Analysseminariet 2004
Vårterminen
Onsdagen 14 januari 13.15-14.15. D. R. Yafaev (Rennes):
Scattering by magnetic fields.
Abstract: We consider the Schr\"odinger operator $H$ in the space
$L_2({\bf R}^d)$ with a magnetic potential $A(x)$ decaying as $|x|^{-1}$
at infinity and satisfying the transversal gauge condition
$ =0$.
Our goal is to study properties of the
scattering matrix $S(\lambda)$ associated to the operator $H$.
In particular, we find the essential spectrum $\sigma_{ess}$ of $S(\lambda)$
in terms of the behaviour of $A(x)$ at infinity.
It turns out that $\sigma_{ess}(S(\lambda))$
is normally a rich subset of the unit circle ${\bf T}$
or even coincides with ${\bf T}$.
We find also the diagonal singularity
of the scattering amplitude (of the kernel of $S(\lambda)$
regarded as an integral operator). In general, the singular part
$S_0$ of the
scattering matrix is a sum of a multiplication operator
and of a singular integral operator. However, if the magnetic field decreases
faster than $ |x|^{-2}$ for $d\geq 3$ (and the total magnetic flux is an
integer times $2\pi$
for $d=2$), then
this singular integral operator disappears. In this case, the scattering
amplitude has only a weak singularity (the diagonal Dirac function is
neglected) in the forward direction and hence scattering is essentially of
short-range nature.
Moreover, we show that, under such assumptions, the absolutely continuous
parts of the operators $S(\lambda)$ and $S_0$ are unitarily equivalent.
An important point of our approach is that
we consider $S(\lambda)$ as a pseudodifferential operator on the unit sphere
and find an explicit expression of its principal symbol in terms of $A(x)$.
Another ingredient is an extensive use (for $d\geq 3$) of a special gauge
adapted to a magnetic potential $A(x)$.
Onsdagen 21 januari 13.15-14.15. N. Uraltseva (St-Petersburg):
On the Lipschitz property of the free boundary for parabolic obstacle
problem.
Abstract: We consider a parabolic obstacle problem with zero constraint.
Without any additional assumptions on a free boundary we prove that near
the fixed boundary, where the homogenious Dirichlet condition holds,
the boundary of the non-coincidence set is a graph of a Lipschitz function.
The result is optimal.
Joint with: D. Apushkinskaya, H. Shahgholian.
Onsdagen 4 februari 13.15-14.15. D. Chelkak (St-Petersburg):
The inverse problem for the harmonic oscillator perturbed by a
potential. Characterization.
Joint with: P. Kargaev, E. Korotyaev.
Onsdagen 11 februari 13.15-14.15. Bernt Wennberg (Chalmers):
On the derivation of a linear Boltzmann equation from a
periodic lattice gas.
(joint work with Valeria Ricci, Rome)
Abstract: One fundamental problem in kinetic theory is to make a rigorous
derivation of the Boltzmann equation as the limit of the dynamics of a
family of particle systems. In the general case, this has been only
partially achieved.
Here we consider the easier problem of deriving the linear Boltzmann
equation from the evolution of a Lorentz gas, i.e. the motion of
non-interacting particles that are specularly reflected on a set
obstacles.
In the talk, I will discuss the case of a completely random distribution
of obstacles (considered by Gallavotti in the 1970's), the case of a
periodic distribution (in this case, there is no Boltzmann equation in the
limit), and a class of intermediary distributions, where it is also
possible to derive the Boltzmann equation.
Onsdagen 18 februari 13.15-14.15. Wei-Min Wang (Orsay):
Quasi-periodic solutions for nonlinear random Schroedinger equation.
(joint work with J. Bourgain)
Abstract:
We construct time quasi-periodic solutions for nonlinear random
Schroedinger equations on a set of random potentials of positive
measure (asymptotically full measure). We use a Newton scheme and
a Lyapunov-Schmidt P and Q decomposition.
The main techniques of this construction are multisccale iteration,
semi-algebraic set and Cartan type of theorem. The quasi-periodic
solutions are near to solutions to the linear equation.
Onsdagen 25 februari 13.15-14.15. Alexandru Aleman (Lund):
Analytic contractions and boundary behavior.
(joint work with S. Richter and C. Sundberg)
Abstract: The Hilbert spaces $P^2(\mu)$ have been studied extensively.
However, a number of problems remain unresolved. For instance, if $\mu$
is supported on the closed unit disk, and if $P^2(\mu)$ consists of
holomorphic functions on the unit disk, we may ask whether the boundary
values of the holomorphic function in terms on non-tangential approach
coincide with the values of the functions in terms of the boundary measure.
We prove that this is the case.
Onsdagen 3 mars 13.15-14.15. Dirk Hundertmark (University of Illinois,
Urbana-Champaign):
Statistical Mechanics and Anderson Localization.
Abstract: We present a family of finite-volume criteria
which cover the regime of exponential decay for certain moments of Green's
functions of operators with random potentials ($=$ disordered systems).
Such decay is a technically convenient characterization of localization
for it is known to imply spectral localization, absence of level
repulsion, and strong dynamical localization, which is stronger than mere
absence of diffusion. The constructive criteria also rule out any fast
power law decay of the Green's functions at mobility edges.
This family of finite-volume criteria is, at least in spirit, very much
similar to well-known finite volume criteria from statistical mechanics,
especially in percolation and spin systems. Our approach is highly
motivated by these results.
Onsdagen 10 mars 13.15-14.15. Jan Boman (Stockholm):
Novikov's inversion formula for the attenuated
Radon transform --- a new approach.
Abstract: The attenuated Radon transform $R_{\rho}$ is a $2$-dimensional
weighted Radon transform defined by
$R_{\rho}f(L) = \int_{L} f \rho_L ds$,
where
$$
\rho_L(x) = \exp\big(-\int_{L(x)} \mu\, ds\big) ,
$$
$\mu(x)$ is a given function with compact support, and $L(x)$ is one of
the components of $L\setminus\{x\}$, the choice being given by the
orientation of $L$. The problem to invert $R_{\rho}$ is important for
a certain kind of medical X-ray examination (Emission Computed Tomography).
An explicit inversion formula for $R_{\rho}$ was given by
Roman G. Novikov (Ark. Mat. 2002); I reported on this result in our
seminar in October 2000. In joint work (to appear in J. Geom.
Anal.) Jan-Olov Strömberg and I prove the same formula for a more
general class of weights with a new, much simpler method.
Onsdagen 17 mars 13.15-14.15. Grigori Rozenblioum (Chalmers):
Spectral properties of some boundary value problems for
Dirac operator with singular potential.
(joint paper with M. S. Agranovich)
Abstract: For computations in atom physics a method of R-matrix
is widely used, mathematically based on eigenfunction expansions
in eigenfunctions of some boundary value problems (R-matrix relates
some components of the (vector) wave function to other ones). For Dirac
operator in the 3-dimensional space, the Coulomb-type electric
potential is, unlike the Schrodinger case, not a relatively
compact operator, and even the definition of the self-adjoint operator
encounters complications. We discuss the definition of
the operator and study the properties of eigenfunctions and eigenvalues
of two boundary value problems, with traditional placing of the spectral
parameter and with spectral parameter in the boundary condition.
For the latter problem the spectrum turns out to be rather unusual,
consisting of two series of eigenvalues, one of which converges to
zero, the other one to infinity.
The R-matrix is expressed via the corresponding eigenfunctions.
Onsdagen 24 mars 13.15-14.15. Natalia Abuzyarova (KTH):
Branch point area methods in conformal mapping.
(joint paper with H. Hedenmalm)
Abstract: The area method is one of the classical methods of the
theory of conformal mappings. We present a further development of this
method, which is based on corollaries of Stokes' theorem for branched covering
surfaces of the Riemann sphere. As an application, precise integral
estimates for the classes $S$ and $\Sigma$ of univalent functions are
obtained; the latter estimate implies a pointwise estimate for the class
$\Sigma$ obtained by Goluzin in the 1940s by extremality methods.
Onsdagen 31 mars 13.15-14.15. Natan Krugljak (Luleå):
Covering Theorems, Almost Optimal Approximation and Interpolation.
Abstract. I plan to discuss the technique that has been developed
in connection to interpolation of Sobolev spaces. We will focus on
Besicovich-type covering theorem, which have surprizingly untrivial proof.
I also hope to discuss some new results on invertibility of operators on
interpolation scales and their applications to the Hardy-type inequalities.
Onsdagen 7 april 13.15-14.15. : utgår p g a påsk.
Onsdagen 14 april 13.15-14.15. Jörg Schmeling (LTH, Lund):
Random coverings, moving targets, an return times.
Abstract. Let $M$ be a Riemannian manifold and $x_n$ a point process. Let also
a requence of radii $r_n>0$ be given. We consider the following questions:
\medskip
$\bullet$ What is the value (a. s.) of
$$dim_H\{x\in M: #\{n:x\in B(x_n,r_n)\}=\infty\}.$$
\medskip
$\bullet$ What is (a. s.) the value of
$$dim_H\{x\in M: #\{n:x\in B(x_n,r_n)\}<\infty\}.$$
\medskip
We address this question when the random process is generated by a dynamical
system on the circle with an invariant measure. We discuss the connection
to the moving target property and to return time statistics. We use
multifractal analysis to analyze linear expanding maps of the circle and
Diophantine approximation for rotations.
Onsdagen 21 april 13.15-14.15. Lars Håkan Eliasson (Paris):
Quasiperiodic Schrödinger operators in two dimensions.
Abstract: In 1975 Dinaburg and Sinai published an article proving that
(under quite general assumptions) the quasi-periodic Schrödinger operator
in 1 Dimension has Floquet solutions and absolutely continuous spectrum.
This was the starting point for the study of this operator which now has
gone on for almost 30 years and still is very active.
Though the number of articles 1D dimension is enormous, very little has
been done in higher dimension. We shall discuss a version of the theorem
of Dinaburg and Sinai in 2 D.
Onsdagen 28 april 13.15-14.15. P. Kurlberg (Chalmers): A local
Riemann hypothesis.
Onsdagen 5 maj 13.15-14.15. N. Varopoulos (Paris): Parabolic
Potential theory in Lipschitz domains.
Abstract: In view of obtaining a central limit theorem in a Lipschitz domain
one proves sharp L2-L2 estimates for the restriction of singular integrals on
the (parabolic) boundary of such a domain.
Onsdagen 12 maj 13.15-14.15. T. Hoffermann-Ostenhof (Wien):
Nodal domain theorems.
Abstract: We discuss generalizations of Courant's nodal theorem.
In addition we report on some open questions concerning
the relation between the spectrum and the nodal domains
of eigenfunctions. This is joint work with A. Ancona and
B. Helffer.
CANCELLED!! Replacement:
Onsdagen 12 maj 13.15-14.15. A. Olofsson (KTH): A von Neumann Wold
decomposition of two-isometries.
EXTRA SEMINAR: Måndagen 17 maj 10.30-11.30. P. W. Jones (Yale):
Title to be announced.
Onsdagen 19 maj 13.15-14.15. M. Solomyak (Rehovot, Israel):
On the spectrum of a family of differential operators appearing in the
theory of irreversible quantum graphs. General theory.
Abstract: We study the spectrum of a family $A_\alpha$ of partial
differential operators, depending on a real parameter $\alpha$. The
differential expression, which defines the action of the operator, does
not involve $\alpha$, it appears only in the boundary condition.
From the point of view of the Perturbation Theory, we are dealing with
the operators, defined via their quadratic forms, and the perturbation is
only form-bounded, but not form-compact with respect to the unperturbed
operator. This situation is rather unusual for this class of problems,
which is reflected in the character of results. In particular, there exists
the "borderline" value $\alpha_0=\sqrt 2$, such that the spectral properties
of $A_\alpha$ for $\alpha < \sqrt2$ and for $\alpha>\sqrt2$ are quite
different ("phase transition").
The quadratic form approach works only for $\alpha<\sqrt2$, for the large
$\alpha$ a different techniques is necessary. The results for large $\alpha$
are obtained in cooperation with S. Naboko.
Onsdagen 26 maj 13.15-14.15. V. Vasyunin (St-Petersburg, Russia):
Bellman function technique: comparision of dyadic and non-dyadic
problems.
Abstract: An outline of the proof of the John -- Nirenberg inequality in the
integral form will be presented. It will be shown, how
the Bellman function method works being applied to this inequality
for the functions from the dyadic BMO and from the usual one.
The sharp constant will be found in both cases.
Onsdagen 2 juni 13.15-14.15. D. Hejhal (Uppsala):
Some computational aspects of Maass waveforms.
Abstract: The talk will describe some recent results in this area.
Particular emphasis will be given to methods of computation
and to some machine experiments aimed at testing certain
asymptotic randomness conjectures of M.V. Berry.
Onsdagen 9 juni 13.15-14.15. B. Simon (CalTech):
The lost proof of Loewner's Theorem.
Abstract:
A real-valued function, F, on an interval (a,b) is called matrix monotone if
F(A) < F(B) whenever A and B are finite matrices of the same order with
eigenvalues in (a,b) and A < B. In 1934, Loewner proved the remarkable
theorem that F is matrix monotone if and only if F is real analytic with
continuations to the upper and lower half planes so that Im F > 0 in the
upper half plane.
This deep theorem has evolved enormous interest over the years and a number
of alternate proofs. There is a lovely 1954 proof that seems to have been
``lost'' in that the proof is not mentioned in various books and review
article presentations of the subject, and I have found no references to the
proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and a
variant of that proof which I've found, which even avoids the need for
estimates, and proves a stronger theorem.
Höstterminen 2004
Onsdagen 25 augusti 13.15-14.15. Peter M. Knopf (Pleasantville, NY):
Boundedness of Positive Solutions of Second-Order Rational Difference
Equations.
Abstract:
We study positive solutions for all
second-order rational difference equations with nonnegative
coefficients of the form
$$ x_{n+1} = {\alpha + \beta x_n + \gamma x_{n-1}\over A
+ Bx_n + C x_{n-1}}.$$
We obtain necessary and sufficient conditions that their positive
solutions are bounded. These results provide solutions to several open
problems and conjectures proposed by M.R.S. Kulenovi{\'c} and G. Ladas.
Onsdagen 8 september 13.15-14.15. Mattias Jonsson (KTH):
Potential theory on trees.
Abstract: For most humans it is pretty clear what a tree is, but for
mathematicians the situation is a little more complicated. I will discuss
quite general trees: basically real line segments welded together in such
a way that no cycles appear, but with no bounds on the branching.
On such trees one can define a natural Laplace operator which effectively
combines the usual Laplacians on the real line and on a (finite)
simplicial tree. In particular, this Laplacian allows us to identify
Borel measures with certain functions on the tree.
Time permitting, I will also discuss some surprising applications of the
analysis to the study of the singularities of plurisubharmonic
functions in dimension two. This is joint work with Charles Favre at
CNRS, Paris.
Onsdagen 15 september 13.15-14.15. : Håkan Hedenmalm (KTH):
Hele-Shaw flow on weakly hyperbolic surfaces.
Abstract: We consider the Hele-Shaw flow that arises from injection of
two-dimensional fluid into a point of a curved surface. The resulting
fluid domains are more or less determined implicitly by a mean value property
for harmonic functions. We improve on the results of Hedenmalm and Shimorin
(J Math Pures Appl, 2002) and obtain essentially the same conclusions while
imposing a weaker curvature condition on the surface. Incidentally, the
curvature condition is the same as the one that appears in Hedenmalm and
Perdomo's paper (J Math Pures Appl, 2004), where the problem of finding
smooth area minimizing surfaces for a given curvature form under a natural
normalizing condition was considered. Probably there are deep reasons behind
this coincidence. This is joint work with A. Olofsson.
Onsdagen 22 september 13.15-14.15. Julius Borcea (SU):
On rational approximation of algebraic functions.
Abstract: We construct a new scheme of approximation of any multivalued
algebraic function $f(z)$ by a sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ of
rational functions. The latter sequence is generated by a recurrence relation
which is completely determined by the algebraic equation satisfied by $f(z)$.
Compared to the usual Pad\'e approximation our scheme has a number of
advantages, such as simple computational procedures that allow us to prove
natural analogs of the Pad\'e Conjecture and Nuttall's Conjecture for the
sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ in the complement
$\mathbb{CP}^1\setminus \mathcal{D}_{f}$, where $\mathcal{D}_{f}$ is the
union of a finite number of segments of real algebraic curves and finitely
many isolated points. In particular, our construction makes it possible to
control the behavior of spurious poles and to describe the asymptotic ratio
distribution of the family $\{r_{n}(z)\}_{n\in \mathbb{N}}$. If time permits
we will also discuss applications of our results to polynomial recursions
related to certain combinatorial problems.
This is joint work with Rikard B\o{}gvad and Boris Shapiro."
Onsdagen 29 september 13.15-14.15. M. Shashahani (IPM, Tehran):
Multi-temporal Wave Equation on Symmetric and Locally Symmetric Spaces.
Abstract:The generalization of the wave equation to symmetric spaces
naturally leads to systems of partial differential equations with
multi-dimensional time. Nevertheless one can develop a spectral and
scattering theory for these systems where the incoming and outgoing
subspaces in the conventional treatment of the wave equation are replaced by
subspaces parametrized by the Weyl group. The methods depend strongly on
representation theory of semi-simple groups and its applications to
analysis on symmetric and locally symmetric spaces.
Onsdagen 6 oktober 13.15-14.15. Dmitry Gioev (KTH):
Introduction to Random Matrix Theory.
Abstract: We will start by explaining briefly the main physical motivation
for Random Matrix Theory (RMT), namely that it suggests a model that
describes the statistical behavior of energy levels of complex systems.
There are three main types of ensembles of random matrices that are
physically motivated: unitary, orthogonal, and symplectic.
We will define the Unitary Ensemble of random matrices, introduce the
basic probabilistic quantities of interest,
and show how these quantities can be expressed in terms of
orthogonal polynomials (OP's).
We will then explain the idea of universality in RMT,
and in particular introduce the appropriate scaling limit.
Universality means that the statistical behavior predicted by RMT
should not depend on a particular choice of distribution of the matrix
elements (which has no physical meaning), but should depend only on the
type of symmetry imposed on the ensemble (in this case, unitary) which is
physically meaningful.
At this point it will be apparent that the proof of universality for
Unitary Ensembles reduces to a study of asymptotics of the OP's.
Such a study is possible due to the fact that the OP's solve a certain
Riemann-Hilbert problem (RHP). Finally, we mention the appropriate RHP,
and the proof of universality for the unitary case.
This talk can serve as a preparation for our second talk to be given on
October 11 (please see
DNA-seminariet).
That second talk is on our recent joint work with Percy Deift
(Courant Institute) on the proof of the Universality Conjecture
for the other two cases, that is for the Orthogonal and Symplectic
Ensembles of random matrices.
Onsdagen den 13 oktober 13.15-14.15. Nikolai Filonov
(St.Petersburg University): Some inequalities between Dirichlet and
Neumann eigenvalues.
Abstract:
We give a simple proof of the following result. Denote by $\lambda_k$
(resp. $\mu_k$) the eigenvalues of the Laplace operator of Dirichlet problem
(resp. Neumann problem) in the domain $\Omega$ which volume
is supposed to be finite. Then $\mu_{k+1} < \la_k$ for all $k$.
Onsdagen den 20 oktober 13.15-14.15. Daniel Schnellmann (ETH):
Weakly expanding skew-products of quadratic maps.
Abstract: We will look at iterations of the quadratic map
f_a(x)=a-x^2, for a between 1 and 2, where after each iteration step a small
perturbation term is added.
More precisely, we will consider the maps
F: S^1 X R -> S^1 X R:
F(\theta,x)=(g(\theta), a-x2+\alpha\sin(2\pi\theta)),
where $g(\theta)=d\theta \mod 1$, and $\alpha$ is a small real number.
If $d\ge16$ is an integer a famous theorem due to Marcelo Viana states that
this system admits two positive Lyapunov exponents at Lebesque almost
every point.
We will look at some general ideas of the proof of this result.
Furthermore we will see how far this approach of Viana can be generalized
to any real valued $d>1$.
Onsdagen den 27 oktober 13.15-14.15. Hans Ringström:
On the asymptotics of solutions to a wave map equation
arising in general relativity.
Abstract: One important open problem in the mathematical study
of classical general relativity is strong cosmic censorship.
The motivation for this problem is as follows. Given initial data,
there is one part of the spacetime which is uniquely determined;
the so called maximal globally hyperbolic development (MGHD).
There are however initial data for which this development is
extendible in inequivalent ways, and consequently one cannot uniquely
determine what spacetime one is in simply by looking at initial data.
In other words, general relativity is not deterministic. The strong
cosmic censorship conjecture then states that for generic initial data,
the MGHD is inextendible. To try to deal with this problem in all
generality is not realistic. Consequently, one considers the same
question in classes of spacetimes with symmetries. I will in this talk
consider a class of spacetimes with a two dimensional group of isometries.
The resulting mathematical problem is then that of analyzing the
asymptotic behaviour of solutions to a system of non-linear hyperbolic
PDE:s in 1+1 dimensions.
Onsdagen den 3 november 13.15-14.15. Benoit Mandelbrot (Yale):
Recent topics in fractals and multifractals.
Onsdagen den 10 november 13.15-14.15. Ignacio Uriarte-Tuero (Yale,
Helsingfors): On Marcinkiewicz integrals and harmonic measure.
Abstract: Jones and Makarov gave sharp density estimates for harmonic
measure using a modified version of Marcinkiewicz integrals called
$\tilde I_{0}$. It was also used by Jones and Smirnov to substantially
advance in the Sobolev and quasiconformal removability problems. We
generalize and slightly change $\tilde I_{0}$ to make it account for
different densities of sets over which to integrate, in particular giving
a different proof than Jones' and Makarov's of its key properties. This
should have applications to the aforementioned Sobolev and quasiconformal
removability problems.
Onsdagen den 17 november 13.15-14.15. Serguei Shimorin (KTH):
Branching point area theorems for univalent functions.
Abstract:
Area methods are a classical tool in the theory of univalent functions.
Such topics as Grunsky, Goluzin, or Schiffer-Tammi inequalities
are in fact different modifications of the Polynomial Area Theorem
which in turn reduces to an appropriate application of the Green
formula.
In the talk, we duscuss a new type of area theorems obtained
by considering branching point compositions with univalent functions.
As a result, we obtain a new series of sharp integral inequalities.
We discuss also branching point versions of Grunsky and Goluzin
intequalities.
Onsdagen den 24 november 13.15-14.15. Magnus Aspenberg (KTH):
The Collet-Eckmann condition for rational functions on the Riemann
sphere.
Abstract:
In 1986 Mary Rees proved in a famous paper that
the set of functions admitting an absolutely continuous
invariant measure in the parameter space
of rational functions of a given degree $d \geq 2$, has positive
Lebesgue measure.
In the talk I will formulate a theorem, which states that the set of
functions satisfying the so called Collet-Eckmann condition, has positive
Lebesgue measure in the parameter space of rational funtions for any
fixed degree $d \geq 2$. M. Rees' theorem is a consequence of this combined
with recent results of J. Graczyk, S. Smirnov and F. Przytycki.
A function is {\em Collet-Eckmann} if there are
constants $C > 0$ and $\gamma > 0$ such that, for every critical point
$c$ whose forward orbit does not contain any other critical point, the
following holds:
$$
|(R^n)'(R(c))| \geq C e^{\gamma n}, \text{for all $n \geq 0$.}
$$
The approach to prove the main theorem is to use a method developed
earlier by M. Benedicks and L. Carleson, where they prove corresponding
results for the quadratic family and families of H\'enon maps.
Another consequence of the main theorem is that
the Julia set is equal to the Riemann sphere for a
set of positive Lebesgue measure in the space of rational functions
for any fixed degree $d \geq 2$.
Onsdagen den 8 december 13.15-14.15. Anders Olofsson (KTH):
Wandering subspace theorems.
Abstract:
Let $H$ be a Hilbert space, and $T$ a bounded linear operator on $H$,
which is bounded from below. We are interested in the following questions:
Is it possible to expand $H$ as the linear span of $T^j(E)$, where
$j=0,1,2,\ldots$ and $E=H\ominus T(H)$?
In the case $T$ is an isometry, the answer is classical. Recently, A. Aleman,
S. Richter, C. Sundberg, and S. Shimorin obtained results when $T$ is not
too far from an isometry. Here, we discuss summability of the expansion in
the new setting.
Onsdagen den 15 december 13.15-14.15. Mikael Passare (SU):
Tropical geometry and (co-)amoebas.