Holomorphic Disks and Low-Dimensional Topology

该建议涉及三维和四维流形的不变量，特别是那些由提议者与ZoltanSzabo合作构建的不变量。这些不变量似乎与规范理论（Donaldson和Seiberg-Witten）不变量密切相关，除了它们具有比它们的规范理论前辈更具有组合性的优势，因此更容易计算。为了说明它们的强度，我们可以对先前用规范理论在四维微分拓扑中得到的一些结果进行修正。这些不变量还具有超越规范理论的新的拓扑应用。例如，它们可以用于对手术产生晶状体空间的结进行新的限制。作者希望进一步扩展这一理论，看看他们可能给拓扑维度3和4进一步的见解。该建议涉及研究三维和四维空间的内在结构。西蒙·唐纳森在80年代早期引入了“量规论技术”，彻底改变了四维空间的研究。这些技术与空间有关，即某些偏微分方程（来自数学物理）的解的空间。然后使用这些解空间的过程属性来深入了解定义方程的底层空间的更精细结构。该提案涉及规避这些（通常难以识别的）辅助解决方案空间的努力，以提供更直接的组合方法来理解三维和四维空间组合在一起的方式。

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.