Low-Dimensional Manifolds

9704490罗研究曲面的拓扑和几何以及3-流形的Heegaard分裂。曲面理论的一个基本工具是由Dehn和瑟斯顿提出的曲面上本质单环的同构类的空间。德恩称这个空间为曲面的算术场。瑟斯顿在研究Teichmueller空间的紧致化时使用了这个空间。许多数学家都知道，同构类的空间具有内在的模结构。罗使用模块化结构重建了Teichmueller空间和测量的层板空间。这使他能够推导出测得的薄片的空间是半实代数的。他目前正在进行曲面映射类组的重建工作。他还试图将这一结果应用于可测分层空间，以研究不可约非Haken 3-流形的Heegaard图。拓扑学、几何学和数学物理中的许多问题都涉及到对曲面上所有简单环的考虑。基本的简单环是一条没有自交的曲线，它不能收缩到曲面上的一点。问题是要理解所有这些环路放在一起。循环之间的基本关系是它们的交集数。罗正在开发一种代数计算来理解空间上的交叉数。这可能最终会通过使用Heegaard曲面理论来更好地理解三维空间。***

9704490 Luo Luo studies topology and geometry of surfaces and Heegaard splittings of 3-manifolds. One of the fundamental tools in surface theory is the space of isotopy classes of essential simple loops on surfaces introduced by Dehn and Thurston. Dehn called the space the arithmetic field of the surface. The space was used by Thurston in his work on the compactification of the Teichmueller space. It has been known by many mathematicians that the space of isotopy classes has an intrinsic modular structure. Luo has used the modular structure to reconstruct the Teichmueller space and the space of measured laminations. This has enabled him to derive that the space of measured laminations is semi-real algebraic. He is currently working on the reconstruction of the mapping class group of the surface. He is also trying to apply the result on the space of measured laminations to study the Heegaard diagrams of irreducible non-Haken 3-manifolds. Many problems in topology, geometry and mathematical physics involve the consideration of all simple loops on a surface. An essential simple loop is a curve without self-crossing that cannot be shrunk to a point on the surface. The problem is to understand all such loops put together. The basic relation between loops is their intersection number. Luo is developing an algebraic calculation to understand the intersection numbers on a space. This may eventually lead to a better understanding of 3-dimensional spaces through the use of Heegaard surface theory. ***