Low-Dimensional Geometry and Topology

点击翻译按钮获取中文摘要

AbstractAward: DMS-0103843Principal Investigator: Feng LuoThe principal investigator will focus on two problems in theTeichmuller theory and 3-manifold topology. In Teichmullertheory, the aim of the investigation is to understand the complexstructure on the Teichmuller space by constructing holomorphicfunctions arising from flat singular metric uniformization of theRiemann surface. We have produced many naturally defined complexvalued functions on the Teichmuller space. The goal is to showthat they are holomorphic. This will give us a betterunderstanding of the complex structure which is of vitalimportance to the Teichmuller theory. In 3-manifold topology, wepropose to show that any non-trivial 3-manifold group has anon-trivial SL(2,F) representation for some field F. We havetranslated the existence problem into a problem concerning howsimple loops propagate in a surface. With the recent advance ofour knowledge on surfaces, one may eventually solve the problemusing surface topology. The existence of SL (2,F)representations will have many important consequences in3-manifold topology.A 3-manifold is a space in which every point has a smallsurrounding similar to our real world. It is an importantmathematical problem to classify all 3-manifolds. One of the maintool developed in recent decades in 3-manifolds theory is to usegeometry. In particular, the geometry of surfaces has been usedvery successfully in understanding the 3-dimensional spaces. Theproposed work addresses the topology of 3-manifolds and thegeometry of surfaces. We attempt to use the symmetry theory(SL(2) representation theory) to understand the fundamental groupof 3-manifolds which is a vital invariant of 3-manifolds. TheSL(2,C) representation theory has been used very successfully inrecent years by many topologists. Our approach seems to be newand uses simple loops on surfaces. The second part of theproposed work addresses the geometry of surfaces. One of the mainproblems on surface geometry is the moduli space problem. Themoduli space problem asks for, for instance, what is the shape ofthe space of all convex polyhedrons which look like a cube. Manygeometric problems are best expressed in terms of the topologyand geometry of the moduli space. The corresponding object forhigh genus surface is the Teichmuller space. In contrasts to thetopology of the Teichmuller space which is well understood forabout 60 years, the geometry of it is much less understood. Ourproposed work is an attempt to understand explicitly the complexanalytic geometry of the Teichmuller space. The explicitdescription of the complex geometry of the Teichmuller space willhave applications not only in mathematics but also in physics,for instance in string theory.