Noncommutative elliptic theory. Examples

We study differential operators with coefficients in noncommutative algebras. As an algebra of coefficients, we consider crossed products corresponding to the action of a discrete group on a smooth manifold. We give index formulas for the Euler, signature, and Dirac operators twisted by projections over the crossed product. The index of Connes operators on the noncommutative torus is computed.     

Noncommutative Polynomials Describing Convex Sets

Foundations of Computational Mathematics - The free closed semialgebraic set $${mathcal {D}}_f$$ determined by a hermitian noncommutative polynomial $$fin {text {M}}_{{delta }}({mathbb...     

On Construction of Characteristic Classes in Noncommutative Geometry

In this paper, we review and compare several constructions of characteristic classes used in noncommutative geometry.     

Aspects of Matrix Theory and Noncommutative Geometry

We review some basic foundations of the matrix model of M-theory. We study the important problem of compactifying the matrix theory in relation to noncommutative geometry. We show that there exist solutions of this problem other than the well-known toroidal solutions of Connes, Douglas, and Schwarz. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 181–190, November, 2005.     

Normed Groups and Their Applications in Noncommutative Differential Geometry

A normed semigroup is a semigroup having both rich topological and rich algebraic structure. This work is devoted to an abstract study of normed semigroups that arise in some problems of noncommutative geometry. The most interesting example, namely, the Abel semigroup , where A is a von Neumann algebra, is considered in detail. The definition of the latter semigroup is based on the notion of stable equivalence of normal elements of W^*-algebras, which generalizes the notion of stable equivalence of projectors. Bibliography: 8 titles.     

Braided algebras and their applications to Noncommutative Geometry

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the “mountain property” for the numerators and denominators of their Poincaré–Hilbert series (which are always rational functions).     

Omega Limit Sets in Positive Cones

The omega limit sets of a nonlinear operator T which is definedon a positive cone and satisfies certain ray-contractive typeconditions are discussed. Under the assumption that the restrictionof T to a compact subset is surjective, the following alternativesare proved: the omega limit set of a point in the cone eitherconsists of a fixed point or forms a 2-cycle. In addition, newproofs and extensions to relevant results are given. 2000 MathematicsSubject Classification 47H07, 47H09.     

Noncommutative Geometry (An Introduction to Selected Topics)

We give an introduction to noncommutative geometry and to some of its applications. Emphasis will be on noncommutative manifolds, notably noncommutative tori and spheres.     

Applications of Connes'' Geodesic Flow to Trace Formulas in Noncommutative Geometry

The “trace formula” of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifoldXis included in the set of periods of the geodesic flow restricted toS*X. Most of the objects involved in this trace formula have analogues in Connes' Noncommutative Geometry. This paper shows, on several significant examples of Noncommutative Geometry, that Connes' definition of geodesic flow leads to statements analogous to the classical trace formula of Chazarain, Duistermaat, and Guillemin.     

Riemannian Geometry of Conical Singular Sets

In this paper, we study a class of singular Riemannian manifolds. The singular set itself is a smooth manifold with a cone-like neighborhood. By imposing a reasonable convergence condition on the metric, we can determine the local geometrical structure near the singular set. In general, the curvature near the singular set is unbounded. We prove that a bounded curvature assumption would have a strong implication on the geometrical and topological structures near the singular set. We also establish the Gauss–Bonnet–Chern formula, which can be applied to the study of singular Eistein 4-manifolds.     

矩阵代数收敛至球面与非交换度量几何

介绍了Rieffel定义的紧致量子度量空间与量子Gromov-Hausdorff距离和近来Latrémolière定义的量子Gromov-Hausdorff邻距,分别讨论了矩阵代数如何在这两种量子距离下收敛至球面.     

Flattening and Subanalytic Sets in Rigid Analytic Geometry

Let K be an algebraically closed field endowed with a completenon-archimedean norm with valuation ring R. Let f: Y X be amap of K-affinoid varieties. In this paper we study the analyticstructure of the image f(Y) X; such an image is a typical exampleof a subanalytic set. We show that the subanalytic sets areprecisely the D-semianalytic sets, where D is the truncateddivision function first introduced by Denef and van den Dries.This result is most conveniently stated as a Quantifier Eliminationresult for the valuation ring R in an analytic expansion ofthe language of valued rings. To prove this we establish a Flattening Theorem for affinoidvarieties in the style of Hironaka, which allows a reductionto the study of subanalytic sets arising from flat maps, thatis, we show that a map of affinoid varieties can be renderedflat by using only finitely many local blowing ups. The caseof a flat map is then dealt with by a small extension of a resultof Raynaud and Gruson showing that the image of a flat map ofaffinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in thezero characteristic case, a Uniformization Theorem for subanalyticsets: a subanalytic set can be rendered semianalytic using onlyfinitely many local blowing ups with smooth centres. As a corollarywe obtain the fact that any subanalytic set in the plane R2is semianalytic. 2000 Mathematical Subject Classification: 32P05,32B20, 13C11, 12J25, 03C10.     

Limit Sets of Free Convergence Groups

We study the limit sets of convergence groups and prove that the limit set of a free discontinuous cocompact convergence group with an invariant component of the discontinuity domain is a discontinuum.