Problems in Low Dimensional Geometry and Topology

点击翻译按钮获取中文摘要

This proposal addresses central problems in low dimensional topology and geometry. The PI plans to investigate the topology of ending laminations space as well as problems related to Dehn surgery on links and taut foliations. In addition to study the topology of low volume hyperbolic 3-manifolds as well as those which are ultra large. The latter have large volume, Cheeger constant bounded below and have strongly irreducible minimal genus Heegaard splittings. The PI also plans to study smooth Schoenflies 4-balls. Three-manifold topology is the study of objects that locally look like the standard three-dimensional space in which we live. Two-manifold topology is the study of surfaces such as the sphere, torus and the surface of higher genus. While objects such as surfaces and standard 3-space are very simple, the class of objects that exist in them is incredibly rich and fascinating. For example, there is knot theory in 3-space and the theory of curves and laminations in surfaces. These theories in turn are fundamentally connected with many other mathematical structures such as non Euclidean geometry. This proposal addresses basic mathematical questions in these areas. A classical fundamental theorem in topology is that any smooth closed curve in the plane, such as the circle, bounds a smooth disc. In fact such a theorem is true in dimension three and unknown in dimension four. The PI plans to investigate this very mysterious question at the heart of smooth 4-dimensional topology.