Athanase Papadopoulos

Athanase Papadopoulos
University of Strasbourg | UNISTRA · Institut de Recherche Mathématique Avancée

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292
Publications
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2,288
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Publications

Publications (292)
Article
In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichmüller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit circle at each point in the tangent space to Teichmüller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asym...
Preprint
We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichm{\"u}ller space can recover the marking and geometry of this marked surface. We then translate...
Article
Full-text available
We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller 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 fo...
Preprint
This is a biography and a report on the work of Vladimir Turaev. Using fundamental techniques that are rooted in classical topology, Turaev introduced new ideas and tools that transformed the field of knots and links and invariants of 3-manifolds. He is one of the main founders of the new topic called quantum topology. In surveying Turaev's work, t...
Preprint
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Given two triangles whose angles are all acute, we find a homeomorphism with the smallest Lipschitz constant between them and we give a formula for the Lipschitz constant of this map. We show that on the set of pairs of acute triangles with fixed area, the function which assigns the logarithm of the smallest Lipschitz constant of Lipschitz maps bet...
Article
On the occasion of Chebyshev’s twohundredth anniversary, I review part of his work, showing that in several respects he was the heir of Euler. In doing this, I consider the works of Euler and Chebyshev on three topics in applied science: industrial machines, ballistics and geography, and then on three topics in pure mathematics: integration, contin...
Preprint
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These are notes on the hyperbolic geometry of surfaces, Teichm{\"u}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,...
Preprint
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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{\'e}tique, des nombres et de la philosophie. Autour de la musique de Walter Zimmermann...
Chapter
We present an overview of some significant results of Thurston and their impact on mathematics.
Preprint
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).
Preprint
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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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.
Preprint
In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{\"u}ller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space to Teichm{\"u}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thursto...
Preprint
Full-text available
This survey will appear in Vol. VII of the Hendbook of Teichm{\"u}ller theory (European Mathematical Society Publishing House, 2020). It is a commentary on Teichm{\"u}ller's paper "Einfache Beispiele zur Wertverteilungslehre", published in 1944, whose English translation appears in that volume. Together with Teichm{\"u}ller's paper, we survey the d...
Preprint
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...
Book
This book consists of 16 surveys on Thurston's work and its later development. The authors are mathematicians who were strongly influenced by Thurston's publications and ideas. The subjects discussed include, among others, knot theory, the topology of 3-manifolds, circle packings, complex projective structures, hyperbolic geometry, Kleinian groups,...
Preprint
Full-text available
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{\"u}ller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasi...
Preprint
Full-text available
This is a commentary on Teichm{\"u}ller's paper Unter-suchungen{\"u}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 loc...
Preprint
Herbert Gr{\"o}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 resu...
Preprint
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 Europ...
Chapter
René Thom discovered several refined topological notions in the writings of Aristotle, especially the biological ones. More generally, he understood that some assertions made by philosophers from Greek antiquity have a definite topological content, even if they were stated more than two and a half millennia before the field of topology was born. He...
Chapter
We describe a path in the history of curvature, starting from Greek antiquity, in the works of Euclid, Apollonius, Archimedes and a few others, passing through the works of Huygens, Euler, and Monge and his students, and ending in the twentieth century at the works of Bonnesen, Fenchel, Busemann, Feller and Alexandrov. Our goal is not to review the...
Chapter
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.
Preprint
We study Thurston's Lipschitzand curve metrics, as well as the arc metric on the Teichmueller space of the torus equipped with hyperbolic metrics eith one boundary component of fixed length. We construct natural Lipschitz maps between two such hyperbolic surfaces that generalize Thurston's stretch maps. The construction is based on maps between ide...
Article
We present three theorems due to Menelaus of Alexandria (1st–2nd century A.D.) that concern spherical triangles. The theorems are extracted from his Spherics, a treatise on spherical geometry which remains unknown to mathematicians. We also comment on these theorems and make relations with modern mathematical works. Menelaus’ treatise contains 91 p...
Article
Full-text available
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 exte...
Preprint
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.
Preprint
Full-text available
Ren{\'e} 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...
Article
Full-text available
Résumé On démontre deux résultats de rigidité pour des groupes d'automorphismes de l'espace ML(S) des laminations géodésiques mesurées d'une surface hyperbolique fermée orientable S et de l'espace PML(S) des laminations géodésiques mesurées projectives de S. Les résultats concernent les automorphismes de ML(S) préservant le nombre d'intersection gé...
Preprint
We prove a rigidity result for the action of the mapping class group on the space of geodesic laminations of a closed hyperbolic surface of genus g $\ge$ 2 equipped with the left Hausdorff topology.
Preprint
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...
Article
Full-text available
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...
Article
Full-text available
This is a biography of Herbert Busemann (1905--1994). The final version will appear in Volume I of the Selected Works of Herbert Busemann (2 volumes, Springer Verlag, to appear in 2017).
Article
We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and th...
Chapter
In this chapter, we review the works of Cauchy and Puiseux on the theory of functions of a complex variable that preceded Riemann’s introduction of what soon became known as Riemann surfaces. The work of the two French mathematicians (especially that of Puiseux) inaugurates a group-theoretic point of view which complements the topological one disco...
Chapter
Riemann’s mathematical papers contain many ideas that arise in 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 there 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 a...
Chapter
Riemann introduced in his doctoral dissertation (1851) the concept of Riemann surface as a new ground space for meromorphic functions and in particular as a domain for a multi-valued function defined by an algebraic equation such that this function becomes single-valued when its is defined on its associated Riemann surface. It took several years to...
Chapter
We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and th...
Article
This is the introduction I wrote for the multi-authored book "From Riemann to differential geometry and relativity", edited by L. Ji, A. Papadopoulos and S. Yamada (Berlin, Springer verlag, 2017). The book consists of twenty chapters, written by various authors. This introduction, besides giving the information on the content of the book, is a quic...
Article
Full-text available
Nicolas-Auguste Tissot (1824--1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the "Tissot indicatrix." This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot in...
Article
Full-text available
We construct one-parameter families of right-angled hexagons in the hyperbolic plane such that each right-angled hexagon belongs to such a family, and between each pair of hexagons in the same family we describe a Lipschitz map that realizes the best Lipschitz constant in its homotopy class relative to the boundary. This produces new geodesics for...
Article
The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, hig...
Book
This book explores the work of Bernhard Riemann and its impact on mathematics, philosophy and physics. It features contributions from a range of fields, historical expositions, and selected research articles that were motivated by Riemann’s ideas and demonstrate their timelessness. The editors are convinced of the tremendous value of going into Rie...

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