Mathematical Sciences: Symmetries of Manifolds, Surgery and Scalar Curvature

本课题的主要目的是研究低维和高维流形的拓扑结构和几何结构。我们将特别注意流形的对称性。所要研究的问题涵盖了很大范围的问题，从关于低维流形的非常具体的问题到关于正弯曲高维流形几何的一般问题。本研究将使用的主要技术包括：规范理论、外科学、同伦理论、变换群和微分几何。流形是非常自然的研究对象。它们是局部类似于欧几里得空间的空间。例如，我们生活的空间（忽略时间）有三个维度，使其成为局部的三维流形，但不一定是全局的，就像欧几里得三维空间一样。考虑到时间，物理空间变成了一个四维流形。无论它是全局的欧几里得四维空间，还是其他可能有界的四维流形，都是一个深奥的宇宙学问题。回答这个问题不仅仅是一个数学问题，但探索各种可能性是一个重要的和适当的数学问题。

The main purpose of this research project is to study the topology and geometry of low and higher dimensional manifolds. Particular attention will be paid to symmetries of manifolds. The problems which will be studied cover a large spectrum of questions, ranging from very specific problems concerning low- dimensional manifolds to general problems concerning the geometry of positively curved higher dimensional manifolds. The main techniques which will be employed in this research include: gauge theory, surgery, homotopy theory, transformation groups and differential geometry. Manifolds are very natural objects to study. They are the spaces which are locally like Euclidean spaces. For example, the space in which we live (ignoring time), has three dimensions, making it a three-dimensional manifold, locally, but not necessarily globally, like Euclidean three-space. Taking time into account, physical space becomes a four-dimensional manifold. Whether it is globally Euclidean four-space or some other, possibly bounded, four-dimensional manifold is a deep question of cosmology. Answering this question is not a matter for mathematics alone, but exploring the various possibilities is an important and properly mathematical matter.