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Athanase Papadopoulos

Publications and source records attributed to Athanase Papadopoulos.

At least 37 records · Page 2Linked to original sources
arXiv2024

The earthquake metric on Teichm{ü}ller space

This is the first paper to systematically study the earthquake metric, an asymmetric Finsler metric on Teichm{ü}ller space introduced by Thurston. We provide proofs for several assertions of Thurston and establish new properties of this metric, among which are incompleteness, asymptotic distance to the boundary and comparisons with the Thurston metric and the Weil--Petersson metric. In doing so, we propose a novel asymmetric generalisation of the notion of completion for symmetric metrics, which we call the FD-completion, and prove that for the earthquake metric the FD-completion and various symmetrised metric completions coincide with the Weil--Petersson completion. We also answer a question of Thurston by giving an interpretation of this metric arising from a global minimisation problem, namely, the earthquake magnitude minimisation problem. At several points of this paper, we formulate a certain number of open problems which will show that the earthquake metric constitutes a promising subject.

math.GT↗
arXiv2024

Geometry on surfaces, a source for mathematical developments

We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results for such structures. Higher-dimensional analogues are also discussed. Some constructions with Riemann surfaces lead, by analogy, to notions that hold for arbitrary fields, and not only the field of complex numbers. The Riemann sphere is also defined using surjective homomorphisms of real algebras from the ring of real univariate polynomials to (arbitrary) fields, in which the field with one element is interpreted as the point at infinity of the Gaussian plane of complex numbers. Several models of the hyperbolic plane and hyperbolic 3-space appear, defined in terms of complex structures on surfaces, and in particular also a rather elementary construction of the hyperbolic plane usingreal monic univariate polynomials of degree two without real roots. Several notions and problems connected with conformal structures in dimension 2 are discussed, including dessins d'enfants, the combinatorial characterization of polynomials and rational maps of the sphere, the type problem, uniformization, quasiconformal mappings, Thurston's characterization of Speiser graphs, stratifications of spaces of monic polynomials, and others. Classical methods and new techniques complement each other. The final version of this paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A. Papadopoulos), Springer Nature Switzerland, 2024.

math.CV↗
arXiv2024

Some footnotes on Thurston's Notes ''The Geometry and Topology of 3-manifolds''

These are a few historical remarks, addenda and references with comments on some topics discussed by Thurston in his notes ''The geometry and topology of three-manifolds''. The topics are mainly hyperbolic geometry, geometric structures, volumes of hyperbolic polyhedra and the so-called Koebe--Andreev--Thurston theorem. I discuss in particular some works of Lobachevsky, Andreev and Milnor, with an excursus in Dante's cosmology, based on the insight of Pavel Florensky. The final version of this paper will appear as a chapter in the book ''In the tradition of Thurston. III'', ed. K. Ohshika and A. Papadopoulos, Springer Nature Switzerland, 2024.

math.GT↗
arXiv2024

The infinitesimal and global Thurston geometry of Teichm{ü}ller space

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{ü}ller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.

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arXiv2023

Café conversations: A tribute to Norbert A'Campo

This is an intrusion in the life and the mathematics of Norbert A'Campo, intended to be a tribute to him and an acknowledgement of his impact on those who know him and his work. The final version of this paper appears in the book ``Essays in Geometry, Dedicated to Norbert A'Campo'', IRMA Lectures in Mathematics and Theoretical Physics, EMS Press, Berlin, 2023.

math.HO↗
arXiv2023

Pavel Florensky and his world

This is an overview of the life and works of Pavel Florensky, an important and singular figure of the period rightly described as the \emph{Silver Age of Russian mathematics}, with a substantial overlap with the \emph{Silver Age of Russian literature, poetry and philosophy}. Florensky is certainly among the great scientists, philosophers, theologians and historians of art of the twentieth century, with a very atypical trajectory of life. He had a mathematical background and his work in philosophy, theology and history of art is imbued with mathematical ideas. Talking about his life and works is also an opportunity to reflect upon the Russian mathematical school of the first third of the twentieth century, its philosophical foundations and the conflicts it underwent. It is also an occasion for discussing poetry, literature and art during the Russian Silver Age. This article will appear as a chapter in the Handbook of the History and Philosophy of Mathematical Practice, edited by Bharath Sriraman, published by Springer, to appear in 2023.

math.HO↗
arXiv2023

On the Teichm{ü}ller space of acute triangles

We continue the study of the analogue of Thurston's metric on the Teichm{ü}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 of the Finsler unit sphere a result on the infinitesimal rigidity of the metric.We give a description of the maximal stretching loci for a family of extreme Lipschitz maps.

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arXiv2023

Timelike Hilbert geometry of the spherical simplex

We prove the following result on the timelike spherical Hilbert geometry of simplices: Let $Δ_2$ be a simplex on the 2-sphere and $\tildeΔ_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair $Δ_2, \tildeΔ_2$ is isometric to a union of six copies of vector spaces equipped with a timelike norm, isometrically and transitively acted upon by the group $\mathbb{R}_{>0}^2 \times \mathbb{Z}_3\times \mathbb{Z}_2$. This is a timelike spherical analogue of a well-known result (due to Busemann) stating that the Hilbert metric of a Euclidean simplex is isometric to a metric induced by a normed vector space. At the same time, this gives a new example of timelike space.

math.DG↗
arXiv2023

Some notes on the impact of Lagrange's memoir "On the construction of geographical maps"

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 particularly interested in geography.

math.DG↗
arXiv2022

The Teichm{ü}ller-Randers metric

In this paper, we introduce a new asymmetric weak metric on the Teichm{ü}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{ü}ller-Randers metric, is an asymmetric deformation of the Teichm{ü}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{ü}ller metric a differential 1-form. We study basic properties of the Teichm{ü}ller-Randers metric. In the case when the 1-form is exact, any Teichm{ü}ller geodesic between two points is a unique Teichm{ü}ller--Randers geodesic between them. A particularly interesting case is when the differential 1-form is (up to a factor) the differential of the logarithm of the extremal length function associated with a measured foliation. We show that in this case the Teichm{ü}ller-Randers metric is incomplete in any Teichm{ü}ller disc, and we give a characterisation of geodesic rays with bounded length in this disc in terms of their directing measured foliations.

math.CV↗
arXiv2022

On Delisle's geographical projection

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, highlighting some important mathematical points, in particular on the metric geometry of surfaces.The final version of this paper will appear in the book \emph{Mathematical Geography in the Eighteenth Century: Euler, Lagrange and Lambert}, ed. Renzo Caddeo and Athanase Papadopoulos, Springer International Publishing, 2022.

math.HO↗
arXiv2022

Ren{é} Thom: From mathematics to philosophy

I discuss Ren{é} 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 and his relationship with the philosophical community. I will also touch upon the connections between Thom's ideas and those of Leibniz, Riemann, Freud and others. The last version of this paper will appear as a chapter in the book Handbook of the History and Philosophy of Mathematical Practice (ed. Bharath Sriraman), Springer.

math.HO↗
arXiv2022

Clairaut, Euler and the figure of the Earth

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 period, based on empirical measurements, they emitted another theory, claiming that on the contrary, the Earth has the form of a spheroid flattened at the equator and elongated at the poles. To find the real figure of the Earth became one of the major questions that were investigated by geographers, astronomers, mathematicians and other scientists in the eighteenth century, and the work done around this question had an impact on the development of all these fields. In this paper, we review the work of the eighteenth-century French mathematician, astronomer and geographer Alexis-Claude Clairaut related to the question of the figure of the Earth. We report on the relation between this work and that of Leonhard Euler. At the same time, we comment on the impact of the question of the figure of the Earth on mathematics, astronomy and hydrostatics. Finally, we review some later mathematical developments that are due to various authors that were motivated by this question. It is interesting to see how a question on geography had such an impact on the theoretical sciences. The final version of this paper will appear in Ganita Bh{ā}r{ā}t{ī}, (Indian Mathematics) the Bulletin of the Indian Society for History of Mathematics.

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arXiv2021

Mathematics and map drawing in the eighteenth century

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, the characteristics of the desired geographical maps were formulated in terms of an appropriate choice of the images of the parallels and meridians, and the mathematical properties required by the map concern the distortion of the maps restricted to these lines. The paper also contains some notes on the general use of mathematical methods in cartography in Greek Antiquity and in the modern period, and on the mutual influence of the two fields, mathematics and geography. The final version of this paper will appear in Ganita Bharati (Indian mathematics).

math.HO↗
arXiv2021

Tangent spaces of the Teichm{ü}ller space of the torus with Thurston's weak metric

In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{ü}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{ü}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichm{ü}ller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.The final version of this paper will appear in Annales Academiæ \ Scientiarum Fennicæ\ Mathematica.

math.CV↗
arXiv2021

Vladimir Turaev, friend and colleague

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, this article will give at the same time an overview of an important part of the intense activity in low-dimensional topology that took place over the last 45 years, with its connections with mathematical physics.

math.HO↗
arXiv2021

Minimal stretch maps between Euclidean triangle?

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 between them is a symmetric metric. We show that this metric is Finsler, we give a necessary and sufficient condition for a path in this metric space to be geodesic and we determine the isometry group of this metric space. This study is motivated by Thurston's asymmetric metric on the Teichm{ü}ller space of a hyperbolic surface, and the results in this paper constitute an analysis of a basic Euclidean analogue of Thurston's hyperbolic theory. Many interesting questions in the Euclidean setting deserve further attention.

math.GT↗