TRR 109: Discretisation in Geometry and Dynamics

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The central goal of the CRC is to pursue research on the discretization of differential geometry and dynamics. In both fields of mathematics, the key objects under investigation are governed by differential equations. Generally, the term “discretization” refers to any procedure that turns a differential equation into difference equations involving only finitely many variables, whose solutions approximate those of the differential equation.In dynamics, it became apparent that obtaining locally high-accurate approximations is not enough if one is interested in the global, qualitative long-term behavior of a dynamical system. A good discretization scheme should therefore preserve important qualitative aspects of the continuous system. For example, if energy is preserved in the continuous system, then the discretized system should also exhibit some sort of energy conservation. Since the modern theory of dynamical systems is formulated in the language of geometry, the subfield that is concerned with such structure-preserving discretizations is called geometric integration.In differential geometry, structure-preserving discretizations turned out to be useful as well. For example, for many special classes of surfaces (such as minimal surfaces or surfaces with constant Gauss curvature) structure-preserving discretizations are known. These types of discrete surfaces are polyhedral surfaces with special properties defined in elementary geometric terms. However, they exhibit the same qualitative behavior as the continuous surfaces, which are governed by nonlinear partial differential equations.The common idea behind these developments in geometry and dynamics is to find and investigate discrete models that exhibit properties and structures characteristic of the corresponding smooth geometric objects and dynamical processes. Refining the discrete models by decreasing the mesh size should of course converge in the limit to the conventional description via differential equations, but in addition the important characteristic qualitative features should already be captured at the discrete level. The resulting discretization should constitute a fundamental mathematical theory, which incorporates the classical analog in the continuous limit.The CRC brings together scientists, who have joined forces in tackling the numerous problems raised by the challenge of discretizing geometry and dynamics.