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

Sleek thought of elliptic operators, or just elliptic conception, has been formed by way of the Atiyah-Singer Index Theorem created forty years in the past. Reviewing elliptic thought over a wide diversity, 32 best scientists from 14 varied nations current contemporary advancements in topology; warmth kernel ideas; 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 ideal to graduate scholars and researchers attracted to cautious expositions of newly-evolved achievements and views in elliptic conception. The contributions are according to lectures awarded at a workshop acknowledging Krzysztof P Wojciechowski's paintings within the concept of elliptic operators.

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

**Example text**

T h e o r e m 4 . 1 . [17] The following gluing formulae hold: detcP2 detcX>2_ . w f c „,, . „ _ a , , ^Id + U + U-1* = 2 -c(o,^)-v ( d e t i ) - 2 d e t ^£i±^±^l) ) fj(V) - fj(Vc_) - f,{Vc+) = ^~. Log det F U mod Z, where C(s,D y ) is the (-function of DY, hy = dim ker Dy, det^ denotes the Fredholm determinant, and Log the principal value logarithm. 1 by other orthogonal projection in the smooth, self-adjoint Grassmannian Gr^XJCD±), which consists of orthogonal projections V± such that V± — C± are smoothing operators and GV± = (Id -V±)G.

Then L\ is a positive operator [19]. We define Li in a similar way. We can now state the general gluing formulae for the spectral invariants of Dirac type operators. 2. [17] The following general gluing formulae hold: A J ^ ' =2 -C(°^)-^(detL)-Met,( v detcZ7^ • detcp2,2 ' •J](detLi) - 2 d e t F ( fj(V) - fj(VVl) - fj(VV2) = -i-LogdetFt/12 2Id + ^+ ^1) \ 4 4 / j ' modZ. Z7TI R e m a r k 4 . 1 . 2 to noncompact manifolds with cylindrical end. We refer to [18] for this result and its proof. 2 (when Vi are the generalized Atiyah-Patodi-Singer spectral projections) has the same form as (or its reduced form modulo Z of) the gluing formulae proved by Hassell, Mazzeo, and Melrose [12], Wojciechowski [34], Bunke [6], Miiller [23], Briining and Lesch [5], Kirk and Lesch [14], Park and Wojciechowski [27].

Mazzeo, and R. B. Melrose, Analytic surgery and the accumulation of eigenvalues, Comm. Anal. Geom. 3 (1995), 115-222. 13. A. Hassell and S. Zelditch, Determinants of Laplacians in exterior domains, IMRN 18 (1999), 971-1004. Gluing formulae of spectral invariants and Cauchy data spaces 37 14. P. Kirk and M. Lesch, The eta invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum Math. 16 (2004), 553-629. 15. Y. Lee, Burghelea-Friedlander-Kappeler's gluing formula for the zetadeterminant and its applications to the adiabatic decompositions of the zetadeterminant and the analytic torsion, Trans.