The Topology and Geometry of Topological Field Theories

拓扑场论在20世纪80年代通过数学家对低维拓扑的研究而兴起，包括Donaldson、Floer、Gromov等人。在努力理解他们的工作时，威滕发现，在第2、3和4维中，这些理论可以被视为一种特殊的超对称量子场论。这种思想的结合已被证明是新数学在各种领域的丰富来源，包括几何、拓扑和代数。与此工作平行，Atiyah和Segal提出了拓扑场理论的公理化，允许他们使用代数技术，特别是同构代数进行研究。最近在这个方向上有大量的工作，这次会议及其相关的研究生研讨会的目的是调查该领域的现状并探索未来的方向。物理学中出现了场论，用来描述亚原子粒子的行为、它们之间的相互作用以及作用在它们身上的力。拓扑场论起源于物理学中的弦理论，以及粒子可能不是离散的物体，而是一维的（如果仍然很小的话）“弦”的观察。这启发了许多重要的数学；特别是在低维物体的几何方面——这个领域在过去的25年里已经完全改变了。这里资助的会议将汇集这一领域和密切相关领域的数学家和数学物理学家，以更好地了解目前最肥沃的互动领域在哪里。会议之前将有一个为期一周的研讨会，旨在让新的研究人员接触到该领域的最新思想和技术。

Topological field theories arose in the 1980s through the work of mathematicians studying low dimensional topology, including Donaldson, Floer, Gromov, and others. In his effort to understand their work, Witten discovered that, in dimensions 2,3 and 4, these theories could be viewed as a special kind of supersymmetric quantum field theory. This combination of ideas has proved to be a fertile source of new mathematics in a variety of fields, including geometry, topology, and algebra. Parallel to this work, Atiyah and Segal presented an axiomatization of topological field theories which allows their study using the techniques of algebra, and in particular of homological algebra. Their has been a great deal of recent work in this direction and the purpose of this conference and its associated graduate student workshop is to survey the state of the field and to explore directions for the future.Field theories arise in physics to describe the behavior of sub-atomic particles, their interactions, and the forces that act on them.Topological field theories arise from string theory in physics and the observation that particles may not be discrete objects, but one-dimensional, if still very small, "strings". This has inspired a great deal of important mathematics; in particular in the geometry of low-dimensional objects -- a field that has been completely transformed in the past twenty-five years. The conference funded here will bring together mathematicians and mathematical physicists in this and closely related areas to gain a better understanding of where the most fertile areas of interaction presently are. The conference will be preceded by a week-long workshop intended to give new researchers to be exposed to the latest ideas and techniques in the field.