# Athanase PapadopoulosUniversity of Strasbourg | UNISTRA · Institut de Recherche Mathématique Avancée

Athanase Papadopoulos

## About

343

Publications

45,014

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

2,785

Citations

## Publications

Publications (343)

We continue the study of the analogue of Thurston’s metric on the Teichmüller space of Euclidean triangles which was started by Saglam–Papadopoulos (Minimal stretch maps between Euclidean triangles, 2022). By direct calculation, we give explicit expressions of the distance function and the Finsler structure of the metric restricted to the subspace...

In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichm{\"u}ller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the...

Teichmüller spaces play a major role in many areas of mathematics and physical science. The subject of the conference was recent developments of Teichmüller theory with its different ramifications that include the classical, the higher, the super and the quantum aspects of the theory.

This is an overview of the life and works of Pavel Florensky, an important and singular figure of the period rightly described as the Silver Age of Russian mathematics, with a substantial overlap with the Silver Age of Russian literature, poetry and philosophy. Florensky is certainly among the great scientists, philosophers, theologians and histori...

We continue the study of the analogue of Thurston's metric on the Teichm{\"u}ller space of Euclidean triangle which was started by Saglam and Papadopoulos in [1].By direct calculation, we give explicit expressions of the distance function and the Finsler structure of the metric restricted to the subspace of acute triangles.We deduce from the form o...

This is a personal view of and commentary on Yuri Manin's relationship with mathematics, philosophy, literature, poetry, art, and culturel. His life epitomized those of the great natural philosophers of yore.

We prove the following result on the timelike spherical Hilbert geometry of simplices: Let $\Delta_2$ be a simplex on the 2-sphere and $\tilde{\Delta}_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair $\Delta_2, \tilde{\Delta}_2$ is isometric to a union of six copies of vector spaces equipped wi...

These are notes on the impact of Lagrange's memoir on the construction of geographical maps. We mention the relations of some ideas and questions introduced in this memoir with other notions that appeared later in the works of several mathematicians, including in particular Chebyshev (19th c.) and Darboux (19th-20th c.), two mathematicians who were...

In this paper, we introduce a new asymmetric weak metric on the Teichm{\"u}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{\"u}ller-Randers metric, is an asymmetric deformation of the Teichm{\"u}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{\"u}ller...

Joseph-Nicolas Delisle was one of the most important scientists at the Saint Petersburg Academy of Sciences during the first period when Euler was working there. Euler was helping him in his work on astronomy and in geography. In this paper, Delisle's geographical projection is presented and Euler's study of this projection isexplained, highlightin...

I discuss Ren{\'e} Thom's approach to philosophy based on his mathematical background. At the same time, I will highlight his connection with Aristotle, his criticism of the modern view of science as a predictive process, his ideas on mathematical education, his position with respect to the French school of mathematics that was dominent in his time...

The sphericity of the form of the Earth was questioned around the year 1687, primarily, by Isaac Newton who deduced from his theory of universal gravitation that the Earth has the form of a spheroid flattened at the poles and elongated at the equator. In France, somepreeminent geographers were not convinced by Newton's arguments, and about the same...

Delisle’s geographical projection is presented and Euler’s study of this projection is explained, highlighting some mathematical points.

In this chapter, we review Euler’s work on geography in relation with that of the French astronomer, geographer, philosopher and naturalist Pierre Louis Moreau de Maupertuis. The latter was the president of the Prussian Academy of Sciences during the period in which Euler worked there, and he exerted some influence on him. More especially, we shall...

In this chapter, we give an overview of Leonhard Euler’s work as a geographer. In particular, we shall linger on it relation with the famous French geographer and astronomer Joseph-Nicolas Delisle, with whom he collaborated during his first stay at the Saint Petersburg Academy of Sciences (1727–1741). Delisle, who played an important role in Euler’...

These are notes on the impact of Lagrange’s memoir on the construction of geographical maps. We mention the relations of some ideas and questions introduced in this memoir with other notions that appeared later in the works of several mathematicians including Chebyshev (19th c.) and Darboux (19th–20th c.), two mathematicians who were particularly i...

In this chapter, we highlight some landmarks in the early history of geography, with a stress on its mathematical component: establishing coordinates, measuring large distances using astronomical data, and drawing geographical maps based on various types of projections from the sphere onto the plane. We shall mention works by several mathematicians...

This article is a personal overview of the work of Dennis Sullivan who was awarded the 2022 Abel prize. It was commissioned by the Bulletin of the (Indian) Mathematics Consortium, and it will appear there.

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...

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...

We consider the mathematical theory of geographical maps, with an emphasis on the eighteenth century works of Euler, Lagrange and Delisle. This period is characterized by the frequent use of maps that are no more obtained by the stereographic projection or its variations, but by much more general maps from the sphere to the plane. More especially,...

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...

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...

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...

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...

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,...

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...

We present an overview of some significant results of Thurston and their impact on mathematics.

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).

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.

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.

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...

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...

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...

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,...

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...

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...

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...

We give a general overview of the influence of Thurston on the French mathematical community and we show how some of the major problems he solved have their roots in the French mathematical tradition. At the same time, we survey some of Thurston’s major results and their impact.

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...

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...

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...

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.
AMS classification:
01-0201A2034-0234-0354-0392B99

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...

In this workshop, various topics in Teichmüller theory and mapping class groups were discussed. Twenty-three talks dealing with classical topics and new directions in this field were given. A problem session was organised on Thursday, and we compiled in this report the problems posed there.

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...

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...

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.

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...

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...

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...