Symmetries of Analytic Curves

Analytic curves are classified w.r.t. their symmetries under a regular Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves; each of them decomposing naturally into symmetry free subcurves mutually and uniquely related by the group action. We conclude that a connected analytic 1-dimensional submanifold is either analytically diffeomorphic to the unit circle or some interval, or that each point (except for at most countably many) admits a symmetry free chart.Comment: 49 page

STARRY: Analytic Occultation Light Curves

We derive analytic, closed form, numerically stable solutions for the total flux received from a spherical planet, moon or star during an occultation if the specific intensity map of the body is expressed as a sum of spherical harmonics. Our expressions are valid to arbitrary degree and may be computed recursively for speed. The formalism we develop here applies to the computation of stellar transit light curves, planetary secondary eclipse light curves, and planet-planet/planet-moon occultation light curves, as well as thermal (rotational) phase curves. In this paper we also introduce STARRY, an open-source package written in C++ and wrapped in Python that computes these light curves. The algorithm in STARRY is six orders of magnitude faster than direct numerical integration and several orders of magnitude more precise. STARRY also computes analytic derivatives of the light curves with respect to all input parameters for use in gradient-based optimization and inference, such as Hamiltonian Monte Carlo (HMC), allowing users to quickly and efficiently fit observed light curves to infer properties of a celestial body's surface map.Comment: 55 pages, 20 figures. Accepted to the Astronomical Journal. Check out the code at https://github.com/rodluger/starr

Definable sets of Berkovich curves

In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.Comment: 53 pages, 1 figure. v2: Section 7.2 on weakly stable fields added and other minor changes. Final version. To appear in Journal of the Institute of Mathematics of Jussie

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.Comment: 19 pages, This is the final version with significant corrections and improvements. To appear in Arkiv f\"or matemati