"Research - Professur Analysis, Lehrstuhl IX, Fakultät für Mathematik, TU Dortmund"));
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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
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.