Steps towards "Quantum Gravity" and the practice of science: will the merger of mathematics and physics work?

The author recalls general tendencies of the "mathematization" of the sciences and derives challenges and tentative obstructions for a successful merger of mathematics and physics on fancied steps towards "Quantum Gravity". This is an edited version of the author's opening words to an international workshop "Quantum Gravity: An Assessment", Denmark, May 17-18, 2008. It followed immediately after the Quantum Gravity Summer School 2008, see http://QuantumGravity.ruc.dk/Comment: To appear as part of a Springer Lecture Notes in Physics publication: "Quantum Gravity - New Paths towards Unification" (B. Booss-Bavnbek, G. Esposito, M. Lesch, Eds.

Spectral Invariants of Operators of Dirac Type on Partitioned Manifolds

We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize various (occasionally overlooked) aspects of rigorous definitions and explain the quite different stability properties. Moreover, we utilize the heat equation approach in various settings and show how these topological and spectral invariants are mutually related in the study of additivity and nonadditivity properties on partitioned manifolds.Comment: 131 pages, 9 figure

Weak Symplectic Functional Analysis and General Spectral Flow Formula

We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain $D_m$ and fixed {\it intermediate} domain $D_W$. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed boundary conditions. Here $D_W$ is the first Sobolev space and $D_m$ the subspace of sections with support in the interior. We express the spectral flow of the operator curve by the Maslov index of a corresponding curve of Fredholm pairs of Lagrangian subspaces of the quotient Hilbert space $D_W/D_m$ which is equipped with continuously varying {\it weak symplectic structures} induced by the Green form. In this paper, we specify the continuity conditions; define the Maslov index in weak symplectic analysis; discuss the required weak inner Unique Continuation Property; derive a General Spectral Flow Formula; and check that the assumptions are natural and all are satisfied in geometric and pseudo-differential context. Applications are given to $L^2$ spectral flow formulae; to the splitting of the spectral flow on partitioned manifolds; and to linear Hamiltonian systems

The mathematization of the individual sciences - revisited

We recall major findings of a systematic investigation of the mathematization of the individual sciences, conducted by the author in Bielefeld some 35 years ago under the direction of Klaus Krickeberg, and confront them with recent developments in physics, medicine, economics, and spectral geometry.Comment: Dedicated to Klaus Krickeberg on his 80th birthday, 21 page

Basic Functional Analysis Puzzles of Spectral Flow

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