arXiv subjects

Athanase Papadopoulos

Publications and source records attributed to Athanase Papadopoulos.

At least 55 records · Page 3Linked to original sources

arXiv2021

Optimal Lipschitz Maps on One-holed Tori and the Thurston Metric Theory of Teichmueller Space

We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston's stretch maps and prove the following: (1) On the Teichmueller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is =< 4 arcsinh(1), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmueller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

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arXiv2021

Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichm{ü}ller space

These are notes on the hyperbolic geometry of surfaces, Teichm{ü}ller spaces and Thurston's metric on these spaces. They are associated with lectures I gave at the Morningside Center of Mathematics of the Chinese Academy of Sciences in March 2019 and at the Chebyshev Laboratory of the Saint Petersburg State University in May 2019. In particular, I survey several results on the behavior of stretch lines, a distinguished class of geodesics for Thurston's metric and I point out several analogies between this metric and Teichm{ü}ller's metric. Several open questions are addressed. The final version of these notes will appear in the book "Moduli Spaces and Locally Symmetric Spaces", edited by L. Ji and S.-T. Yau, International Press, 2021.

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arXiv2021

Math{é}matiques, musique et cosmologie : A partir du Tim{é}e de Platon

Based on Plato's Timaeus, we present some reflections on music, cosmology and mathematics and their mutual influence.The article is dedicated to the composer Walter Zimmermann. The final version of this article will appear in the volume "Les jeux subtils de la po{é}tique, des nombres et de la philosophie. Autour de la musique de Walter Zimmermann", ed. P. Michel, M. Andreatta and J.-L. Besada, Hermann, Paris, 2021.

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arXiv2020

A glimpse into Thurston's work

We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book "In the tradition of Thurston: Geometry and topology", edited by K. Ohshika and A. Papadopoulos (Springer, 2020).

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arXiv2020

Map drawing and foliations of the sphere

I will consider some questions related to Euler's work on cartography and its consequences, in which the foliations of the sphere by meridians and parallels play important roles.

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arXiv2020

Messiaen et les mathématiques

We review some of Olivier Messiaen's use of mathematics in his composition and his theoretical writings. The final version of this paper appeared in the book Twentieth-Century Music and Mathematics, R. Illiano (ed.), Brepols, Turnhout, 2019.

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arXiv2020

Value distribution theory and Teichm{ü}ller's paper "Einfache Beispiele zur Wertverteilungslehre"

This survey will appear in Vol. VII of the Hendbook of Teichm{ü}ller theory (European Mathematical Society Publishing House, 2020). It is a commentary on Teichm{ü}ller's paper "Einfache Beispiele zur Wertverteilungslehre", published in 1944, whose English translation appears in that volume. Together with Teichm{ü}ller's paper, we survey the development of value distribution theory, in the period starting from Gauss's work on the Fundamental Theorem of Algebra and ending with the work of Teichm{ü}ller. We mention the foundational work of several mathematicians, including Picard, Laguerre, Poincar{é}, Hadamard, Borel, Montel, Valiron, and others, and we give a quick overview of the various notions introduced by Nevanlinna and some of his results on that theory.

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arXiv2020

A note on Nicolas-Auguste Tissot: At the origin of quasiconformal mappings (to appear in Vol. VII of the Handbook of Teichm{ü}ller Theory)

Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name ``Tissot indicatrix'', and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of the earth) by the projection that realizes the geographical map. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distorsion of mappings between general surfaces. His ideas are close to those that are at the origin of the work on quasiconformal mappings that was developed several decades after him by Gr{ö}tzsch, Lavrentieff, Ahlfors and Teichm{ü}ller. Gr{ö}tzsch, in his papers, mentions the work of Tissot, and in some of the drawings he made for his articles, the Tissot indicatrix is represented. Teichm{ü}ller mentions the name Tissot in a historical section in one of his fundamental papers in which he points out that quasiconformal mappings were initially used by geographers. The name Tissot is missing from all the known historical reports on quasiconformal mappings. In the present article, we report on this work of Tissot, showing that the theory of quasiconformal mappings has a practical origin. The final version of this article will appear in Vol. VII of the Handbook of Teichm{ü}ller Theory (European Mathematical Society Publishing House, 2020).

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arXiv2019

A Commentary on Teichm{ü}ller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the \emph{Handbook of Teichmüller theory}

This is a commentary on Teichm{ü}ller's paper Unter-suchungen{ü}ber konforme und quasikonforme Abbildungen (Inves-tigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry , in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a qua-siconformal map. The two results find wide applications in value distribution theory, where they allow the efficient use of moduli of doubly connected domains and of quasiconformal mappings. Te-ichm{ü}ller's paper also contains a thorough development of the theory of conformal invariants of doubly connected domains.The final version of this paper will appear in Vol. VII of the \emph{Handbook of Teichm{ü}ller theory} (European Mathematical Society Publishing House, 2020).

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arXiv2019

TeichmUller's work on the type problem (To appear in Vol. VII of the Handbook of Teichm{ü}ller theory, 2020)

The type problem is the problem of deciding, for a simply connected Riemann surface, whether it is conformally equivalent to the complex plane or to the unit dic in the complex plane. We report on Teichm{ü}ller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal map-pings to the type problem) (1937) and Untersuchungen{ü}ber kon-forme und quasikonforme Abbildungen (Investigations on con-formal and quasiconformal mappings) (1938). They concern simply connected Riemann surfaces defined as branched covers of the sphere. At the same time, we review the theory of line complexes, a combinatorial device used by Teichm{ü}ller and others to encode branched coverings of the sphere. In the first paper, Teichm{ü}ller proves that any two simply connected Riemann surfaces which are branched coverings of the Riemann sphere with finitely many branch values and which have the same line complex are quasiconformally equivalent. For this purpose, he introduces a technique for piecing together quasi-conformal mappings. He also obtains a result on the extension of smooth orientation-preserving diffeomorphisms of the circle to quasiconformal mappings of the disc which are conformal at the boundary. In the second paper, using line complexes, Teichm{ü}ller gives a type criterion for a simply-connected surface which is a branched covering of the sphere, in terms of an adequately defined measure of ramification, defined by a limiting process. The result says that if the surface is ''sufficiently ramified'' (in a sense to be made precise), then it is hyperbolic. In the same paper, Te-ichm{ü}ller answers by the negative a conjecture made by Nevan-linna which states a criterion for parabolicity in terms of the value of a (different) measure of ramification, defined by a limiting process. Teichm{ü}ller's results in his first paper are used in the proof of the results of the second one.The final version of this paper will appear in Vol. VII of the Handbook of Teichm{ü}ller theory (European Mathematical Society Publishing House, 2020).

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arXiv2019

On five papers by Herbert Gr{ö}tzsch

Herbert Gr{ö}tzsch is the main founder of the theory of quasicon-formal mappings. We review five of his papers, written between 1928 and 1932, that show the progress of his work from conformal to quasiconformal geometry. This will give an idea of his motivation for introducing quasicon-formal mappings, of the problems he addressed and on the results he obtained concerning these mappings. The final version of this paper will appear in Vol. VII of the Handbook of Teichm{ü}ller theory.

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arXiv2019

W. P. Thurston and French mathematics

We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appear in the Surveys of the European Mathematical Society.

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arXiv2019

Timelike Hilbert and Funk geometries

A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $ρ$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, $ρ(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between pairs of distinct points in a triple $x,y,z$, whenever these distances are defined, satisfy the so-called \emph{time inequality}, which is a reverse triangle inequality $ρ(x,y)+ρ(y,z)\leq ρ(z,y)$. In the 1960s, Herbert Busemann developed an axiomatic theory of timelike spaces and of locally timelike spaces. His motivation comes from the geometry underlying the theory of relativity, and he tried to adapt to this setting his geometric theory of metric spaces, namely, his theory of $G$-spaces (geodesic spaces). The classical example he considers is the $n$-dimensional Lorentzian space. Two other interesting classes of examples of timelike spaces he introduced are the timelike analogues of the Funk and Hilbert geometries. In this paper, we investigate these two geometries, and in doing this, we introduce variants of them, in particular the timelike relative Funk and Hilbert geometries, in the Euclidean and spherical settings. We describe the Finsler infinitesimal structure of each of these geometries (with an appropriate notion of Finsler structure) and we display the interactions among the Euclidean and timelike spherical geometries. In particular, we characterize the de Sitter geometry as a special case of a timelike spherical Hilbert geometry. The final version of this paper will appear in Differential Geometry and its Applications.

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arXiv2019

Convexity in Greek antiquity

We consider several appearances of the notion of convexity in Greek antiquity, more specifically in mathematics and optics, in the writings of Aristotle, and in art. The final version of this article will appear in the book `Geometry in History', ed. S. G. Dani and A. Papadopoulos, Springer Verlag, 2019.

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arXiv2018

Bijections of geodesic lamination space preserving left Hausdorff convergence

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

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arXiv2018

Topology and biology: From Aristotle to Thom

Ren{é} Thom discovered several refined topological notions in the writings of Aristotle, especially the biological. More generally, he considered that some of the assertions of the Greek philosophers have a definite topological content, even if they were written 2400 years before the field of topology was born. In this article, we expand on these ideas of Thom. At the same time, we highlight the importance of biology in the works of Aristotle and Thom and we report on their conception of mathematics and more generally on science. The final version of this article will appear in the book \emph{Geometry in history} (ed. S. G. Dani and A. Papadopoulos), Springer Verlag, 2019.

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arXiv2018

Homeomorphisms and intersection number

We prove two rigidity results for automorphism groups of the spaces ML(S) of measured laminations on a closed hyperbolic surface S and PML(S) of projective measured laminations on this surface. The results concern the homeomorphisms of ML(S) that preserve the geometric intersection between laminations and the homeomorphisms of PML(S) that preserve the zero sets of these intersection functions.

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arXiv2017

Physics in Riemann's mathematical papers

Riemann's mathematical papers contain many ideas that arise from physics, and some of them are motivated by problems from physics. In fact, it is not easy to separate Riemann's ideas in mathematics from those in physics. Furthermore, Riemann's philosophical ideas are often in the background of his work on science. The aim of this chapter is to give an overview of Riemann's mathematical results based on physical reasoning or motivated by physics. We also elaborate on the relation with philosophy. While we discuss some of Riemann's philosophical points of view, we review some ideas on the same subjects emitted by Riemann's predecessors, and in particular Greek philosophers, mainly the pre-socratics and Aristotle. The final version of this paper will appear in the book: From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017.

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