Knotting and Linking Phenomena in Topology

拓扑中的打结和连接现象

• 批准号: 9803694
• 负责人: Tim Cochran
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803694&HistoricalAwards=false
• 关键词: Knotting; Linking; Phenomena; Topology

9803694 Cochran Dr. Cochran will investigate two areas. He will provide major new information about which knotted circles in 3-space are slice knots, i.e., bound embedded 2-disks in the 4-ball, and he will study invariants of 3-dimensional manifolds that can be computed algorithmically (invariants of finite type). As regards the first, in collaboration with Peter Teichner and Kent Orr, he has succeeded in finding an infinite sequence of geometric relations on knots that are successive approximations to the knots bounding an embedded disk in the 4-ball. These relations can be expressed in terms of both Freedman-Casson-Whitney towers and the language of surgery theory. He will investigate algebraic invariants that reflect these relations. He will provide a bridge between surgery theory, the techniques of Freedman-Casson, and the invariants of Casson and Gordon. He will also produce the first examples of knots that have vanishing Casson-Gordon invariants but are not slice knots. In the second area, in collaboration with Paul Melvin, he will focus on applications of their new theory of finite type invariants to existing problems about 3-manifolds, and to finding the topological significance of specific new invariants that he has defined from the quantum invariants. Dr. Cochran will study complex new phenomena of the knotting and linking of spatial objects. The study of the complex 3-dimensional geometric "shapes" assumed by (for example) closed DNA, carbon compounds, viruses and proteins is central to several frontiers of mathematical biology and pharmacology. For example, if the 3-dimensional structure of a virus is known and if a protein can be found with a complementary shape and chemistry, then this protein can be used as the basis of an anti-viral drug. Much of such research proceeds in an ad hoc fashion because of our ignorance of the 3-dimensional geometry of proteins. Dr. Cochran will focus on very new, "higher order" knotting phenomena. He seeks new geometric and quantitative descriptions of phenomena. ***

小行星9803694 科克伦博士将研究两个领域。 他将提供关于3空间中哪些打结的圆是切片结的主要新信息，即，边界嵌入2盘在4球，他将研究不变量的三维流形，可以计算算法（不变量的有限型）。 至于第一，在合作与彼得Teichner和肯特奥尔，他已经成功地找到了一个无限序列的几何关系的结是连续近似的结界定嵌入磁盘在4球。 这些关系可以用Freedman-Casson-Whitney塔和手术理论的语言来表达。 他将调查代数不变量，反映这些关系。 他将提供手术理论之间的桥梁，弗里德曼卡森的技术，卡森和戈登的不变量。 他还将产生第一个例子的结消失卡森戈登不变量，但不是切片结。 在第二个领域，在与保罗梅尔文合作，他将专注于应用他们的新理论的有限型不变量现有的问题约3流形，并找到拓扑意义的具体新的不变量，他已定义的量子不变量。 Cochran博士将研究空间物体的打结和连接的复杂新现象。 对封闭DNA、碳化合物、病毒和蛋白质所呈现的复杂三维几何“形状”的研究是数学生物学和药理学的几个前沿领域的核心。 例如，如果病毒的三维结构是已知的，并且如果可以发现具有互补形状和化学性质的蛋白质，则该蛋白质可以用作抗病毒药物的基础。 由于我们对蛋白质的三维几何结构的无知，许多此类研究都是以临时的方式进行的。 Cochran博士将专注于非常新的“高阶”打结现象。 他寻求新的几何和定量描述的现象。 ***