Research at the chair of Analysis

Topics

Random differential equations: Properties of solutions and spectral features of differential equations with random coefficients, in particular of elliptic and parabolic type. Finite difference analogs of such partial differential equations.
Harmonic analysis: Applications of harmonic analysis for partial differential equations, unique continuation estimates, uncertainty relations, Carleman estimates, spectral inequalities.
Control theory of the heat equation and related equations on unbounded and large bounded domains. Observability criteria.
Spectral Theory of selfadjoint operators, second order elliptic differential operators on Euclidean spaces and manifolds, Laplace(-Beltrami) operators, Schrödinger operators, discrete Schrödinger operators and more general finite difference operators on lattices and general groups and graphs.
Perturbation Theory of eigenvalues and spectral bands: Estimates on movement of eigenvalues, lifting of eigenvalues under semi-positive perturbations. Analogous estimates for edges of the essential spectrum.
Schrödinger operators: Selfadjointness, spectral theory, behaviour of eigensolutions, low-lying eigenvalues, Allegretto-Piepenbrink theory for general operators.
Random operators modelling disordered systems: Random Schrödinger operators, Anderson models, random Hamiltonians with correlations, quantum percolation models, general percolation Hamiltonians.
Mathematical theory of Anderson localisation: Proof of localisations via multiscale analysis and the fractional moment method, Wegner estimates, Lifschitz tails and initial scale estimates, decoupling properties of random operators, non-monotone parameter dependence of operators.
Ergodic Theory, in particular as a tool to study the integrated density of states. Banach-space valued ergodic theorems, uniform approximation via ergodic theory.
Percolation theory: Percolation on Cayley graphs, sharpness of the phase transition, distribution of the cluster size, Laplacians on percolation clusters.
Asymptotic aspects of geometric group theory: Random walks on Cayley graphs, quasi-isometries, Laplacians on graphs, their spectral distribution function, eigenfunctions.

Projects

Ongoing and completed projects

Random Schrödinger operators with breather potentials as a paradigmatic model for non-linear influence of randomness (since 2018)
GRK 2131: High-dimensional Phenomena in Probability - Fluctuations and Discontinuity (joined project of UA Ruhr since 2015; participation since 2017)
PhD project: Uniform convergence of random functions on amenable groups
Multiscale version of the Logvinenko-Sereda Theorem (since 2015)

Unique continuation principles and equidistribution properties of eigenfunctions (2013-2016)
Zufällige und periodische Quantengraphen Joint with Daniel Lenz and Peter Stollmann. (2009-2016)
Heisenberg fellowship: Schrödinger operators (2009-2010)
Spektrale Eigenschaften von periodischen und zufaelligen Operatoren auf kombinatorischen Graphen (2008-2012)
Extension VE 253 2-2 of the of the Emmy Noether Project.
Estimates on spectral gaps for quantum waveguide Schrödinger operators (2006-2010)
Emmy Noether Junior research group: Spectral properties of random Schroedinger operators and random operators on manifolds and graphs (2004-2010)
Research fellowship: Analysis of spectral properties of solid-state Schrödinger operators (2002-2004)

Publications

See here for a list of publications by members of the chair.

Talks and posters

See here for a list of talks and posters by members of the chair.

Kontakt

Adresse

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund

Sie finden uns auf dem sechsten Stock des Mathetowers.

Sekretariat

Janine Textor (Raum M 620)

Tel.: (0231) 755-3063
Fax: (0231) 755-5219
Mail: janine.textor@tu-dortmund.de
Bürozeiten:
Di. und Do. von 8 bis 12 Uhr
Home Office:
Mo. und Fr. von 8 bis 12 Uhr