Coordination Funds

该课程结合了微分几何、几何拓扑学和全局分析方面的研究。跨越和超越这些学科的边界，它涉及几何拓扑环境中的收敛和极限，以及无限大物体的渐近性质。总体主题大致可以分为收敛、紧凑和刚性三个横截面主题。收敛的例子出现在Gromov-Hausdorff极限和几何演化方程中。几何不变量、拓扑不变量和解析不变量在极限下的行为是最重要的问题。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。极限还可以用来构造几何和拓扑学中的渐近不变量，如单纯体积不变量或L2不变量。紧化反映了几何对象在适当的曲率条件下的渐近性质。从拓扑学、微分几何学、算子代数和概率学等方面的方法在本研究中发挥了作用。在Riemannian和Lorentzian环境下，重要的问题是Laplace或Dirac算子的边值问题，以及非紧流形上的谱几何和布朗运动。此外，连续形变刚性是许多几何和拓扑分类问题的本质。它出现在几何环境中，通常是在负曲率存在的情况下，也出现在拓扑甚至代数环境中。刚性也是将无限群和更一般的粗空间的解析、几何和同调不变量联系起来的同构猜想的基础。优先方案支持单独的研究项目和协调的研究活动。这些活动将确保研究方向的连贯性，确定有希望的跨学科研究路线，鼓励建立新的研究合作，并实现性别平等措施。

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.