FRG: Topological Invariants of 3 and 4-Manifolds

塞尔曼·阿克布鲁这是数学科学分部重点研究小组(FRG)颁发的奖项，http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htmThe的目的是为源于塞贝格-威腾理论的不变量找到纯粹的拓扑(连续或组合)定义。其中包括4维流形的基本类和3维流形的Oszvath-Szabo Floer同调。最近的进展，特别是最近两年Ozsvath和Szabo的惊人结果，使Seiberg-Witten理论看起来更接近拓扑学，这个小组的共同努力可能会完成这项任务。这一提议属于规范理论和弦理论这两个更大的主题，它们与理论物理密切相关，并受到它们的推动，特别是试图统一自然力。低维边界理论是n-1维量子场理论的数学类比，即n维空间和1维时间对应于n维流形之间的(n-1)维边界。这些边界主义可以有额外的结构，例如奇数维的接触结构和偶数维的辛结构。在理解这类物理学的数学基础方面取得的进步，有助于更好地理解物理学。

DMS-0244622Selman AkbulutThis is a Division of Mathematical Sciences Focused Research Group (FRG) award made under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htmThe goal of this proposal is to find purely topological (continuous orcombinatorial) definitions of the invariants arising out ofSeiberg-Witten theory. These include the basic classes for4-dimensional manifolds, and the Oszvath-Szabo Floer homology for3-manifolds. Recent progress, in particular the striking results ofOzsvath and Szabo in the last two years, has brought Seiberg-Wittentheory seemingly close to topology, and a concerted effort by thisgroup may be able finish the task.This proposal falls under the larger topics of gauge theory and stringtheory, which are closely connected to and motivated by theoreticalphysics, in particular the attempt to unify the forces of nature.Low-dimensional bordism theory is the mathematical analogue of n+1dimensional quantum field theories, i.e. an n-dimensional space and1-dimensional time correspond to an (n+1)-dimensional bordism betweentwo n-dimensional manifolds. These bordisms can have extra structure,e.g. contact structures in odd dimensions and symplectic structures ineven dimensions. Progress in understanding the mathematicalunderpinnings of this sort of physics feeds back into betterunderstanding of the physics.