Real pinor bundles and real Lipschitz structures

We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin structures based on certain groups which we call real Lipschitz groups. In the fiberwise-irreducible case, we classify the latter in all dimensions and signatures. As a simple application, we show that the supersymmetry generator of eleven-dimensional supergravity in "mostly plus" signature can be interpreted as a global section of a bundle of irreducible Clifford modules if and $\textit{only if}$ the underlying eleven-manifold is orientable and spin.Comment: 94 pages, various tables and diagram

Minimal stretch maps between hyperbolic surfaces

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve. Cataclysms are introduced, generalizing earthquakes by permitting more violent shearing in both directions along a fault. Cataclysms provide useful coordinates for Teichmuller space that are convenient for computing derivatives of geometric function in Teichmuller space and measured lamination space.Comment: 53 pages, 11 figures, version of 1986 preprin

Quantum-Gravity Analysis of Gamma-Ray Bursts using Wavelets

In some models of quantum gravity, space-time is thought to have a foamy structure with non-trivial optical properties. We probe the possibility that photons propagating in vacuum may exhibit a non-trivial refractive index, by analyzing the times of flight of radiation from gamma-ray bursters (GRBs) with known redshifts. We use a wavelet shrinkage procedure for noise removal and a wavelet `zoom' technique to define with high accuracy the timings of sharp transitions in GRB light curves, thereby optimizing the sensitivity of experimental probes of any energy dependence of the velocity of light. We apply these wavelet techniques to 64 ms and TTE data from BATSE, and also to OSSE data. A search for time lags between sharp transients in GRB light curves in different energy bands yields the lower limit $M \ge 6.9 \cdot 10^{15}$ GeV on the quantum-gravity scale in any model with a linear dependence of the velocity of light $~ E/M$. We also present a limit on any quadratic dependence.Comment: This version is accepted for publication in Astronomy & Astrophysics. The discussion and introduction are extended making clear why the wavelet analysis should be superior to straight cross-correlation analysis. More details on compiled data are elaborated. 18 pages, 9 figures, A&A forma

Malliavin Calculus for regularity structures: the case of gPAM

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accomodates a robust, and purely deterministic, translation operator, in $L^2$-directions, between "models". In the concrete context of the generalized parabolic Anderson model in 2D - one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.Comment: Minor revision of [v1]. This version published in Journal of Functional Analysis, Volume 272, Issue 1, 1 January 2017, Pages 363-41

Foundations for a theory of emergent quantum mechanics and emergent classical gravity

Quantum systems are viewed as emergent systems from the fundamental degrees of freedom. The laws and rules of quantum mechanics are understood as an effective description, valid for the emergent systems and specially useful to handle probabilistic predictions of observables. After introducing the geometric theory of Hamilton-Randers spaces and reformulating it using Hilbert space theory, a Hilbert space structure is constructed from the Hilbert space formulation of the underlying Hamilton-Randers model and associated with the space of wave functions of quantum mechanical systems. We can prove the emergence of the Born rule from ergodic considerations. A geometric mechanism for a natural spontaneous collapse of the quantum states based on the concentration of measure phenomena as it appears in metric geometry is discussed.We show the existence of stable vacua states for the quantized matter Hamiltonian. Another consequence of the concentration of measure is the emergence of a weak equivalence principle for one of the dynamics of the fundamental degrees of freedom. We suggest that the reduction of the quantum state is driven by a gravitational type interaction. Such interaction appears only in the dynamical domain when localization of quantum observables happens, it must be a classical interaction. We discuss the double slit experiment in the context of the framework proposed, the interference phenomena associated with a quantum system in an external gravitational potential, a mechanism explaining non-quantum locality and also provide an argument in favour of an emergent interpretation of every macroscopic time parameter. Entanglement is partially described in the context of Hamilton-Randers theory and how naturally Bell's inequalities should be violated.Comment: Extensive changes in chapter 1 and chapter 2; minor changes in other chapters; several refereces added and others update; 192 pages including index of contents and reference

Periodic solutions of o.d.e. systems with a lipchitz non linearity

In this report, we address differential systems with Lipschitz non linearities; this study is motivated by the subject of vibrations of structures with unilateral springs or non linear stress-strain law close to the linear case. We consider existence and solution with fixed point methods; this method is constructive and provides a numerical algorithm which is under study. We describe the method for a static case example and we address periodic solutions of differential systems arising in the vibration of structures

A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations

AbstractWe introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions