The equivariant family index theorem in odd dimensions

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes–Chern character of an equivariant Dirac spectral triple. The formula that we obtain is used in the first paper of the series. The computation has two main steps. The first step is the justification that the CM cocycle represents the Connes–Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly–Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner–Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we compute the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.

We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex \((\tilde {b}, d)\); the boundary \(\tilde {b}\) is called topological Hochschild boundary. These ideas combine Connes’ work (Noncommutative geometry, Academic Press, 1994) with ideas of the articles by Teleman and Teleman (St Cerc Math Tom 18(5):753–762, Bucarest, 1966; Rev Roumaine Math Pures Applic 12:725–731, 1967; Central European journal of mathematics. C*-algebras and elliptic theory II. Trends in mathematics, 251–265, Birkhauser, 2008; Local3 index theorem. arXiv: 1109.6095v1, math.KT, 28 Sep. 2011; textitLocal Hochschild homology of Hilbert-Schmidt operators on simplicial spaces. arXiv hal-00707040, Version 1, 11 June 2012; Local algebraic K-theory, 26 Aug 2013 - arXiv.org > math > arXiv:1307.7014, 2013; The local index theorem, HAL-00825083, arXiv 1305.5329, 22 May 2013; \(K_ {i}^{loc}(\ mathbb {C}) \), i = 0, 1 10 set 2013 - arXiv:1309.2421v1 [math.KT] 10 Sep 2013).

In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes Weiping Zhang’s index theorem for sub-signature operators.

We report on the construction of secondary invariants in connection with the Atiyah-Singer index theorem for families, and the Theorem of Riemann-Roch-Grothendieck. The local families index theorem plays an important role in the construction. In complex geometry, the corresponding objects are the analytic torsion forms and the analytic torsion currents. These objects exhibit natural functorial properties with respect to composition of maps. Gillet and Soulé have used these objects to prove a Riemann-Roch theorem in Arakelov geometry. Also we state a Riemann-Roch theorem for flat vector bundles, and report on the construction of corresponding higher analytic torsion forms.

Local atiyah-singer index theorem for families of dirac operators

Superconnection and family Bergman kernels

The noncommutative geometry of graph [formula omitted]-algebras I: The index theorem

In this paper, we give a local Riemann-Roch theorem for certain almost Hermitian manifolds by a canonical connection and the theory of almost Hermitian structure of null eccentricity. In particular, we obtain a local Riemann-Roch theorem for all closed almost complex manifolds of dimension 4. Also, by using the heat equation method, we give an analytic approach to a local index theorem for generalized Dirac operators.

A performance index of local dexterity stiffness index (LDSI) is proposed to consider both kinematic dexterity and static rigidity characteristics of a parallel kinematic machine (PKM). Based on the proposed index, the static performance of an Exe-Variant PKM module is predicted in a quick manner. Since the proposed LDSI is a dimensional index, a dimensionless global index is further presented to optimize the geometric parameters of the PKM module. An optimal searching algorithm is used to obtain reasonable radii of the moving platform and the fixed base of Exe-Variant PKM.

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

Two dynamical indicators, the local dimension and the extremal index, used to quantify persistence in phase space have been developed and applied to different data across various disciplines. These are computed using the asymptotic limit of exceedances over a threshold, which turns to be a generalized Pareto distribution in many cases. However, the derivation of the asymptotic distribution requires mathematical properties, which are not present even in highly idealized dynamical systems and unlikely to be present in the real data. Here, we examine in detail the issues that arise when estimating these quantities for some known dynamical systems. We focus on how the geometry of an invariant set can affect the regularly varying properties of the invariant measure. We demonstrate that singular measures supported on sets of the non-integer dimension are typically not regularly varying and that the absence of regular variation makes the estimates resolution dependent. We show as well that the most common extremal index estimation method is not well defined for continuous time processes sampled at fixed time steps, which is an underlying assumption in its application to data.

Brownian Chen series and Atiyah–Singer theorem

On considere des conditions suffisantes pour qu'un theoreme d'indice local soit vrai pour un operateur de Dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de Levi-Civita L

The structure of smooth algebras in Kapranov's framework for noncommutative geometry

Dedication Preface Historical perspective Conference participants Conference photo Conference timetable 1. On flatness and the Ore condition J. A. Beachy 2. Localization in general rings, a historical survey P. M. Cohn 3. Noncommutative localization in homotopy theory W. G. Dwyer 4. Noncommutative localization in group rings P. A. Linnell 5. A non-commutative generalisation of Thomason's localisation theorem A. Neeman 6. Noncommutative localization in topology A. A. Ranicki 7. L2-Betti numbers, isomorphism conjectures and noncommutative localization H. Reich 8. Invariants of boundary link cobordism II. The Blanchfield-Duval form D. Sheiham 9. Noncommutative localization in noncommutative geometry Z. Skoda.