TRR 109: Discretisation in Geometry and Dynamics

CRC的中心目标是对微分几何和动力学的离散化进行研究。在这两个数学领域，研究的关键对象都是由微分方程式支配的。一般说来，离散化是指将一个微分方程化成只涉及有限多个变量的差分方程组，其解近似于该微分方程解的任何过程。在动力学中，如果一个人对动力系统的整体、定性的长期行为感兴趣，那么得到局部高精度的近似显然是不够的。因此，一个好的离散化方案应该保留连续系统的重要定性方面。例如，如果能量在连续系统中是守恒的，那么离散系统也应该表现出某种形式的能量守恒。由于现代动力系统理论是用几何的语言表述的，所以与这种保持结构的离散有关的子域称为几何积分。在微分几何中，保持结构的离散也被证明是有用的。例如，对于许多特殊类型的曲面(如极小曲面或常高斯曲率曲面)，结构保持离散化是已知的。这些类型的离散曲面是具有以基本几何术语定义的特殊性质的多面体曲面。然而，它们表现出与由非线性偏微分方程组控制的连续曲面相同的定性行为。这些几何和动力学发展背后的共同思想是寻找和研究呈现相应光滑几何对象和动力学过程的属性和结构特征的离散模型。通过减小网格尺寸来精化离散模型当然应该收敛到传统的通过微分方程描述的极限，但另外重要的特征定性特征应该已经在离散水平上被捕捉到。由此产生的离散化应该构成一个基本的数学理论，它将经典的类比结合到连续的极限中。CRC将科学家聚集在一起，他们联手解决了离散几何和动力学带来的众多问题。

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.