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36
Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q, Asian
 Journal of Mathematics
, 1998
"... Abstract. It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, we prove that Grothendieck’s correspondence between dessins d’enfants and Belyi morphisms is a special case of this c ..."
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Cited by 37 (9 self)
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Abstract. It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, we prove that Grothendieck’s correspondence between dessins d’enfants and Belyi morphisms is a special case of this correspondence through an explicit construction of Strebel differentials. For a metric ribbon graph with edge length 1, an algebraic curve over Q and a Strebel differential on it is constructed. It is also shown that the critical trajectories of the measured foliation that is determined by the Strebel differential recover the original metric ribbon graph. Conversely, for every Belyi morphism, a unique Strebel differential is constructed such that the critical leaves of the measured foliation it determines form a metric ribbon graph of edge length 1,
A Circle Packing Algorithm
 Computational Geometry: Theory and Applications
"... . A circle packing is a configuration P of circles realizing a specified pattern of tangencies. We describe an efficient algorithm for computing the radii of packings in euclidean and hyperbolic geometry, discuss its performance, and illustrate some recent applications. Introduction A circle packin ..."
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Cited by 32 (2 self)
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. A circle packing is a configuration P of circles realizing a specified pattern of tangencies. We describe an efficient algorithm for computing the radii of packings in euclidean and hyperbolic geometry, discuss its performance, and illustrate some recent applications. Introduction A circle packing is a configuration P of circles realizing a specified pattern of tangencies. As such, it enjoys dual natures  combinatoric in the pattern of tangencies, encoded in an abstract "complex" K, and geometric in the radii of the circles, represented by a radius "label" R. As an early example, Figure 1 displays a simple complex K and a circle packing having its combinatorics. More substantial packings may involve over 100,000 circles. Figure 1. A simple packing example Key words and phrases. circle packing, conformal geometry, discrete Dirichlet problem. The second author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance. 2 CHARLES R. COL...
On Associators and the GrothendieckTeichmüller Group I
, 1998
"... . We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1, Dr2]) between associators (for quasitriangular quasiHopf algebras) and (a variant of) the GrothendieckTeichmuller group becomes simple and natural, leading to a simplication of Drinfel'd's original work. In p ..."
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Cited by 25 (4 self)
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. We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1, Dr2]) between associators (for quasitriangular quasiHopf algebras) and (a variant of) the GrothendieckTeichmuller group becomes simple and natural, leading to a simplication of Drinfel'd's original work. In particular, we reprove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a notnecessarilyrational associator exists. Contents 1. Introduction 1 1.1. Reminders about quasitriangular quasiHopf algebras 1 1.2. What we do 2 1.3. Acknowledgement 4 2. The basic denitions 4 2.1. Parenthesized braids and GT 4 2.2. Parenthesized chord diagrams and GRT 8 3. Isomorphisms and associators 11 4. The Main Theorem 15 4.1. The statement, consequences, and rst reduction 15 4.2. More on the group \ GRT 15 4.3. The second reduction 18 4.4. A cohomological interlude 19 4.5. Proof of the semiclassical hexagon equation 20...
Uniformizing Dessins And Belyi Maps Via Circle Packing
, 1997
"... Grothendieck's theory of Dessins d'Enfants involves combinatorially determined affine, reflective, and conformal structures on compact surfaces. In this paper the authors establish the first general method for uniformizing these dessin surfaces and for approximating their associated Belyi meromo ..."
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Cited by 18 (6 self)
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Grothendieck's theory of Dessins d'Enfants involves combinatorially determined affine, reflective, and conformal structures on compact surfaces. In this paper the authors establish the first general method for uniformizing these dessin surfaces and for approximating their associated Belyi meromorphic functions. The paper begins by developing a discrete theory of dessins based on circle packing. This theory is surprisingly faithful, even at its coarsest stages, to the geometry of the classical theory, and it displays some new sources of richness; in particular, algrebraic number fields enter the theory in a new way. The paper goes on to show that the discrete dessin structures converge to their classical counterparts under a hexagonal refinement scheme. In addition, since the discrete objects are computable, circle packing provides opportunities both for routine experimentation and for large scale explicit computation. A range of examples up to genus 4 is given in the pape...
The Approximation of Conformal Structures via Circle Packing
 In Computational Methods and Function Theory 1997, Proceedings of the Third CMFT conference
"... . This is a pictorial tour and survey of circle packing techniques in the approximation of classical conformal objects. It begins with numerical conformal mapping and the conjecture of Thurston which launched this topic, moves to approximation of more general analytic functions, and ends with recent ..."
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Cited by 12 (1 self)
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. This is a pictorial tour and survey of circle packing techniques in the approximation of classical conformal objects. It begins with numerical conformal mapping and the conjecture of Thurston which launched this topic, moves to approximation of more general analytic functions, and ends with recent work on the approximation of conformal tilings and conformal structures. x1 Introduction A circle packing is a configuration of circles with a specified pattern of tangencies. The regular hexagonal or "penny" packing in the plane  every circle tangent to six others  is certainly familiar to everyone, and the literature contains a smattering of other examples stretching back to the ancient Greeks. But I offer this as a first illustration of the type of packings we will discuss. S 2 Circle Packing 1 Circle packing has its beginning as a distinct topic with applications to 3manifolds in Thurston's Notes [20]. Its connections to analytic functions, the subject of this survey, can b...
From Dynamics on Surfaces to Rational Points on Curves
 BULLETIN AMS
, 1999
"... Introduction Fermat's last theorem states that for n 3 the equation X n + Y n = Z n (1.1) has no integer solutions with X;Y; Z 1. Inspiring generations of work in number theory, its proof was finally achieved by Wiles. A qualitative result, Finite Fermat, was obtained earlier by Faltings; ..."
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Cited by 8 (3 self)
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Introduction Fermat's last theorem states that for n 3 the equation X n + Y n = Z n (1.1) has no integer solutions with X;Y; Z 1. Inspiring generations of work in number theory, its proof was finally achieved by Wiles. A qualitative result, Finite Fermat, was obtained earlier by Faltings; it says the Fermat equation has only a finite number of solutions (for each given n, up to rescaling). This paper is an appreciation of some of the topological intuitions behind number theory. It aims to trace a logical path from the classification of surface diffeomorphisms to the proof of Finite Fermat. The route we take is the following. x2. The isotopy classes of surface diffeomorphisms f : S ! S form the mapping class group Mod(S). Thurs
Belyi Functions for Archimedean Solids
 Discrete Math
, 1996
"... The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semiregular maps which co ..."
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Cited by 8 (0 self)
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The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semiregular maps which correspond to the socalled Archimedean solids. R#sum# La notion de fonction de Belyi est un outil technique qui relie la combinatoire des cartes (c'est#dire, des graphes plong#s sur des surfaces) avec la th#orie de Galois, la th#orie des nombres alg#briques et la th#orie des surfaces de Riemann. Dans cet article nous calculons les fonctions de Belyi pour une classe des cartes semireguli#res, correspondant # ce qu'on appelle les solides d'Archim#de. 1 Introduction The title of the present paper attempts to link together traditional and contemporary mathematics. The name of Archimedes represents tradition; that of Belyi, one of the most recent advances in Galois theory, known (even i...
Sextic Coverings Of Genus Two Which Are Branched At Three Points
, 2002
"... this paper, we will let Q denote the subfield of algebraic numbers in the complex field C ..."
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Cited by 7 (1 self)
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this paper, we will let Q denote the subfield of algebraic numbers in the complex field C
Noncommutative matrix integrals and representation varieties of surface groups in a finite group, Annales de l’Institut Fourier 55
, 2005
"... Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the ..."
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Cited by 6 (2 self)
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Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers. The purpose of this paper is to establish Feynman diagram expansion formulas for noncommutative matrix integrals over a finitedimensional real or complex von Neumann algebra. An interesting case is the real or complex group algebra of a finite group. Using the graphical expansion formulas, we give a new proof of the classical formulas for the number
Simple Loops on Surfaces and Their Intersection Numbers
, 1997
"... Given a compact orientable surface \Sigma, let S (\Sigma) be the set of isotopy classes of essential simple loops on \Sigma. We determine a complete set of relations for a function from S (\Sigma) to Z to be a geometric intersection number function. As a consequence, we obtain explicit equations i ..."
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Cited by 6 (4 self)
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Given a compact orientable surface \Sigma, let S (\Sigma) be the set of isotopy classes of essential simple loops on \Sigma. We determine a complete set of relations for a function from S (\Sigma) to Z to be a geometric intersection number function. As a consequence, we obtain explicit equations in R S (\Sigma) and P (R S (\Sigma) ) defining Thurston's space of measured laminations and Thurston's compactification of the Teichmuller space. These equations are not only piecewise integral linear but also semireal algebraic.