Higher spectral flow and an entire bivariant JLO cocycle

Given a smooth fibration of closed manifolds and a family of generalised Dirac operators along the fibers, we define an associated bivariant JLO cocycle. We then prove that, for any $\ell \geq 0$, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the $C^{\ell+1}$ topology and functions on the base manifold with the $C^\ell$ topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fr\'echet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow

Geodesic fields for Pontryagin type C0C^0-Finsler manifolds

Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $x\in M$ and $TM=\{(x,y);x\in M;y \in T_xM\}$ be its tangent bundle. A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow \mathbb [0,\infty)$ such that $F(x,\cdot): T_xM \rightarrow [0,\infty)$ is an asymmetric norm. In this work we introduce the Pontryagin type $C^0$-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin's maximum principle for the problem of minimizing paths. We define the extended geodesic field $\mathcal E$ on the slit cotangent bundle $T^\ast M\backslash 0$ of $(M,F)$, which is a generalization of the geodesic spray of Finsler geometry. We study the case where $\mathcal E$ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by $\mathcal E$ than by a similar structure on $TM$. Finally we show that the maximum of independent Finsler structures is a Pontryagin type $C^0$-Finsler structure where $\mathcal E$ is a locally Lipschitz vector field.Comment: 41 pages, 4 figure

The functional analytic foundation of Colombeau algebras

Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a functional analytic approach placing these algebras in a unifying hierarchy, which clarifies their structural properties as well as their relation to each other.Comment: 31 pages; updated section on sheaf propertie