Floer homology and low-dimensional topology

Florer同调和低维拓扑

• 批准号: 1207812
• 负责人: Joshua Greene
• 金额: $ 15.56万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207812&HistoricalAwards=false
• 关键词: Floer; homology; low; dimensional; topology

The principal investigator (PI) will pursue a collection of concrete problems in low-dimensional topology, using a combination of traditional topological techniques, Heegaard Floer homology, combinatorics, and whatever else might come in handy. As a sample list, the PI hopes to address the questions of which alternating knots have unknotting number one; which alternating knots bound smoothly slice disks in the four-ball (joint work with Brendan Owens); and which 3-manifolds admit a strong Heegaard diagram (joint work with Adam Levine and John Luecke). In each of these projects, Heegaard Floer homology provides important information and sets up a challenging combinatorial problem, which in turn requires modern methods to solve.Low-dimensional topology is concerned with the properties of curves, surfaces, and 3- and 4-dimensional spaces -- objects that we can visualize, though sometimes with a bit of effort. On one hand, this field has a rich tradition in the sciences: it owes a lot of its development to physics, both classical and modern; it impacts biology, where the knotting of DNA plays a significant role; and it interacts with fields all across mathematics. On the other hand, it has a distinctly visual nature that is reflected in the arts: for example, through Celtic knots, motifs in ancient architecture, and Escher's prints. The goal of the PI's proposal is to study some attractive and simply-stated problems in low-dimensional topology. In fact, many of these problems focus on the mathematical properties of Celtic (a.k.a. alternating) knots. While some of these problems have been around for many decades, they have only recently become accessible through the advent of sophisticated techniques from nearby fields (Floer homology and combinatorics). The interplay between these different fields fascinates the PI.

首席研究员（PI）将使用传统的拓扑技术，Heegaard Floer同源性，组合术以及其他方便的方式来解决低维拓扑中的具体问题集合。 作为示例列表，PI希望解决哪些交替结的问题排名第一。在四球中（与Brendan Owens的联合工作）交替结合平滑的磁盘； 3-manifolds承认了强大的Heegaard图（与Adam Levine和John Luecke的联合合作）。 在每个项目中，Heegaard Floer同源性提供了重要的信息，并设置了一个具有挑战性的组合问题，进而需要现代方法来解决。高维拓扑与曲线，表面以及3-维空间的特性有关，我们可以将其视而不见，有时可以付出一些努力。 一方面，这个领域在科学中具有丰富的传统：它的发展归功于古典和现代的物理学。它影响生物学，其中DNA的打结起着重要作用。它与数学跨数学的领域相互作用。 另一方面，它具有明显的视觉性质，在艺术中反映出：例如，通过凯尔特结，古代建筑中的图案和埃舍尔的印刷品。 PI提议的目的是研究低维拓扑的一些有吸引力且简单的问题。 实际上，其中许多问题集中在凯尔特人（又称交替）结的数学特性上。 尽管其中一些问题已经存在了数十年，但由于最近从附近的田野（Floer同源性和组合学）出现了复杂的技术，它们才能进入。 这些不同领域之间的相互作用使PI着迷。