Coordination Funds

该计划结合了微分几何，几何拓扑和全球分析的研究。跨越和超越这些学科的边界，它关注的是几何拓扑设置中的收敛和极限，以及无限大小的对象的渐近性质。总体主题可以大致分为三个横截面主题收敛，紧化和刚性。收敛的例子出现在Gromov-Hausdorff极限和几何演化方程。几何、拓扑和解析不变量在极限下的行为具有根本的意义。通常极限空间是非光滑的，因此需要适当地推广曲率或谱不变量等概念。极限也可以用来构造几何和拓扑中的渐近不变量，如单纯体积或L2-不变量。紧化反映了几何对象在适当曲率条件下的渐近性质。本文运用了拓扑学、微分几何、算子代数和概率论等方法。重要的问题是拉普拉斯或Dirac型算子的边值问题，无论是在黎曼和洛伦兹的设置，以及谱几何和布朗运动的非紧流形。除了连续变形刚性是必不可少的许多分类问题的几何和拓扑。它出现在几何环境中，通常是在负曲率的存在下，以及拓扑甚至代数设置中。刚性也是无限群和更一般的粗糙空间的解析、几何和同调不变量的同构关系的基础。优先方案支持个别研究项目和协调的研究活动。这些活动将确保研究方向的一致性，确定有希望的跨学科研究方向，鼓励建立新的研究合作，并实现两性平等措施。

This programme combines research in differential geometry, geometric topology, and global analysis. Crossing and transcending the frontiers of these disciplines it is concerned with convergence and limits in geometric-topological settings and with asymptotic properties of objects of infinite size. The overall theme can roughly be divided into the three cross-sectional topics convergence, compactifications, and rigidity.Examples of convergence arise in Gromov-Hausdorff limits and geometric evolution equations. The behaviour of geometric, topological and analytic invariants under limits is of fundamental interest. Often limit spaces are non-smooth so that it is desirable to generalize notions like curvature or spectral invariants appropriately. Limits can also be used to construct asymptotic invariants in geometry and topology such as simplicial volume or L2-invariants.Compactifications reflect asymptotic properties of geometric objects under suitable curvature conditions. Methods from topology, differential geometry, operator algebras and probability play a role in this study. Important issues are boundary value problems for Laplace or Dirac type operators, both in the Riemannian and Lorentzian setting, as well as spectral geometry and Brownian motion on non-compact manifolds.Besides continuous deformations rigidity is essential for many classification problems in geometry and topology. It appears in geometric contexts, typically in the presence of negative curvature, and in topological and even algebraic settings. Rigidity also underlies isomorphism conjectures relating analytic, geometric and homological invariants of infinite groups and more general coarse spaces.The priority programme supports individual research projects and coordinated research activities. These activities will ensure a coherence of research directions, identify promising lines of interdisciplinary research, encourage the establishment of new research cooperations, and realize gender equality measures.