PRR Project

Summary

During the last decade hypercomplex analysis got a large boost mainly due to two developments. On the operator-theoretical side the development of the so-called s-spectrum, a non-commutative version of the spectrum which takes advantage of the underlying geometry, with their associated functional calculi allowed for the development of spectral theory for operators over non-commutative structures like Clifford algebras. On the more applied side the development of frame-based algorithms based on sparsity principles gave a large boost to reconstruction methods in diffraction tomography where one needs to deal with large amounts of data as well as in 3D electric impedance tomography. These developments lead to different topics which currently receive a lot of attention and to which the group of complex and hypercomplex analysis contributed heavily.Therefore, we are looking for a candidate with a strong background in Analysis and Numerical methods. In particular, the candidate should be familiar with the theory of Dirac operators, Clifford algebras, partial differential equations, fractional differential operators, inverse problems, as well as in discrete function theories, discrete differential calculus, and numerical analysis (finite difference methods).The candidate will be joining the group on complex and hypercomplex analysis and strengthen its research in this area.In particular, the candidate is expected to work on the following topics:- Spectral theory for Clifford-algebra-valued operators, with special emphasize on perturbation theory;- Theory of monogenic functions for parabolic Dirac operators or Dirac operators of Levy-Leblond type;- Strichartz estimates and hyperbolic equations, including Dirac operators in the context of Minkowski space-time;- Function-theoretical methods in Sub-Riemannian geometries based on Grushin manifolds and niltpotent Lie groups and their application in quantum computing and tomography;- Clifford-algebra-valued discrete pseudo-differential operators and numerical methods;- Applications to the Calderón problem and 3D electric impedance tomography with complex conductivities, in particular the study of the uniqueness problem in 3D for bounded conductivities as well as in the development of respective reconstruction methods;- Study of fractional Dirac operators, in particular Dirac operators with SU(n)-symmetries with special focus on the case of SU(3) symmetries with a view to applications in quarks and supersymmetry;Furthermore, the prospective candidate is expected to collaborate in:- training and supervision activities, including supervision of PhD students as well as post-doctoral researchers; - enlarging international contacts and collaborations; - supervision and coordination of scientific projects, including infrastructure or computational applications; - development of outreach activities and cooperation with the society.