Mathematical Sciences: Differential Geometry and Geometric Analysis

三位研究者将研究非负弯曲流形的结构、测大地线、最小实Kaehler子流形、有界曲率坍缩、拉普拉斯的绝热极限和小特征值、黎曼流形和半单群的刚性性质等问题。这些研究者将分析理解所有弯曲流形问题的各个方面。自微分几何一百多年前提出以来，这个普遍问题一直是微分几何的中心主题。具体来说，这些研究者将研究测地线沿弯曲流形的行为。在球面上，大圆是唯一的测地线。但在其他更复杂的表面上，测地线可以呈现出许多令人惊讶的特性。测地线的研究对宇宙学有特殊的应用，因为根据爱因斯坦的理论，我们每个人都沿着测地线在四维时空中运动。

Three investigators will study problems involving the structure of non-negatively curved manifolds, geodesics, minimal real Kaehler submanifolds, collapse with bounded curvature, adiabatic limits and small eigenvalues of the Laplacian, and rigidity properties of Riemannian manifolds and semisimple groups. These investigators will analyze various aspects of the problem of understanding all curved manifolds. This general problem has been the central theme of differential geometry since its conception over one hundered years ago. Specifically, these investigators will study the behavior of geodesics along curved manifolds. On the surface of a sphere, great circles are the only geodesics. But on other more complex surfaces, geodesics can take on many surprising properties. The study of geodesics has particular application to cosmology since according to Einstein's theories, each of us moves through four-dimensional space-time along a geodesic.