The domination monoid in henselian valued fields

We study the domination monoid in various classes of structures arising from the model theory of henselian valuations, including RV-expansions of henselian valued fields of residue characteristic 0 (and, more generally, of benign valued fields), p-adically closed fields, monotone D-henselian differential valued fields with many constants, regular ordered abelian groups, and pure short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type reductions to suitable fully embedded families of sorts in quite general settings, and full computations in concrete ones.Comment: 35 pages. Minor revisio

Dynamical Origin of the Lorentzian Signature of Spacetime

It is suggested that not only the curvature, but also the signature of spacetime is subject to quantum fluctuations. A generalized D-dimensional spacetime metric of the form $g_{\mu \nu}=e^a_\mu \eta_{ab} e^b_\nu$ is introduced, where $\eta_{ab} = diag\{e^{i\theta},1,...,1\}$. The corresponding functional integral for quantized fields then interpolates from a Euclidean path integral in Euclidean space, at $\theta=0$, to a Feynman path integral in Minkowski space, at $\theta=\pi$. Treating the phase $e^{i\theta}$ as just another quantized field, the signature of spacetime is determined dynamically by its expectation value. The complex-valued effective potential $V(\theta)$ for the phase field, induced by massless fields at one-loop, is considered. It is argued that $Re[V(\theta)]$ is minimized and $Im[V(\theta)]$ is stationary, uniquely in D=4 dimensions, at $\theta=\pi$, which suggests a dynamical origin for the Lorentzian signature of spacetime.Comment: 6 pages, LaTe

Quasilinear continuity equations of measures for bounded BV vector fields

The focus of interest here is a quasilinear form of the conservative continuity equation d/dt v + D·(f(v, ·) v) = 0 (in R^N× ]0, T[) together with its measure-valued distributional solutions. On the basis of Ambrosio’s results about the nonautonomous linear equation, the existence and uniqueness of solutions are investigated for coefficients being bounded vector fields with bounded spatial variation and Lebesgue absolutely continuous divergence in combination with positive measures absolutely continuous with respect to Lebesgue measure. The step towards the nonlinear problem here relies on a further generalization of Aubin's mutational equations that is extending the notions of distribution-like solutions and "weak compactness" to a set supplied with a countable family of (possibly non–symmetric) distance functions (so–called ostensible metrics)

On the Construction of Simply Connected Solvable Lie Groups

Let $\omega_\mathfrak{g}$ be a Lie algebra valued differential $1$-form on a manifold $M$ satisfying the structure equations $d \omega_\mathfrak{g} + \frac{1}{2} \omega_\mathfrak{g}\wedge \omega_\mathfrak{g}=0$ where $\mathfrak{g}$ is solvable. We show that the problem of finding a smooth map $\rho:M\to G$, where $G$ is an $n$-dimensional solvable Lie group with Lie algebra $\mathfrak{g}$ and left invariant Maurer-Cartan form $\tau$, such that $\rho^* \tau= \omega_\mathfrak{g}$ can be solved by quadratures and the matrix exponential. In the process we give a closed form formula for the vector fields in Lie's third theorem for solvable Lie algebras. A further application produces the multiplication map for a simply connected $n$-dimensional solvable Lie group using only the matrix exponential and $n$ quadratures. Applications to finding first integrals for completely integrable Pfaffian systems with solvable symmetry algebras are also given.Comment: 22 pages. Fixed typos from version 1, and added more details in the example

A pp-adic analogue of Borel's theorem

We prove that Shimura varieties of abelian type satisfy a $p$-adic Borel-extension property over discretely valued fields. More precisely, let $\mathsf{D}$ denote the rigid-analytic closed unit disc and $\mathsf{D}^{\times} = \mathsf{D} \setminus \{0\}$, let $X$ be a smooth rigid-analytic variety, and let $S(G,\mathcal{H})_{\mathsf{K}}$ denote a Shimura variety of abelian type with torsion-free level structure. We prove every rigid-analytic map defined over a discretely valued $p$-adic field $\mathsf{D}^{\times} \times X \rightarrow S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{an}}$ extends to an analytic map $\mathsf{D} \times X \rightarrow (S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{BB}})^{\textrm{an}}$, where $S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{BB}}$ is the Baily-Borel compactification of $S(G,\mathcal{H})_{\mathsf{K}}$. We also deduce various applications to algebraicity of analytic maps, degenerations of families of abeloids, and to $p$-adic notions of hyperbolicity. Along the way, we also prove an extension result for Rapoport-Zink spaces