Prof. Dr. Rainer Schimming: Personal statements, Research, Publications

Education:

• 1964-71 Student of Mathematics at Universität Leipzig.
• 1971 Ph.D. in Leipzig. Dissertation advisor: Prof. Dr. Paul Günther
• 1975-76 Postdoctoral research student at University of Kiev (Ukraine)
• 1979 Habilitation in Leipzig
• 1986 Sabbatical semester at Einstein-Laboratorium Potsdam

Employment:

• 1971-81 Assistent and Oberassistent at Universität Leipzig
• 1981-92 Dozent (Assistant Professor) at Universität Greifswald
• since 1992 Privatdozent
• since 1994 Akademischer Rat
• since 31.01.1997 außerplanmäßiger Professor

Professional activities:

• Research on field equations of mathematical physics on manifolds, general theory of relativity, special interests for applied mathematics and philosophy of science
• Lectures and Seminars for students of mathematics, physics, chemistry, ... on the whole of analysis, including differential equations and functional analysis, linear and general algebra, geometry, including differential geometry, probability theory, numerical analysis.
• Special courses on Lie theory, gauge field theory, solitons, general relativity, ...
• Coordinator of a permanent research seminar ``Mathematical Physics''
• Supervisor of 3 doctoral students, 3 postdoctoral students, many diploma works
• Reviewer for ``Mathematical Reviews'' and ``Zentralblatt für Mathematik''
• Referee for various scientific journals, evaluations of many dissertations
• Research project of Deutsche Forschungsgemeinschaft with Dr. I. Avramidi 1996-1999.

Supervised dissertation works:

• Tankred Hirschmann: Zum Cauchyproblem quasilinearer Evolutionsgleichungen bzw. hyperbolischer Differentialgleichungen der Mathematischen Physik. Ernst-Moritz-Arndt-Universität Greifswald 1987
• Saad Zagloul Rida Ahmed: Powers of Differential Operators and Explicit Differential Polynomials. Freie Universität Berlin 1996
• Markus Heyerhoff: Die frühe Geschichte der Solitonentheorie. Ernst-Moritz-Arndt-Universität Greifswald 1997
• Elke Dallmer: Beiträge zur Struktur von Lie-Algebren. Anwendung auf Konstante Yang-Milles-Potentiale. Ernst-Moritz-Arndt-Universität Greifswald 1999
• Fathy Ibrahim Abdel-Bassier: On the Structure of Higher-Order GravitationalField Equations on n-Dimensional Spacetimes. Minia University, El-Minia (Egypt) 2002

Selected invited talks:

• Universities of Kiev, Moscow, Minsk, 1976
• University and Banach Center Warsaw (Poland), 1974, 1979, 1983
• University of Brno (Czechia), 1976, 1985, 1986
• International Conference ``Group Theoretical Methods in Mechanics'', Novosibirsk (Sibiria), 1978
• Mathematisches Forschungsinstitut Oberwolfach, 1990, 1993
• University of Odense (Denmark), 1990, 1992
• University of Aberdeen, 1991
• Universities of Kopenhagen, Lund, Köln, Innsbruck, Mannheim, Heidelberg, 1992
• Max-Planck-Institut Bonn, 1992
• International Conference ``KdV' 95'', Amsterdam, 1995

Memberships:

1. - Deutsche Mathematiker-Vereinigung since 1990
2. - Deutsche Physikalische Gesellschaft since 1990
3. - American Mathematical Society since 1993
4. - London Mathematical Society since 2001

SELECTED PUBLICATIONS

 1.

Let (M,g) be a plane-wave spacetime of even dimension n. If , , then the De Rham wave equation for p-forms satisfies Huygens' principle. The like, if , , then the p-form Maxwell equations satisfy Huygens principle [1].

 2.

The recursive structure of the Hadamard coefficients to the p-form De Rham-Laplace equations leads to the ``fantastic cancellation'' of McKean and Singer and to a new simple proof of the Gauss-Bonnet-Chern integral theorem [3].

 3.

Necessary conditions (in tensorial form) as well as sufficient conditions (=classes of examples) for Huygens' principle (HP) to a hyperbolic operator , acting on sections of a vector bundle, are derived. The existence of a logarithm-free elementary solution for any type of L, elliptic, hyperbolic, ultrahyperbolic, or holomorphic, is studied as a natural generalization of HP [4,12,16].

 4.

Cauchy's problem to the Bach field equations of general relativity is studied: existence and uniqueness of a solution, extension theorem, the constraints as nonlinear elliptic equations [6].

 5.

An existence theorem for the Bach-Einstein gravitational field equations is proved: these admit a one-parameter family of non-conformally-flat centrally symmetric static solutions [7].

 6.

An algebraic algorithm, suited for computer implementation, for the Taylor expansion of Hadamard's coefficients is presented. The algorithm also works, with modifications, for other geodesic transport equations [4,12,15].

 7.

An explicit formula for the Korteweg-de Vries hierarchy is found, together with other results on this hierarchy [10,18].

 8.

The volume of truncated light cones in a Lorentzian manifold contains the information whether or not the manifold is flat or is Ricci-flat [11].

 9.

A natural generalization of harmonic manifolds is proposed: harmonic Laplace-like operators. This new concept leads to interesting necessary conditions and sufficient conditions [13].

10.

The Helmholtz exceptional numbers of an even-dimensional harmonic manifold are introduced as the numbers for wich the Helmholtz equation admits a logarithm-free elementary solution. These exceptional numbers are found for the symmetric spaces of rank one [16].