# Talks and abstracts:

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.*