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Research in the group Mathematical Physics

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  • Theoretische Grenzen und algorithmische Verfahren verteilter komprimierender Abtastung
    (Third Party Funds Single)
    Term: 01-01-2019 - 31-12-2021
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    The theoretical limits of distributed compressive sensing are studied by
    tools from both information theory and statistical physics. The investigations
    cover both noise-free and noisy distributed compressive sensing. The theoretical insights
    are 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 from statistical mechanics, and also, to propose theoretically optimal approaches for sampling and low complexity. The proposed research will lead to improved performance of reconstruction algorithms for distributed compressive sensing, e.g. higher compression rates and/or higher fidelity of reconstruction.
  • Analysis and Implementation of Highly Compressive Sensing
    (Third Party Funds Single)
    Term: 01-03-2016 - 28-02-2019
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    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 analysis will rely on the replica method. Particular emphasis will be put on the implications of replica symmetry breaking.The main objectives of the project are:1) to find a system of saddle point equations that describe the replica symmetry breaking solution to the compressive sensing problem,2) to prove rigorous one-sided bounds for these solutions by adapting Guerra's arguments for the Sherrington-Kirkpatrick spin glass model to spin glass model of compressive sensing and, hereby, justifying the correctness of the replica method in compressive sensing,3) to numerically solve the system of saddle point equations, and 4) to determine which practical algorithms work well for compressive sensing at high compression ratios.
  • Applied index theory for quantum and classical systems
    (Third Party Funds Single)
    Term: 01-01-2016 - 31-12-2018
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    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 of boundary states or point defects from bulk invariants. The proposal aims to implement this program in situations which have not been tackled before like interacting spin systems, photonic crystals and lattices of classical springs, and also to further develop the index approach to scattering systems and topological materials.
  • EXIST-Gründerstipendium: FoodOptimizer ist ein wissenschaftliches Ernährungs- und Symptomtagebuch
    (Third Party Funds Single)
    Term: 01-10-2015 - 30-09-2016
    Funding source: Bundesministerium für Wirtschaft und Technologie (BMWi)
    FoodOptimizer ist ein wissenschaftliches Ernährungs- und Symptomtagebuch, das eine systematische Analyse und Diagnoseunterstützung zur Aufdeckung von  Nahrungsmittelallergien und -unverträglichkeiten anbietet.
  • Dynamik, Spektralanalyse und Streutheorie für zeitumkehrinvariante Systeme
    (Third Party Funds Single)
    Term: 01-06-2010 - 30-06-2013
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    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 Spintronics ermöglichen könnten. Es handelt sich dabei um delokalisierte Randzustände, die durch eine topologische Invariante gegen Anderson-Lokalisierung geschützt sind. Das Ziel des Problembereiches II ist es, das gesamte Arsenal der Streutheorie für periodische Hintergrundoperatoren zu entwickeln. Hauptanwendung wäre dann ein Beweis von neuartigen Levinson-Theoremen, die einen Zusammenhang zwischen der Anzahl der gebundenen Zustände und der totalen Streuphase herstellen.