Knot concordance: Fifty years since Fox and Milnor, June 2008

Knot 索引：Fox 和 Milnor 五十年，2008 年 6 月

• 批准号: 0813619
• 负责人: Daniel Ruberman
• 金额: $ 3.4万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0813619&HistoricalAwards=false
• 关键词: Knot; concordance; Fifty; years; since

Fifty years ago, in a 1957 research announcement for the summer AMS meeting, R. H. Fox and J. Milnor, began their influential collaboration with the title, `Singularities of 2-spheres in 4-space and equivalence of knots.' Here they introduced the seminal idea that the concordance class of the link of a singularity obstructs its removal. Both concordance of knots, and the motivating goal of understanding singularities remain central to topology and algebraic geometry. A conference at Brandeis University will be held on June 2-5, 2008, to bring together a variety of researchers and students in geometric topology whose work connects to this fundamental idea. The conference will fertilize new research directions by encouraging mathematical interaction and collaboration. A substantial number of young investigators will be invited to give them exposure, broaden their perspective, and allow them to get to know each other and the more senior members of these fields. There will be approximately twenty invited addresses, and ample time will be set aside for interaction among the attendees. The conference will also honor the memory of Jerome Levine, a pioneer and key contributor to the field.A knot is a non-intersecting closed curve in three dimensional space. By closed, one means that if one travels along the curve, originating from any point on the curve, one eventually returns to the point of origin. A foundational subject to the field of topology, knot theory interacts with important areas in geometry, biology, and physics as well. The Knot concordance group measures wrinkles, or singularities, of surfaces in four dimensional space, and through this connection, enlightens our understanding of four dimensional shapes. Despite intense effort over the last 50 years, these groups have not been fully computed. Recently, new techniques from low-dimensional topology, including gauge theory, non-commutative algebra, higher- order linking theory, and quantum topology, have resulted in great advances in the study of knot concordance, revealing new structures in the concordance group. The conference will promote interactions between researchers working on these aspects of knot concordance and engender future research on this fundamental problem.

50年前，在1957年夏季AMS会议的一份研究报告中，R。H.福克斯和J.米尔诺，开始了他们有影响力的合作的标题，“奇异的2球在4空间和等价的结。“在这里，他们引入了一个开创性的想法，即奇点链接的和谐类阻碍了它的移除。 纽结的和谐性和理解奇点的动机目标仍然是拓扑学和代数几何的核心。 2008年6月2日至5日将在布兰代斯大学举行一次会议，汇集各种研究人员和几何拓扑学的学生，他们的工作与这一基本思想有关。 会议将通过鼓励数学互动和合作来丰富新的研究方向。 将邀请大量年轻的研究人员，让他们接触，扩大他们的视野，并让他们了解对方和这些领域的更资深的成员。 将有大约20个邀请演讲，并将留出充足的时间供与会者之间的互动。 会议还将荣誉杰罗姆莱文，一个先驱和关键贡献者的领域。一个结是一个不相交的封闭曲线在三维空间。 闭合意味着，如果从曲线上的任何点开始沿着曲线沿着行进，则最终返回到原点。 纽结理论是拓扑学领域的基础学科，它与几何学、生物学和物理学的重要领域相互作用。 纽结一致性组测量四维空间中表面的褶皱或奇点，并通过这种联系启发我们对四维形状的理解。尽管在过去的50年里付出了巨大的努力，这些群体还没有被完全计算出来。 近年来，规范理论、非交换代数、高阶联结理论和量子拓扑等低维拓扑学的新方法使纽结和谐的研究取得了很大进展，揭示了和谐群的新结构。 这次会议将促进研究人员之间的互动，这些方面的结一致性和产生未来的研究这个基本问题。