Wolfgang Arendt (Ulm): Spectrum, diffusion and inverse problems
Abstract:
We start by Weyl's formula and give some information on the inverse
spectral theory, 50 years after Kac's famous question.
The problem will be analyzed in the spirit of intertwining semigroups.
This leads to a positive inverse result which shows that diffusion determines the domain.
But it also sheds new light on the counter examples.
Dirichlet, Neumann and Robin boundary conditions are discussed.
Also the spectrum of the Dirichlet-to-Neumann operator is investigated for which
Weyl's formula is valid: its spectrum determines the surface.
References:
W. Arendt: Does diffusion determine the body? Crelle's J. 2002
W. Arendt, R. Nittka, W. Peter, F. Steiner: Weyl's law. In: Mathematical
Analysis of Evolution, Information and Complexity. W. Arendt, W. Schleich,
eds. Wiley, Weinheim 2009, p.1-71.
W. Arendt, M. Biegert, T. ter Elst: Diffusion determines the manifold. Crelle's J. 2012
W. Arendt, T. ter Elst, J. Kennedy: Analytical aspects of isospectral drums. Oper. Matrices 8 (2014) 255-277
W. Arendt, T. ter Elst: Eigenvalues and ultracontractivity: Weyl's law for
the Dirichlet-to-Neumann operator. Integral Equations and Operator Theory, to
appear.
Michael Baake (Bielefeld): Dynamical systems of number-theoretic origin in the theory of
aperiodic order
Abstract:
Reguler model sets (a special class of cut and project sets), which go
back to Yves Meyer (1972) in mathematics and to
Peter Kramer (1982) in physics, form a versatile class of structures with
amazing harmonic properties. These sets are also known as mathematical
quasicrystals, and include the famous Penrose tiling with fivefold symmetry as
well as its various generalisations to other non-crystallographic symmetries.
They are widely used to model the structures discovered in 1982
by Dan Shechtman (2011 Nobel Laureate in Chemistry).
More recently, also systems such as the square-free integers or the visible
lattice points have been studied in this context, leading to the theory of
weak model sets. This is an extension of the class of regular model sets
that was also briefly considered by Meyer and by Schreiber in the 1970s,
but has not seen any systematic investigation. Due to the connection
with B-free integers and lattice systems, which are of renewed interest
in the light of Sarnak's research program around M"obius orthogonality,
weak model sets are now being studied in more detail by several groups.
This talk will review some of the developments, and introduce important
concepts from the field of aperiodic order, with focus on spectral aspects.
Hansjörg Geiges (Köln): Transversely holomorphic flows and contact circles
Abstract:
In this talk I shall report on joint work with
Jesùs Gonzalo concerning contact circles on 3-manifolds.
Such circular (or even spherical) families of contact forms
arise naturally in various contexts, e.g. in the
construction of hyperkähler metrics via
the Gibbons-Hawking ansatz.
The emphasis in the talk will be on dynamical aspects
of contact circles. I shall place the moduli theory
of contact circles in the context of transversely
holomorphic flows. This leads to a dynamical
classification of contact circles. Along the way,
we come across a generalised Gauss-Bonnet theorem.
Werner Kirsch (Hagen): Twisted waves, Lifshitz tails, and squared potentials
Abstract:
We consider a waveguide which is randomly twisted. We investigate
the density of states for energies near the bottom of the spectrum.
This is joint work with David Krejcirik and Georgi Raikov.
Andreas Knauf (Erlangen): Molecular resonances and regularization
Abstract:
In physical applications of the semiclassical theory of resonances,
one needs techniques to cope with the Coulomb singularity.
These are based on classical regularization.
We give an overview over results obtained up to now and open questions.
Gerhard Knieper (Bochum): Geodesic flows on closed surfaces with zero topological entropy
Abstract:
Topological entropy is a measure for the complexity of dynamical systems on compact spaces.
In particular, in low dimensions ( 2d for diffeomorphisms and 3d for flows) positive topological entropy is equivalent to the existence
of a horseshoe. This implies non-zero Lyapunov exponents on an invariant cantor set and exponential growth rate of periodic orbits.
In this talk we will consider geodesic flows on closed surfaces with zero topological entropy and show that in some cases the dynamics has features
similar to integrable systems.
Matthieu Léautaud (Paris-7): Quantitative unique continuation and intensity of waves in the
shadow of an obstacle
Abstract:
The question of global unique continuation is the following:
Does the observation of the wave intensity on a little subdomain during
a time interval (0,T) determine the total energy of the wave? In an
analytic context, this question was solved in 1949 by the well-know
Holmgren-John theorem; in the "smooth case", it was finally tackled by
Tataru-Robbiano-Zuily-Hörmander in the nineties. After a review of these
results, we shall describe the quantitative unique continuation estimate
associated to the qualitative theorem of
Tataru-Robbiano-Zuily-Hörmander, that is, give the optimal logarithmic
stability result. In turn, this estimate yields the optimal a priori
bound on the penetration of waves into the shadow region, as well as the
cost of approximate controls for the wave equation. This is joint work
with Camille Laurent.
Stéphane Nonnenmacher (Orsay): Spectral correlations for randomly perturbed nonselfadjoint operators
Abstract:
This is a joint work with Martin Vogel (Orsay).
We are interested in the spectrum of semiclassical nonselfadjoint operators.
Due to a strong pseudospectral effect, a tiny
perturbation can dramatically modify the spectrum of such an operator.
Hager & Sjöstrand have thus considered adding small random pertubations, and proved
that the eigenvalues of the perturbed operator typically spread over the classical
spectrum, satisfying a probabilistic Weyl's law in the semiclassical limit.
Beyond this Weyl's law, we investigate the correlations between the eigenvalues, at
microscopic distances. In the case of 1-dimensional operators, these
correlations depend on the structure of the energy
shell of the unperturbed operator (a finite set of points), and of the
type of perburbation (random matrix vs. random potential), but otherwise
enjoy a form of universality, where the central object is the Gaussian Analytic
Function (GAF), a family of random entire functions.
The GAF was originally introduced in the context of Quantum Chaos in the 1990s, in
order to describe the statistical properties of 1D chaotic eigenfunctions.
In the present model the GAF (and its variants) rather arise through the spectral
determinant of our randomly perturbed operator.
Norbert Peyerimhoff (Durham): Finitely supported eigenfunctions and jumps of the IDS in the
Kagome lattice
Abstract:
The Kagome lattice is a semiregular tiling of the plane by
triangles and hexagons. In contrast to the three regular
planar tilings by triangles, squares and hexagons, the corresponding
graph Laplacian does admit finitely supported
eigenfunctions. As a consequence, the associated Integrated Density of
States (IDS) has jumps. I think that the Kagome lattice
is an ideal example to illustrate general spectral phenomena. With
special focus on this example, I will try to introduce
all the relevant concepts in this talk. The presentation is based on a
joint paper with Daniel Lenz, Olaf Post, and Ivan Veselic.
Furthermore, the Kagome lattice is one example of the 11 Archimedean
tilings and, in a joint ongoing project with Matthias Taeufer,
we study the IDS of these Archimedean tilings with the help of an
explicit integral formula for the IDS.
Alfonso Sorrentino (Roma, Tor Vergata): Integrability and spectral properties of Birkhoff billiards
Abstract:
A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where “the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered”.
Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation.
Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures.
In this talk I shall focus on several of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture) and to the possibility of inferring dynamical information on the billiard map, from its length spectrum (i.e., the collection of lengths of its periodic orbits).
This talk is based on joint works with G. Huang and V. Kaloshin.
Steven Zelditch (Northwestern): Ergodic geodesic flow and nodal sets of eigenfunctions
Abstract:
An open problem is whether every compact Riemannian
manifold possesses a sequence of eigenfunctions for which the number
of nodal domains tends to infinity. In fact, this was only known for the standard
sphere, torus and a few other simple examples until recently. The main result
in my talk is that the number of nodal domains tends to infinity along almost the
entire sequence of eigenfunctions on a non-positively curved surface with
concave boundary (joint work with Junehyuk Jung). For closely related negatively
curved `real Riemann surfaces' I can show that the number of nodal domains
grows like the logarithm of the eigenvalue. That is based on quantum ergodic
restriction theorems and small scale QE results of Hezari-Riviere and X. Han.
TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund
You find us on the 6th floor of the Math tower.
Janine Textor (room M 620)
Tel.: (0231) 755-3063
Fax: (0231) 755-5219
Mail: janine.textor@tu-dortmund.de