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

Sleek conception of elliptic operators, or just elliptic thought, has been formed via the Atiyah-Singer Index Theorem created forty years in the past. Reviewing elliptic conception over a vast diversity, 32 prime scientists from 14 diverse international locations current contemporary advancements in topology; warmth kernel innovations; spectral invariants and slicing and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics.The first of its type, this quantity is preferrred to graduate scholars and researchers attracted to cautious expositions of newly-evolved achievements and views in elliptic conception. The contributions are in line with lectures awarded at a workshop acknowledging Krzysztof P Wojciechowski's paintings within the idea of elliptic operators.

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2 (1999), 423-444. Received by the editors September 15, 2005 ; Revised January 4, 2006 Part II Topological Theories This page is intentionally left blank Analysis, Geometry and Topology of Elliptic Operators, pp. 41-62 © 2006 World Scientific Publishing Co. T H E BEHAVIOR OF T H E ANALYTIC I N D E X U N D E R NONTRIVIAL EMBEDDING DAVID BLEECKER Department of Mathematics University of Hawaii Honolulu, HI 96822, USA bleeckerQmath. hawaii. edu Dedicated to Krzysztof P. Wojciechowski on his 50th birthday The Atiyah-Singer index formula states that the analytic index of an elliptic pseudo-differential operator equals the topological index of the K-theoretic class of its asymptotic symbol.

Remark 2,2. , ker A M = {0} implies that ker 72. = {0}. Hence, under this condition, all the operators occurring in (3) have trivial kernels. Without this condition, we have an additional term on the right side of (3). 3. When we assume that A M has the following product form over a collar neighborhood U = Y x [—1, l ] u of Y, 28 Jinsung Park where u denotes the variable of the normal direction to Y and Ay is a Laplace type operator over Y, we can obtain the exact value of C(Y) as in 19], [15], [28], C(F) = 2 - c ( ° ' A y ) - f l y .

Loya and J. Park, Decomposition of the zeta-determinant for the Laplacian on manifolds with cylindrical end, Illinois J. Math. 48, no. 4 (2004), 1279-1303. 17. P. Loya and J. Park, On the gluing problem for the spectral invariants of Dirac operators, Adv. , to appear. 18. P. Loya and J. Park, On the gluing problem for Dirac operators on manifolds with cylindrical ends, J. Geom. Anal. 15 (2005), 285-319. 19. P. Loya and J. Park, The comparison problem for the spectral invariants of Dirac type operators, Preprint, 2004.