Low-Dimensional Geometry and Topology

Abstract Award: DMS-0103843 Principal Investigator: Feng Luo The principal investigator will focus on two problems in the Teichmuller theory and 3-manifold topology. In Teichmuller theory, the aim of the investigation is to understand the complex structure on the Teichmuller space by constructing holomorphic functions arising from flat singular metric uniformization of the Riemann surface. We have produced many naturally defined complex valued functions on the Teichmuller space. The goal is to show that they are holomorphic. This will give us a better understanding of the complex structure which is of vital importance to the Teichmuller theory. In 3-manifold topology, we propose to show that any non-trivial 3-manifold group has a non-trivial SL(2,F) representation for some field F. We have translated the existence problem into a problem concerning how simple loops propagate in a surface. With the recent advance of our knowledge on surfaces, one may eventually solve the problem using surface topology. The existence of SL (2,F) representations will have many important consequences in 3-manifold topology. A 3-manifold is a space in which every point has a small surrounding similar to our real world. It is an important mathematical problem to classify all 3-manifolds. One of the main tool developed in recent decades in 3-manifolds theory is to use geometry. In particular, the geometry of surfaces has been used very successfully in understanding the 3-dimensional spaces. The proposed work addresses the topology of 3-manifolds and the geometry of surfaces. We attempt to use the symmetry theory (SL(2) representation theory) to understand the fundamental group of 3-manifolds which is a vital invariant of 3-manifolds. The SL(2,C) representation theory has been used very successfully in recent years by many topologists. Our approach seems to be new and uses simple loops on surfaces. The second part of the proposed work addresses the geometry of surfaces. One of the main problems on surface geometry is the moduli space problem. The moduli space problem asks for, for instance, what is the shape of the space of all convex polyhedrons which look like a cube. Many geometric problems are best expressed in terms of the topology and geometry of the moduli space. The corresponding object for high genus surface is the Teichmuller space. In contrasts to the topology of the Teichmuller space which is well understood for about 60 years, the geometry of it is much less understood. Our proposed work is an attempt to understand explicitly the complex analytic geometry of the Teichmuller space. The explicit description of the complex geometry of the Teichmuller space will have applications not only in mathematics but also in physics, for instance in string theory.