Computing Teichm\"{u}ller Maps between Polygons

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichm\"{u}ller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $\epsilon$ of the dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.Comment: 28 pages, 6 figure

Zero Lyapunov exponents of the Hodge bundle

By the results of G. Forni and of R. Trevi\~no, the Lyapunov spectrum of the Hodge bundle over the Teichm\"uller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichm\"uller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure.Comment: 47 pages, 10 figures. Final version (based on the referee's report). A slightly shorter version of this article will appear in Commentarii Mathematici Helvetici. A pdf file containing a copy of the Mathematica routine "FMZ3-Zariski-numerics_det1.nb" is available at this link here: http://w3.impa.br/~cmateus/files/FMZ3-Zariski-numerics_det1.pd

Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes

We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are derived from the parametrisation of the moduli spaces by the bundle of measured geodesic laminations over Teichm\"uller space of S and can be viewed as analytic continuations of the shear coordinates on Teichm\"uller space. In terms of these coordinates the gravitational symplectic structure takes a particularly simple form, which resembles the Weil-Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichm\"uller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichm\"uller space, corresponding to different values of the curvature.Comment: 40 pages, 6 figure

Structure and Diagenesis in Upper Carboniferous Tight Gas Reservoirs in NW Germany

Upper Carboniferous sandstones are important tight gas reservoirs in Central Europe. This field-based study, conducted in a km-scale reservoir outcrop analog (Piesberg quarry, Lower Saxony Basin, NW Germany), focused on the diagenetic control on spatial reservoir quality distribution. Geothermometers were used to characterize a fault-related thermal anomaly. A prototype workflow based on terrestrial laser scanning is presented, which allowed for the automated detection and analysis of fractures

Higher spin JT gravity and a matrix model dual

We propose a generalization of the Saad-Shenker-Stanford duality relating matrix models and JT gravity to the case in which the bulk includes higher spin fields. Using a $\textsf{PSL}(N,\mathbb{R})$ BF theory we compute the disk and generalization of the trumpet partition function in this theory. We then study higher genus corrections and show how this differs from the usual JT gravity calculations. In particular, the usual quotient by the mapping class group is not enough to ensure finite answers and so we propose to extend this group with additional elements that make the gluing integrals finite. These elements can be thought of as large higher spin diffeomorphisms. The cylinder contribution to the spectral form factor then behaves as $T^{N-1}$ at late times $T$, signaling a deviation from conventional random matrix theory. To account for this deviation, we propose that the bulk theory is dual to a matrix model consisting of $N-1$ commuting matrices associated to the $N-1$ conserved higher spin charges. We find further evidence for the existence of the additional mapping class group elements by interpreting the bulk gauge theory geometrically and employing the formalism developed by Gomis et al. in the nineties. This formalism introduces additional (auxiliary) boundary times so that each conserved charge generates translations in those new directions. This allows us to find an explicit description for the $\textsf{PSL}(3,\mathbb{R})$ Schwarzian theory for the disk and trumpet and view the additional mapping class group elements as ordinary Dehn twists, but in higher dimensions.Comment: 44 pages, 7 figure

Cutting sequences on Bouw-Moeller surfaces : an S-adic characterization.

Résumé. On considère un codage symbolique des géodésiques sur une famille de surfaces de Veech (surfaces de translation riches en symétries affines) récemment découverte par Bouw et Möller. Ces surfaces, comme l’a remarqué Hooper, peuvent être réalisées en coupant et collant une collection de polygones semi-réguliers. Dans cet article, on caractérise l’ensemble des suites symboliques (“suites de coupage”) qui correspondent au codage de trajectoires linéaires, à l’aide de la suite des côtés des polygones croisés. On donne une caractérisation complète de l’adhérence de l’ensemble des suites de coupage, dans l’esprit de la caractérisation classique des suites sturmiennes et de la récente caractérisation par Smillie-Ulcigrai des suites de coupage des trajectoires linéaires dans les polygones réguliers. La caractérisation est donnée en termes d’un système fini de substitutions (connu aussi sous le nom de présentation S-adique), réglé par une transformation unidimensionnelle qui ressemble à l’algorithme de fraction continue. Comme dans le cas sturmien et dans celui des polygones réguliers, la caractérisation est basée sur la renormalisation et sur la définition d’un opérateur combinatoire de dérivation approprié. Une des nouveautés est que la dérivation se fait en deux étapes, sans utiliser directement les éléments du groupe de Veech, mais en utilisant un difféomorphisme affine qui envoie une surface de Bouw-Möller vers sa surface “duale”, qui est dans le même disque de Teichmüller. Un outil technique utilisé est la présentation des surfaces de Bouw-Möller par les diagrammes de Hooper. ABSTRACT. We consider a symbolic coding for geodesics on the family of Veech surfaces (translation surfaces rich with affine symmetries) recently discovered by Bouw and Möller. These surfaces, as noticed by Hooper, can be realized by cutting and pasting a collection of semi-regular polygons. We characterize the set of symbolic sequences (cutting sequences) that arise by coding linear trajectories by the sequence of polygon sides crossed. We provide a full characterization for the closure of the set of cutting sequences, in the spirit of the classical characterization of Sturmian sequences and the recent characterization of Smillie-Ulcigrai of cutting sequences of linear trajectories on regular polygons. The characterization is in terms of a system of finitely many substitutions (also known as an S-adic presentation), governed by a one-dimensional continued fraction-like map. As in the Sturmian and regular polygon case, the characterization is based on renormalization and the definition of a suitable combinatorial derivation operator. One of the novelties is that derivation is done in two steps, without directly using Veech group elements, but by exploiting an affine diffeomorphism that maps a Bouw- Möller surface to the dual Bouw-Möller surface in the same Teichmüller disk. As a technical tool, we crucially exploit the presentation of Bouw-Möller surfaces via Hooper diagrams

An evaluation on the clinical outcome prediction of rotor detection in non-invasive phase maps

[EN] Phase maps obtained from Electrocardiographic imaging (ECGI) have been used in the past for rotor identification and ablation guidance in atrial fibrillation (AF). In this study, we propose a new rotor detection algorithm and evaluate its potential use for prediction of pulmonary vein isolation (PVI) success. The mean precision and recall of the algorithm were evaluated by using manually annotated ECGI phase maps and resulted in 0.82 and 0.75, respectively. Phase singularities and rotors were then quantified on ECGI signals from 29 patients prior to PVI. A significantly higher concentration of phase singularities (PSs) in the pulmonary veins in patients with a successful PVI was found. Our results suggest that rotorrelated metrics obtained from ECGI derived phase maps contain relevant information to predict clinical outcome in PVI patients.This work was supported by PersonalizeAF project. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skodowska-Curie grant agreement No 860974.Fambuena-Santos, C.; Hernández-Romero, I.; Molero-Alabau, R.; Climent, AM.; Guillem Sánchez, MS. (2021). An evaluation on the clinical outcome prediction of rotor detection in non-invasive phase maps. 1-4. https://doi.org/10.22489/CinC.2021.2511