Aperiodic order: spectral theory, combinatorics, and dynamics

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The central object of study are Schrodinger operators with potentialsdisplaying aperiodic order. In one dimension there have been recentadvances in the understanding of their spectral and quantum dynamicalproperties, particularly in the case of the Fibonacci potential andrelated models, so-called Sturmian potentials, which are the standardmodels of one-dimensional quasicrystals. It is the goal of the proposedresearch to extend the theory to larger classes of potentials in onedimension and to tackle the higher dimensional case. A crucial tool in onedimension is the trace map, an energy-indexed dynamical system which canbe used to characterize and study the spectrum of the operators. Alongwith combinatorial partition results and Gordon-type criteria one canobtain good bounds on generalized eigenfunctions from which one can deducespectral and quantum dynamical consequences. It appears feasible that thisapproach is applicable to potentials beyond the class of Sturmianpotentials -- sufficiently low complexity should suffice to inducepartitions and trace maps. In higher dimensions the main goal is to findan analog of Gordon's criterion which can serve as a link betweencombinatorics and spectral theory.The mathematics of aperiodic order is a young emerging field that hassparked a lot of research activity since the mid-nineties. Researchersfrom disciplines as diverse as spectral theory, group theory, dynamicalsystems, combinatorics, and algebraic topology have found a commonground that was motivated by the discovery of quasicrystals in 1984and the subsequent reconsideration of the nature of order and orderedstructures. By now, quite a number of structural models for quasicrystalshave been proposed. Joint efforts are being undertaken to investigatetheir properties and shed light on why quasicrystals exist, how they form,why they are stable. Regarding their electronic transport properties, itis expected that quasicrystals may exhibit anomalous behavior. It istherefore planned to study transport properties of Sturmian and relatedmodels from this perspective.