Pseudoholomorphic curves, orbifolds, and group actions

在过去的二十年里，格罗莫夫的伪全纯曲线理论一直处于数学许多重大进展的中心。本项目旨在将Gromov的伪全纯曲线的思想和技术应用于流形上的群作用，这在技术上相当于研究群作用的商空间中的伪全纯曲线。这样的商空间，称为轨道空间，一般具有奇点，奇点对应于群作用的不动点。特别地，本课题拟系统地研究四维流形上的一类光滑有限群作用，其中包括非奇异代数曲面的有限阶自同构，这是代数几何中一个长期研究的课题。该项目还试图对五维球体上的某种类型的圆作用进行分类，这涉及到对球体上的爱因斯坦度量的研究，这是黎曼几何中一个至关重要的主题。对称在数学中的重要性早已被认识到。对称研究的关键问题是理解空间中在对称下固定的点的集合的结构。这个项目的中心思想是观察到，当研究四维空间的对称性时（例如，我们生活的宇宙），人们通常可以通过研究四维空间中某种类型的刚性二维子空间（如肥皂泡）来提取关于对称性不动点的有用信息。

DMS-0603932Weimin Chen Gromov's pseudoholomorphic curve theory has been at the center of many great advances in mathematics in the past twenty years. This project seeks to apply Gromov's ideas and techniques of pseudoholomorphic curves in the context of group actions on manifolds, which technically amounts to studying pseudoholomorphic curves in the quotient space of the group action. Such a quotient space, called an orbifold, may have singularities in general, which correspond to the fixed points of the group action. Particularly, this project proposes to systematically study a certain class of smooth finite group actions on four-dimensional manifolds, which includes finite order automorphisms of nonsingular algebraic surfaces, a subject which has been long studied in algebraic geometry. The project also seeks to classify a certain type of circle actions on the five-dimensional sphere involved in the study of Einstein metrics on the sphere, which is a subject of central importance in Riemannian geometry.The importance of symmetry in mathematics has been long recognized. Acrucial issue in the study of symmetry is to understand the structureof the set of points in the space which are fixed under the symmetry.The central new idea in this project is the observation that when studying symmetries of a four-dimensional space (for instance, the universe in which we live), one can often extract useful information about the fixed points of a symmetry by studying a certain type of rigid two-dimensional subspaces (like a soap bubble) in the four-dimensional space.