Holomorphic Disks and Low-Dimensional Topology

DMS-0234311 Peter Ozsvath该提案涉及三维和四维流形的不变量，特别是提案者与ZoltanSzabo合作构建的不变量。这些不变量似乎与规范理论（唐纳森和塞伯格-威滕）不变量密切相关，除了它们具有比规范理论前辈更组合的优点，因此更容易计算。为了说明它们的强度，我们可以重新证明以前用规范理论得到的四维微分拓扑中的一些结果。这些不变量也有超越规范理论的新的拓扑应用。例如，他们可以用来给新的限制，纽结谁sursesurgently给透镜空间。作者希望进一步扩展这一理论，看看他们可能会给三维和四维拓扑学带来什么样的进一步见解。这一提议涉及研究三维和四维空间的内在结构。四维空间的研究是由西蒙唐纳森在80年代初通过他的"规范理论技术"的介绍革命。这些技术与某些偏微分方程（来自数学物理）的解的空间相关联。这些解决方案空间的课程属性然后被用来给洞察到underlyingspaces上的方程被定义的更精细的结构。该提案涉及努力绕过这些（通常很难识别）辅助空间的解决方案，以更直接的组合方法来理解三维和四维空间的方式。

DMS-0234311Peter OzsvathThe proposal deals invariants for three- and four-dimensional manifolds,specifically those which the proposer constructed in collaboration ZoltanSzabo. These invariants appear to be closely related to gauge-theoretic(Donaldson and Seiberg-Witten) invariants, except that they have theadvantage of being more combinatorial in nature than their gauge-theoreticpredecessors, and hence easier to calculate. To illustrate their strength,one can reprove some results in four-dimensional differential topologywhich were obtained previously using gauge theory. These invariants alsohave new topological applications which go beyond gauge theory. Forinstance, they can be used to give new restrictions on knots whosesurgeries give lens spaces. The author hopes to further extend thistheory, to see what further insight they may give to topology indimensions three and four.The proposal deals with studying the intrinsic structure of three- andfour-dimensional spaces. The study of four-dimensional spaces wasrevolutionized by Simon Donaldson in the early 80's by his introduction of"gauge-theoretic techniques". These techniques associate to a space thespace of solutions of certain partial differential equations (which comefrom mathematical physics). Course properties of these solution spaces arethen used to give insight into the finer structure of the underlyingspaces on which the equations are defined. The proposal deals with efforts at circumventing these (often difficult to identify) auxiliary spaces of solutions, to give more directly combinatorial approaches to understanding the way three- and four-dimensional spaces fit together.