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Habilitation à diriger des recherches

Discrete Complex Analysis

Abstract : My present interest is in Discrete Differential Geometry, especially applied to Computer Graphics, but it stems from Discrete Complex Analysis and Integrable Models, which has been my main subject of study during the past 6 years. The intention is to translate the best part of the theory of surfaces and complex analysis to the era of computers and discrete surfaces. This XIXth century theory, paved the way to the world of engineering marvels of the XXth century. Its discrete counterpart would be a real benefit for many different subjects of industrial interest. I have developped the theory of Discrete Complex Analysis and Discrete Riemann Surfaces as a tool to tackle issues in Exactly Solvable Models in Statistical Mechanics. The main idea is to try to see, in an exactly solvable model, a Finite Conformal Field Theory, without having to go to the thermodynamic limit. This life long project, set by my advisor Daniel Bennequin, was given positive partial answers in my PhD thesis: criticality in the Ising model can be seen at the finite level as compatibility with discrete conformality. The links with integrability extend to discrete integrable models, that I investigated in the Technical Universtiy Berlin with A. Bobenko. We found a quadratic counterpart to this linear theory, building the two first steps of a hierarchy of discrete integrable systems.
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https://tel.archives-ouvertes.fr/tel-00439782
Contributor : Christian Mercat <>
Submitted on : Tuesday, December 8, 2009 - 3:22:53 PM
Last modification on : Friday, May 17, 2019 - 11:36:50 AM
Long-term archiving on: : Thursday, October 18, 2012 - 10:21:07 AM

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  • HAL Id : tel-00439782, version 1

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Christian Mercat. Discrete Complex Analysis. Mathematics [math]. Université Montpellier II - Sciences et Techniques du Languedoc, 2009. ⟨tel-00439782⟩

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