Analysis, Geometry And Topology of Elliptic Operators: by Matthias Lesch, Bernhelm Booss-Bavnbek, Slawomir Klimek,

By Matthias Lesch, Bernhelm Booss-Bavnbek, Slawomir Klimek, Weiping Zhang

Smooth concept of elliptic operators, or just elliptic thought, has been formed via the Atiyah-Singer Index Theorem created forty years in the past. Reviewing elliptic concept over a extensive variety, 32 top scientists from 14 various nations current fresh advancements in topology; warmth kernel concepts; spectral invariants and slicing and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics. the 1st of its variety, this quantity is perfect to graduate scholars and researchers drawn to cautious expositions of newly-evolved achievements and views in elliptic thought. The contributions are in keeping with lectures offered at a workshop acknowledging Krzysztof P Wojciechowski's paintings within the concept of elliptic operators.

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Extra info for Analysis, Geometry And Topology of Elliptic Operators: Papers in Honor of Krysztof P. Wojciechowski

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If the tangential operator is not invertible there is no canonical Atiyah-Patodi-Singer boundary condition for D. The positive spectral projection of B is not in Gr^ c (B). Rather one has to choose a Lagrangian subspace V C ker B and put Pv •= l(o,oc)(B)+U v , where Hy denotes the orthogonal projection onto V. Then Py e G r ^ ( B ) . The boundary condition given by Py is called a generalized Atiyah-PatodiSinger boundary condition. Gilkey asked how the eta-invariant depends on V. I did some explicit calculations on a cylinder which let me guess the correct formula.

Wojciechowski, Scattering theory and adiabatic decomposition of the (^-determinant of the Dirac Laplacian, Math. Res. Lett. 9 (2002), no. 1, 17-25. 27. J. Park and K. P. Wojciechowski, Adiabatic decomposition of the ^-determinant and Scattering theory, Michigan Math. DG/0111046 . 28. J. Park and K. P. Wojciechowski, Adiabatic decomposition of the zetadeterminant and the Dirichlet to Neumann operator, J. Geom. Phys. 55 (2005), 241-266. 29. J. Park and K. P. Wojciechowski, Agranovich-Dynin formula for the zetadeterminants of the Neumann and Dirichlet problems, Spectral geometry of manifolds with boundary and decomposition of manifolds, 109-121, Contemp.

Except writing papers with my supervisor this was my first mathematical collaboration. It was done completely by fax and email; Krzysztof and I met for the first time more than a year after the paper had been finished. In [16] Krzystof and I proved a special case of the following result. The result as stated is a consequence of the Scott-Wojciechowski Theorem as was shown in [14], Sec. 4. The Scott-Wojciechowski Theorem will be explained below. T h e o r e m 4 . 1 . Let P,Q € G r ^ ( 5 ) . Then rj(Dp) - 7j(DQ) = logdet F ($(P)$(Q)*) modZ.

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