Discrete conformal mappings and Riemann Surfaces: Theory and Applications
The general idea of discrete differential geometry is to find and investigate discrete models that exhibit properties and structures characterisitic of the corresponding smooth geometric objects. This is a challenging problem, since equivalent points of view in the smooth setting may lead to a number of inequivalent treatments in the discrete setting. We will illustrate the paradigm of structure-preserving discretizations on the example of conformal maps by showing how simple definitions lead to surprisingly rich theories.
Overview
Abstract
The general idea of discrete differential geometry is to find and investigate discrete models that exhibit properties and structures characterisitic of the corresponding smooth geometric objects. This is a challenging problem, since equivalent points of view in the smooth setting may lead to a number of inequivalent treatments in the discrete setting. We will illustrate the paradigm of structure-preserving discretizations on the example of conformal maps by showing how simple definitions lead to surprisingly rich theories. We establish a connection between conformal geometry for triangulated surfaces, the geometry of ideal hyperbolic polyhedra and discrete uniformization of Riemann surfaces. Applications in geometry processing and computer graphics will be demonstrated. We will also show excerpts from our new computer-animated movie entitled "conform!". It will be demonstrated that the difference between the continuous and discrete models is hardly noticeable. Our aim is to convince you that this new branch of mathematics is both (literally) beautiful and useful.
Brief Biography
Alexander Bobenko is a professor of mathematics at the Technical University of Berlin. He received his PhD in mathematical physics from Steklov Mathematical Institute in St. Petersburg, Russia in 1985. His fields of interest include geometry, mathematical physics and their applications, in particular discrete differential geometry, integrable systems and visualization. He is an author of several mathematical books and animation films. Bobenko is a director of the DFG Collaboration Center "Discretization in Geometry and Dynamics".
Film Screening
How is modern mathematical teamwork carried out?
The multiple award-winning film The Discrete Charm of Geometry by Ekaterina Eremenko will screen on April 3rd after the CEMSE Dean's Distinguished Lecture Discrete conformal mappings and Riemann Surfaces: Theory and Applications by Prof. Alexander I. Bobenko, Technische Universität Berlin. Following the screening, Prof. Bobenko will be available for a Q&A session.
