^{n} \K,
where K is a compact convex set in R^{n} Ì C^{n} for a plurisubharmonic function growing logarithmically
on C^{n} and tending to zero along K. The solution we are
considering is therefore the SiciakZaharjuta extremal function
associated to K. We use the method of characteristics to describe
the solution. We describe a variational problem for Robin constants
associated to holomorphic disks passing through the hyperplane at
infinity in CP^{n}, which is in a sense dual to the
KobayashiRoyden functional on disks used to define the infinitesimal
Kobayashi distance. We use Lempert's work along these lines for the
exteriors of strictly convex sets in C^{n}, passing to a
limit of a sequence of approximating domains. The characteristic
curves of the extremal function for the limit real convex set always
exist and we show they are given by quadrics in CP^{n}. The
variational formulation and elementary geometry enable us to analyze
the extremal functions fairly explicitly, especially under weak
regularity conditions on K. An application is given to polynomial
approximation in higher dimensions.
This is joint work with Norman Levenberg (Indiana University) and
Sione M'au (Aukland University).
 JOE CALLAGHAN, University of Toronto
A Green's function for thetaincomplete polynomials
[PDF] 
Let K be a compact subset of ndimensional complex space and f a
continous function on K. For 0 < q < 1, define a
qincomplete polynomial as any polynomial P with no terms of
degree smaller than q times the degree of P. We seek
necessary and sufficient conditions on f and K, for f to be the
uniform limit on K, of qincomplete polynomials. Our
approach is to define an appropriate Green's function for
qincomplete polynomials. Then the solution to the
approximation problem can be phrased in terms of the MongeAmpere
measure of this function. More precisely, this type of approximation
will be possible exactly when f vanishes outside of the support of
this measure.
 FRED CHAPMAN, Waterloo
A Sufficient Condition for Uniform Convergence of a New Class
of Newton Interpolation Series in Two Complex Variables
[PDF] 
The name "Geddes series" refers to an extensive catalogue of new
classes of series expansions for interpolating and approximating
multivariate functions. The catalogue includes classes such as
GeddesTaylor series, GeddesFourier series, and a surprisingly
large number of other classes. The general Geddes series scheme was
invented by the presenter and named in honor of his thesis supervisor
in 2003.
The simplest class of Geddes series is the GeddesNewton series,
which interpolates a function of two variables on the lines of a
twodimensional grid; when either variable is held fixed, a
GeddesNewton series reduces to a generalized Newton interpolation
series in the free variable. Grids with infinitely many lines
generate GeddesNewton series with infinitely many terms. We
conjecture that if the original function is analytic in two complex
variables over a sufficiently large region, the resulting
GeddesNewton series converges uniformly to the original function on
every sufficiently small compact set containing all the grid lines.
We also conjecture that the rate of convergence is linear or
superlinear.
This talk describes our progress to date in proving these conjectures.
To that end, we will present a new contour integral remainder formula,
new rigorous error estimates, and a new sufficient condition for
uniform convergence at a linear or superliner rate. All that remains
is to prove that this sufficient condition is always satisfied. We
will also use Maple to present two applications which demonstrate the
rapid convergence of GeddesNewton series in practice:
(1) the fast and accurate evaluation of certain kinds of
multiple integrals in four real dimensions, and
(2) the uniform approximation of special functions by
elementary functions.
This talk presents joint research with Keith Geddes.
 DAN COMAN, Department of Mathematics, Syracuse University, Syracuse, NY
132441150
Invariant currents and dynamical Lelong numbers
[PDF] 
Let f be a polynomial automorphism of C^{k} of degree
l, whose rational extension to P^{k} maps the
hyperplane at infinity to a single point. Given any positive closed
current S on P^{k} of bidegree (1,1), we show that the
sequence l^{n}(f^{n})^{*} S converges in the sense of
currents on P^{k} to a linear combination of the Green
current T_{+} of f and the current of integration along the
hyperplane at infinity. We give an interpretation of the coefficients
in terms of generalized Lelong numbers with respect to an invariant
dynamical current for f^{1}. The results are joint work with
Vincent Guedj.
 BRUCE GILLIGAN, University of Regina, Regina, SK S4S 0A2
Kaehler homogeneous manifolds
[PDF] 
Let G be a connected complex Lie group and H a closed, complex
subgroup of G. In this talk we will assume that the complex
homogeneous manifold X : = G/H is Kähler. Kähler homogeneous
manifolds X are completely understood if X is compact or the
metric is Ginvariant. The situation is also understood if the
group G is semisimple, solvable, or a direct product S×R of
its radical R with a maximal semisimple subgroup S of G.
Attempts to construct examples of noncompact manifolds X homogeneous
under a nontrivial semidirect product G = S \ltimes R with a not
necessarily Ginvariant Kähler metric motivated this work.
In this setting the Sorbit S/SÇH in X is Kähler. Thus
SÇH is an algebraic subgroup of S. The Kähler assumption on
X ought to imply the Saction on the base Y of any homogeneous
fibration X ® Y is algebraic too. Natural considerations allow a
reduction to the case where H=G is a discrete subgroup and
there is a homogeneous fibration X = G/G® G/I = : Y with
I^{°} an abelian, normal subgroup of G and the fiber
I^{°}/(I^{°} ÇG) a Cousin group, i.e., a
complex Lie group with no nonconstant holomorphic functions. We prove
an algebraic condition does hold in the homogeneous manifold Y = [^(G)]/[^(G)], where [^(G)] : = G/I^{°} and
[^(G)] : = I/I^{°}. Namely, we show that an element
[^(g)] Î [^(G)] of infinite order lying in a semisimple
subgroup [^(S)] of [^(G)] is an obstruction to the existence of
a Kähler metric on X. So if X is Kähler, then [^(S)] Ç[^(G)] is finite.
As a consequence, if the group [^(G)] is a linear algebraic group
whose radical [^(R)] is a vector group and the representation of
[^(S)] on [^(R)] is linear with no nonzero invariant vector,
then G/G cannot be Kähler. An example of such a group
[^(G)] is the affine group of C^{n} for n > 1.
 NIHAT GOGUS, Department of Mathematics, 215 Carnegie Hall, Syracuse
University, Syracuse, NY 13244, USA
Fusion and Localization of Plurisubharmonic Functions
[PDF] 
We address the problem of continuity of plurisubharmonic envelopes. A
bounded domain D is called cregular if the plurisubharmonic
envelope of every continuous function on [`(D)] extends continuously
to [`(D)]. Using Gauthier's Fusion Lemma, we show that a domain is
locally cregular if and only if it is cregular.
 IAN GRAHAM, University of Toronto, Toronto, ON
The CaratheodoryCartanKaupWu theorem on an
infinitedimensional Hilbert space
[PDF] 
Let W be a bounded convex domain in a separable Hilbert space.
Let f : W® W be a holomorphic mapping with a fixed
point p. We give a criterion, in terms of triangularizability and
spectral properties of df_{p}, for f to be biholomorphic. This is
joint work with Joseph A. Cima (Chapel Hill), KangTae Kim (Pohang
University of Science and Technology, Korea), and Steven G. Krantz
(Washington University, St. Louis).
 DANIEL JUPITER, Department of Mathematics, Texas A&M University, College
Station, TX 778433368
Global Approximation of CR Functions on CR Manifolds
[PDF] 
We discuss the problem of globally approximating CR functions on CR
manifolds. Some known results in this direction will be mentioned, as
will obstructions to solving the problem.
We present a new result, joint with Al Boggess (Texas A&M):
approximation on BloomGraham model graphs.
 FINNUR LARUSSON, University of Western Ontario, London, Ontario
The SiciakZahariuta extremal function as the envelope of
disc functionals
[PDF] 
The SiciakZahariuta extremal function of a subset X of complex
affine space, also known as the pluricomplex Green function with
logarithmic growth at infinity, is the supremum of all entire
plurisubharmonic functions of minimal growth that are negative on X.
In the case when X is convex, there is a disc formula for the
SiciakZahariuta extremal function due to Lempert. We will describe
new joint work with Ragnar Sigurdsson in which Lempert's formula is
generalized to arbitrary open sets.
 EUGENE POLETSKY, Syracuse University, Syracuse, NY 13244, USA
Relative Disk Envelopes
[PDF] 
Let X be a complex manifold, Y be an open subset of X and let
f be an upper semicontunuous function on Y. Consider the space
H(X,Y) of all analytic disks in X whose boundaries lie in Y. On
this space we introduce an equivalence relation: two analytic disks
are equivalent if their centers coincide and they can be connected by
a continuous curve in H(X,Y). We show that on the set Y¢ of
equivalence classes there is a local homeomorphism r into X
that defines on Y¢ a structure of a complex manifold.
We define the relative disk envelope of f on X as the infimum
of the integrals of f over the boundaries of all analytic disks
in H(X,Y) with centers at z_{0} Î X and boundaries in Y. As the
result we get a function on Y¢ which is plurisubharmonic.
This approach immediately generates many geometric questions that will
be also discussed.
 RASUL SHAFIKOV, University of Western Ontario
Uniformization of domains with spherical boundary
[PDF] 
In this talk I will discuss the problem of uniformization of Stein
domains with spherical boundaries.
 ZBIGNIEW SLODKOWSKI, University of Illinois at Chicago, Department of Math.,
Stat. and Comp. Sci., Chicago, IL 60302
Complex surfaces with real analytic plurisubharmonic
exhaustion function
[PDF] 
This paper is a continuation of the earlier work of the authors,
Minimal kernels of a class of weakly complete spaces
(J. Funct. Anal. 210(2004), 125147). A minimal kernel
(defined in that paper) of a weakly complete manifold was shown to be
a union of a family of pairwise disjoint compact pseudoconcave sets.
If the manifold X admits a real analytic exhaustion function and has
complex dimension two, then either the kernel is empty (and X is
Stein), or the kernel is the union of an isolated sequence of compact
complex curves (and X can be obtained by blowups of a Stein space),
or the kernel is equal to X, which has to be foliated (in a relaxed
sense of the word) by compact complex curves and/or compact Levi flat
hypersurfaces (with possible singularities). The details of this
structure are most transparent when the exhaustion function has only
isolated critical points.
This is joint work with Giuseppe Tomassini.

