Analysseminariet 2003
Vårterminen
Lokal: 3721
Onsdagen 5 februari 13.15-15.00. Pär Kurlberg (Chalmers):
Eigenfunctions of quantized cat maps.
Onsdagen den 12 februari 13.15--14.15. Per Sjölin (KTH):
A theorem of Antonov on convergence of Fourier series.
Onsdagen den 19 februari 13.15. Yolanda Perdomo (Lund):
Mean value surfaces with prescribed curvature form.
Onsdagen den 26 februari 13.15. Haakan Hedenmalm (KTH):
Composition operators on a Hilbert space of Dirichlet series.
Onsdagen den 5 mars 13.15. Jan-Erik Björk (SU): Rigid bodies
and the Kovalevsky gyroscope.
Onsdagen den 12 mars 13.15. Utgår p g a internat på
Wijks kursgård.
Onsdagen den 19 mars 13.15. Alexandru Aleman (Lund): Uniform
spectral radius and compact Gelfand transform.
Onsdagen den 26 mars 13.15. Hans Rullgård (Stockholm): An
explicit inversion formula for the exponential Radon transform using data
from 180 degrees.
Onsdagen den 2 april 13.15. Peter Sjögren (Göteborg):
Maximal operators and spectral multipliers for the Ornstein-Uhlenbeck
semigroup.
Onsdagen den 9 april 13.15. Eero Saksman (Helsingfors, Jyvaeskylae):
The boundary correspondence of the Nevanlinna counting function on the
unit disk.
Onsdagen den 16 april 13.15. Victor Shulman (Ryssland):
Radical Banach algebras and radicals in Banach algebras.
Onsdagen den 23 april 13.15. Seminariet tar påskledigt.
Onsdagen den 30 april 13.15. Seminariet tar valborgsledigt.
Onsdagen den 7 maj 13.15. Kurt Johansson (KTH): The Aztec
diamond and the Airy process.
Onsdagen den 14 maj 13.15. Nail Ibragimov (Karlskrona):
Lie group analysis of nonlinear problems.
Torsdagen den 22 maj 10.15. Leopold Flatto: Lap counting function
of linear mod one and tent maps.
Höstterminen
Onsdagen 3 september 13.15-14.15. Håkan Hedenmalm (KTH):
Weighted Bergman spaces and integral means spectra of conformal maps.
Abstract: The classical theory of conformal maps of simply connected domains
goes back to the work of Koebe and Bieberbach. They found sharp estimates for
the pointwise behavior of the derivative of a conformal mapping in the
class S or Sigma. However, their work does not answer the question of the
behavior of the derivative in the mean, which we take to mean the possible
growth of the integral of |f'|^t along the circles |z|=r. This growth
determines a function, b_f(t), the integral means spectrum of f, and the
supremum of this function over all f is denoted by B(t), the universal
integral means spectrum. Some estimates of B(t) were found by Makarov,
Carleson, Pommerenke, and Bertilsson. Here, we present a novel method which
leads to much better estimates of B(t) than what has been possible with any
of the previously applied methods. The improvement is that we are able to
use essentially the full power of the classical area type methods.
This work is the result of collaboration with S. Shimorin.
Onsdagen 10 september 13.15-14.15. Timo Weidl (Stuttgart):
On the Discrete Spectrum of a Pseudo-Relativistic Two-Body Pair Operator.
Abstract: The Herbst operator $\sqrt{-\Delta+1}-V$ is often used in
pseudo-relativistic quantum mechanics. We show, that one should be careful
when applying this model. In particular, the number of bound states
of the two-body pair Herbst operator depends on the relative velocity of the
system with respect to the observer.
We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete
spectrum and calculate spectral asymptotics for
the eigenvalue moments and the local spectral density in the
pseudo-relativistic limit.
Onsdagen 17 september 13.15-14.15. Mikhail Sodin (Tel Aviv):
Nodal domains of harmonic functions and Laplace-Beltrami
eigenfunctions.
We show that harmonic functions in the unit disc with a bounded number of
nodal domains (i.e., domains where the function does not change its sign)
obey a version of `Harnack's inequality'.
We apply this to asymptotic geometry of eigenfunctions for the
Laplace-Beltrami operator on a compact (two-dimensional) surface. In
particular, we establish a connection between the length of the nodal line
and the mean doubling index of an eigenfunction. This is a report on work
in progress with Leonid Polterovich.
Onsdagen 24 september 13.15-14.15. Ari Laptev (KTH): Absolutely
continuous spectrum of Schrödinger operators.
We consider a class of potentials for which the absolutely continuous
spectrum of the corresponding multidimensional Schrödinger operator
is essentially supported by [0,infty). Our main theorem states that this
property is preserved for slowly decaying potentials provided that there
are some oscillations with respect to one of the variables.
Onsdagen 1 oktober 13.15-14.15. Vladimir Kostov (Nice): The
Deligne-Simpson problem.
We consider the problem for which $(p+1)$-tuples of conjugacy classes
$c_j\subset gl(n,C)$ (resp. $C_j\subset GL(n,C)$) do there exist irreducible
(i.e. without proper invariant subspace) tuples of matrices $A_j\in c_j$
such that $A_1+\ldots +A_{p+1}=0$ (resp. $M_j\in C_j$,
$M_1\ldots M_{p+1}=I$). The matrices $A_j$ (resp. $M_j$) are interpreted
as matrices-residua of Fuchsian systems
$dX/dt=(\sum_{j=1}^{p+1}A_j/(t-a_j))X$ on Riemann's sphere (resp. as the
matrices of their monodromy operators).
Onsdagen 8 oktober 13.15-14.15. Peter Wilcox Jones (Yale):
Martingales with bounded square functions.
Onsdagen 15 oktober 13.15-14.15. Andreas Strömbergsson (Uppsala):
Rate of convergence for horocycle flows.
I will talk about the horocycle flow on the unit tangent bundle of a
non-compact hyperbolic surface of finite area. It was proved by Dani and
Smillie (1984) that every non-closed orbit of this flow goes
asymptotically equidistributed with respect to the Liouville volume
measure. This result was later vastly generalized by Ratner to unipotent
flows on arbitrary Lie groups, 1991.
I will discuss how to obtain an effective rate of convergence result in
the case of the horocycle flow. The bounds I will give depend on the small
eigenvalues of the Laplacean and on the rate of excursion into cusps for
the geodesic corresponding to the given initial point. I will show that in
a certain sense the bounds obtained are the best possible, for any given
initial point.
Onsdagen 22 oktober 13.15-14.15. Kyril Tintarev (Uppsala):
A singular elliptic problem on a half-space.
We consider isoperimetric elliptic problems on unbounded domains. Many of
such problems lack a minimizer, for example the Hardy and the limit-exponent
Sobolev inequality on the half space. We show that minimizers exist for an
interpolation between these inequalities. A similar result of E.Lieb for
the case of ${\bf R}^n\setminus\{0\}$ uses a reduction to radially symmetric
functions. In case of half-space, we turn to the enhanced version of
Banach-Alaoglou theorem in presence of invaraint transformations, the weak
convergence decomposition (WCD) lemma, employing natural automorphisms of
the half-space.
Onsdagen 29 oktober 13.15-14.15. Anders Karlsson (KTH):
Harmonic functions on homogeneous graphs: uniform radial variation and
the Dirichlet problem.
Onsdagen 5 november 13.15-14.15. Öyvind Björkås
(Bodö, Norge):
Fast wavelet methods for boundary integral equations.
I will talk about a wavelet method for computing singular integrals
along curves in the plane, which can be used for solving the boundary
integral equation that arises from the Dirichlet problem for the
Laplacian. The method includes ideas from fast multipole methods and
non-standard representations of integral operators.
The wavelet bases used in the method are combinations of multiwavelets
and Daubechies' compactly supported, differentiable wavelets. I will
outline the structure of the bases, and discuss how to use them to
factor and compress the operators to obtain a fast method. A numerical
example will be given, with some comments on the implementation.
Onsdagen 12 november 13.15-14.15. Vladimir Kapustin
(St-Petersburg): Boundary values of Cauchy type integrals.
The results of A. G. Poltoratski and A. B. Aleksandrov about
non-tangential boundary values of pseudocontinuable functions from $H^2$
on sets of zero Lebesgue measure are applied to operators in
$L^2$-spaces on the unit circle. We prove that for arbitrary measures
$\mu, \nu$ on the unit circle, any operator $L2(\mu)\to L2(\nu)$ whose
commutator with multiplication by $z$ is a rank one operator is a sum of
an operator of multiplication and a Cauchy type transform in the sense
of angular boundary values.
Onsdagen 19 november 13.15-14.15. Anders Öberg (Uppsala):
Square summability of g-functions and uniqueness of g-measures.
We prove uniqueness of $g$-measures for $g$-functions satisfying quadratic
summability of variations. Our result is in contrast to the situation of,
e. g., the one-dimensional Ising model with long-range interactions,
since $\ell_1$-summability of variations is required for general potentials.
We illustrate this difference with some examples. To prove our main result
we use a product martingale argument.
Onsdagen 26 november 13.15-14.15. Kari Astala (Helsingfors):
Calderon's problem in inverse tomography and complex analysis.
Consider the equation Div ( a(x) Grad u )= 0, in a domain $V \subset R^n$
which is bounded and has connected complement. We assume that
$a(x) \in L^\infty$ is real valued and is bounded away from 0.
In 1980 A.P. Calderon posed the question if the boundary measurements
or the operator $\Lambda_a$, mapping the Dirichlet boundary values of
$u$ to the Neumann boundary values $\partial_n u$, determines the
coefficient $a(x)$ inside the domain $V$. This inverse problem is also
known as the Electrical Impedance Tomography.
In this talk we give a positive answer to Calderon's question in dimension
two; the work is joint with Lassi Päivärinta (Helsinki). The proof is
based on arguments using complex analysis, PDE's and quasiconformal methods.
Onsdagen 3 december 13.15-14.15. Gaven Martin (New Zealand):
The identification of the two generator arithmetic lattices of
hyperbolic 3-space.
Onsdagen 10 december 13.15-14.15. Tadeusz Iwaniec (Syracuse):
WEAKLY DIFFERENTIABLE MAPPINGS BETWEEN MANIFOLDS.
Reporting joint work with P. Hajlasz, J. Maly and J. Onninen
We shall discuss Sobolev classes of weakly differentiable
mappings between compact Riemannian manifolds without boundary. The point
to make here is that we shall not impose any topological conditions on the
manifolds. The central themes are:
1) Approximation by smooth mappings
2) Integrability of the Jacobian determinant and the degree formula.
In 1) we characterize (essentially all) Sobolev type classes in which
smooth mappings are dense. The novelty of our approach is that we are able
to detect tiny sets (called web like structure) on which the mappings in
question are continuous.
In 2) The integrability theory of the Jacobian determinants in a manifold
setting is really different than one might a priori expect based on the
Euclidean case. To our surprise, the case when the target manifold admits
only trivial cohomology (like spheres) is more complex than the case
with nontrivial cohomology. The necessity of the cohomological constraints
on the target manifold is a new phenomenon in the regularity theory of the
Jacobian determinats and other null Lagrangians.
The presentation will be accessible for a general audience; pictures will
serve as "proofs".
Onsdagen 17 december 13.15-14.15. Vadim Kaloshin (CalTech):
How often does a surface diffeomorphism have infinitely many sinks?
Abstract: Consider the space of $C^r$ diffeomorphisms (smooth invertible
selfmaps) of a compact surface $M$ (e.g. $S2$ or $T2$) Diff$^r(M)$ with
$r\geq 2$. A sink of $f:M \to M$ is a periodic point $x \in M$ which
attract all points from its neighbourhood (as in your kitchen). Points
attracted to $x$ called basin of attraction of $x$. In 60-th Thom
conjectured that a generic diffeomorphism can not have infinitely many
coexisting sinks. Indeed, each sink has an open basin of attraction and
infinitely many of those seems too much. In 70-th Newhouse constructed an
open set of diffeomophisms $N \subset \textup{Diff}^r(M)$ such that
generic diffeomorphism in $N$ does have infinitely many coexisting sinks.
It is an amazing phenomenon, called Newhouse phenomenon. It disproves
Thom's conjecture and significant obstacle to describe ergodic properties
of surface diffeomorphisms. We shall discuss this phenomenon and closely
related results of Benedicks-Carleson, Mora-Viana, Wang-Young,
Morreira-Yoccoz. The main result indicates in some sense this phenomenon
has "probability zero". This is a particular case of so-called Palis
conjecture. This is a joint work with Anton Gorodetsky.