
L.
Vega
April 20, 15:00
Un resultado de unicidad para
la ecuación de Schrödinger
Abstract: Presentaremos un trabajo
realizado en colaboración con L. Escauriaza, C. Kenig,
y G. Ponce en el que se demuestra que una solución de la
ecuación de Schrödinger con un potencial acotado que
decaiga mas rápido que una gaussiana en dos tiempos distintos
ha de ser cero. Este resultado generaliza el bien conocido de
G.H. Hardy que afirma que una funcion tal que ella y su transformada
de Fourier decaigan mas rápido que una gaussiana ha de
ser trivial. 
J.
Bennett
April 21, 11:30
Multilinear Kakeya and
kplane transform estimates
Abstract: We will discuss certain
selfsimilarity properties of the classical multilinear BrascampLieb
inequalities and their kplane transform generalisations. 
J.
Garnett
April 25, 11:30
Problems on analytic capacity, Lipschitz harmonic
capacity and bilipschitz invariance
April 27, 11:30
Interpolating Blaschke products and
approximation problems

C.
Pérez
April 26, 12:00
On the two weight
problem for the Hilbert transform
Abstract: In this talk we plan
to survey on some recent work with D. Cruz Uribe and J.M. Martell
where we prove twoweight norm inequalities for CalderónZygmund
singular integrals that are sharp for the Hilbert transform. As
an application we give new results for the Sarason conjecture
on the product of unbounded Toeplitz operators on Hardy spaces. 
May 3, 11:30
Extremal mappings of finite distortion
May 4, 11:30
Random quasiconformal mappings
Abstract: In these two talks we
discuss recent topics in quasiconformal mapping in two dimension.
The first, joint work with Iwaniev and Martin, considers mapping
which minimize the mean of the distortion function and related
questions in minimizing the Dirichlet energies among suitable
classes of homeomorphisms.
If time permits we also cover results for mapping of finite distortion,
needed for the second talk.
The second talk, joint work with Rohde, Sackman and Tao, discusses
solutions to the Beltrami equations when the coefficients are
chosen randomly.

A.
Volberg
May 8, 15:00
Asymptotics of orthogonal
polynomials beyond the scope of SzegoKolmogorovKrein theorem and
Koosis type weighted estimates of the Hilbert transform
Abstract: We will show some new
asymptotics of orthogonal polynomials when the measure is a small
perturbation of the measure in SzegoKolmogorovKrein theorem.
The reasoning is based on Koosis type estimates. Then we derive
the asymptotics of orthogonal polynomials in case of LARGE perturbation
of SzegoKolmogorovKrein theorem. But this large perturbation
should be rather specific. We show the relation with the problem
of finding the sum rules for Jacobi matrices and with works of
Killip, Simon, Deift.
May 9, 15:00
Global regularity for critically
dissipative quasigeostrophic equation
Abstract: In this joint work with
A. Kiselev and F. Nazarov it is shown that the solution of quasigeostrophic
equation with critical order of dissipation is smooth for any
time if the initial data is smooth. There is no assumption on
the smallness of initial data.

S.
Treil
May 10, 11:30 & May 11, 11:30
Two weight estimates and perturbations of
selfadjoint operators
Abstract: In the first talk I am
going to present a special regularization of the Cauchy integral,
appearing in the theory of rank one perturbations of selfadjoint
operators. I will show how the interplay between operator theory
and harmonic analysis allows to obtain new result in both areas.
In particular, as a corollary of an elementary theory of two weight
estimates I will get a new result in the perturbation theory,
namely, a sufficient condition of the disappearance of the singular
spectrum.
The second talk will be devoted to the twoweight estimates of
the socalled "well localized" operators, which include
all martingale transforms (Haar multipliers), as well as the socalled
Haar shift (the latter operator is especially important, because
taking the average of Haar shifts over all dyadic lattices one
gets a nonzero multiple of the Hilbert Transform). I will present
a necessary and sufficient condition of Sawyer type for the twoweight
estimates for such welllocalized operators. This part is based
on a joint work with F. Nazarov and A. Volberg. 
L.
Grafakos
May 16, 15:00
L^2 boundedness
versus L^p boundedness for CalderonZygmund singular integral
operators
Abstract: We study the L^p boundedness
of CalderonZygmund singular integral operators on $R^n$ given
by convolution with a kernel of the form $\Omega(x/x)x^{n}$,
where $\Omega$ is even and integrable function on the sphere $S^{n1}$
with meanvalue zero. In particular, we are interested in the
question as to whether an $L^2$ bounded such operator must necessarily
be $L^p$ bounded for $1<p\neq2<\infty$. This work is joint
with Petr Honzik and Dmitry Ryabogin. 
May 17, 12:00
Identifiability
at the boundary for the vector potential for a magnetic Schrödinger
operator
Abstract: We consider a magnetic
Schrödinger operator $ L_W = \sum (D_j + W_j)^2$ in a $C^1$domain,
$\Omega$, in $\reals^n$ with $n\geq 3$. We assume that the vector
potential is continuous. We let $\Lambda _W$ denote the DirichlettoNeumann
map associated to $L_W$. We show how to recover the tangential
component of $W$ at the boundary from the DirichlettoNeumann
map. Our method is constructive and provides a stability estimate.
This work is joint with Mikko Salo. 
S.
Hofmann
May 18, 15:00
Stability of layer potentials and $L^{2}$
Solvability of boundary value problems for divergence form elliptic equations with
complex $L^{\infty}$ coefficients
Abstract: We consider divergence
form elliptic operators of the form $L=\dv A(x)\nabla$, defined
in $\mathbb{R}^{n+1} = \{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\}$,
where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$,
uniformly elliptic, complex and $t$independent. We show that
for such operators, boundedness and invertiblity of the corresponding
layer potential operators on $L^2(\mathbb{R}^{n}) = L^2(\partial\mathbb{R}_{+}^{n+1})$,
is stable under complex, $L^{\infty}$ perturbations of the coefficient
matrix. Using a variant of the $Tb$ Theorem, we also prove that
the layer potentials are bounded and invertible on $L^2(\mathbb{R}^n)$
whenever $A(x)$ is real and symmetric (and thus, by our stability
result, also when $A$ is complex, $\Vert AA_{1}\Vert_{\infty}$
is small enough and $A_{1}$ is real, symmetric, $L^{\infty}$ and
elliptic). In particular, we establish solvability of the Dirichlet
and Neumann (and Regularity) problems, with $L^2$ (resp. $\dot{L}^2_1)$
data, for small complex perturbations of a real symmetric matrix.
Previously, these sorts of $L^2$ results for complex (or even
real but nonsymmetric) coefficients were known to hold only for
perturbations of constant matrices, or in the special case that
the coefficients $A_{j,n+1}=0=A_{n=1,j}$, $1\leq j\leq n$. In
the latter case, the result is equivalent to the analytic perturbation
theory for Kato's square root operators. Joint work with M. Alfonseca, P. Auscher,
A. Axelsson and S. Kim 
May 23, 15:00
Weighted norm inequalities in absence of
kernels
Abstract: This is joint work
with J.M. Martell. We present an abstract machinery allowing
to obtain weighted norm inequalities for operators in absence
of kernel estimates. Applications for elliptic operators on
$R^n$ or Riesz transforms on noncompact complete manifolds
will be given.

S.
Kurylev
May 24, 12:00 & May 25, 15:00
Dirac operators with boundary
data: index theorems and inverse problems
Abstract: We consider index theorems
and inverse problems for the Diractype operators associated with
Clifford connections The main assumption is the existence of chirality.
This makes it possible to introduce a superstructure on the Dirac
bundle. The superstructure makes it possible to sometimes introduce
the decomposition of the Dirac operator, with proper boundary
conditions, into two Fredholm operators. It also makes it possible
to control the propagation of waves in each channel. We use as
boundary data for the inverse problem in the form of either the
Cauchy data of the solution of the non stationary Dirac equation,
or the boundary spectral data for the corresponding selfadjoint
elliptic operator. we then obtain formulae connecting these data
with the index of the Dirac operator and develop a method to reconstruct
the Dirac bundle up to a natural group of transformations. Joint
work with M.Lassas 
