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

Glossy concept of elliptic operators, or just elliptic conception, has been formed by means of the Atiyah-Singer Index Theorem created forty years in the past. Reviewing elliptic thought over a wide variety, 32 top scientists from 14 diversified nations current contemporary advancements in topology; warmth kernel suggestions; spectral invariants and slicing and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics.The first of its style, this quantity is perfect to graduate scholars and researchers drawn to cautious expositions of newly-evolved achievements and views in elliptic idea. The contributions are in accordance 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 Krzysztof P Wojciechowski**

**Sample text**

Now the question is how we define U using K±. We here observe that H(T>±) is a Lagrangian subspace in L2(Y, So) with respect to the symplectic form (G , ). Hence, there is the unitary operator over L2(Y,S~) which transforms H(V+) to H(D-), that is, describes the difference of them. From this reasoning, we can see that the operator K_K^ X does this since (x, K_X) = (x, (/t_/t^1)«;_(-a;) for (X,K±X) £H(D±). But, we actually need to find the unitary operator which transforms the Cauchy data space of V*^.

P. Wojciechowski, Elliptic boundary problems for Dirac operators, Birkhauser, Basel, 1993. 8. J. Briining and M. Lesch, On the eta-invariant of certain non-local boundary value problems, Duke Math. J. 96 (1999), 425-468, dg-ga/9609001. 9. , On boundary value problems for Dirac type operators: I. Regularity and self-adjointness, J. Funct. Anal. PA/9905181. 10. A. Calderon, Boundary value problems for elliptic equations, Outlines of the joint Soviet-American symposium on partial differential equations (Novosibirsk), 1963, pp.

The case of a singular tangential C o m m . M a t h . P h y s . 1 0 9 (1995), 315-327. The ^-determinant and the additivity of the n-invariant on the self-adjoint Grassmannian, C o m m . M a t h . P h y s . 2 0 1 (1999), 4 2 3 - Received by the editors September 14, 2005; revised January 5, 2006 Analysis, Geometry and Topology of Elliptic Operators, pp. 23-38 © 2006 World Scientific Publishing Co. re. kr Dedicated to Krzysztof P. Wojciechowski on his 50th birthday We review the gluing formulae of the spectral invariants - the ^-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator.