ARTHUR E. FISCHER Professor Emeritus B.A., Columbia University Ph.D., Princeton University M.D., Albert Einstein College of Medicine

Email: aef_at_ucsc_dot_edu

RESEARCH INTERESTS

Arthur Fischer's research interests are diverse but are mainly in the fields of differential geometry and general relativity. His main research tools are global nonlinear analysis which involves nonlinear partial differential equations and infinite-dimensional manifolds. His research often intertwines various areas of physics and mathematics.

One of the main themes of Professor Fischer's research is concerned with the structure of various moduli spaces of Riemannian structures on a fixed closed manifold and with the moduli space of Einstein structures on a spacetime. His major results in this area have been carried out jointly with Professor Jerrold Marsden of the California Institute of Technology and Professor Vincent Moncrief of Yale University. In the Riemannian case, they have analyzed the structure of various subsets of metrics with given scalar curvature. In the Lorentz case, the case of interest in general relativity, they have analyzed the structure of the space of Einstein spacetimes.

In other areas of geometry and relativity, Professor Fischer has been involved with the introduction and development of the concept of linearization stability of nonlinear partial differential equations. This concept, in turn, has been applied to develop an adjoint Hamiltonian formulation of the Einstein field equations as well as to answer several questions in Riemannian geometry. With this adjoint formulation, the dynamical form of the Einstein evolution equations can be written in a way which is very concise and which reveals several salient structural features of the Einstein equations.

Professor Fischer is also involved in research regarding the theory of TeichmŸller space, research which has been carried out jointly with Professor Anthony Tromba of the University of California, Santa Cruz. They have been successful in using a purely geometrical approach to the classical theory of TeichmŸller space and thus have been able to rewrite parts of TeichmŸller theory using Riemannian geometry. This work, in turn, has been used in the study of the (2+1)-form of Einstein's equations and, more recently, has been used in the study of the (3+1)-form of Einstein's equations.

Professor Fischer is also working with Professor Moncrief on a project that transforms Einstein's dynamical equations of general relativity to a reduced Hamiltonian form. As an application of these results, they relate the Einstein flow to the topology of the spacelike hypersurface, which in turn relates to geometrization conjectures regarding three-manifold topologies. Their current interest is to relate the Einstein flow more explicitly to the structure of three-manifolds, hoping to show intrinsic relations between the Einstein equations, cosmic topologies, and the structure of three-manifolds.

SELECTED PUBLICATIONS

Fischer, A (1992), Riemannian maps between Riemannian manifolds, in Mathematical Aspects of Classical Field Theory, proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, Seattle, Washington, July 20-26, 991, Contemporary Mathematics, volume 132, edited by M Gotay, J Marsden, and V Moncrief, American Mathematical Society, Providence, Rhode Island, pp. 331-366, MR93k:58035.

Fischer, A, and Moncrief, V (1994), Reducing Einstein's equations to an unconstrained Hamiltonian system on the cotangent bundle of Teichmuller space, in Physics on Manifolds, Proceedings of the International Colloquium in honour of Yvonne Choquetbruhat, Paris, June 3-5, 1992, edited by M Flato, R Kerner, and A Lichnerowicz, Kluwer Academic Publishers, Boston, pp. 11-151, MR95a:58037.

Fischer, A, and Moncrief, V (1994), The structure of quantum conformal superspace, in Global Structure and Evolution in General Relativity, Proceedings of the First Samos Meeting on Cosmology, Geometry and Relativity, held at Karlovassi, Samos, Greece, September 5-7, 1994, edited by S Cotsakis and G Gibbons, Lecture Notes in Physics, volume 460, Springer-Verlag, Berlin, ISBN 3-540-60751-X, pp. 111-173, MR97g:83015.

Fischer, A (1996), A theory of cosmological models, Fields Institute Communications 7, American Mathematical Society, Providence, pp. 107-153.

Fischer, A (1996), Riemannian submersions and the regular interval theorem of Morse theory, in Annals of Global Analysis and Geometry, vol.14, pp. 263-300.

Fischer, A, and Moncreif, V (1996), A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom, General Relativity and Gravitation 28, pp. 207-219, MR97a5020a.

Fischer, A, and Moncrief, V (1996), Quantum conformal superspace, General Relativity and gravitation 28,pp. 221-237, MR97a:58020b.

Fischer, A, and Moncrief, V (1997), Hamiltonian reduction of Einstein's equations of general relativity, in Nuclear Physics B (Proceedings Supplement) 57, in Proceedings of the Second meeting on Constrained Dynamics and Quatum Gravity QG96, Santa Margherita Ligure, Italy, 17 Spetember 1996, edited by J Nelson, NorthHolland, Elsevier Science B.V., The Netherlands, pp. 142-161, abstract at http://www.elsevier.nl/ge-jng/29/35/28/36/9/26/abstrac.html, full article at http://www.elsevier.nl/gejng/29/35/28/36/show/Products/NPE/toc.htt#iss:1-3, MR99a:83005.

Fischer, A, and Moncrief, V (1998), The reduced Hamiltonian of general relativity and the o-constant of conformal geometry, in Mathematical and Quantum Aspects of Relativity, held at Pythagoreon, Samos, greece, 31 August -4 Septmeber 1998, edited by S Cotsakis and G W Gibbons, Lecture Notes in Physics, Volume 537, Springer-Verlag, Berlin, ISBN3-540-66865-9. ISSN 0075-8450, pp. 70-101, abstract at http://link.springer.de/link/service/series/2669/bibs/0537/05370070.pdf, MR1843034.

Fischer, A, and Moncrief, V (1999), The Einstein flow, the sigma-constant, and geometrization of three-mainfolds, Classical and Quantum Gravity 16, pp. L79-L87, http://www.iop.org/PEL/abstract/cq1601112 and http://www.iop.org/PEL/article/cq1601112/full, MR2000m:53099.

Fischer, A, and Moncrief, V (1999), Hamiltonian reduction of Einstein's equations and the geometrization of three-manifolds, in Equadiff 99, International Conference on Differential Equations, Berlin, August 17, 1999, Volume 1, edited by Bernold Fiedler, Konrad Groger, and Jurgen Sprekels, World Scientific, Singapore, ISBN 981-02-4988-8, pp. 279-282.

Fischer, A, and Moncrief, V (1999), The reduced Hamiltonian of Einstein's equations of general relativity and the sigma-constant of conformal geometry, Lecture Notes in Physics, Springer, New York, to appear.

Fischer, A, and Moncrief, V (2000), Hamiltonian reduction, the Einstein flow, and collapse of 3-manifolds, in Constrained Dynamics and Quantum Gravity 1999, Abstracts of the Third Meeting on Constrained Dynamics and Quantum Gravity, QC99, Villasimius, Italy, 13-17 September 1999, edited by V De Alfaro, J Nelson, M Cadoni, M Cavaglia and A T Filippov, Nuclear Physics B (Proceedings Supplement) 88, North Holland, Elsevier Science B.V., The Netherlands, ISSN 0920-5632, pp. 83-102, abstract at http://www.elsevier.nl/gej0nj/29/35/28/93/21/31/abstract.htm., full article at http://link.springer.de/link/service/series/2669/papers/0537/05370070.pdf, MR1843034.

Fischer, A, and Moncrief, V (2001), Reduced Einstein equations- An expanded view, in Proceedings of the Ninth Marcel Grossman Meeting, Rome, July 2-8, 2000, University of Rome "la Sapienza", World Scientific, Singapore, to appear http://www.icra.it/MG/mr9/Proceedings.html.

Fischer, A, and Moncrief, V (2001), Reduced Einstein equations- A Brief Summary, in Proceedings of the Ninth Marcel Grossman Meeting, Rome, July 2-8, 2000, University of Rome "la Sapienza", edited by Vahe G Gurzadyan, Robert Jantzen, and Remo Ruffini, World Scientific, Singapore, to appear http://www.icra.it/MG/mr9/Proceedings.html.

Fischer, A, and Moncrief, V (2001), The reduced Einstein equations and the conformal volume collapse of 3-manifolds, Classical and Quantum Gravity, to appear.

Fischer, A, and Moncrief, V (2002), Conformal volume collapse of 3-manifolds and the reduced Einstein flow, in Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J.E. Marsden, edited by P Holmes, P K Newton, and A Weinstein, Springer-Verlang, pp. 1-60, to appear.

Fischer, A, and Moncrief, V (2002), Nonautonomous dynamical systems and the phase portrait of the reduced Einstein equations, to appear.

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