Knots and 3-manifolds

低维拓扑处于不断变化的状态。佩雷尔曼的影响？他在该领域的工作尚未被完全吸收。几何化猜想是许多悬而未决的问题的潜在动机，评估其中哪些仍然存在，并且仍然有趣，是社区中正在进行的一个过程。当然还有很多工作要做，现在确实有机会更清晰和深入地理解所有三维流形的空间结构。本提案中讨论的三个问题将有助于这一理解。第一个问题的目标是阐明一个特定流形的不可约Heegaard分裂集的整体结构。第二个问题是一个经典的链接手术问题。它提出了一个问题，即什么时候可以通过对3-球面中的一个链环进行外科手术来获得S^1×S^2的n个拷贝的连通和。这与嵌入S^4中的光滑3-球面的舍恩弗莱猜想直接相关。第三个问题是更具投机性的，并试图获得有关的信息之间的关系的几何不变量的嵌入曲线在3空间。 提案人与各级学生合作，传达数学和科学研究的兴奋，包括本提案中概述的研究。她既指导和教在加州大学戴维斯分校宇宙计划，一个为期一个月的住宅暑期计划，高中学生在数学和科学，她的课程集中在低维拓扑和结理论，拟议的研究主题的领域。 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。 这是一个令人兴奋的时刻，在该领域，作为数学家开始吸收的影响佩雷尔曼？的著名著作几何化猜想。 当然还有很多工作要做，现在确实有机会更清晰和深入地理解所有三维流形的空间结构。 这个建议讨论了三个具体的问题，在该领域的Heegaard分裂，结手术，结不变量。 这些问题的解决方案将揭示所有3-流形之间的关系，以及三维空间如何位于四维空间中。

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.