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By V. Villani (Ed.)

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L²-invariants: theory and applications to geometry and K-theory

In algebraic topology a few classical invariants - comparable to Betti numbers and Reidemeister torsion - are outlined for compact areas and finite staff activities. they are often generalized utilizing von Neumann algebras and their strains, and utilized additionally to non-compact areas and limitless teams. those new L2-invariants comprise very attention-grabbing and novel details and will be utilized to difficulties coming up in topology, K-Theory, differential geometry, non-commutative geometry and spectral conception.

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In the decomposition H'(M, O(End(e))) ~_ Hi(M, O(End°(C))) + HI(M, O) of the tangent space at D" E He(E, h), the holomorphic sectional curvature vanishes in the direction of Hi(M, 0). The question remains whether it is positive in the direction of Hl( M, O( End°( ED") )). Although we can compute the curvature of 2~I(E, h) explicitly using the Gauss equation for CR-submersions [8], the fact that the holomorphic sectional curvature is nonnegative can be understood intuitively from the general principle that the holomorphic curvature decreases with a holomorphic subbundle and increases with a quotient bundle.

REFERENCES 1. W. Casselman, Canonical eztensions of Harish-Chandra modules to representations of G, Preprint. 2. W. -S. I. Piatetski-Shapiro and P. Sarnak, Poincard series for SO(n, 1), In preparation. 35 3. R. S.. 4. V. Kuznetsov, Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture. Sums of Kloosterman sums~ Math. USSR Sbornik 39 (1981)~ 299-342. 5. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. I. (1965), 1-15. 6. R. Wallach, Asymptotic ezpansions of generalized matriz entries of representations of real reductive groups.

38 2 Deformation of Surfaces Infinitesimal and Punctured Compact Riemann Deformation Assume t h a t X, B are real analytic manifolds and that ,r : X --* B is a surjective, proper, smooth real analytic map. th. th. for two such charts the w-components depend hotomorphically on each other for fixed z-components. T h e atlas `4 makes every fiber Xt := ~r-i(t), t E B a compact complex submanifold of X . We assume that every X~ is a Riemann surface of genus p. th. ,s,~(t) are disjoint. 1 X is called a real analytic deformation of Riemanu surfaces of genus p with m distinguished points.

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