Analysseminariet 2007,
löpande planering
Vårterminen
Onsdagen den 17 januari 13.15-14.15. Alfred Louis (Saarbrücken) :
Inverse Problems: Theory and Applications.
Abstract:
Onsdagen den 24 januari 13.15-14.15. Anders Öberg (Uppsala) :
Convergence of the transfer operator for probability weights.
Abstract:
I will talk about convergence of the iterates of the transfer operator
under square summability of variations of the probability weight functions
that specify simple dynamical systems. In addition I will mention some
unsolved problems in the area and I will also give a background of the area
as such. The talk will be intended for a general analysis audience.
Onsdagen den 31 januari 13.15-14.15. Håkan Hedenmalm (KTH) :
A trifle on planar Beurling and Fourier transforms.
Abstract:
We study the Beurling and Fourier transforms on subspaces of
$L^2({\mathbb C})$ defined by an invariance property with respect to the
root-of-unity group. This leads to generalizations of these transformations
acting unitarily on weighted $L^2$-spaces over $\mathbb C$.
Onsdagen den 7 februari 13.15-14.15. Marta Sanz-Sole (Barcelona) :
Three-dimensional stochastic waves.
Abstract:
We shall study the joint H\"older continuity in the time and space
variables for the sample paths of the solution of a stochastic wave equation
in spatial dimension $d=3$ with nonlinear coefficients. The driving noise is
white in time and with a spatially homogeneous covariance defined by the
product of a Riesz kernel and a smooth function.
Our results rely on a detailed analysis of the stochastic integral used in
the formulation of the equation. For covariances given by Riesz kernels,
we show that the H\"older exponents that we obtain are optimal.
This is a joint work with Robert C. Dalang, EPFL, Switzerland.
Onsdagen den 14 februari 13.15-14.15. Kurt Johansson (KTH) :
Directed last-passage percolation and random matrices.
Abstract:
Certain dirceted last-passage or directed polymer models in two-dimensions
have intertesting and important connections to random matrix measures.
The last-passage times are defined by certain recursion relations with noise
and we can also think of them as multi-dimensional Markov chains. I will
discuss some cases where the transition function for these Markov chains
can be written down explicitly. This gives a new proof of certain earlier
results and is related to some recent work by Jon Warren.
Onsdagen den 21 februari 13.15-14.15. Håkan Andreasson (Chalmers) :
Sharp bounds on the compactness of static objects in
spherically symmetric spacetimes.
Abstract:
Onsdagen den 28 februari 13.15-14.15. Alexander Vinogradov (Moscow,
Salerno): An invitation to Secondary Calculus.
Abstract:
The intention is to explain very informally what is that Calculus and
to show some results, applications and perspectives.
Onsdagen den 7 mars 11.00-12.00. Stanislav Smirnov (Genf) :
Ising model and loop soups.
Abstract:
Onsdagen den 7 mars 13.15-14.15. Kunio Yoshino (Tokyo) :
On the generating function of eigenvalues of Daubechies localization
operator.
Abstract: I will talk about applications of of generating function of
eigenvalues of Daubechies operator. The main topics are:
(i) Reconstruction formula for symbol function.
(ii) Integral representation of Daubechies operator in Bargmann-Fock space.
(iii) A special class of Daubechies operators.
Onsdagen den 14 mars 13.15-14.15. Björn Gustafsson, Vladimir Tkachev
(KTH) :
The resultant on compact Riemann surfaces and the exponential
transform.
Abstract:
We introduce a notion of resultant of two
meromorphic functions on a compact Riemann surface and indicate
its relationship to some more other objects, for example
the polynomial giving the algebraic dependence between the two
meromorphic functions. The concept is discussed from three distinct
perspectives: complex analysis, elimination theory, and operator
theory.
Recall that the function of two complex
variables defined by
$$
E_\Omega (z,\bar{w}) =\exp[-\frac{1}{\pi}\int_\Omega
\frac{dA(\zeta)}{(\zeta -z)(\bar{\zeta} -\bar{w})}].
$$
where $dA(\zeta)$ stands for the Lebesgue planar measure, is called the
exponential transform of a bounded domain $\Omega$. It is well known that
$E_\Omega (z,\bar{w})$ is as an infinite determinant (Carey, Pincus, 1974);
on the other hand, $E_\Omega (z,\bar{w})$ is rational provided $\Omega$ is
a quadrature domain (Putinar, 96). As a particular application, we show
that the exponential transform of a quadrature domain in the
complex plane is expressed in terms of the resultant of two
meromorphic functions on the Schottky double of the domain.
Onsdagen den 21 mars 13.15-14.15. Per Sjölin (KTH) :
Estimates for multiparameter maximal operators of Schr&oml;dinger type.
Abstract:
For $a>1$ and $f$ belonging to the Schwarz space we set
$$S_t f(x)=\int_{-\infty}^{+\infty}e^{ix\xi}e^{it|\xi|^a}\hat f(\xi)d\xi.$$
We also set
$$M^*f(x)=\sup_{0
Onsdagen den 28 mars 13.15-14.15. Alexandru Aleman (Lund) :
Volterra invariant subspaces of $H^p$.
Abstract: A complete description is obtained for the
subspaces of the Hardy space $H^p$ ($p\geq 1$) that are invariant
under the Volterra integral operator. We then show that this result
can be applied to derive complete characterizations of such
subspaces in a large class of Banach spaces of analytic functions in
the unit disc containing the usual Bergman and Dirichlet spaces.
This is joint work with Boris Korenblum.
Onsdagen den 4 april: seminariet tar påsklov.
Onsdagen den 11 april: Hans Rullgård (SU) :
Electron tomography. A short overview of methods and challenges.
Abstract:
This CIME "problem-seminar" will deal with the mathematical research
problems that arise in the joint work with Sidec Technologies on
electron tomography.
Already in 1968 one recognized that the transmission electron microscope
could be used in a tomographic setting as a tool for structure
determination of macromolecules. However, its usage in mainstream
structural biology has been limited and the reason is mostly due to the
incomplete data problems that leads to severe ill- posedness of the inverse
problem. Despite these problems its importance is beginning to increase,
especially in drug discovery.
From a mathematical point of view, the reconstruction problem in electron
tomography amounts to the solution of an inverse scattering problem. To
solve this inverse problem there are two major challenges that must be
dealt with. The first is to develop an accurate model of the process of
image formation in the transmission electron microscope, which in the
terminology of inverse problems is the determination of the forward
operator. The second is to choose a suitable regularization method for
the inverse problem.
In the model for image formation, the electron-specimen interaction is
modelled as a diffraction tomography problem and the picture is completed
by adding a description of the optical system of the transmission electron
microscope. We highlight some of the limitations of this model and the
numerical problems that arise when one attempts at a numerical
implementation of the forward operator. Next we turn our attention to the
inverse problem which is very difficult mainly due to the devastating
combination of very noisy data and the severe ill-posedness (due to the
limited data problems). Restrictions in the data acquisition geometry leads
to limited angle tomographic data and therefore implies that the conditions
for stable reconstruction are not fulfilled. Moreover, only a subregion of
the specimen is subject to electron exposure, so we are dealing with local
tomographic data which leads to non-uniqueness. The severe ill- posedness
means that a regularization method must used to obtain reconstructions of
any practical value at all, and a good reconstruction is likely to require
a carefully chosen regularization. Moreover, the non-uniqueness is best
understood using microlocal analysis which allows us to explain those
singularities that are stably visible from the limited data given by the
data collection protocol in the electron microscope. Open mathematical
problems related to regularisation theory and microlocal analysis will be
mentioned.
Finally, if time permits, we provide some examples of reconstructions from
electron tomography and demonstrate some of the biological interpretations
that one can make.
Onsdagen den 18 april: Yacin Ameur (KTH) : Strichartz estimates for
the time-dependent Schr\"odinger equation with a singular potential.
Abstract: We consider the time-dependent Schr\"odinger equation with a
singular potential of a very particular form, namely $V=c/|x|^2$ where
$c$ is a non-negative constant. We shall
discuss so-called Strichartz-estimates which claim that certain
weighted space-time $L2$ norms of the solution can be estimated in
terms of Sobolev norms of the initial condition. Our results
generalize known estimates from the free case $c=0$. This is joint
work with Bj\"orn Walther.
Onsdagen den 25 april: Jonas Hägg (KTH) : Gaussian fluctuations
in the Airy and GUE point processes.
Abstract:
The Gaussian unitary ensemble (GUE) is a classical random matrix.
It is an $n \times n$ Hermitian matrix such that the $n^2$ real and
imaginary parts, on and above the diagonal, are independent and
$N \left( 0, \frac{1 + \delta_{ij}}{4} \right)$ distributed.
The seminar will concern the $n \rightarrow \infty$ limit
distribution of the $k$th largest eigenvalue of the GUE.
Onsdagen den 2 maj: Alexander Soshnikov (UC Davis) : On spectral
radius of Wigner random matrices with non-symmetrically distributed
entries.
Abstract:
In the first part of the talk, I will show that the spectral radius of an
$N\times N$ random symmetric matrix with i.i.d. bounded centered but
non-symmetrically distributed entries is bounded from above by
$ 2 \*\sigma + o( N^ {-6/11+\varepsilon}), $
where $\sigma2 $ is the variance of the matrix
entries and $\varepsilon $ is an arbitrary small positive number. Our bound
improves the earlier results by Z.F\"{u}redi and J.Koml\'{o}s (1981), and
Van Vu (2005). Our approach heavily relies on combinatorial considerations.
This is a joint paper with Sandrine Peche.
In the second part of the talk, I will discuss a resolvent approach to study
the universality at the edge of the spectrum.
Onsdagen den 9 maj: Pavel Kurasov (Lunds TH) : Schr\"odinger
operators on graphs and geometry.
Abstract:
Schr\"odinger operators on finite compact metric graphs are considered so
far with standard boundary conditions at the vertices. It is proven that the
asymptotics of the eigenvalues determine the Euler characteristic of the
underlying metric graph in the case of essentially bounded potentials.
Possible extensions of this result to more general boundary conditions and
potentials as well as to difference operators are discussed.
Onsdagen den 16 maj: Johan Andersson (SU) : On a power sum problem.
Abstract:
I will talk about some of my recent results on the Tur\'an power sum method.
In particular I will show how to obtain the correct asymptotics in the
inf max problem (see arXiv:math/0609271)
$$\inf_{ |z_k| \geq 1 } \, \max_{\nu=1,\ldots,n^2}
\left|\sum_{k=1}^n z_k^\nu \right| = \sqrt n+O(n^{0.2625}).
$$
The method of proof uses a combination of a probabalistic argument and an
explicit arithmetical construction. I will also discuss some related
problems, such as what happens when the interval $\nu=1,\ldots,n^2$ is
replaced by other intervals.
Torsdagen (!!!) den 24 maj: Meiyu Su (Long Island U) :
Earthquake maps determined by Thurston unbounded earthquake measures
suppoted on geodesic laminations in the hyperbolic plane D.
Abstract:
We obtain an analogue in earthquake theory to the David's theorem on the
solution of the Beltrami differential equation.
More precisely, let $\mathbb{D}$ be the unit disk. For every Borel
measurable function $\mu $ on $\mathbb{D}$ satisfying the
condition
$$||\mu ||_{\infty }=\inf _{{\rm supp}(K_F)} \sup _{z\in
{\rm supp}(\mu)}|\mu (z)|\le k,$$
for a constant $00$ and any geodesic arc $\beta $ transversal to
$\mathcal{L} $ of hyperbolic length $\le 1$ and sufficiently close
to the boundary in the Euclidean metric, then there exists an
earthquake map $(E, \mathcal{L})$ such that $\sigma$ is the
earthquake measure induced by $E$.
Onsdagen den 30 maj 13.15-14.15. Alan McIntosh (Canberra) :
The Square Root Problem of Kato for Elliptic Operators: Survey,
Solution and Sequel.
Abstract:
About 1960 Tosio Kato, during his investigation of the evolution of
physical systems, was led to pose a key question about the square
roots of elliptic partial differential operators. A positive answer
to his question implies that the square root is stable under small
perturbations, this being useful in solving related hyperbolic
equations with time-varying coefficients.
The one-dimensional problem was solved by Coifman, McIntosh and Meyer
in 1982, along with the boundedness of the Cauchy integral on
Lipschitz curves. It was only in 2001 that the question was fully
answered by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian.
I will survey this development, and indicate some of the ideas which
led to the final solution.
Subsequently these ideas have been adapted to obtain quadratic
estimates for perturbations of the Hodge-Dirac operator $D = d + d^*$,
where $d$ denotes the exterior derivative acting on differential
forms, thus considerably extending their applicability. This is
joint work with Andreas Axelsson and Stephen Keith.
Höstterminen
Onsdagen den 5 september 13.15-14.15. Brett D. Wick (Vanderbilt and KTH)
: Multiparameter Riesz Commutators.
Abstract:
An important question in analysis to determine the behavior of
an operator associated to a symbol purely from data about the symbol. For
example, a multiplication operator is bounded if and only if the symbol is
bounded.
We will be interested in determining necessary and sufficient conditions
on the symbol which will imply that a commutator between a multiplication
operator and a singular integral operator is bounded. This commutation
allows for cancellation to play a role and is a significant feature in
determining conditions on the symbol.
In addition to the history behind this problem, connections with real and
complex analysis in one and several variables and interpretations of these
results in operator theory and function theory will be highlighted.
Finally, recent results will be discussed.
This is joint work with Michael Lacey, Stefanie Petermichl and Jill Pipher.
Onsdagen den 12 september 13.15-14.15. Bertrand Duplantier (Saclay) :
SLE and Quantum Gravity .
Abstract:
We shall describe some geometrical properties of conformally invariant
scaling curves in the plane, i.e., SLE curves, and the multifractal
properties of their harmonic measure. These properties have natural
``quantum gravity'' aspects, corresponding to the embedding of the curves
in a random lattice, i.e., in a random Riemannian metric.
The quantum gravity perspective on conformal random geometry will be
examplified in the cases of Brownian and self-avoiding paths (SLE8/3) and
percolation (SLE6).
Onsdagen den 19 september 13.15-14.15. Richard Miles (KTH) :
Algebraic dynamical systems.
Abstract:
Dynamical systems arising from automorphisms of compact abelian groups are
a familiar testing ground for ideas in ergodic theory, as well as being of
interest in their own right. In 1989, B. Kitchens and K. Schmidt
introduced commutative algebra as a tool for studying such systems,
leading to substantial developments in this area. We review some results
from the subsequent `algebraic dictionary' of dynamical properties
together with some newer concepts for algebraic actions of $Z^d$, $d>1$.
Onsdagen den 26 september 13.15-14.15. Peter Jones (Yale) :
On Some Problems in Geometry and Spectral theory.
Abstract: We discuss some problems related to Diffusion Geometry and
the related geometry of domains, manifolds, and data sets.
Onsdagen den 3 oktober 13.15-14.15. Martin Andersson (IMPA) :
Physical measures in partially hyperbolic dynamical systems.
Abstract (Swedish): The theory of differentiable dynamical systems has
had great success by viewing these systems from a stochastic perspective.
The most important question is whether these systems have physical measures,
and if so, how these measures are affected by small external perturbations.
A measure $\mu$ is said to be physical for $f:M\mapsto M$ if there is a set
$B(\mu)$ of positive Lebesgue measure such that
\[
\frac{1}{n} \sum_{k=0}^{n-1} \varphi(f^k(x)) \rightarrow \int \varphi d\mu
\]
for all $x \in B(\mu)$ and all continuous functions $\varphi : M
\mapsto \mathbb{R}$.
It is known since a long time that hyperbolic (Axiom A) systems have finite
number of physical measures, which together describe the statistics of
iterations $x, f(x), f2(x), \ldots $ for Lebesgue almost every
$x$. Furthermore these depend continuously on $C2$ perturbations in
the system (stochastic stability).
The existence of physical measures has not been proved for typical
partially hyperbolic dynamical systems. In this seminar we will
discuss an open set of partially hyperbolic dynamical systems with so
called mostly contracting central direction, for which physical
measures do indeed exist. In this connection we run into some new
phenomena such as bifurcatins of physical measures. In particular
these systems do not have to be statistically stable but we shall see
that most are anyway.
Onsdagen den 10 oktober 13.15-14.15. Nicole Mihalache Ciurdea (KTH) :
Properties of the critical orbits associated to the geometry of
Julia sets.
Abstract:
If all recurrent critical orbits in the Julia set of rational maps (without
parabolic cycles) are Collet-Eckmann, then all its Fatou components are
H\"older domains. This generalizes results of Carleson, Jones, Yoccoz, and
Graczyk, Smirnov about the geometry of Fatou/Julia sets.
The proof of each aforementioned result will be overviewed during the talk.
Onsdagen den 17 oktober 13.15-14.15. Yacin Ameur (KTH) :
Berezin quantisation and nearly Gaussian measures.
Abstract:
I will discuss a central limit theorem for the weighted Berezin transform
associated with a rather general weight in the plane.
The theorem has applications to the theory of random normal matrices, and
the work presented here is part of a joint project with H. Hedenmalm and
N. Makarov. The project aims at generalizing known results from the Hermitian
case to the more general case of normal matrices.
Onsdagen den 24 oktober 13.15-14.15. Annemarie Luger (Vienna and Lund) :
Generalized Nevanlinna functions as a tool for singular differential
operators.
Abstract:
We use the hydrogen atom operator in order to demonstrate how
Generalized Nevanlinna functions naturally appear when considering
Sturm-Liouville operators with singular potentials. Here the
classical picture from the regular case can be recovered, however,
with different properties.
Onsdagen den 31 oktober 13.15-14.15. Ioannis Parissis (Georgia Tech) :
.
Abstract:
Onsdagen den 7 november 13.15-14.15. Brett Wick (USC and KTH):
Bounded Analytic Projections, Holomorphic Vector Bundles and the
Corona Problem.
Abstract: A simple lemma of N. Nikolski connects the existence of a
bounded analytic projection with the Operator Corona Problem (existence
of a bounded analytic left inverse for an operator-valued function).
So, to solve the Corona problem we give a sufficient condition to gurantee
the existence of a bounded analytic projection onto a holomorphic family of
generally infinite dimensional subspaces (a holomorphic sub-bundle of a
trivial bundle). This sufficient condition is also necessary in the case
of finite dimension or codimension of the bundle, so as corollaries of
the main result we obtain new results about the Operator Corona Problem.
In particular, we find a new sufficient condition and a complete solution
in the case of finite codimension. This is joint work with S. Treil.
OBS! Tisdagen den 13 november 13.15-14.15. Pekka Koskela
(Jyväskylä) : Dimension distortion under mappings of
exponentially integrable distortion.
Abstract:
It is well-known that a planar quasiconformal (or more generally
quasiregular) mapping sends sets of Hausdorff-dimension strictly
less than two to similar sets. By results of Astala on the higer regularity
of quasiconformal mappings, one actually has a sharp formula for the
dimension distortion. We discuss generalized dimension distortion
under mappings of exponentially integrable distortion, especially in
connection with higher regularity.
Onsdagen den 14 november 13.15-14.15. Mildred Hager (Lund) :
Eigenvalue asymptotics for randomly perturbed non-selfajoint operators.
Abstract:
Onsdagen den 21 november 13.15-14.15. Joakim Arnlind (KTH) :
Representation theory of $C$-algebras for spheres and tori via graphs
and dynamical systems.
Abstract: To construct fuzzy analogues of a manifold $\Sigma$, a
common procedure is to replace the (commutative) function algebra on
$\Sigma$ by a noncommutative algebra $\mathcal{A}$. This can be done
in many ways, but for manifolds endowed with a symplectic form, there
is a natural correspondence, relating Poisson-brackets of functions to
commutators in $\mathcal{A}$.
We introduce $C$-algebras as fuzzy analogues of compact surfaces of
arbitrary genus, and for a class of spheres and tori we completely
characterize the representation theory of the corresponding
$C$-algebras. To achieve this, a graph method has been developed that
leads to a classification of representations in terms of ``loops'' and
``strings''. Moreover, to every surface there is an associated
dynamical map $s:\mathbb{R}2\to\mathbb{R}2$ whose periodic orbits
and $N$-strings one has to understand in order to construct explicit
matrix representations.
Onsdagen den 28 november 13.15-14.15. Paul Malliavin (Paris) :
.
Abstract:
Onsdagen den 5 december 13.15-14.15. Jacek Graczyk :
Cohomological inequality and smoothing operators.
Abstract:
In the work with Duncan Sands it is proved that an analytical conjugacy
class of every smooth map of the interval with all critical points
non-flat and all periodic points repelling contains a map with negative
Schwarzian derivative. For circle maps one needs an additional
"integrability" condition. The main idea of the proof is to find an
analytical solution of a cohomological inequality. The dynamics supply us
with a natural but only essentially bounded measurable solution.Using
smoothing techniques one can promote this "rough" solution to an analytic
one.
During the talk I will concentrate on analytical aspects of the problem.
Onsdagen den 12 december 13.15-14.15. Jan Boman (SU) :
Unique continuation of microlocally analytic distributions and
injectivity theorems for the ray transform.
Abstract:
Let $L_k$, $k = 1, 2, \ldots$, be an infinite family of distinct hyperplanes
in $\mathbf R^n$ such that $\lim_{k\to\infty} L_k = L_0$ (in the topology
of the manifold of hyperplanes). It is easy to prove that if $u$ is a
$C^{\infty}$ function that vanishes on all $L_k$, then $u$ is flat along
$L_0$ in the sense that the derivatives of $u$ of all orders vanish on
$L_0$. If $u$ is only a locally integrable function or a distribution and
the wave front set $WF(u)$ of $u$ is disjoint from the set $N^*(L_0)$ of
conormals of $L_0$ in the open set $U \subset \mathbf R^n$, then the
restriction of $u$ and all its distribution derivatives to $L_0\cap U$
and $L_k\cap U$ are well defined if $k$ is large enough, and we may ask
if $u$ must be flat along $L_0\cap U$ if $u$ vanishes on $L_k\cap U$ for
all $k\ge 1$. The answer is yes (under a certain additional condition).
Using this theorem and a vanishing theorem for microlocally real analytic
distributions that I gave in 1992 (C.\ R.\ Acad.\ Sci.\ Paris 1992,
p.\ 1231-1234) I will give new proofs of results of B\'elisle, Mass \'e,
and Ransford on injectivity for ray transforms.