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，特别是提案人与ZoltanSzabo合作构建的那些不变量。这些不变量似乎与规范理论(Donaldson和Seiberg-Witten)不变量密切相关，除了它们的优点是本质上比它们的规范理论前身更具组合性，因此更容易计算。为了说明它们的强弱，我们可以反驳以前用规范理论得到的四维微分拓扑中的一些结果。这些不变量还具有超越规范理论的新的拓扑应用。例如，它们可以用来对外科手术提供镜片空间的结节进行新的限制。作者希望进一步扩展这个理论，看看他们对三维和四维空间的拓扑学有什么进一步的见解。建议研究三维和四维空间的内在结构。上世纪80年代初，西蒙·唐纳森引入了规范理论技术，使四维空间的研究发生了革命性的变化。这些技术将某些偏微分方程解的空间(来自数学物理)与空间联系起来。这些解空间的课程性质随后被用来洞察在其上定义方程的底层空间的更精细结构。该提案涉及绕过这些(通常难以识别)解的辅助空间的努力，以给出更直接的组合方法来理解三维和四维空间如何结合在一起。

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.