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Michael Kunzinger - Research

Research Interests



Nonlinear Functional Analysis, Algebras of Generalized Functions

Starting with my doctoral thesis a main focus of my scientific interests lies on the field of algebras of generalized functions, in particular Colombeau algebras. Colombeau Algebras are differential algebras containing the space of distributions as a linear subspace and the space of smooth functions as a faithful subalgebra. Owing to the fact that in these algebras both differentiation and a wide class of nonlinear operations can be carried out unrestrictedly while at the same time there are strong consistency results with classical distributional operations are available, a rich theory for solving nonlinear partial differential equations has been developed in this setting.

[J2] provides a characterization of Colombeau generalized functions by their point values on generalized points which constitutes the key to extending a number of classical geometric concepts to generalized functions (graph of a generalized function, flow of ODEs involving generalized functions).

Following this, Michael Grosser, Roland Steinbauer, James Vickers and I, jointly with J.P. Wilson were working intensively on a restructuring of the theory of algebras of generalized functions. Using calculus in convenient vector spaces in the sense of Andreas Kriegl and Peter Michor we achieved an intrinsic formulation of the theory of algebras of generalized functions on differentiable manifolds preserving the main characteristics of the local theory: Embedding of the space of distributions as a linear subspace, embedding of the space of smooth functions as a faithful subalgebra, embedding commutes with Lie derivatives with respect to arbitrary smooth vector fields, generalized tensor calculus. These results (parts of which appear in [B2], [J8], [P3], [P4]) also play a central role in [B3].

Based on regularity theory for algebras of generalized functions (developed by Michael Oberguggenberger), joint with Günther Hörmann we have carried out an analysis of the interaction between nonlinear operations, singularities and differentiation from a microlocal point of view (in particular: propagation of singularities in nonlinear PDEs involving generalized functions). In [J7], microlocal properties of basic operations (multiplication and pullback) in the Colombeau algebra are analyzed. Although multiplication of distributions can be carried out unrestrictedly in the algebra, the wave front set of the product displays high sensitivity to the configuration of the factors' singularity structure. In case the wave front sets of the factors are in favorable position an extension of the classical inclusion relation (in the distributional setting) due to Hörmander is proven. Explicit examples are given showing that the inclusion relation will break down even in the Colombeau algebra if Hörmander's condition is violated.

[J33], jointly with Michael Grosser, Roland Steinbauer, and James Vickers extends the construction of [J8] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby obtain a universal algebra of generalized tensor fields canonically containing the space of distributional tensor fields. The canonical embedding of distributional tensor fields also commutes with the Lie derivative. This construction provides the basis for applications of algebras of generalized functions in nonlinear distributional geometry and, in particular, to the study of spacetimes of low differentiability in general relativity.

Jointly with Shantanu Dave and Günther Hörmann, in [J35] and [P8] we study regularizations of Schwartz distributions on a complete Riemannian manifold M. These approximations are based on families of smoothing operators obtained from the solution operator to the wave equation on M derived from the metric Laplacian. The resulting global regularization processes are optimal in the sense that they preserve the microlocal structure of distributions, commute with isometries and provide sheaf embeddings into algebras of generalized functions on M.

In [J30], Annegret Burtscher and I show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring of generalized numbers in this unified setting.

An independent line of research is initiated, jointly with Shantanu Dave in [J36]: We introduce a (bi)category whose objects can be functorially assigned spaces of distributions and generalized functions. In addition, these spaces of distributions and generalized functions possess intrinsic notions of regularity and singularity analogous to usual Schwartz distributions on manifolds. The objects in this category can be obtained from smooth manifolds, noncommutative spaces, or Lie groupoids.

Jointly with Günther Hörmann and Sanja Konjik, in [J37] and [P9] we study the underlying symplectic linear algebra on modules of the ring of Colombeau generalized numbers as well as the resulting non-smooth symplectic geometry.

A very fruitful line of cooperation started when Paolo Giordano joined the Faculty of Mathematics in 2010. Initially we worked on the algebraic foundations of his theory of Fermat reals ([J31]). After this, we introduced some new topologies on Colombeau generalized numbers and laid the foundations for an intrinsic differentiation theory in [J34]. Jointly with Hans Vernaeve we introduced a far-reaching generalization of Colombeau's construction that is based on the minimal logical requirements to obtain well-defined set-theoretic functions on generalized numbers that enjoy satisfactory differentiability properties, the so called Generalized Smooth Functions, see [J40].



Non-smooth Differential Geometry

One of my main research interests is differential geometry in the presence of singularities, or, more generally, non-smooth differential geometry in a broad sense. In applications of algebras of generalized functions to geometric questions there are a number of instances where a concept of generalized functions taking values in a differentiable manifold is called for (e.g., geodesics of generalized metrics, flow of generalized vector fields, ...). A new type of generalized functions meeting these requirements is introduced in [J11]. Typical singularities that can be modelled in this approach are jump discontinuities. Generalized vector bundle homomorphisms, also introduced in [J11] enable the definition of tangent maps to such generalized mappings, thereby allowing for modelling even stronger (delta-type) singularities (occurring naturally as derivatives of jump discontinuities) in the vector components of the resulting homomorphisms. These results are extended to a functorial theory of manifold-valued nonlinear generalized functions in [J13], and in [J32] to full Colombeau algebras (jointly with Eduard Nigsch). See also [J16] for a survey.

In [J9], [J11], [J12] Roland Steinbauer and I have worked out a theory of nonlinear distributional geometry in the framework of the so-called special variant of Colombeau's construction. We establish point value characterizations of generalized sections of vector bundles and derive algebraic characterizations (as modules over the ring of smooth resp. Colombeau functions) of spaces of generalized sections. Applications of this approach include nonsmooth Hamiltonian mechanics, semi-Riemannian geometry in the presence of singularities and General Relativity. Flows of generalized vector fields (as special instances of generalized diffeomorphisms) are the focus of [J15], joint with Michael Oberguggenberger, Roland Steinbauer, and James Vickers.

A theory of generalized connections, including first applications to singular solutions of Yang-Mills theory is developed (together with Roland Steinbauer and James Vickers) in [J17]. In [J23] we study sheaf and embedding properties of manifold-valued Colombeau generalized functions. As an application we introduce a space of manifold-valued distributions and study its basic properties.

In [J38], Roland Steinbauer, Milena Stojkovic, James Vickers and I use comparison geometry to show that the exponential map of any C^{1,1}-semi-Riemannian metric locally is a bi-Lipschitz homeomorphism. This work, inspired by an open problem of Piotr Chrusciel marked the starting point of several investigations into questions of causality theory and singularity theorems in General Relativity, see below.



Symmetry Groups of Differential Equations

Group Analysis for Generalized Functions

The aim of this project (parts of which were carried out as project P-10472MAT of the Austrian Science Foundation as well as in my doctoral thesis) is to extend Lie's theory of symmetry groups of differential equations to algebras of generalized functions. In this way an investigation of symmetry properties of weak resp. distributional solutions (delta waves, shocks, etc.) or, more generally, Colombeau solutions of PDEs is made possible. Work towards this goal has been organized roughly in two phases: first of all, a number of consistency properties for the transfer of classical symmetry groups into the generalized functions setting have been achieved. In this context a (purely classical) factorization result for Lie point symmetries has turned out to be particularly helpful as it allows the transfer of large classes of symmetries to generalized solutions.

In a second phase the infinitesimal methods of Lie group analysis of PDEs themselves have been extended to the Colombeau setting. This process was enabled by the above-mentioned pointvalue characterization on the one hand and by an application of the theory of ODEs in Colombeau algebras (developed recently by R. Hermann and Michael Oberguggenberger) on the other. Results of these phases are collected in [J6], [P1], [P2], as well as in my doctoral thesis, Lie transformation groups in Colombeau algebras.

Together with N. Dapic and S. Pilipovic I then developed a unified theory of symmetry group analysis in the presence of singularities comprising three possible solution concepts of differential equations involving generalized functions:

  • distributional solutions
  • weak solutions
  • solutions in algebras of generalized functions
(see [J10]). Jointly with S. Konjik, we then developed a global approach to symmetry group analysis in algebras of generalized functions (based on [J11], [J13], and [J15]), presented in [J21]. A study of group invariants of Lie transformation groups in the Colombeau setting is given in [J20].

In [J25], jointly with Sanja Konjik and Michael Oberguggenberger we propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of Nöther's theorem.

Classical Group Analysis

This is an ongoing research collaboration with Roman Popovych.

In [J27], jointly with R. Popovych and H. Eshragi, we extend and enhance the framework of group classification based on the new notions of conditional equivalence group and normalized class of differential equations. Based on these techniques, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schroedinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schroedinger equations with modular nonlinearities and potentials and some subclasses of this class. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods.

In [J24], jointly with N. Ivanova, we investigate conservation laws and potential symmetries for the class of linear $(1+1)$-dimensional second-order parabolic equations. We employ the technique of admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. Criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and the application of multiple dual Darboux transformations. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.

[J26] shows that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables.

In [J28], the notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+1)-dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in-depth prior study of the corresponding singular vector fields.

[J29] analyzes the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution. Earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited. This leads to a no-go theorem on determining equations for operators of generalized conditional symmetry. It is also shown that up to certain equivalences there exists a one-to-one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions.



Partial Differential Equations

The main focus of my activities in this field so far has been on modelling singularities in nonlinear field equations. Jointly with Günther Hörmann we treat simplified versions of the Maxwell-Lorentz system of electrodynamics with singular sources. This leads to weakly hyperbolic nonlinear systems which we investigate by the method of regularized derivatives. We derive existence and uniqueness results and calculate distributional limits of solutions with delta-type initial values ([J1], [J5]).

In collaboration with G. Rein, R. Steinbauer, and G. Teschl we have considered kinetic equations in mathematical physics. In ([J14]) we study an ensemble of classical particles coupled to a Klein-Gordon field. We prove existence of global weak solutions to the resulting relativistic Vlasov-Klein-Gordon field for initial data satisfying a size restriction. We extend this study in [J18] to prove local existence of classical solutions and to develop a continuation criterion for solutions. In [J22], together with I. Kmit and R. Steinbauer we investigate spherically symmetric solutions of the Vlasov-Poisson system in the context of algebras of generalized functions. This allows to model highly concentrated initial configurations and provides a consistent setting for studying singular limits of the system.



General Relativity

Due to the inherent nonlinearities of the field equations, studying singularities in general relaivity by means of classical distributional methods soon runs into serious conceptual problems. In order to cope with such problems in a mathematically rigorous fashion one has to resort to some kind of distributional geometry allowing to carry out nonlinear operations with singular objects. As was already mentioned above the development of such a theory has been one of the main projects of our research group.

Together with R. Steinbauer we have studied geodesic and geodesic deviation equations in singular space times. In [J3] a satisfactory solution concept for such equations was developed and applied to the case of pp-Waves (plane fronted waves with parallel rays). For describing such waves Roger Penrose had used a discontinuous coordinate transformation turning the distributional form of the metric into a continuous form. While mathematically ill-defined (involving undefined nonlinear operations on distributions), this procedure provides physically equivalent descriptions of the situation (particle trajectories in both pictures coincide).

The key to understanding this behaviour lies in realizing that the Penrose transformation actually transports points of space time along (discontinuous) geodesics of the original (distributional) metric. By means of a global univalence result of Gale and Nikaido we were able to prove that the transformation is actually the distributional `shadow' of a Colombeau coordinate transformation. This also introduces a possibility of handling `discontinuous diffeomorphisms', see [J4]. The necessary tools for describing such transformations (which furnish an example of manifold-valued generalized functions) are presented in [J11].

In [J39], Roland Steinbauer, Milena Stojkovic, James Vickers and I used regularization methods to extend much of classical causality theory to Lorentzian metrics of regularity C^{1,1}. These results were fundamental in our further work on singularity theorems for metrics of this regularity. In [J41], [J42] we extend the validity of the classical singularity theorems of Hawking and of Penrose to C^{1,1}-metrics.



Locally Convex Spaces

My activities in this area have concentrated on the investigation of certain classes of locally convex spaces that lie `between' barrelled and Baire spaces (Baire-like spaces). Apart from questions of classifying such spaces I am mainly interested in structure theory of strict inductive limits of locally convex spaces based on this classification. Research topics include intrinsic problems such as metrizability and normability of (LF)-spaces and classification of (LB)-spaces as well as applications to locally convex spaces in distribution theory. These investigations have resulted in the book [B1], cf. also the corresponding review in Bull. AMS 32 (3) (1995), 354-357.
Last modified January 4, 2016 by Michael Kunzinger.