Parametrised databases of surfaces based on Teichmüller theory

• Armando Rodado Amaris ^[1] ; Gina Lusares ^[2]
  1. [1] Universidad de Los Lagos

     Universidad de Los Lagos

     Osorno, Chile

     
  2. [2] Universidad de Valparaíso

     Universidad de Valparaíso

     Valparaíso, Chile

     
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 18, Nº. 1, 2016, págs. 69-88
• Idioma: inglés
• DOI: 10.4067/S0719-06462016000100006
• Enlaces
  • Texto completo
• Resumen
  • español

    Proponemos un nuevo marco teórico para construir bases de datos de superficies con rica estructura matemática. Nuestro enfoque está basado en ideas que vienen de teoría de espacios de Teichmüller y espacios módulares de superficies de Riemann cerradas, y el problema de encontrar una descomposición celular canónica y explícita de estos espacios. Las bases de datos construidas usando nuestro enfoque tendrán una estructura gráfica subyacente, la que se puede construir a partir de un solo grafo por movimientos de expansión y contracción.

  • English

    We propose a new framework to build databases of surfaces with rich mathematical structure. Our approach is based on ideas that come from Teichmüller and moduli space of closed Riemann surfaces theory, and the problem of finding a canonical and explicit cell decomposition of these spaces. Databases built using our approach will have a graphical underlying structure, which can be built from a single graph by contraction and expansion moves.

• Referencias bibliográficas
  • Albat, F,Müller, R. Free-Form Surface Construction in a Commercial CAD/CAM System, Mathematical of Surfaces XI. 11th IMA International Conference...
  • Rodado, A. J.R. Weierstrass Points and Canonical Cell Decompositions of the Moduli and Teichmuller Spaces of Riemann Surfaces of Genus Two.
  • Amaris, A.J.R,Cox, M.P. (2014). A Flexible Theoretical Representation of The Temporal Dynamics of Structured Populations as Paths on Polytope...
  • AMPL. A Modeling Language for Mathematical Programming.
  • Bobenko, A,Springborn, B. (2003). Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356. 659-689
  • Franz, M. Convex: a Maple package for convex geometry.
  • Farkas, H. M,Kra, I. (1991). Riemann Surfaces, Graduate Texts in Mathematics 71. Springer-Verlag. New YorkHeidelbergBerlin.
  • Grünbaum, B. (1967). Convex Polytopes. Pure and Applied Mathematics.
  • Kuusalo, T,Näätänen, M. (1995). Geometric Uniformization in Genus 2. Annales Academiae Scientiarum Fennicae , Series A.I. Mathematica. 20....
  • Kuusalo, T,Näätänen, M. Weierstrass points of extremal surfaces in genus 2.
  • Harris, J,Morrison, I. (1998). Moduli of Curves, Graduate Texts in Mathematics. Springer-Verlag.
  • Hass, A,Susskind, P. The Geometry of the Hyperelliptic Involutions in Genus Two. Proceeding of the American Mathematical Society. 105.
  • Marden, A,Rodin, B. (1990). On Thurston's formulation and proof of Andreev's Theorem. Computational Methods and Function Theory. 1435....
  • McCarthy, J. D. (1995). Weierstrass points and Z2 homology. Topology and its Applications. 63. 173-188
  • Mcshane, G. (2004). Weierstrass Points and Simple Geodesics. Bull. London Math. Soc. 36. 181-187
  • Milnor, J. Hyperbolic Geometry: The First 150 Years. Bulletin of The American Mathematical Society. 6.
  • Penner, R. (1987). The decorated Teichm¨uller space of punctured surfaces. Comm. Math. Phys. 113. 299-339
  • Papadopoulus, A. (2007). Handbook of Teichmuller Theory, Volume I. iRMA Lectures in Mathematics and Theoretical Physics. 11.
  • Ratcliffe, J. (1994). Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. 149.
  • Springborn, B. Variational Principles for Circles Packing. 1.
  • Stepheson, K. Circle Packing: A Mathematical Tale. Notices of the AMS. 50.
  • Thurston, W. P. (1997). Three-Dimensional Geometry and Topology, I.
  • De Verdier´e, Y. C. (1991). Une principe variationnel pour les empilements de cercles. Invent. Math. 104. 655-669
  • Wang, Yalin,Dai, Wei,Gu, Xianfeng,Chan, Tony F,Yau, Shing-Tung,Toga, Arthur W,Thompson, Paul M. Teichmuller Shape Space Theory and its Application...
  • Zeng, W,Shi, R,Wang, Y,Gu, X. (2013). Teichmuller Shape Descriptor and its Application to Alzheimer's Disease Study. International Journal...
  • Ziegler, G. (1994). Lectures on Polytopes. Springer-Verlag.