Mathematical Sciences: Problems in Knot Theory and Low- Dimensional Topology: Applications of Quasipositive Knots and Surfaces

数学科学：结理论和低维拓扑问题：拟正结和曲面的应用

• 批准号: 9504832
• 负责人: Lee Rudolph
• 金额: $ 7.5万
• 依托单位: Clark University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504832&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Knot; Theory

9504832 Rudolph Recent research on manifolds of dimension 3 and 4 both draws upon, and has striking consequences for, the theory of complex curves in complex surfaces. Topologically, complex curves in the unit ball in complex 2-space are simply represented as quasipositive surfaces, which are combinatorially defined and readily manipulable. Earlier, Dr. Rudolph combined quasipositivity with results from gauge theory to obtain estimates showing that many knots are much more complicated (in certain specific senses) than classical knot invariants can detect; he is now working to extend such estimates (using results from monopole theory). Other work in progress includes development of skein theory for Legendrian knots, an attack on a question of Harer about fibered links (are they all stably Hopf-plumbed?), and (with Michel Boileau) a study of Stein-fillable 3-manifolds. Knots and links are superficially simple objects which turn up in contexts as diverse as physics, theoretical meteorology, and biology. Mathematically, a rich source of knots is the theory of equations in a small number of variables (the simplest mathematical link, which looks like two rings in a piece of chain, comes from the simple equation xy = 0). Knot theory--the study of aspects of the geometry of knots and links which are insensitive to continuous deformations--has always had a lot to give to, and take from, the qualitative side of the theory of equations. Quasipositive knot theory takes advantage of extra structure in certain equations to obtain stronger results about their associated knots; potential applications range from quantum gravity to unraveling the molecular structure of DNA. ***

小行星9504832 最近对3维和4维流形的研究都借鉴了复杂曲面中的复杂曲线理论，并对其产生了显著的影响。 在拓扑上，复2-空间中单位球上的复曲线被简单地表示为拟正曲面，它们是组合定义的，并且易于操作。 早些时候，鲁道夫博士将准正性与规范理论的结果结合起来，得到了一些估计，这些估计表明许多纽结比经典纽结不变量所能检测到的要复杂得多（在某些特定意义上）;他现在正在努力扩展这些估计（使用规范理论的结果）。 其他正在进行的工作包括勒让德结的绞理论的发展，对Harer关于纤维链接的问题的攻击（它们都是稳定的Hopf-plumbed吗？），和（与米歇尔Boileau）的研究斯坦可填充3-流形。 结和链是表面上简单的对象，它们出现在物理学、理论气象学和生物学等不同的背景中。 在数学上，节点的丰富来源是少量变量的方程理论（最简单的数学链接，看起来像一条链中的两个环，来自简单的方程xy = 0）。 纽结理论--研究对连续变形不敏感的纽结和链环的几何学方面--总是有很多东西要给予和借鉴方程理论的定性方面。 准正纽结理论利用某些方程中的额外结构来获得关于其相关纽结的更强结果;潜在应用范围从量子引力到解开DNA的分子结构。 ***