Geometric analysis on non-compact and singular spaces

几何分析是对几何条件下偏微分方程的研究。一方面，人们可以研究波动方程或热方程等微分方程在弯曲空间上，并观察几何和拓扑结构如何影响这些方程的解。另一方面，几何分析在某些几何问题的解中也起着重要的作用，典型的是在给定空间上搜索具有均匀曲率等良好性质的黎曼度量（一种几何）。事实上，很多时候，人们通过求解偏微分方程得到这样的度规，例如，人们可以想到广义相对论中的爱因斯坦方程。由于奇点在科学和日常生活中无处不在，从破波到广义相对论中的黑洞概念，因此考虑更普遍的弯曲空间允许奇点是很自然的。这将是拟议研究计划的重点。也就是说，我们将学习各种偏微分方程，一些来自物理，一些来自几何问题，在潜在的弯曲空间有奇点或无穷端点的情况下。即使对于简单的奇点，比如锥的尖端或边缘，这也会带来许多严重的困难，但也会带来有趣的新现象。很好地理解奇异点对某些偏微分方程解的影响，反过来又有助于检测奇异点的存在，例如在医学成像中。我们将探索三个研究方向。在一个方向上，我们将研究几何方程在奇点附近的解的行为，在同样的情况下，这将给奇点一个更好的几何描述。在另一个方向上，我们将尝试通过研究波动方程，拉普拉斯函数的谱，以及由这个谱构建的更一般的不变量来“听到”奇点的存在。最后，通过几何方程，我们将研究奇点的存在如何影响底层空间的拓扑结构，反之亦然。

Geometric analysis is about the study of partial differential equations in geometric settings. On one hand, one can study differential equations like the wave equation or the heat equation on curved spaces and look how the geometry and the topology influence the solutions to these equations. On the other hand, geometric analysis also plays an important role in the solutions of certain geometric problems, typically the search on a given space of a Riemannian metric (a geometry) with nice properties like a uniform curvature. Indeed, quite often, one obtains such metrics by solving a partial differential equation, for instance one can think of Einstein's equation in the theory of general relativity. Since singularities are omnipresent in science and every day life, from breaking waves to the notion of black holes in general relativity, it is very natural to consider more generally curved spaces admitting singularities. This will be the focus of the proposed research program. Namely, we will study various partial differential equations, some coming from physics, some from geometric problems, in situations where the underlying curved space has singularities or infinite ends. Even for simple singularities like the tip of a cone or an edge, this can bring many serious difficulties, but also interesting new phenomena. A good understanding of the influence of singularities on the solutions to certain partial differential equations can in turn be useful to detect the presence of singularities, for instance in medical imaging. We will explore three directions of research. In one direction, we will study the behavior of solutions to geometric equations near a singularity, which at the same occasion shall give a better geometric description of that singularity. In another direction, we will try to 'hear' the presence of a singularity by studying the wave equation, the spectrum of the Laplacian, and more generally invariants built out of this spectrum. Finally, through geometric equations, we will study how the presence of singularities influence the topology of the underlying space and vice versa.