Analysseminariet 2005,
löpande planering
Vårterminen
Onsdagen den 26 januari 13.15-14.15. Dmitry Beliaev (KTH):
Harmonic measure on random sets.
Abstract:
Many problems in complex analysis can be reduced to the evaluation of the
{\it universal spectrum}: the supremum of multifractal spectra of harmonic
measures for all planar domains. Its exact value is still unknown, with
very few estimates available. We describe related problems and available
estimates from above. Then we discuss in more detail estimates from below,
describing the search for a fractal domain which attains the maximal
possible spectrum.
Onsdagen den 2 februari 13.15-14.15. Serguei Kisliakov (St-Petersburg):
Double singular integrals: interpolation and correction.
Abstract:
The orthogonal projection of L2 on the two-dimensional torus
onto the two-dimensional Hardy space H2 is a prototypic example
of a double singular integral. Though in general double
singular integrals are difficult to handle, the above one can be
analyzed rather thoroughly by a specific trick. It turns out that the
same trick applies to certain operators of one-dimensional Fourier
analysis. This leads to new interpolation and correction theorems. My
aim in the talk is to discuss such theorems for the Hardy-Littlewood
square function in dimension 1. A short survey of the background material
will also be given. (This is a joint work with D. S. Anisimov).
Onsdagen den 9 februari 13.15-14.15. Hjalmar Rosengren (Chalmers):
Harmonic analysis on the Sklyanin algebra.
Abstract:
Quantum groups arose in the 1980's as an algebraic framework for studying
solvable models in statistical mechanics. Typically, such models are
described by elliptic functions. Most research on quantum groups has
focused on the trigonometric (or "q") limit case, while the more general
elliptic quantum groups have proved more difficult to study. We will
discuss some topics of harmonic analysis on the Sklyanin algebra, which is
the simplest elliptic quantum group. In particular, we will describe a
connection with elliptic hypergeometric series, a very new class of
special functions. We will also discuss the proof of a conjecture of
Sklyanin from 1983, concerning Sklyanin algebra invariant integrals on the
torus. The talk will be very elementary, and no previous knowledge of
quantum groups will be assumed.
Onsdagen den 16 februari 13.15-14.15. Carl Sundberg (UT Knoxville):
Nontangential limits in $P^t(\mu)$ spaces and the index of invariant
subspaces.
Abstract: We show that if $P^t(\mu)$ is irreducible and $\mu$ is supported
in the closed unit disk, then either (1) we have no boundary mass of $\mu$,
no boundary values a. e., and shift invariant subspaces of arbitrary index,
or (2) we have positive boundary mass, boundary values a. e. on the
support of $\mu$ on the unit circle, and invariant subspaces of index $1$
only ($0$ in the degenerate case). At a critical point in the argument,
the work of X. Tolsa on analytic capacity is used.
Onsdagen den 23 februari 13.15-14.15. Andrzej Szulkin (SU): Eigenvalue
and boundary value problems for equations involving the $p$-Laplace operator.
Abstract: Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$
and let $\Delta_{p}u := \mathrm{div}(|\nabla u|^{p-2}\nabla u)$ be the
$p$-Laplacian, for $p$ between $1$ and $\infty$. We consider two problems:
1. The eigenvalue problem $-\Delta_{p}u = \mu|u|^{p-2}u$ in $\Omega$,
$u=0$ on $\partial\Omega$. If $p=2$, it is well known that the
spectrum $\sigma(-\Delta)$ consists of a sequence of eigenvalues
$\mu_{k}\to\infty$. We consider the case $p\ne 2$ which turns out to
be much more difficult and is only partially understood.
2. The boundary value problem $-\Delta_{p}u = f(x,u)$ in $\Omega$,
$u=0$ on $\partial\Omega$, where $f(x,u)/(|u|^{p-2}u) \to \lambda_{0}$
as $u\to 0$ and $f(x,u)/(|u|^{p-2}u) \to \lambda_{\infty}$ as
$|u|\to\infty$. We discuss the existence of solutions $u\ne 0$ under
the assumption that the interval with the endpoints $\lambda_{0}$ and
$\lambda_{\infty}$ intersects the spectrum $\sigma(-\Delta_{p})$.
Onsdagen den 2 mars 13.15-14.15. Eero Saksman (Jyväskylä):
On the quasi-conformal Jacobian problem.
Abstract:
It is a well known question to describe the
Jacobians of quasiconformal homeomorphisms of
the Euclidean space to itself. We describe the problem
and discuss some work done on it (joint with J. Heinonen
and Mario Bonk, Ann Arbor).
Onsdagen den 9 mars 13.15-14.15. Pavel Kurasov (LTH Lund):
Can one hear the shape of a graph?
Abstract: The inverse spectral problem for the Laplace operator on a finite
metric graph is investigated. It is shown that this problem has a
unique solution for graphs with rationally independent edges and
without vertices having valence 2. To prove the result trace
formula connecting the spectrum of the Laplace operator with the
set of periodic orbits for the metric graph is established. This is joint
work with M. Nowaczyk.
Onsdagen den 16 mars 13.15-14.15. Håkan Hedenmalm:
Quantum Hele-Shaw flow.
Abstract: We discuss the quantum Hele-Shaw flow, a random measure
process in the complex plane introduced by the physicists Wiegmann,
Zabrodin, et al. This process arises in the theory of electronic
droplets confined to a plane under a strong magnetic field, as well as
in the theory of random normal matrices. We extend a result of Elbau
and Felder to general external field potentials, and also show
that if the potential is $C^2$-smooth, then the quantum Hele-Shaw flow
converges, under appropriate scaling, to the classical (weighted)
Hele-Shaw flow, which can be modeled in terms of an obstacle problem.
This is joint work with N. Makarov.
Onsdagen den 24 mars 13.15-14.15. Utgår p g a påsk.
Onsdagen den 30 mars 13.15-14.15. Michael Benedicks (KTH):
Coexistense of attractors.
Abstract: For a dynamical system there is a notion of attractors due to
Milnor. A Milnor attractor is roughly an attractor that attracts a positive
Lebesgue measure set in phase space. There is also the notion of
minmal Milnor attractors. If there is an ergodic measure such there is
a set of initial points in phase space with the property that the
corresponding Birkoff sums converge to the measure, the support of this
measure is a Milnor attractor. There can be infinitely coexisting Milnor
attractors --- both stable periodic orbits and "strange attractors" but
this is generally believed to be a rare phenomena (a conjecture of
J.~Palis). I will describe some previous results in this field and also
a recent result of M.~Tsujii about finitely many coexisting attractors for
partially hyperbolic maps of the twodimensinal torus.
Onsdagen den 6 april 13.15-14.15. Anders Szepessy (KTH):
Stochastic hydrodynamic limits of Stochastic Ising models.
Abstract: Even small noise can have substantial influence on the
dynamics of differential equations, e.g. for nucleation and coarsening
in phase transformations. The aim of this talk is to present an accurate
model for the noise in macroscopic differential equations, related to
phase transformations/reactions, derived from more fundamental microscopic
Master equations.
I will show that localized spatial averages, with width $\epsilon$, of
solutions to stochastic Ising models with long range interaction, of
width $\mathcal{O}(1)$, are approximated with error
$\mathcal{O}(\epsilon2 +(\gamma/\epsilon)^{2d})$ in distribution
by a solution of an Ito stochastic differential equation, with drift as
in the mean field model and a small diffusion coefficient of order
$(\gamma/\epsilon)^{d/2}$, generating noise with spatial correlation
length $\epsilon$, where $\gamma$ is the distance between neighboring
spin sites on a uniform periodic lattice in $\rset^d$.
The proof is simple and based on two ideas:
no law of large numbers is applied, instead the proof uses
$\mathcal{O}((\gamma/\epsilon)^{2d})$ consistency with the
Kolmogorov-backward equation from a Chapman-Enskog expansion; and the
long range interaction yields smoothing and contributes with the
$\mathcal{O}(\epsilon2)$ error.
Onsdagen den 13 april 13.15-14.15. Håkan Hedenmalm (KTH):
Real zero polynomials and Polya-Schur type theorems.
Abstract: Consider $P$, the linear space of all polynomials, which is
an algebra if we add the operation of multiplication. Also, consider $P_R$,
the multiplicative semigroup of all polynomials with only real zeros. We
adjoin the zero polynomial to $P_R$. Let $T$ be a linear operator on $P$.
The question is, which $T$ preserve $P_R$? The classical Gauss-Lucas theorem
asserts that differentiation is such an operator. When $T$ is required to
commute with (a) differentiation, or (b) multiplicative differentiation,
theorems ascribed to (a) Polya and Benz, and (b) Polya and Schur, completely
settles the question. The general question, however, remains unsolved.
Here, we consider the problem under the added hypothesis that $T$ commute
with inverted plane differentiation.
This is joint work with A. Aleman and D. Beliaev.
Onsdagen den 20 april 13.15-14.15. Stefan Rauch-Wojciechowski (LiU):
Phase space, invariant manifolds and stability properties of
asymptotic solutions of the Tippe Top.
Abstract: The Tippe Top has a shape of a truncated sphere with a peg
attached to the flat surface.
When spun sufficiently fast on its spherical bottom the tippe top
turns up and continues
motion on the peg. This behaviour takes place for wide range of
parameters and of initial conditions.
I shall analyse the structure of phase space through a sequence of
invariant manifolds and give a description of what happens for all
initial conditions and all values parameters. Then I show that, due to
the gliding friction, all solutions tend (in the sense of the LaSalle´
theorem) to an asymptotic manifold consisting of periodic solutions.
These solutions are Liapunov stable but are not asymptotically stable
although every solution approaches one these periodic trajectories.
I shall demonstrate the motion of the tippe top.
Onsdagen den 27 april 13.15-14.15. N. Kruglyak (Luleå):
Covering theorems and singular integrals in limiting cases.
Onsdagen den 4 maj 13.15-14.15. : M. Skriganov (St-Petersburg):
Harmonic analysis on totally disconnected groups and irregelarities
of point distributions.
Abstract: We shall study point distributions in the multi-dimensional unit
cube which possess the structure of finite abelian groups with respect to
certain p-ary arithmetic operations. Such disrtibutions can be thought of
as finite subgroups of a compact totally disconnected group of Cantor
type. We shall apply the methods of harmonic analysis on these groups to
estimate very precisely the discrepancy for such distributions.
The paper is availible on the site: www.pdmi.ras.ru
Onsdagen den 11 maj 11.00-12.00 (obs!). E. Doubtsov (St-Petersburg):
An Uncertainty Principle on the complex sphere.
Abstract. Let $m$ be a measure on the complex sphere.
Denote by $m_{pq}$ the projection of $m$ on $H(p,q)$,
the space of complex spherical harmonics.
We obtain new quantitative versions of the following general
Uncertainty Principle:
if the polynomials $m_{pq}$ are sufficiently small, then
$m$ is absolutely continuous with respect to Lebesgue measure
on the sphere. Also, we discuss the sharpness of such results.
Onsdagen den 11 maj 13.15-14.15. P. W. Jones (Yale): The Gaussian
Free Field, Random Homeomorphisms, and Random Loops.
Onsdagen den 18 maj 13.15-14.15. A. B. Aleksandrov (St-Petersburg):
Approximation by M. Riesz's kernels in $L^p$ for $p<1$.
Abstract: Let $\alpha>0$. Denote by ${\frak X}_\alpha({\mathbb R}^n)$ the
$\mathbb R$-linear span of the scalar
Riesz kernels $\{\frac1{|x-a|^\alpha}\}_{a\in{\mathbb R}^n}$.
Denote by ${\frak Y}_\alpha({\mathbb R}^n)$ the ${\mathbb R}$-linear span of
the vector (${\mathbb R}^n$-valued) Riesz kernels
$\{\frac1{|x-a|^{\alpha+1}}(x-a)\}_{a\in{\mathbb R}^n}$.
Let ${\frak X}^p_\alpha({\mathbb R}^n):={\frak X}_\alpha ({\mathbb R}^n)
\cap L^p({\mathbb R}^n)$ and
${\frak Y}^p_\alpha ({\mathbb R}^n):={\frak Y}_\alpha ({\mathbb R}^n)
\cap L^p({\mathbb R}^n,{\mathbb R}^n)$.
Let $X^p_\alpha ({\mathbb R}^n)$ and $Y^p_\alpha ({\mathbb R}^n)$ denote
the closure of ${\frak X}^p_\alpha ({\mathbb R}^n)$ and
${\frak Y}^p_\alpha ({\mathbb R}^n)$ in $L^p({\mathbb R}^n)$ and
$L^p({\mathbb R}^n,{\mathbb R}^n)$ respectively.
Note that $X^p_\alpha ({\mathbb R}^n)=\{0\}$ and
$Y^p_\alpha({\mathbb R}^n)=\{0\}$ if $\alpha p\ge n$.
Denote by $L^p_0({\mathbb R}^n)$ the closure in $L^p({\mathbb R}^n)$ of the
set $\big\{\varphi\in{\cal S}({\mathbb R}^n):\int_{{\mathbb R}^n}
\varphi=0\big\}$,
where ${\cal S}({\mathbb R}^n)$ denotes the space of rapidly decreasing
smooth functions on ${\mathbb R}^n$. Denote by
$L^p_0({\mathbb R}^n,{\mathbb R}^n)$ the closure in
$L^p({\mathbb R}^n,{\mathbb R}^n)$ of the set
$\{\nabla\varphi:\varphi\in{\cal S}({\mathbb R}^n)\}$.
Clearly, $L^p_0({\mathbb R}^n)=L^p({\mathbb R}^n)$ for $p\not=1$, and
$L^1_0({\mathbb R}^n)=\{f\in L^1({\mathbb R}^n):\int_{{\mathbb R}^n} f=0\}$.
Moreover,
$L^p_0({\mathbb R}^n,{\mathbb R}^n)=L^p({\mathbb R}^n,{\mathbb R}^n)$ for
$p<1$.
It is not difficult to prove that $X^p_\alpha (\R^n)=L^p_0(\R^n)$ and
$Y^p_\alpha (\R^n)=L^p_0(\R^n,\R^n)$ if $1\le p<+\infty$ and $\alpha p
{\bf 1.} When is $X^p_\alpha ({\mathbb R}^n)=L^p({\mathbb R}^n)$?
\vskip3pt
{\bf 2.} When is $Y^p_\alpha({\mathbb R}^n)=L^p({\mathbb R}^n,{\mathbb R}^n)$?
Onsdagen den 25 maj 13.15-14.15. L. H. Eliasson (Paris): KAM for
NLS.
Abstract:
We shall discuss the non-linear Schr\"odinger equation with perodic
boundary conditions in dimension $d$. This is an infinite-dimensional
Hamiltonian system and one central problem is the perturbation theory
for lower-dimensional tori = quasi-periodic solutions, usually known as
KAM.
The difficulties in applying KAM in infinite dimensions are substantial
and
become larger with increasing $d$.
For NLS the case $d=1$ was solved in the late 80's by Kuksin and,
later, Bourgain. The case $d=2$ was solved in the early 90's by Bourgain,
but by an
approach (known as the Craig-Wayne scheme) that provides less information
than KAM. We shall discuss these issues and, if times admits,
report on a recent work (with Kuksin) that aims to solve the problem
for any $d$.
Onsdagen den 1 juni 13.15-14.15. N. Makarov (CalTech): Laplacian
random walks on Riemann surfaces.
Onsdagen den 7 september 13.15-14.15. A. V. Sobolev (Sussex) :
Eigenvalue distribution for the perturbed Laplace operator on the
torus.
Abstract: One studies the eigenvalue counting function for the operator
$H = -\Delta + V$ with a continuous real-valued function $V$,
on the two-dimensional torus $\mathbb T$.
The Berry-Tabor hypothesis claims that large eigenvalues of
quantum Hamiltonians
with a completely integrable underlying classical system,
must be distributed according to the Poisson law.
In particular, this hypothesis
applies to the operator $H_0 = -\Delta$ on $\mathbb T$.
However, at present only partial results towards the proof of this
conjecture are available.
The aim of my talk is to show that at large energies the statistical
distributions of discrete spectra
of $H$ and $H_0$ coincide irrespectively of the distribution
law for the spectrum of $H_0$.
Onsdagen den 14 september 13.15-14.15. A. Baranov (St-Petersburg, KTH):
Completeness of systems of reproducing kernels in model subspaces.
Abstract:
Onsdagen den 21 september 13.15-14.15. A. Borichev (Bordeaux):
Sampling and interpolation in radial weighted spaces of analytic
functions.
Abstract:
Onsdagen den 28 september 13.15-14.15. R. Beals (Yale): Exact
solutions of some linear PDE: (almost) elliptic and (almost) hyperbolic.
Abstract:
Onsdagen den 5 oktober 13.15-14.15. D. Jakobson (McGill): Estimates
from below for the spectral function of the Laplacian.
Abstract:
Onsdagen den 12 oktober 13.15-14.15. K. Johansson (KTH):
From Gumbel to Tracy-Widom.
Abstract: The Gumbel distribution is a classical
extreme value distribution which occurs when one
considers the maximum of say N independent Gaussian
random variables for large N. The Tracy-Widom distribution
is the asymptotic distribution for the largest eigenvalue
of a large random Hermitian matrix, and it can also be thought
of as a kind of extreme value distribution. I will discuss
the possiblity of interpolating between these two distributions
in families of random matrix ensembles.
In particular I will discuss a family of determinantal processes
which interpolate between a Poisson process with density
$\exp(-x)$ and the Airy kernel point process. Both these
processes have a last particle with the Gumbel and the
Tracy-Widom distribution respectively. These kind of transition
ensembles have been studied mainly in the bulk of the spectrum,
motivated by problems in for example quantum chaos, where one
is interested in transitions from Poissonian statistics
(integrable dynamics) to random matrix statistics (chaotic
dynamics).
Onsdagen den 19 oktober 13.15-14.15. D. Chelkak (St-Petersburg):
Conformal mappings in the spectral theory for Schr\"odinger
operator with periodic matrix potentials.
Abstract: We consider the Schr\"odinger operator on the real line with
a $N\times N$ matrix real-valued periodic potential. The spectrum of
this operator is absolutely continuous and consists of intervals
separated by the gaps. We define the Lyapunov function (which is
analytic on the $N$-sheeted Riemann surface) and some conformal
mapping (averaged quasimomentum), which properties are similar to the
scalar ($N=1$) case. The Lyapunov function has (real or complex)
branch points, which we call resonances. We show that there exist two
types of gaps in the spectrum: i) "usual" gaps, which endpoints are
periodic and anti-periodic eigenvalues, ii) "resonance" gaps, which
endpoints are resonances (real branch points). We determine the
asymptotics of the periodic, anti-periodic spectrum and the resonances
at high energy in terms of the Fourier coefficients of the potential
and obtain some a priori estimates of the gap lengths.
Based on the joint work with E.Korotyaev (Berlin).
Onsdagen den 26 oktober 13.15-14.15. E. Korotyaev (Berlin):
.
Abstract: We consider the operator $d4/dx4 +V$ on the real line
with a real periodic potential $V$.
The spectrum of this operator is absolutely continuous and consists of
intervals separated by gaps. We define a Lyapunov function
which is analytic on a two sheeted Riemann surface. On each
sheet, the Lyapunov function has the same properties
as in the scalar case, but it has
branch points, which we call resonances. We prove the existence of real as
well as non-real resonances for specific potentials. We determine the
asymptotics of the periodic and anti-periodic spectrum and of the
resonances at high energy. We show that there exist two type of
gaps: 1) stable gaps, where the endpoints are periodic and
anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the
endpoints are resonances (i.e., real branch points of the Lyapunov
function above the bottom of the spectrum). We also show that
the periodic and anti-periodic spectrum together determine the spectrum of
our operator. Finally, we show that for small potentials the spectrum in
the lowest band has multiplicity 4 and the bottom of the spectrum is a
resonance, and not a periodic (or anti-periodic) eigenvalue.
Onsdagen den 2 november 13.15-14.15. L. Peng (Peking): Localization
operators based on Weyl transforms.
Abstract: This is a new look at the time-frequency localization operators
from the point of view of group representation. The connections with several
other active areas are established, such as $H^*$-algebra, Weyl transform,
paracommutator, and compensated compactness. Furthermore, a series of new
results are derived.
Onsdagen den 9 november 13.15-14.15. I. Mitrea (U Virginia):
Global Optimization Techniques for Singular Integrals.
Abstract: We survey recent progress in the direction of understanding
the spectra of integral operators which arise naturally in the
context of elliptic boundary problems in non-smooth domains.
The focus is the Spectral Radius Conjecture, for which we present
both positive and negative results, some of which have been obtained
via computer aided proofs.
Onsdagen den 16 november 13.15-14.15. H. Holden (Trondheim):
The Hunter-Saxton equation .
Abstract: The Hunter-Saxton equation is a nonlinear partial differential
equation that has been used as a simple model for liquid crystals. It
can be written as $(u_t+u u_x)_x=\frac12 (u_x)2$. We survey some of
the known results for the equation. In particular, we study various
finite difference approximations, and show that they converge to a
solution of the equation. This is joint work with K. H. Karlsen and
N. H. Risebro, both from University of Oslo.
Onsdagen den 23 november 13.15-14.15. A. Kupiainen (Helsingfors):
Fourier Law and Boltzmann equation.
Abstract: We review the problem of heat conduction in Hamiltonian systems
and discuss the derivation of Fourier's law from a truncated set
of equations for the stationary state of a mechanical system coupled to
boundary noise.
Onsdagen den 30 november 13.15-14.15. T. Kolsrud (KTH):
Position Dependent NLS Hierarchies: Involutivity, Commutation Relations,
Renormalisation and Classical Invariants.
Abstract: We consider a family of explicitly position dependent hierarchies
$(I_n)_0^\infty$, containing the NLS
(non-linear Schr\"odinger) hierarchy. All $(I_n)_0^\infty$ are involutive and
fulfill $\mathsf DI_n=nI_{n-1}$, where $\mathsf D=D^{-1}V_0$, $V_0$ being the
Hamiltonian vector field $v\frac {\delta}{\delta v}-u\frac {\delta}{\delta u}$
afforded by the common ground state $I_0=uv$. The construction requires
renormalisation of certain function parameters.
It is shown that the `quantum space' $\mathbb C[I_0, I_1, ...]$ projects down
to its classical counterpart $\mathbb C[p]$, with $p=I_1/I_0$, the momentum
density. The quotient is the kernel of $\mathsf D$. It is identified with
classical semi-invariants for forms in two variables.
Onsdagen den 7 december 13.15-14.15. J. Steif (Göteborg):
Critical Dynamical percolation, exceptional times, and harmonic
analysis of boolean functions.
Abstract: Suppose that each of the vertices in the triangular grid in the
plane is ``open" with probability 1/2 independently. It is known since
the work of Harris (1960) that the set of open sites does not have an
infinite connected component. In dynamical percolation, the sites
randomly flip between the states open and closed according to independent
(Poisson) clocks. We show that dynamical percolation has a set of
exceptional times at which an infinite open connected component exists.
This contrasts with the fact that at any fixed time almost surely all
components are finite. One of the tools used is a new inequality relating
the Fourier coefficients of a boolean function with the existence of a
randomized algorithm that calculates the function but is unlikely to
examine any specific input bit. The proof also uses recent results concerning
critical exponents for critical percolation. This is joint work with
Oded Schramm.
Onsdagen den 14 december 13.15-14.15. A. Constantin (Lund):
.
Abstract: