Homology of pseudodifferential operators on manifolds with fibered cusps

The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.

     

K-Theory of Suspended Pseudo-Differential Operators

We compute the K-theory groups of Melrose's algebra of 1-suspended pseudo-differential operators. The boundary map in the six-term long exact sequence turns out to be related to both the eta invariant of Melrose and to the index of elliptic operators. The proof is based on a new identity between the formal trace and the Wodzicki residue trace on the suspended algebra.Partially supported by the European Commission RTN HPRN-CT-1999-00118 Geometric Analysis.     

Towards ideal hexagonal self‐ordering of TiO2 nanotubes

The present work reports on key factors that influence the degree of order in anodic TiO₂ nanotube layers. We show that the anodization voltage and the Ti purity are of crucial importance for the ideality of self‐organization within the nanotube layers and that repeated anodization can significantly improve hexagonal ordering. Optimizing each factor significantly reduces the variation in the average pore diameter and strongly reduces the areal density of polygon ordering/packing errors. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in [Zhe Chang, Xin Li, Phys. Lett. B 668 (2008) 453] and [Zhe Chang, Xin Li, Phys. Lett. B 676 (2009) 173], respectively). They consider that “In Finsler manifold, there exists a unique linear connection – the Chern connection … It is torsion freeness and metric compatibility …”. There are well-known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections are not metric compatible. For instance, the Chern's one (being with zero torsion and “weak” compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.