Research in hyperbolic geometry and mapping class groups

自从瑟斯顿关于几何化猜想的工作以来，曲面和三维流形上的双曲结构的研究一直是低维拓扑学和几何学的中心话题。瑟斯顿提出了许多猜想，这些猜想一直是这一领域的推动力，不仅是因为它们本身的兴趣，而且因为对这些猜想的研究产生了新的数学，在许多数学领域产生了广泛的影响。在过去的十年里，瑟斯顿的许多原始猜想都被解决了，给这个领域引入了新的技术和新的问题。主要研究者将使用这些新工具来回答关于双曲三维流形的进一步问题，特别是将研究双曲三维流形空间的拓扑，试图回答诸如这些空间是否局部连通，如果不是，局部连通性在哪里失效等问题。双曲3-流形的研究产生了重大影响的一个领域是映射类群的研究。这可能是自然产生的群中不是李群中的格子的最简单的例子，几何群论者已经对它进行了大量的研究。PI将应用在双曲三维流形研究中发展起来的许多工具，如曲线复形，来研究关于映射类群的问题。最明显的是，我们生活的空间是一个三维流形，数学的一个基本目标是更好地理解现实世界。在更抽象的层面上，3-流形有一个深刻而复杂的结构，它与数学的许多其他领域相连。这个项目将集中在双曲3-流形上。当看最简单的例子时，首先看起来双曲三维流形是相当罕见的。然而，更深层次的分析表明，在自然意义上，双曲三维流形是最普遍的三维流形类型。虽然3-流形本身是理解双曲3-流形的拓扑对象，但它们很快就会被引向代数、几何和分析。通过研究双曲3-流形，我们对整体数学有了更多的了解。

Since Thurston's work on the geometrization conjecture, the study of hyperbolic structures on surfaces and 3-manifolds has been a central topic in low-dimensional topology and geometry. Thurston made a number of conjectures that have been a driving force in the field, not only due to their intrinsic interest but because the study of these conjectures has produced new mathematics that has had wide ranging impact in many fields of mathematics. In the previous decade many of Thurston's original conjectures have been solved, introducing both new techniques and new problems into the field. The principal investigator will use these new tools to answer further questions about hyperbolic 3-manifolds and in particular will study the topology of spaces of hyperbolic 3-manifolds attempting to answer questions such as if these spaces are locally connected and if not where locally connectivity fails. One area where the study of hyperbolic 3-manifolds has had great influence is in the study of mapping class groups. This is perhaps the simplest example of a naturally occurring group that is not a lattice in a Lie group and has been much studied by geometric group theorists. The PI will apply many of the tools developed in the study of hyperbolic 3-manifolds, such as the curve complex, to study problems about the mapping class group.Three-manifolds are a central object of mathematical study for several reasons. The most obvious is that the space we live in is a 3-manifold and one basic goal of mathematics is to gain a better understanding of the real world. On a more abstract level, 3-manifolds have a deep and intricate structure that connects to many other areas of mathematics. This project will focus on hyperbolic 3-manifolds. When looking at the simplest examples it first appears that hyperbolic 3-manifolds are quite rare. However, a deeper analysis shows that, in a natural sense, hyperbolic 3-manifolds are the most prevalent type of 3-manifolds. While 3-manifolds themselves are topological objects to understand hyperbolic 3-manifolds them one is quickly led to algebra, geometry and analysis. By studying hyperbolic 3-manifolds we learn more about mathematics as a whole.