C*-algebras, positive scalar curvature, and the Novikov conjecture (original) (raw)

An application of Ramanujan graphs to C * -algebra tensor products, II

Séminaire de théorie spectrale et géométrie, 1996

L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Twenty Four Manuscripts in the Virtual Library of the Faculty of Mathematics in Belgrade

The aim of this paper is to present 24 almost forgotten handwritten manuscripts that were recently found in the library of the Mathematical Institute of the Serbian Academy of Sciences and Arts. These writings belong to the prominent professors of the Belgrade University who lectured pure and applied mathematics at the beginning of the previous century: Dimitrije Danić (1862-1932), two manuscripts on applied mathematics; Kosta Stojanović (1867-1921), two manuscripts on physics and geometry; Mihailo Petrović (1868-1943), 14 manuscripts on mathematics, mostly on differential calculus and algebra; Milutin Milanković (1879-1958), five writings on applied mathematics, physics, and theoretical mechanics and one manuscript by an unknown author. Digital copies of these writings are deposited in the Virtual Library of the Faculty of Mathematics of the University of Belgrade.

Unitary representations of the Virasoro algebra and a conjecture of Kac

Compositio Mathematica, 1988

L'accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The theory of Kaluza-Klein-Jordan-Thiry revisited

Annales De L Institut Henri Poincare-physique Theorique, 1990

L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Math-Net.Ru as a digital archive of the Russian mathematical knowledge from the XIX century to today

Lecture Notes in Computer Science, 2013

The main goal of the project Math-Net.Ru is to collect scientific publications in Russian and Soviet mathematics journals since their foundation to today and the authors of these publications into a single database and to provide access to full-text articles for broad international mathematical community. Leading Russian mathematics journals have been comprehensively digitized dating back to the first volumes.

C*-algebras and Elliptic Theory

Trends in Mathematics, 2006

The paper "Index Theory for Generalized Dirac Operators on Open Manifolds" by J. Eichhorn is devoted to the index theory on open manifolds. In the first part of the paper, a short review of index theory on open manifolds is given. In the second part, a general relative index theorem admitting compact topological perturbations and Sobolev perturbations of all other ingredients is established. V. Nazaikinskii and B. Sternin in the paper "Lefschetz Theory on Manifolds with Singularities" extend the Lefschetz formula to the case of elliptic operators on the manifolds with singularities using the semiclassical asymptotic method. In the paper "Pseudodifferential Subspaces and Their Applications in Elliptic Theory" by A. Savin and B. Sternin the method of so called pseudodifferential projectors in the theory of elliptic operators is studied. It is very useful for the study of boundary value problems, computation of the fractional part of the spectral AtiyahPatodiSinger eta invariant and analytic realization of topological K-groups with finite coefficients in terms of elliptic operators. In the paper "Residues and Index for Bisingular Operators" F. Nicola and L. Rodino consider an algebra of pseudo-differential operators on the product of two manifolds, which contains, in particular, tensor products of usual pseudo-differential operators. For this algebra the existence of trace functionals like Wodzickis residue is discussed and a homological index formula for the elliptic elements is proved. B. Bojarski and A. Weber in their paper "Correspondences and Index" define a certain class of correspondences of polarized representations of C *-algebras. These correspondences are modeled on the spaces of boundary values of elliptic operators on bordisms between two manifolds. In this situation an index is defined. The additivity of this index is studied in the paper. Noncommutative aspects of Morse theory: In the paper "New L2-invariants of Chain Complexes and Applications" by V.V. Sharko homotopy invariants of free cochain complexes and Hilbert complex are studied. These invariants are applied to calculation of exact values of Morse numbers of smooth manifolds. A. Connes and T. Fack in their paper "Morse Inequalities for Foliations" outline an analytical proof of Morse inequalities for measured foliations obtained by them previously and give some applications. The proof is based on the use of a twisted Laplacian. Riemannian aspects: The paper "A Riemannian Invariant, Euler Structures and Some Topological Applications" by D. Burghelea and S. Haller discusses a numerical invariant associated with a Riemannian metric, a vector field with isolated zeros, and a closed one form which is defined by a geometrically regularized integral. This invariant extends the ChernSimons class from a pair of two Riemannian metrics to a pair of a Riemannian metric and a smooth triangulation. They discuss a generalization of Turaevs Euler structures to manifolds with non-vanishing Euler characteristics and introduce the Poincare dual concept of co-Euler structures. The duality is provided by a geometrically regularized integral and involves the invariant mentioned above. Euler structures have been introduced because they permit to remove the ambiguities in the definition of the Reidemeister torsion. Similarly, co-Euler structures can be used to eliminate the metric dependence of

On the Novikov conjecture

1993

Signatures of quadratic forms play a central role in the classification theory of manifolds. The Hirzebruch theorem expresses the signature σ (N)∈ Z of a 4k-dimensional manifold N4k in terms of the L-genus L (N)∈ H4∗(N; Q). The 'higher signatures' of a manifold M with fundamental group π1 (M)= π are the signatures of the submanifolds N4k⊂ M which are determined by the cohomology H∗(Bπ; Q).

The first eigenvalue of the Laplacian on manifolds of nonnegative curvature

Proceedings of Symposia in Pure Mathematics, 1975

für Mathematik in den Naturwissenschaften Leipzig

mis.mpg.de

We prove that the Seidel morphism of (M × M ′ , ω ⊕ ω ′) is naturally related to the Seidel morphisms of (M, ω) and (M ′ , ω ′), when these manifolds are monotone. We deduce a condition for loops of Hamiltonian diffeomorphisms of the product to be homotopically non trivial. This result was inspired by and extends results obtained by Pedroza [P]. All the symplectic manifolds we consider in this note are closed. A symplectic manifold (M, ω) is strongly semi-positive if (at least) one of the following conditions holds (c 1 denotes the first Chern class c 1 (T M, ω)): (a) there exists λ ≥ 0, such that for all A ∈ π 2 (M), ω(A) = λc 1 (A), (b) c 1 vanishes on π 2 (M), (c) the minimal Chern number N (c 1 (π 2 (M)) = N Z) satisfies N ≥ n − 1. Under this assumption, Seidel introduced [S] a group morphism: q M :π 1 (Ham(M, ω)) −→ QH * (M, ω) × , where QH * (M, ω) × denotes the group of invertible elements of QH * (M, ω), the quantum homology of (M, ω). We recall that the identity of the group QH * (M, ω) × is the fundamental class of M , which is denoted [M ]. As usual, Ham(M, ω) denotes the group of Hamiltonian diffeomorphisms of (M, ω) andπ 1 (Ham(M, ω)) is a covering of π 1 (Ham(M, ω)) which will be defined below. The inclusions of Ham(M, ω) and Ham(M ′ , ω ′) in Ham(M × M ′ , ω ⊕ ω ′) induce a map between the respective fundamental groups: ([g], [g ′ ]) → [g, g ′ ], where [g, g ′ ] stands for [(g, g ′)], the homotopy class of the loop (g, g ′). The extension of this map to the coveringsπ 1 is straightforward. We denote it by i :π 1 (Ham(M, ω)) ×π 1 (Ham(M ′ , ω ′)) −→π 1 (Ham(M × M ′ , ω ⊕ ω ′)). We also denote by κ Q : QH * (M, ω) ⊗ QH * (M ′ , ω ′) −→ QH * (M × M ′ , ω ⊕ ω ′) the inclusion given by Künneth formula and the compatibility of the Novikov rings with the cartesian product (see §1 for definitions). Let (M, ω) and (M ′ , ω ′) be strongly semi-positive symplectic manifolds and let φ ∈π 1 (Ham(M, ω)) and φ ′ ∈π 1 (Ham(M ′ , ω ′)). When (M × M ′ , ω ⊕ ω ′) is strongly semi-positive (this is not necessarily the case, see discussion in Remark 8), one can, on one hand, compute the images of φ and φ ′ via the respective Seidel's morphisms and then see the result as an element in QH * (M × M ′ , ω ⊕ ω ′) × via κ Q. On the other hand, one can compute the image of i(φ, φ ′), via the Seidel morphism of the product.

Partial *-algebras of closed operators and their commutants. II: Commutants and bicommutants

C*-algebras, positive scalar curvature, and the Novikov conjecture (original) (raw)

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