Aspects of Spectral Theory

15-19 Jan 2006
Erwin Schrödinger Institute
Boltzmanngasse 9, A-1090 Wien


Titles and (some) Abstracts:

Wolf-Jürgen Beyn: Homoclinic tangencies, bifurcations, and symbolic dynamics

It is well known that transversal homoclinic points in finite dimensional dynamical systems (with continuous as well as discrete time) lead to Cantor like invariant subsets on which chaotic dynamics prevails. Famous theorems by Smale, Shilnikov and Conley characterize the dynamics in terms of a specific subshift of finite type from symbolic dynamics. In this talk we discuss parametrized dynamical systems that exhibit a so called homoclinic tangency, i.e. a first order contact of stable and unstable manifolds of an equilibrium. Such systems mark the onset of two branches of transversal homoclinic points and hence two sets with symbolic dynamics. Despite a series of geometrical results by Takens and coworkers the problem of a complete characterization of this onset in terms of symbolic dynamics remains open. We report about a partial result that allows to reduce the dynamics mear the homoclinic tangency to a set of bifurcation equations that are indexed symbolically. For certain examples this local theory allows to interpret the results of numerical calculations obtained by global continuation.

Friedrich Götze: Quadratische Formen und Spektraltheorie

Gerhard Knieper: Geschlossene Geodätische auf Riemannschen Mannigfaltigkeiten nicht positiver Krümmung

Thomas Kriecherbauer: Über Kombinatorik und Zufallsmatrizen

Bernd Metzger: Tauber-Theorie für das parabolische Andersonmodell

Peter Müller: On Mott's formula for the ac-conductivity in the Anderson model

We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schroedinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the ac-conductivity is bounded by $ C \nu2 (\log \frac 1 \nu)^{d+2}$ at small frequencies $\nu$. This is to be compared to Mott's formula, which predicts the leading term to be $ C \nu2 (\log \frac 1 \nu)^{d+1}$. This is joint work with Abel Klein and Olivier Leneoble.

Christoph Richard: Entropie der Rautenparkettierung mit periodischen Randbedingungen

Klaus Schmidt: Mahler measure and entropies of certain ergodic Z^d-actions

Michael Stolz: Random matrices, symmetric spaces, and mesoscopic physics

Ivan Veselić: Spectral properties of Anderson-percolation Hamiltonians on graphs

Peter Zeiner: Mehrfache Koinzidenzen in 2 und 3 Dimensionen




last modified on 11 Jan 2006