Abstracts: 

David Auckly 
Title: A framework for analysis on noncompact quasiconformal 4-manifolds.

Abstract: This talk will report on analytical parts of a project joint 
with V. Kapovitch. An Alexandrov metric is a special structure that is
studied by differential geometers. The existance question for Alexandrov
metrics leads to the need for a gauge theory that would be applicable to
non-compact quasiconformal 4-manifolds. In order to define a gauge
theoretic moduli space, one needs to have a function space that
compactly embeds into the continous functions, and is well defined when
one considers the overlap maps. In 4 dimensions, the Sobolev space W^4_1,
does not embed into the continous functions. On the other hand, the
spaces, W^p_1 are not preserved by quasiconformal overlap maps for p>4.
We define a weighted modified Sobolev space sitting between W^4_1 and each
W^p_1. As an application of the weighted, modified Sobolev space, we will
prove that any solution to a certain elliptic partial differential
equation that grows slower than a given exponential function, must in
fact decay exponentially.
 

Sun-Sig Byun 
Title: Geometric approach to W^1,p estimates.

Abstract: We consider a Neumann problem for divergence form elliptic
equations with discontinuous coefficients. We will prove a W^1,p
esimate by a geometric approach which employs varified Vitali's
covering lemma, Hardy-Littlewood maximal function and compactness
method. Our methods can be easily extended to parabolic
equations.
 

Luca Capogna 
Title: Wave maps with target in the Heisenberg group

Abstract: This is a joint project with Jalal Shatah (Courant/NYU)
concerning  well posedness of the Cauchy problem for  wave maps with
target in the Heisenberg group. Such maps are solution of systems of
quasilinear wave equations and satisfy differential constraints in the 
form of transport equations with rough coefficients. We can show local
well-posedness for a large class of initial data.
 

Thierry de Pauw
Title: Nearly flat almost monotone measures are big pieces of Lipschitz graphs.

Abstract: Mass minimizing integral currents and stationary varifolds have strong
regularity properties. For instance their support contains an open dense
set which is an embedded smooth submanifold of the ambient space. The
question asked in this talk is to what extent these regularity properties
depend only upon some monotonicity property. We show that monotonicity
implies partial regularity of the supports.
 

Brian Hollenbeck 
Title: Best Constants for Operators Involving the Hilbert Transform.

Abstract: We calculate sharp constants in inequalities of the form, $\|Sf\|_{L^p} \le
C_p\|f\|_{L^p}$, $1< p < \infty$, where $f$ is a complex-valued function and
$S$ is an operator involving the Hilbert transform, $H$, and the
identity operator, $I$.  This is equivalent to finding the $L^p$-norm of
$S$.
In particular, we calculate the norm of $aI + bH$, where $a, b \in \R$.
We also prove $\|(I + iH)f\|_{L^p} \le 2 \csc \pip \|f\|_{L^p}$ for
$1 < p < \infty$.  This immediately
gives the norm of the Riesz projection to be $\csc \pip$, solving a conjecture
made by Gohberg and Krupnik in 1968 and a problem posed by Pe\l czy\'nski
in 1985.
 

Yaozhong Hu 
Title: On Logaritmic Sobolev and some other inequalities.

Abstract: In this talk, I will present the Nelson's 
hypercontractivity inequality, Logarithmic
Sobolev inequality and the motivation
for this inequalities. I will also present 
a more general inequality which includes 
logarithmic Sobolev inequality, Poincare
inequality as its particular cases.
A simple proof of this general inequality
will also be given.  

Some related inequalities such as Meyer's inequality,
Interpolation inequality, correlation inequality
will also presented. The correlation conjecture 
will also be presented.  
 

Frank Jochmann 
Title: Asymptotic behavior of solutions to nonlinear polarization  models.

Abstract: This talk is concerned with the Maxwell-Bloch system and the anharmonic
oscillator model  describing the electromagnetic field
in polarizable media. The main subject is the asymptotic behavior of the
solutions to these models, in particular decay properties  and
covergence to stationary states.
 

Lev Kapitanski 
Title: $S^3$ and $S^2$ nonlinear $\sigma$-models.

Abstract: The Skyrme model (1961) was one of the first attempts to describe 
elementary particles as localized in space solutions of nonlinear PDEs. 
The fields take their values in SU(2)=S^3 and stabilize at spatial infinity.
Thus, the configuration space splits into different sectors (homotopy classes) 
with a constant integer topological charge (the degree) in each sector. 
Faddeev's model (1975) was designed to provide additional internal structure 
(knottedness) to the localized solutions. The fields take their values 
in the two-dimensional sphere and the topological charge is the Hopf invariant.
I will discuss some old and new results for these models.
 

Gocha Lepsveridze 
Title: The Rate of Growth of Integral Means from Orlicz Clases.

Abstract:  Let $f\in L({\bf R}^n )$ be any function. 
For every $x\in {\bf  R}^n$
we consider integral means $1/|I|\int_{I} f$, where
$I$ is an $n$ dimensional interval
in ${\bf R}^n$. We  obtain the certain  weak type
maximal inequalities from which
are derived some exact  estimates on growth order of 
these means for functions from 
Orlicz classes  $L\Phi(L)({\bf R}^2)$ .   In
particular,
for any function  $f\in L\log_+^{\beta}({\bf
R}^2),0<\beta<1$  the expression
$\frac{1}{|I|}\int_I
f/\log^{1-\beta}\Big(\frac{1}{M(I)}\Big)$ tends to $0$
as
$\diam(I)\to 0$, where $M(I)$ denotes the length of
the biggest side of the interval $I$.
Estimates of growth order of Multiple  Fourier series
are implied.
 

Mircea Martin
Title: Multidimensional Generalizations of Alexander's Inequality. 

Abstract: Alexander's inequality says that if
$\Omega$ is a compact set in the complex plane and
$C(\Omega)$ is the Banach algebra of complex-valued
continuous functions on $\Omega$, then
$$ {\rm dist}_{C(\Omega)}[\bar{z},R(\Omega)]\leq\left[\frac{1}{\pi}{\rm
area}(\Omega)\right]^{1/2}, $$
where $\bar{z}$ is the complex conjugate coordinate
function, and $R(\Omega)$ stands for the uniform closure in
$C(\Omega)$ of rational functions that are analytic on open
neighborhoods of $\Omega$.
This inequality with many interesting applications is
just a quantitative form of the classical Hartogs-Rosenthal
theorem: Whenever ${\rm area}(\Omega)=0$, it follows that
$R(\Omega)$ is a subalgebra of $C(\Omega)$ that contains $z$
and $\bar{z}$, and by the Stone-Weierstrass theorem one gets
$R(\Omega)=C(\Omega)$.
We will present several proper multidimensional
generalizations of Alexander's inequality in the framework
of Clifford analysis. The compact space $\Omega$ is now a
subset of $\mathbb{R}^{m+1}$, $m\geq1$, and instead of
$C(\Omega)$ we take the Banach algebra
$C(\Omega,\frak{A}_m)$ of $\frak{A}_m$-valued continuous
functions on $\Omega$, where $\frak{A}_m$ is the Clifford algebra
with $m$ generators. The analog of $R(\Omega)$, denoted by
$R(\Omega,\frak{A}_m)$, is defined as the uniform closure in
$C(\Omega,\frak{A}_m)$ of functions Clifford-analytic on
open neighborhoods of $\Omega$.
As a direct generalization of Alexander's inequality, we
will show that $$ {\rm dist}_{C(\Omega,\frak{A}_m)}[\bar{x},R(\Omega,\frak{A}_m)]\leq
A_m[{\rm vol}(\Omega)]^{1/(m+1)}, $$
where $\bar{x}$ is the Clifford conjugate of the identity
function on $\mathbb{R}^{m+1}$, $A_m$ is a universal
constant that only depends on $m$, and ${\rm vol}(\Omega)$
is the Lebersgue measure of $\Omega$ in $\mathbb{R}^{m+1}$.
Actually, this result will be derived from a more general
inequality that estimates the distance in
$C(\Omega,\frak{A}_m)$ from an
arbitrary smooth $\frak{A}_m$-valued function to
$R(\Omega,\frak{A}_m)$. In particular, that general
inequality will imply
$R(\Omega,\frak{A}_m)=C(\Omega,\frak{A}_m)$, whenever
${\rm vol}(\Omega)=0$, so, once more, we end up with a
quantitative form of the Hartogs-Rosenthal theorem, but now
this theorem is in the setting of Clifford analysis.
 

Gabriel Nagy 
Title: Idemptotents in finite AW* factors

Abstract: The talk addresses a problem posed by Kaplansky in the 1950's,
which conjectures that an AW* factor is a von Neumann algebra.
In connection with this question, we prove that the quasitrace of an 
idempotent in an AW* factor of type II_1 is equal to the dimension 
function of its left (or right) support. Based on this result, we discuss 
some linear algebraic reformulations of Kaplansky's conjecture. 
 

Richard Rochberg 
Title: Hankel and Schrodinger Forms on Dyadic Trees

Abstract:  Hankel forms on dyadic trees can be viewed as discrete models
for Hankel forms on the Dirichlet space or as discrete models for
Schrodinger forms.  I will describe how these discrete models are related
to the classical questions and will describe boundedness criteria for the
discrete forms. The criteria for those forms to be in Schatten-Von
Neumann classes is not known.
 

Sharon Schaffer Vestal 
Title: Using Functional Analysis to relate a wavelet GMRA to a multiwavelet MRA

Abstract: It is well-known that wavelets have an associated subspace structure 
calleda multiresolution analysis (MRA).  There are other wavelets, minimally
supported frequency (MSF) wavelets, which are associated with a generalized
multiresolution analysis (GMRA).  We will present a theorem that links the
two structures and give examples illustrating this relationship.
 

Eric Weber
Title: Frame Representations of Groups and Sampling Theory.

Abstract:  We consider unitary representations of Abelian groups that give
rise to a frame sequence.  By analyzing the group we can get information
regarding the corresponding analysis operator; in particular, we have a
way of "parametrizing" the range.  Such information is significant for
multiplexing schemes.  We then demonstrate how this can be applied to
sampling theory.
 

Shihshu Walter Wei 
Title: On the structure of minimal submanifolds in nonpositively curved manifolds.

Abstract: We provide a topological obstruction for a complete submanifold
with a specific uniform bound involving Ricci curvature to be minimally immersed
in any complete simply-connected manifold of nonpositive sectional curvature.
We prove that such minimal submanifolds of dimension greater
than two have only one topological end.  The proof uses the Liouville
theorem for bounded harmonic functions on minimal submanifolds of this sort
due to Yau, and also adapts a technique of Cao-Shen-Zhu to show the
existence of nonconstant bounded harmonic functions based on the Sobolev inequality of Hoffman-Spruck.  This extends the work
of Yau. The same phenomena occur in a wider class of $n$-submanifolds with bounded mean
curvature in an $L^n$ sense. By improving the techniques in Cao-Shen-Zhu, one can obtain the
topological conclusion in the intrinsic settings. These generalize and
unify the structure theorems in the extrinsic settings.
 

Karen Yagdjian
Title: Parametric resonance and global solutions to nonlinear hyperbolic equations.

Abstract: We show  how  parametric resonance can affect global existence of 
solutions to the Cauchy problem for  nonlinear hyperbolic equations.  Namely 
we give some examples of nonlinear hyperbolic equations and systems such that 
for arbitrary small smooth initial data, and for arbitrary large space dimension 
there are blowing up solutions.