Knots and 3-manifolds

低维拓扑处于不断变化的状态。佩雷尔曼？S的工作对场上的影响还没有完全被消化。几何化猜想是许多突出问题的根本动机，评估这些问题中的哪些仍然存在并仍然有趣，现在正在社区中进行。当然，还有很多工作要做，事实上，现在有机会更清晰和更深入地理解所有三重流形的空间结构。本提案中讨论的三个问题将有助于这一理解。第一个问题的目的是阐明指定流形的不可约Heegaard分裂集的整体结构。第二个问题是一个经典的链接外科问题。它询问什么时候可以通过对3-球面上的一个链接进行手术来获得S^1×S^2的n个副本的连通和。这与S[4]中关于光滑3-球面的勋飞猜想有直接关系。第三个问题更具投机性，它试图得到关于3-空间中嵌入曲线的几何不变量之间关系的信息。提倡者与各个层次的学生一起工作，传达数学和科学研究的兴奋，包括本提案中概述的研究。她在加州大学戴维斯分校的宇宙计划中担任导演和教师，这是一个为高中生提供的为期一个月的数学和科学暑期住宿计划，她的课程重点是低维拓扑和纽结理论领域，这是拟议研究的主题。她还与博士生广泛合作，研究与提案中描述的问题密切相关的问题。低维拓扑学是对2、3和4维空间的研究。这在这个领域是一个令人兴奋的时刻，因为数学家们开始吸收佩雷尔曼？S关于几何化猜想的著名工作的影响。当然，还有很多工作要做，事实上，现在有机会更清晰和更深入地理解所有三重流形的空间结构。这项提案讨论了该领域中的三个具体问题，即Heegaard分裂、纽结手术和纽结不变量。这些问题的解决将阐明所有3维流形之间的关系，以及3维空间如何位于4维空间中。

Low-dimensional topology is in a state of flux. The impact of Perelman?s work on the field has not been fully assimilated. The Geometrization Conjecture was the underlying motivation for many outstanding problems, and evaluating which of these remain, and remain interesting, is a process now underway in the community. Certainly there remains much to be done, and indeed there is now an opportunity to understand the structure of the space of all 3-manifolds with greater clarity and depth. The three problems discussed in this proposal will contribute to that understanding. The goal of the first problem is to shed light on the overall structure of the set of irreducible Heegaard splittings of a specified manifold. The second problem is a classical link surgery question. It asks when it is possible to obtain a connect sum of n copies of S^1×S^2 by surgery on a link in the 3-sphere. This is directly related to the Schoenflies conjecture for smooth 3-spheres imbedded in S^4. The third problem is more speculative, and seeks to obtain information about relations among geometric invariants of an imbedded curve in 3-space. The proposer works with students at every level to convey the excitement of research in mathematics and science, including the research outlined in this proposal. She both directs and teaches in the UC Davis Cosmos program, a month-long residential summer program for high school students in math and science, with her course focused on the fields of low-dimensional topology and knot theory, the topic of the proposed research. She also works extensively with PhD students on problems closely related to those described in the proposal.Low-dimensional topology is the study of spaces of 2, 3 and 4 dimensions. It is an exciting moment in the field, as mathematicians begin to assimilate the impact of Perelman?s famous work on the Geometrization Conjecture. Certainly there remains much to be done, and indeed there is now an opportunity to understand the structure of the space of all 3-manifolds with greater clarity and depth. This proposal discusses three specific problems in the field, in the areas of Heegaard splittings, knot surgery, and knot invariants. The solutions to these problems will shed light on the the relationship among all 3-manifolds, and indeed on how 3-dimensional spaces can lie inside 4-dimensional space.