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A brief description of my research topics
Algebras of generalized functions and their applications
Most of my research is centered around the theory of algebras of generalized functions
also known as Colombeau Algebras. These are sheaves of differential algebras which contain the
vector space of Schwartz distributions as a subspace and the space of smooth functions as a subalgebra,
hence provide a framework for multiplying distributions which benefits from maximal compatibility
with classical analysis (read more...)
In particular, I'm interested in all geometric aspects of this theory and its applications, most of all
in general relativity.
Michael Kunzinger and myself, partly in collaboration with James Vickers and Michael Oberguggenberger have developped the foundations of global analysis as well as (semi-)Riemannian geometry with generalized functions in Colombeau's so-called special setting ([J6], [J8], [J9], [J12], [J13], [J18]), which we called non-linear distributional geometry. Some classical questions which we had encountered when dealing with flows of singular vector fields have been solved in collaboration with Hermann Schichl ([J15]).
Together with Michael Grosser and again Michael Kunzinger and James Vickers I have developped a diffeomorphism invariant full Colombeau algebra on differentiable manifolds ([J5])---a task which needed restructuring some of the foundations of the theory in Euclidean space and which has been achieved in collaboration with Eva Farkas ([M1]). The idea of this research is explained in [P6] while an overview is presented in [P8]. Finally in [P9] I have reported on a significant shift of focus which is indispensable when dealing with the tensor case (see below).
A couple of applications to general relativity which I have studied in part together with various coauthors are: impulsive pp-waves ([J1-J3], [P2-3], [P5]), the Penrose junction conditions ([J4]), the (distributional-)Schwarzschild geometry ([J7]), and expanding impulsive gravitational waves ([J11]). The topic of linear and nonlinear distributional gravity has been reviewed in [P7] and more recently in [J16] (jointly with James Vickers).
Significant earlier parts of this research have been collected in the monograph
[M2] with
Michael Grosser, Michael Kunzinger and Michael Oberguggenberger.
Kinetic theory
I have also done some work in nonlinear PDE, more precisely in
collissionless models in kinetic theory. Together with Gerhard Rein, Michael Kunzinger and
Gerald Teschl I have studied the Vlasov-Klein-Gordon system:
In [J10] we have proven the existence of
local weak solutions, while in [J14] we have
derived local classical solvability plus a continuation criterion. Again with Michael Kunzinger and
Irina Kmit I have studied singular solutions of the Vlaosov-Poisson system
([J17]) with the aim of looking at the singular limits of
the VP-system, i.e., the Euler equations and the n-body problem.
Gravitational waves
Quite early in my career I developped some interest in gravitational wave detection, which resulted
in two fun-papers [P1] and [P4]
where together with some fellow students as well as Karsten Danzmann we put forward the idea of a
space-borne gravitational wave detector much larger than LISA. Of course such a project is not at all feasable.
Most recent research and an outlook to future projects
Most recently I have again done research in generalized functions and their applications.
One big challenge in the theory of diffeomorphism invariant full Colombeau algebras has been to construct spaces of generalized tensor fields. By the Schwartz impossibility result the obvious approach of taking the tensor product of the full algebra on a manifold with classical tensor fields is ruled out and one has to take a different route. By introducing a suitable class of transport operators (an idea put forward by James Vickers long time ago) and letting generalized fields depend on them Michael Grosser, Michael Kunzinger, James Vickers and myself could finally construct such spaces in [Pre1]. Further tasks are now to define correctly the covariant derivative and to lay the foundations of semi-Riemannian geometry in this framework.
Together with Eberhard Mayerhofer and James Grant I was concerend with the solution of the wave equation on weakly singluar space-times. More precisely we were looking at Laplace-Beltrami operators of Lorentzian metrics of low regularity and indeed in [J19] we could prove local existence and uniqueness of solutions in generalized functions for a large class of "locally bounded'' metrics.
Some further prospects of this work which are currently under investigation in collaboration with my Ph.D. student Clemens Hanel are regularity issues of the above mentioned solutions as well as generalizations to nonlinear wave equations. There are strong interrelations to current work by Günther Hörmann and his Ph.D. student Christian Spreitzer.Recently there has been renewed interest in (classical=linear) distributional general relativity. The `` maximal reasonable'' class of distributional metrics of Geroch and Traschen has been rederived by Le Floch and Mardare. I have also written a small article [P10] about this so-called GT-class of metrics clarifying some details which have not been made explicit so far. Also together with James Vickers I have proven compatibility of the GT-framework and nonlinear distributional geometry in the range where both of them apply ([J20]).
Further investigations on the description of hypersufrace singularities are currently undertaken together with my diploma student Nastasia Grubic.