The question of how to define nonlinear operations in the context of L. Schwartz' theory of distributions in a consistent way is of fundamental importance in fields like quantum field theory, (semi) Riemannian geometry with non-smooth metric, linear partial differential equations with singular coefficients as well as nonlinear different ial equations (stochastic or deterministic) with singular initial and boundary data. A common approach to this problem is given by algebras of generalized functions in the sense of J. F. Colombeau, which rest on asymptotic approximations of distributions through suitable sequences of smooth functions.,

In [13] I developed a construction of nonlinear generalized tensor fields with applications in singular differential geometry. This was made possible by a fundamental restructuring [8] of the theory of nonlinear generalized functions, making essential use of L. Schwartz' theory of vector-valued distributions [12,6]. This made it possible to study the curvature of distributional (semi-)Riemannian metrics and related quantities rigorously, which is not possible using classical methods.

Moreover, I studied structural properties of algebras of generalized functions [19,20,16,11,9,2,18,22], which marked the beginning of a comprehensive structure theory of such algebras. In particular this encompassed point value characterizations [5] and applications in singular differential geometry [14,10,7,4,3]. The most important example in this context is the rigorous calculation of the curvature of distributional metrics [24,28,27].

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One of my recurring interests is the study of locally convex spaces, in particular (vector valued) distribution theory. This approach allow, for example, to generalized and unify results on convolvability and regularization of distributions [17] and through the study of the Laplace-transformation of distributions [23,26] has applications to fundamental solutions. Further structural investigations of distribution spaces led to a new, simpler description of certain topologies on these [25,30], a representation of dual spaces of topological tensor products [15,21] as well as a unified sequence space representation of spaces of functions and distributions [26], which summarized and generalized previous results in this direction using methods of time-frequence analysis.

References

Lately I have been investigating hypocoercive evolution equations, i.e., evolution equations which despite lacking coercivity show exponential decay because of the interplay of a semidissipative and a konservative part. In this context I am looking (together with the group of Anton Arnold) at generalizing certain notions, in particular the hypocoercivity index, and corresponding results to the setting of infinite-dimensional Hilbert spaces.

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