Knotting and Linking Phenomena in Topology

9803694科克伦科克伦博士将调查两个领域。他将提供关于三维空间中哪些结圆是切片结的主要新信息，即在4球中嵌入2盘的绑定，他将研究可以用算法计算的三维流形的不变量（有限型不变量）。关于第一个，在与Peter Teichner和Kent Orr的合作中，他成功地找到了结点上的无限几何关系序列，这些结点是4球中嵌入圆盘的结点的连续逼近。这些关系可以用弗里德曼-卡森-惠特尼塔和外科理论的语言来表达。他将研究反映这些关系的代数不变量。他将在外科理论、弗里德曼-卡森的技术以及卡森和戈登的不变量之间架起一座桥梁。他还将制作第一个具有消失的卡森-戈登不变量但不是切片结的结的例子。在第二个领域，他将与Paul Melvin合作，专注于将他们的有限型不变量新理论应用于现有的关于3流形的问题，并从量子不变量中找到特定的新不变量的拓扑意义。科克伦博士将研究空间物体的结结和连接的复杂新现象。对封闭DNA、碳化合物、病毒和蛋白质等复杂的三维几何“形状”的研究是数学生物学和药理学几个前沿领域的核心。例如，如果已知一种病毒的三维结构，并且可以找到一种具有互补形状和化学成分的蛋白质，那么这种蛋白质就可以用作抗病毒药物的基础。由于我们不了解蛋白质的三维几何结构，许多这样的研究都是以一种临时的方式进行的。科克伦博士将专注于非常新的“高阶”打结现象。他寻求对现象的新的几何和定量描述。＊＊＊

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. ***