The Topology and Geometry of Topological Field Theories

拓扑场论产生于20世纪80年代，由唐纳森、弗洛尔、格罗莫夫等数学家研究低维拓扑学而产生。在努力理解他们的工作时，威腾发现，在2维、3维和4维，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各种领域新数学的肥沃来源，包括几何、拓扑学和代数。与这项工作平行的是，阿提亚和西格尔提出了拓扑场论的公理化，这使得他们可以使用代数，特别是同调代数的技巧来研究。他们最近在这个方向上做了大量的工作，这次会议及其相关的研究生研讨会的目的是调查场的状态并探索未来的方向。场论产生于物理学，描述亚原子粒子的行为，它们的相互作用，以及作用在它们上面的力。拓扑场论产生于物理中的弦理论和观察到的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。这启发了许多重要的数学；特别是在低维物体的几何学方面--这个领域在过去25年里已经完全改变了。在这里资助的会议将把这一领域和密切相关领域的数学家和数学物理学家聚集在一起，以更好地了解目前交互作用最丰富的领域在哪里。在会议之前，将举行为期一周的研讨会，旨在让新的研究人员接触到该领域的最新想法和技术。

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.