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）将追求低维拓扑中的具体问题的集合，使用传统拓扑技术，Heegaard Floer同源性，组合数学以及其他任何可能派上用场的方法。 作为一个样本列表，PI希望解决以下问题：哪些交替结具有解结编号1;哪些交替结在四球中光滑地绑定切片盘（与Brendan Owens联合工作）;以及哪些3-流形允许强Heegaard图（与Adam Levine和John Luecke联合工作）。 在这些项目中，Heegaard Floer同调提供了重要的信息，并建立了一个具有挑战性的组合问题，这反过来又需要现代方法来解决。低维拓扑涉及曲线，曲面和3-和4-维空间的性质-我们可以可视化的对象，尽管有时需要一点努力。 一方面，这个领域在科学中有着丰富的传统：它的发展在很大程度上归功于物理学，包括古典和现代物理学;它影响生物学，DNA的打结在生物学中起着重要作用;它与整个数学领域相互作用。 另一方面，它具有明显的视觉性质，反映在艺术中：例如，通过凯尔特结，古代建筑中的图案，以及布莱塞的版画。 PI提案的目标是研究低维拓扑中一些有吸引力的简单问题。 事实上，这些问题中的许多问题都集中在凯尔特人的数学性质上（也称为凯尔特人）。交替）结。 虽然其中一些问题已经存在了几十年，但直到最近才通过附近领域（Floer同源性和组合学）的复杂技术的出现而变得可用。 这些不同领域之间的相互作用使PI着迷。