Analysseminariet 2006,
löpande planering
Vårterminen
Onsdagen den 1 februari 13.15-14.15. Ari Laptev (KTH) : Hardy
inequalities for many particles.
Abstract: We prove some inequalities of Hardy type for many particles.
In particular, we shall show how introducing Aharonov-Bohm magnetic fields
could give such inequalities for two-dimensional particles. It turns out that
2D Hardy inequalities hold also for fermions.
:
Onsdagen den 8 februari 13.15-14.15. Anton Baranov (St-Petersburg, KTH):
Stability of bases and frames of reproducing kernels in model spaces.
Abstract:
We study the bases and frames of reproducing kernels in the model subspaces
$K^2_{\Theta}=H^2\ominus \Theta H^2$ of the Hardy class $H2$ in the upper
half-plane. The main problem under consideration is stability of a basis of
reproducing kernels
$k_{\lambda_n}(z)=(1-\overline{\Theta(\lambda_n)}\Theta(z))
/(z-\overline\lambda_n)$
under ``small'' perturbations of the points $\lambda_n$.
We propose an approach to this problem based on
the recently obtained estimates of derivatives in the spaces
${K^2_{\Theta}}$ and produce estimates
of admissible perturbations generalizing
certain results of W.S. Cohn and E. Fricain.
Onsdagen den 15 februari 13.15-14.15. Jens Hoppe (KTH):
Quantum Riemann Surfaces.
Abstract:
Harmonic homogenuous polynomials in 3 commuting variables,
upon substitution of N-dimensional representations of su(2)
for the commuting variables, can be used to define a map
from functions on the 2-sphere to NxN matrices, that sends
Poisson-brackets into commutators. For the torus, an
analogous (but very different) construction is known.
We propose a solution to the longstanding problem of
finding a concrete way to treat all Riemann Surfaces
in a unified way.
Onsdagen den 22 februari 13.15-14.15. Per Sjölin (KTH): Maximal
estimates for solutions to the nonelliptic Schrödinger equation.
Abstract: Maximal estimates are studied for solutions to an initial value
problem for the nonelliptic Schrödinger equation. A result of Rogers,
Vargas, and Vega is extended.
Onsdagen den 1 mars 13.15-14.15. Oleg Viro (Uppsala):
Homology is a refinement of summation.
Abstract: A notion of categorification will be presented by the upgrade of
the Jones polynomial to Khovanov homology. The Jones polynomial of a classical
knot is considered as a Kauffman state sum. An idea of cateorification applied
to a sum is to figure out natural cancellations in the sum, cook up out of
them a differential acting between summands such that the sum would be equal
to a shorter alternating sum of the Betti numbers. The homology groups appear
to be more interesting than the original sum. Can this happen to a sum in
analysis? No prior knowledge of homology is assumed.
Onsdagen den 8 mars 13.15-14.30. Stas Smirnov (Genève):
2D Ising model and conformal invariance.
Abstract: We will discuss why 2D Ising model at critical temperature has
a conformally invariant scaling limit.
Onsdagen den 15 mars 13.15-14.15. Thomas Hoffmann-Ostenhof (ESI Wien):
Nodal domains and spectral optimal partitions.
Abstract:
Onsdagen den 22 mars 13.15-14.15. Julius Borcea (SU): On the
classification of hyperbolicity and stability preservers.
Abstract: A linear operator $T$ on $\mathbb{C}[z]$ is called
hyperbolicity-preserving or an HPO for short if $T(P)$ is hyperbolic
whenever $P\in \mathbb{C}[z]$ is hyperbolic, i.e., it has all real zeros.
One of the main challenges in the theory of univariate complex polynomials
is to describe the monoid $\mathcal{A}_{HP}$ of all HPOs. This reputably
difficult problem goes back to P\'olya-Schur's well-known characterization
of multiplier sequences of the first kind, that is, HPOs which are diagonal
in the standard monomial basis of $\mathbb{C}[z]$. P\'olya-Schur's result
(Crelle, 1914) generated a vast literature on this subject and related
topics at the interface between analysis, operator theory and algebra but
so far only partial results under rather restrictive conditions have been
obtained. In this talk I will report on the progress towards complete
solutions to both this problem and its analog for stable polynomials as
well as their multivariate versions made in an ongoing series of papers
jointly with Petter Br\"and\'en and Boris Shapiro.
Onsdagen den 29 mars 13.15-14.15. Alexandru Aleman (Lund):
Derivation-Invariant Subspaces of $C^\infty$.
Abstract:
We consider the differentiation operator on the space of infinitely
differentiable functions on an interval.
We show that the spectrum of the restriction of this operator to an
invariant subspace can be either the whole complex plane, or a discrete
subset of it, or the empty set. A typical example of an invariant subspace
where this operator has void spectrum is the set of all functions in the
space which vanish identically on an interval. We prove that every
differentiation invariant subspace with this property must have this form.
The interval in question may reduce to a point in which case the subspace
consists of the functions that vanish together with all their derivatives
at that point.
This reports on joint work with B. Korenblum.
Onsdagen den 5 april 13.15-14.15. Nils Dencker (Lund):
Solvability and the Nirenberg-Treves Conjecture.
Abstract: In the 50's, Ehrenpreis and Malgrange proved that all constant
coefficient linear partial differential equations are solvable.
The consensus at that time was that all linear PDE´s were solvable.
Therefore, it came as a surprise when Hans Lewy in 1957 constructed a
non-solvable complex vector field, in fact, the image is a set of the
first category. The vector field is a natural one; it is the
Cauchy-Riemann operator on the boundary of a strictly
pseudo-convex domain.
A rapid development in the 60's lead to the conjecture by Nirenberg
and Treves in 1969: that condition ($\Psi$) is necessary and
sufficient for solvability of (pseudo-)differential operators of
principal type. This is a condition on sign changes of the imaginary
part of the principal symbol along the bicharacteristics of the real
part. Thus, it only depends on the the highest order term of the
operator.
The Nirenberg-Treves conjecture has recently been proved.
We shall present the background and the ideas of the proof,
which will appear in Annals of Mathematics, 163:3, 2006.
Onsdagen den 12 april 13.15-14.15. Anders Olofsson (Bordeaux):
Operator model theory for $n$-hypercontractions.
Abstract:
Let $\mathcal H$ be a general Hilbert space and $n\geq1$ an integer. A
bounded linear operator $T\in{\mathcal L}({\mathcal H})$ is called an
$n$-hyper\-contraction if
$$
\sum_{k=0}^m(-1)^k\binom{m}{k}T^{*k}T^k\geq0\quad\text{in}\
{\mathcal L}({\mathcal H})
$$
for $1\leq m\leq n$. A result of Agler says that an operator
$T\in{\mathcal L}({\mathcal H})$ is an $n$-hypercontraction if and only if
it is part of an operator of the form $S_n^*\oplus U$, where $S_n$ is a shift
operator on a certain vector-valued standard weighted Bergman space on
the unit disc and $U$ is an isometry. We give an explicit construction
modeling a general $n$-hypercontraction as part of an operator of this form.
We also show that this construction has certain canonical features.
Onsdagen den 19 april 13.15-14.15. Tanja Berkvist och Jan-Erik Björk
(SU): Asymptotics of polynomial eigenfunctions for exactly-solvable
differential operators.
Abstract: We will discuss asymptotic properties of zeros of polynomial
eigenfunctions for degenerate exactly-solvable differential operators.
We show that for all such operators the root of the unique $n$th degree
eigenpolynomial with the largest modulus tends to infinity when $n\to\infty$,
as opposed to the non-degenerate case which we have treated previously. We
introduce a scaling factor which in computer experiments results in a
compactly supported limiting measure for the asymptotic zero distribution
of the eigenpolynomials. Conjecturally its support is the union of a finite
number of analytic curves in the complex plane.
Onsdagen den 26 april 13.15-14.15. Tero Kilpeläinen
(Jyväskylä): Removable sets for p-Laplacian type
equations.
Abstract: We survey old and new topics related to removable singularities
for p-Laplacian type equations.
Onsdagen den 3 maj 14.30-15.30 (Obs tiden!). Paul Malliavin (Paris):
Energy dissipation towards higher modes of stochastic Euler fluid
dynamics.
Onsdagen den 10 maj 13.15-14.15. Pavel Kurasov (Lund):
Wigner-von Neumann Perturbations of a Periodic Potential:
Spectral Singularities in Bands.
Abstract: The celebrated Wigner-von Neumann potential provides an example of
a one-dimensional Schr\"odinger operator
$$ - \frac{d2}{dx2} + V(x), \; \; \; \mbox{acting in} \; \; \; L_2
(\mathbb R), $$
with an eigenvalue embedded into the absolutely continuous
spectrum $ [0, \infty).$ This phenomenon is related with the
vanishing of the spectral density at the point where the embedded
eigenvalue may occur.
In the current lecture we are going to investigate the same
phenomena in the case of a periodic background potential. The
absolutely continuous spectrum has band-gap structure in this case
and we are going to study the possibility to obtain eigenvalues
embedded into the absolutely continuous spectrum by adding
Wigner-von Neumann type potentials. The asymptotics of the
solution to the generalized eigenfunction equation will be
investigated. It will be proven that a subordinated solution and
therefore an embedded eigenvalue may occur at the points of the
absolutely continuous spectrum satisfying a certain resonance
(quantization) condition between the frequencies of the
Wigner-von-Neumann perturbation, the frequency of the background
potential and the corresponding quasimomentum. This is a joint
work with S. Naboko.
Onsdagen den 17 maj 13.15-14.15. Peter W. Jones (Yale):
Eigenfunction Local Coordinates and the Local Riemann Mapping
Theorem.
Abstract: One idea in the exciting new area of Diffusion Geometry is to
use certain eigenfunctions as new local coordinates on a data set (e.g. a
collection of documents). These coordinates are surprisingly robust under
perturbation of the underlying sets and have been empirically observed to
provide local coordinates on rather large patches. In this talk we discuss
the mechanism that explains this robustness: it turns out to be a "hidden"
(i.e. simple, though previously unobserved) feature of the Riemann Mapping
Theorem for simply connected planar domains that is quite general. This
simple feature also works on manifolds of arbitrary dimension. The idea is
to use standard estimates for the Heat Kernel to pick eigenfunctions
providing local coordinates on a large Ball, and the diameter of that ball
is optimal up to universal constants. (In more technical language, which
will be explained in English, the main result is an analogue of the
Distortion Theorems for conformal mappings.)
We will look at simple estimates for the Heat Kernel. Here we are allowed
to choose between the Dirichlet Heat Kernel (related to absorbing random
walk) or the Neumann version (related to random walk that reflects off the
boundary). It is the second case that occurs in Diffusion Geometry. By
reexamining the statement of the Riemann mapping theorem and using standard
facts from study of the Laplace operator we will be led to an algorithm for
picking out d eigenfunctions that provide "robust" (i.e. Bi-Lipschitz) local
coordinates in a large neighborhood of a given point in a smooth manifold of
dimension d. The simple model to think of is the d dimensional unit cube or
torus, where the sine or Fourier eigenfunctions easily give rise to robust
local coordinate systems. (Notice that on the circle the two eigenfunctions
sine and cosine provide local coordinate systems in different patches.
Neither one by itself gives a global coordinate patch.) We also discuss
the setting of Diffusion Geometry on a finite set where there are again
"heat" eigenfunctions.
Onsdagen den 24 maj 13.15-14.15. Evgeny Kuznetsov (Luleå):
Smooth and Nonsmooth Calderón-Zygmund Type Decompositions for Morrey
Spaces.
Abstract:
We will show how to construct Calder\'on-Zygmund type decompositions for
the Morrey spaces Mor$_{\lambda }$ for $\lambda\in(0,1]$.
Moreover, for $\lambda >1-\frac{1}{n}$ it is possible to construct a
smooth analogue of the Calder\'on-Zygmund decomposition. The limitation
on $\lambda $ is connected to the property of Whitney cubes: the sum of
the volumes of Whitney cubes to the power $\lambda $ is equal to infinity
whenever $\lambda\in[ 0,1-1/n]$.
Onsdagen den 7 juni 13.15-14.15. Steven R. Bell (Purdue):
New ways to see that almost any domain is almost a quadrature
domain.
Abstract:
I will describe connections between the Bergman kernel and quadrature
domains that allow me to shape quadrature domains as if they were putty
in my hands. I will also define some new classes of quadrature domains
that arise in these considerations that will lead me to nominate some ideal
candidates to play the role of the unit disc, but in a multiply connected
setting.
Höstterminen
Onsdagen den 6 september 13.15-14.15. Håkan Hedenmalm och Alan Sola
(KTH):
Norm expansion along a zero variety in ${\mathbb C}^d$.
Abstract:
The reproducing kernel function of a weighted Bergman space over domains in
$\C^d$ is known explicitly in only a small number of instances.
Here, we introduce a process of orthogonal norm expansion along a subvariety
of codimension $1$, which also leads to a series expansion of the reproducing
kernel in terms of reproducing kernels defined on the subvariety. The
problem of finding the reproducing kernel is thus reduced to the same kind of
problem when one of the two entries is on the subvariety. A complete expansion
of the reproducing kernel may be achieved in this manner. We carry this out
in dimension $d=2$ for certain classes of weighted Bergman spaces over the
bidisk (with the diagonal $z_1=z_2$ as subvariety) and the ball
(with $z_2=0$ as subvariety), as well as for a weighted Bargmann-Fock space
over $\C^2$ (with the diagonal $z_1=z_2$ as subvariety). This reports on joint
work with S. Shimorin.
Onsdagen den 13 september 13.15-14.15. Hans Ringström (KTH):
On stability of cosmological models with accelerated expansion.
Abstract:
The Lorentz manifolds used by physicists to model
the universe nowadays are ones with accelerated expansion. A
question that then arises is if such models are stable. I will describe
a matter model, a so called non-linear scalar field, that causes
accelerated expansion, and state some stability results concerning
it. The essential problem that results is to prove future global
existence of solutions to a non-linear hyperbolic PDE, in other words
an analysis problem, and I will try to spend most of my time on the
analysis. However, in order to be able to formulate what is meant by
stability, it is necessary to describe the geometric background.
Onsdagen den 20 september 13.15-14.15. Anders Karlsson (KTH):
A generalization of a theorem of Varopoulos on the
existence of bounded harmonic functions.
Abstract:
We consider random walks on finitely generated groups
(or Brownian motion on manifolds which cover a compact
Riemannian manifold) and prove that if Liouville's
theorem holds, then all linear drift of the walk must
come from a real additive character. This extends
Varopoulos' theorem which in the case of symmetric walks
of finite support, states that positive linear drift
implies the existence of bounded harmonic function. The
proof is based on a noncommutative law of large numbers
and entropy considerations. Joint work with Ledrappier.
Onsdagen den 27 september 13.15-14.15. Nikolai Kuznetsov (St-Petersburg):
Two-dimensional steady waves on water of finite depth: modified
Bernoulli's equation and its applications.
Abstract:
The nonlinear two-dimensional problem of arbitrary bounded steady waves on
water of finite depth is considered, and a new formulation is proposed for
this classical problem. For this purpose averaging procedure is applied to
the velocity potential over vertical cross-sections of the water domain,
which leads to modified Bernoulli's equation. The latter involves the
difference between the potential and its average along with the free surface
elevation. Several applications of new equation are presented. First,
necessary conditions for the existence of non-trivial solutions to the
general steady-wave problem are obtained. (Earlier, these conditions,
that have the form of bounds on the Bernoulli constant and other wave
characteristics, were established only for the particular problem concerning
simplest periodic waves known as Stokes waves.) Second, it is shown that
there exists the exact upper bound such that if the free-surface profile is
less than this bound only at infinity (positive, negative, or both), then a
specific asymptotic behaviour of the profile is guaranteed at the same
infinity (or both infinities). Third, a new integral property of arbitrary
steady waves is obtained.
The talk is based on the results obtained in the framework of a joint
research project with Prof. Vladimir Kozlov (Linköping).
Onsdagen den 4 oktober 13.15-14.15. Liangyi Zhao (KTH):
On Primes in Quadratic Progressions.
Abstract:
It is due to Dirichlet that a linear polynomial with integer coefficients
represents infinitely many primes if and only if the coefficients are
co-prime. No similar statement is known for any polynomial of higher
degree. It has been conjectured by G. H. Hardy and J. E. Littlewood, with
asymptotic formula, that any quadratic polynomial that may conceivably
present infinitely many primes indeed does. In this talk, I will present
certain approximation to this very difficult problem and almost-all result
regarding primes presented by different quadratic polynomials. I will also
give sketches of the proofs and talk about the possibilities for
improvements. These results are joint works with Stephan Baier.
Onsdagen den 11 oktober 13.15-14.15. Jan-Olov Strömberg (KTH):
Affine structure on the double Hilbert transform and the inversion
of the Radon transform.
Abstract:
The standard method for inverting the Radon transform is
based on the Fourier transform representation of functions and make use of
the Hilbert transform. The set of lines are often represented in
some kind of polar coordinates.
When Novikovs' inversion formula of the attenuated Radon transform
appeared a few years ago, the proof also involved rather intricate
complex analysis on the unit disc.
By using affine coordinate representations of the sets of lines,
and double Hilbert transforms, we get a different inversion formula.
This inversion formula can also be extended to the attenuated case.
Only elementary tools are used in the proof. In fact, its main
ingredience is taught in mathematics on junior high school level.
Onsdagen den 18 oktober 13.15-14.15. Denis Gaydashev (KTH):
Universality and renormalization for Siegel disks.
Abstract:
We will describe our study, both numerical and rigorous, of one
of the central open questions in one-dimensional renormalization theory
-- the conjectural universality of golden-mean Siegel disks. We present an
approach to the problem based on the so called cylinder renormalization.
Numerical implementation of this approach relies on our constructive
proof of the classical Measurable Riemann Mapping Theorem. We will
describe our progress toward a computer assisted proof of the
Hyperbolicity Conjecture in this setting.
Onsdagen den 25 oktober 13.15-14.15. Victor Ivrii (Toronto):
Magnetic Schrödinger Operator: Geometry, Classical and Quantum Dynamics
and Spectral Aymptotics.
Abstract:
I consider even-dimensional Schroedinger operator with small Planck parameter
and a large coupling parameter, and discuss connections between the geometry
of magnetic field, classical and quantum dynamics of the corresponding
movements and the remainder estimate in the spectral asymptotics. I assume
that magnetic field is generic and consider both generic and general
potentials.
Onsdagen den 1 november 13.15-14.15. Eero Saksman (Jyväskylä):
On quantitative estimates for vector-valued singular integrals.
Abstract:
(joint with S. Geiss and S. Montgomery-Smith) Well-known
results of Burkholder, McConnelly, and Bourgain relate boundedness
of vector-valued singular integrals to boundedness of certain martingale
transforms. In the talk we investigate quantitative versions of these
results.
Onsdagen den 8 november 13.15-14.15. Yanyan Li (Rutgers):
Some Liouville theorem and gradient estimates.
Abstract:
The classical Liouville theorem says that a positive entire harmonic
function must be a constant. We give a fully nonlinear version of it.
This extension enables us to establish local gradient estimates of
solutions to general conformally invariant fully nonlinear elliptic
equations of second order. This talk will start from a proof of the
classical Liouville theorem using only the comparison principle and the
invariance of harmonicity under Mobius transformations and scalar
multiplications. We will then outline the proof of the comparison
principle used in establishing the new Liouville theorem.
Finally we outline the proof of the gradient estimates via the Liouville
theorem.
Onsdagen den 15 november 13.15-14.15. Denis Gaydashev (KTH):
Renormalization of Hamiltonian flows.
Abstract:
We will give an overview of recent results in renormalization of Hamiltonian
flows in $T^2$ cross $R^d$.
We will describe how renormalization provides an alternative to the KAM
theory, and, in particular, allows for a rigorous analysis of the problem
of much interset in real dynamics: the break up of the invariant tori for
Hamiltonian flows.
Onsdagen den 22 november 13.15-14.15. Maria Esteban (Paris):
Estimates for best constants in Hardy-like inequalities for multipolar
potentials.
Abstract:
In this talk I will present recent results obtained in collaboration with
Roberta Bosi and Jean Dolbeault about how to estimate the best constants
in Hardy-like inequalities corresponding to potentials with more than one
singularity and this both for the Schödinger and the Dirac operators.
In both cases these estimates yield in fact lower bounds for the lowest
eigenvalue of the operators $H-W$ where $W$ is the sum of terms of the
type $ \mu_i / |x-x_i|^2$ if $H=-\Delta$ and $W$ is the sum of Coulomb
potentials $\nu_i / |x- x_i|$ when $H$ denotes the Dirac operator. This
type of results is relevant in the analysis of relativistic and non
relativistic quantum mechanical systems (molecules and crystals).
Onsdagen den 29 november 13.15-14.15. Håkan Hedenmalm (KTH):
Boundary properties of Green functions in the plane.
Abstract:
We study the boundary properties of the Green function of bounded simply
connected domains in the plane. Essentially, this amounts to studying
the conformal mapping taking the unit disk onto the domain in question.
Our technique is inspired by a 1995 paper of Jones and Makarov.
The main tools are an integral identity as well as a uniform Sobolev
imbedding theorem. The latter is in a sense dual to the exponential
integrability of Marcinkiewicz-Zygmund integrals.
We also develop a Grunsky identity, which contains the information
of the classical Grunsky inequality. This Grunsky identity is the case $p=2$
of a of a more general Grunsky identity for $L^p$ spaces.
This reports on joint work with Anton Baranov.
Onsdagen den 6 december 13.15-14.15. Pär Kurlberg (KTH):
Lower bounds on the order of some pseuderandom number generators.
Abstract:
Given coprime integers b and n, let ord(b,n) be the
multiplicative order of b modulo n. The length of the periods of some
popular pseuderandom number generators (the power generator, the
linear congruential generato, and the Blum-Blum-Shub generator) turns
out to be related to ord(b,n) for apropriately chosen b and n. (Note
that the case n=p, where p is prime is related to Artin's primitive
root conjecture.) We will give lower bounds on ord(b,n) for b fixed
and n ranging over certain subsets of the integers, e.g., the set of
primes, the set of "RSA moduli" (i.e., products of two primes), the
full set of integers, and the images of these sets under the
"Carmichael lambda function". Assuming the generalized Riemann
hypothesis, we can show that the order is essentially maximal for
almost all n in the above mentioned subsets. We can also give weaker
unconditional bounds. The lower bounds in the case of RSA moduli
shows that certain "cycling attacks" on the RSA crypto system are
ineffective.
Onsdagen den 13 december 13.15-14.15. Alexander Borichev (Marseille):
A uniqueness theorem for the Korenblum space.
Abstract:
Onsdagen den 20 december 13.15-14.15. Anders Karlsson (KTH):
The heat equation on regular graphs and applications.
Abstract:
I will discuss explicit formulas for heat kernels on regular
graphs and explain how this leads into Bessel functions.
Applications consist of, on the one hand, old and new formulas
for sums and integrals of Bessel functions and, on the other hand,
expressions for the number of cycles, paths, and spanning
trees in regular graphs.