|
Speaker: Marco Abate, Universita di Pisà
Title: Dynamics of homogeneous vector fields and meromorphic connections
Abstract:
Time-1 maps of homogeneous vector fields provide important examples of
holomorphic maps tangent to the identity;
for instance Camacho has proved that in dimension 1 every holomorphic
germ tangent to the identity is locally
topologically conjugated to such a time-1 map. In this talk I shall
present some powerful techniques useful for
the study of the dynamics of homogeneous vector fields in dimension
two (and possibly higher),
reducing the study of integral curves of the vector field to the study
of geodesics for
a meromorphic connection on the projective line. Two main results are:
a Poincaré-Bendixson theorem
describing the recurrence behavior of such geodesics, and hence of the integral curves;
and a list of formal and holomorphic local normal forms allowing a
detailed study of the dynamics
nearby the poles of the connection. As a consequence we get new
examples showing unexpected behaviors,
and we get a complete description of the dynamics for a large family
of such fields---and thus of the corresponding
time-1 maps, providing the first complete description of the dynamics
in a full neighbourhood of the origin
for a large class of holomorphic germs tangent to the identity in
dimension two. (Joint work with F. Tovena)
Speaker: David Barrett, University of Michigan
Title: Variational problems for Fefferman's measure
Abstract:
This talk will present some basic results for the maximal
hypersurface and isoperimetric problems associated to Fefferman's measure for
strongly pseudoconvex hypersurfaces. These problems are invariant
with respect to biholomorphic maps with constant Jacobian, but the
isoperimetric problem in particular turns out to give
rise to a new global (unrestricted) biholomorphic invariant.
This is joint work with Chris Hammond.
Speaker: Florian Bertrand, University of Wisconsin
Title: Complete Hyperbolicity of some domains in almost
complex manifolds
Abstract: I will present some recent results on the local
complete hyperbolicity of some pseudoconvex domains of finite type in
almost complex manifolds. More precisely, I will consider domains
satisfying some local properties and I will study the asymptotic
behaviour of pseudoholomorphic discs near the boundary of such domains.
Speaker: Sebastien Boucksom, CNRS-Université Paris 7 Institut de Mathematiques
Title: A variational approach to complex Monge-Ampère equations
Abstract: I will describe joint work with Berman, Guedj and Zeriahi where we show
that Monge-Ampère equations in a big cohomology class of a compact Kaehler
manifolds can be solved by a variational method independently of Yau's
theorem. Our formulation yields a natural several variable extension of the
classical logarithmic energy of a measure and applies as well to singular
Kaehler-Einstein metrics on Fano and general type manifolds.
Speaker: Zeljko Cuckovic, University of Toledo
Title:
Compactness of Hankel operators and boundary geometry on
pseudoconvex domains in Cn
Abstract: We study compactness of Hankel operators on smooth bounded
pseudoconvex domains using an important tool from several complex
variables, namely the dbar-Neumann problem. We show that
for any symbol that is smooth up to the boundary, compactness of the
corresponding Hankel operator depends on the behavior of the symbol on
the analytic structure in the boundary of the domain. This is a joint
work with Sonmez Sahutoglu.
Speaker: John d'Angelo, University of Illinois
Title: Iterated Commutators of Complex Vector Fields
Abstract: Let L be a (1,0) vector field on a CR
manifold of hypersurface type. Assume that the Levi form on L
vanishes. We compare several different finite type conditions
involving L and analyze what role pseudoconvexity plays in the
story.
Speaker: Jeff Diller, University of Notre Dame
Title: Polynomial maps of C2 with small topological degree
Abstract:
A meromorphic map on a compact complex surface is said to have `small
topological degree' if its iterates expand preimages of curves more
quickly than those of points. In a recent series of papers with
Dujardin and Guedj, we have carried out a program for constructing and
analyzing a so-called equilibrium measure for most such maps. There
are, however, a couple of technical hypotheses that prevent our results
from holding in perfect generality. These all happen to be satisfied in
the particular case of maps that are polynomial on C2, so in this
talk, I will focus on the polynomial case in order to explain what we
have done.
Speaker: Pak Tung Ho, Purdue University
Title: The CR Yamabe Flow
Abstract:
The Yamabe problem is the following: Given a Riemannian metric
g on a compact manifold of dimension n ≥ 3, find
a metric conformal to g such that its scalar curvature is
constant. The Yamabe flow was introduced by Richard Hamilton to
tackle the Yamabe problem. The Yamabe problem can also be posed for CR
manifolds. Let M be a compact strictly pseudoconvex CR manifold with
contact form θ. The CR Yamabe problem is to find a contact form
conformal to θ such that its Webster scalar curvature is constant. In
this talk, I will describe how the Yamabe flow can be generalized to
the setting of CR Yamabe problem. I will discuss the long time
existence and convergence of the CR Yamabe flow.
Speaker: Suzanne Hruska, University of Wisconsin-Milwaukee
Title: Topology of Fatou components for endomorphisms of
CP2: Linking with the Green's current
Abstract:
Little is known about the global topology of Fatou components for
holomorphic endomorphisms f: CP2 ->
CP2. We
develop a type of linking number between closed loops in the Fatou set
of f with the Green's current T, which forms
the complement of the
Fatou set. Using these linking numbers we establish that many classes
of endomorphisms have Fatou components with infinitely generated first
homology; for example, polynomial endomorphisms of
CP2 for which the restriction to the line at
infinity is hyperbolic and has disconnected Julia set, and polynomial
skew products of CP2 such that the vertical Julia
set in an appropriate slice is disconnected. We conclude with a some
of concrete examples and questions for further study. This is joint
work with R. Roeder.
Speaker: Mattias Jonsson, University of Michigan
Title:The behavior of polynomial mappings at infinity
Abstract:
Polynomial mappings of C2 constitute basic examples
of higher-dimensional complex dynamical systems. Given
a polynomial mapping, one tries to construct dynamically
interesting objects, such as currents and measures.
However, when trying to construct these objects, one runs
into the problem that C2 is not compact. Sometimes one
can get away with adding a line at infinity, to obtain projective
space P2, but in general the induced dynamical system
on P2 will have bad properties. I will discuss the possible
behavior of infinity and how one can find compactifications
of C2 that are well adapted to the dynamics. The talk is
based on joint work with Charles Favre.
Speaker: Margaret Stawiska, University of Kansas
Title: Robin functions on toric manifolds
Abstract: We define Robin functions on toric manifolds, which
generalize classical Robin functions (joint with M. Branker).
Speaker: Brian Street, University of Wisconsin
Title: Multi-parameter Carnot-Carathéodory geometry
Abstract:
We discuss multi-parameter Carnot-Carathéodory geometry.
In particular, we discuss questions motivated by PDEs defined
by vector fields (and associated singular integrals).
It is our hope that these results will play a role in
studying operators which arise in several complex variables.
Speaker: Dror Varolin, SUNY Stony Brook
Title: Interpolation from hypersurfaces in Cn
Abstract:
I will discuss a weighted-L2 extension theorem from hypersurfaces in
Cn to the ambient space. The theorem gives sufficient conditions for
extension. I will explain why at least one of these conditions
is not necessary. Along the way I will discuss some open questions.
Speaker: Elizabeth Wulcan, University of Michigan
Title: Stabilization of monomial maps
Abstract: The construction of many dynamical objects associated with rational
maps (such as invariant currents) requires the induced maps on
cohomology to be compatible with iteration. I will discuss the problem
of finding a model where
(f n)*=(f *)n
when f is a monomial map.
This is joint work in progress with Mattias Jonsson.
Speaker: Steve Zelditch, Johns Hopkins University
Title: Geodesics in the space of Kaehler metrics and the
complex homogeneous Monge Ampère equation
Abstract:
There is a natural Riemannian metric on the infinite dimensional space of Kaehler metrics
on a complex manifold M in a fixed class which makes it formally an
infinite dimensional symmetric space (Mabuchi, Semmes, Donaldson). The
equation for geodesics of metrics is the complex homogeneous
Monge-Ampère equation on R x M where R is an annulus. Various people
have studied the boundary problem for geodesics. My talk is about the
initial value problem. It is an ill-posed Cauchy problem for the HCMA.
We will present some general results on the lifespan of solutions and
give more detailed results for toric varieties and Abelian
varieties. Joint work with Yanir Rubinstein.
|