Low-dimensional Topology and Geometry

摘要奖：DMS-0103511首席研究员：弗朗西斯博纳洪该项目提出研究几个几何问题，在二维和三维。一个共同的主题是，这些问题都涉及双曲几何，无论是作为一种工具来理解widerrange问题或作为一个主题本身的兴趣。该提案的第一部分是首席研究员在过去几年中开发的研究的自然延伸。在纯双曲方面，提出了研究三维流形上双曲结构的凸核，并进一步发展了一种基于测地流技术的曲面上双曲结构的研究方法。在拓扑方面，提出了利用双曲几何的方法来分析曲面上的简单闭曲线。特别是，该建议的目标之一是确定空间的简单closedcurves在一个表面上的分形维数。第二部分的建议是基于令人兴奋的新的acquitures这将连接两个方面的理论打结曲线在3-维空间迄今已很少相互作用，即拓扑量子场论和双曲几何结补。该项目建议攻击这些结构，如果这些被证明，进一步发展这样建立的连接。拟议的研究集中在双曲几何和拓扑之间的相互作用。在低维拓扑学中，人们试图分析2维和3维空间的可能形状。特别是，它包括作为一个子领域纽结理论，其目标是了解所有可能的方式，其中一串可以在空间打结;纽结理论的技术已经成功地应用于分析DNA的重组和复杂分子结构的打结。双曲几何显然是非常不同的。这是一个非欧几里德几何这是介绍了在早期的世纪，以测试的内部一致性公理的classicalgeometry开发的欧几里得和其他希腊数学家。一个意想不到的连接建立在二十世纪七十年代，通过开创性的工作条例草案瑟斯顿谁表明thathyperbolic几何可以成功地用于解决problemin拓扑。例如，有一个与每条打结曲线相关的数字，称为“双曲线体积”，并且可以通过当前软件相当容易地计算出来。如果两条打结曲线具有不同的双曲体积，则保证其中一条曲线不可能变形为另一条曲线。一幅新的图景开始浮现，在数学物理技术中，一条打结曲线的双曲体积出人意料地出现了。数学物理技术最初是为了预测高能粒子的行为而设计的。该提案的主要部分旨在澄清这一情况，并期望拓扑学、双曲几何和数学物理之间的交叉融合将导致这三个领域的进步。

AbstractAward: DMS-0103511Principal Investigator: Francis BonahonThe Project proposes to study several geometric problems indimension 2 and 3. A common theme is that these problems allinvolve hyperbolic geometry, either as a tool to understand widerrange problems or as a topic of interest in itself. The firsthalf of the proposal is a natural extension of the researchdeveloped by the Principal Investigator in the past few years. Onthe purely hyperbolic side, it proposes to study convex cores ofhyperbolic structures on 3-dimensional manifolds, and to furtherdevelop an approach to hyperbolic structures on surfaces which isbased on the technique of geodesic currents. On the moretopological side, it proposes to take advantage of methods ofhyperbolic geometry to analyze simple closed curves onsurfaces. In particular, one of the objectives of the proposal isto determine the fractal dimension of the space of simple closedcurves on a surface. The second part of the proposal is based onexciting new conjectures which would connect two aspects of thetheory of knotted curves in 3--dimensional space which so farhave had very little interaction, namely topological quantumfield theory and hyperbolic geometry on knot complements. Theproject proposes to attack these conjectures and, if these areproved, to further develop the connections so established.The proposed research is focused on the interplay betweenhyperbolic geometry and topology. In low-dimensional topology,one tries to analyze the possible shapes for spaces of dimensions2 and 3. In particular, it includes as a subfield knot theory,where the goal is to understand all possible ways in which astring can be knotted in space; techniques of knot theory havesuccessfully been applied to analyze the recombination of DNA andthe knotting of complex molecular structures. Hyperbolic geometryis apparently very different. It is a non-euclidean geometrywhich was introduced in the early nineteenth century, in order totest the internal consistency of the axioms of the classicalgeometry developed by Euclid and other Greek mathematicians. Anunexpected connection was established in the nineteen seventies,through ground breaking work of Bill Thurston who showed thathyperbolic geometry could be successfully used to solve problemsin topology. For instance, there is a number associated to eachknotted curve, called its "hyperbolic volume" and which can becomputed fairly easily by current software. If two knotted curveshave different hyperbolic volumes, one is guaranteed that it isimpossible to deform one curve to the other. A new picture is nowbeginning to emerge, where the hyperbolic volume of a knottedcurve unexpectedly occurs in techniques of mathematical physicsoriginally designed to predict the behavior of high energyparticles. The main part of the proposal is aimed at clarifyingthis picture, with the expectation that the cross-fertilizationbetween topology, hyperbolic geometry and mathematical physicswill lead to advances in each of these three fields.