Projects

Topological invariants and their index theory, the bulk-boundary correspondence and the more recently introduced spectral localizer are well-established mathematical concepts for disordered topological insulators and are also influential for numerical studies of such materials. This proposal is about extending prior results and techniques to systems with crystalline defects, disordered semimetals and topological metals, as well as non-hermitian topological systems stemming from (leaky and driven) photonics and metamaterials. Another part of the proposal aims at a deeper understanding of scattering on such topological systems.

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The first goal of index theory is to relate topological invariants to indices of Fredholm operators. The most famous result in this direction is the Atiyah-Singer index theorem, but there exist far reaching non-commutative generalizations. While there is a general theory, such index theorems have to be established case by case in applications. The second goal of index theory is to connect invariants and indices of problems related via exact sequences. For example, this allows to read off the topology…

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The theoretical limits of distributed compressive sensing are studied bytools from both information theory and statistical physics. The investigationscover both noise-free and noisy distributed compressive sensing. The theoretical insightsare utilized to design approximate message passing algorithms for joint recovery of large distributed compressive sensing networks with feasible computational complexity. These algo-rithms enable us to verify the non-rigorous results obtained by the replica method…

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The project will investigate the performance limits of compressive sensing and propose practical algorithms that approach these performance limits closely.The project will focus on the regime of high compression ratios where L1-norm regularization is suboptimal.Compressive sensing will be investigated from the viewpoint of statistical physics and considered as a particular instance of a spin glass system.Both average distortion and minimax distortion will be addressed as objective functions (Hamiltonians).The…

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Das Forschungsvorhaben kann in zwei Problembereiche unterteilt werden:I Untersuchung der Topologie, der Dynamik und der Spektren von ungeordneten zeitumkehrinvarianten Systemen mit ungeradem Spin II Topologische und spektrale Aspekte der Streutheorie in Medien mit einem periodischen Hintergrundpotential oder an HyperflächenDas Hauptziel im Problembereich I ist eine detaillierte mathematische Analysis von Spin- Randströmen, die zum Beispiel in Graphenschichten auftreten und in Zukunft so genannte S…

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Im Rahmen diese Projektes wurden verschiedene Fragestellungen aus dem Bereich der Fest-
körperphysik ungeordneter Systeme rigoros analysiert. Die meisten Fortschritte wurden bei
der kontrollierten Störungstheorie für quasi-eindimensionale zufällige Medien gemacht, die es
nun erlauben, auch so genannte Anomalien zu untersuchen. Dies erlaubt insbesondere auch
Verbindungen zur Theorie der vollen Zufallsmatrizen herzustellen. Es wurde Delokalisierung
für bestimmte quasi-eindimensionale unge…

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