LOW-DIMENSIONAL MANIFOLDS AND HIGH-DIMENSIONAL CATEGORIES

摘要奖：DMS-1041217主要研究者：Ian Agol，Robion C.该奖项将为“低维流形和高维类别”会议提供资金。这个标题有点过时，因为低维度和高维度实际上是相同的：3，特别是4。会议庆祝迈克尔弗里德曼的60岁生日和一些当前的数学成就和挑战的影响弗里德曼的工作。对于5维及更高维的流形，有一个完善的分类方案，即所谓的外科理论，其中包括著名的s-配边定理，该定理从同伦数据中产生流形之间的同构。 5以下的维数更难，因为回旋的空间更小，因此，配边定理失败。 对于n-范畴，n大于2的维数是困难的，因为描述n-球可以被切割、胶合和旋转的各种方式的组合关系是复杂的。因此，维度3和维度4代表了两个学科的共同边界，尽管这个边界是从不同的方向接近的。本次会议的主要目标之一是促进四维流形专家与更高范畴和量子场论专家之间的交流。Michael Freedman对四维空间（流形）的研究和分类做出了开创性的贡献，这是拓扑学研究的中心话题，与代数，几何和物理学有联系。 物理学的思想，特别是量子场论的思想，意味着应该有某种结构来描述四维空间的拓扑，即不依赖于空间的长度或角度的度量的内在性质。在数学上，这些理论是用范畴论的代数语言来表述的，需要在这一领域有实质性的新发展。会议的目的是将3维和4维流形的专家与范畴论和量子场论的专家聚集在一起，探讨这些主题之间的相互作用。会议还将探讨其他几何结构，如断裂莱夫谢茨纤维化、接触和辛结构、光滑结构和规范理论。

AbstractAward: DMS-1041217Principal Investigator: Ian Agol, Robion C. KirbyThis award will provide funding for a conference "Low-dimensionalmanifolds and high-dimensional categories". The title is somewhatironic, in that the low and high dimensions are in fact the same:3 and especially 4. The conference celebrates the occasion ofMichael Freedman's 60th birthday and some of the currentmathematical achievements and challenges influenced by Freedman'swork. For manifolds of dimension 5 and higher, there is awell-established classification scheme, the so called surgerytheory which includes the celebrated s-cobordism theorem thatproduces diffeomorphisms between manifolds from homotopytheoretic data. Dimensions below 5 are more difficult becausethere is less room to maneuver and, as a consequence, thes-cobordism theorem fails. For n-categories, dimensions ngreater than 2 are difficult because of the complexity of thecombinatorial relations which describe the various ways n-ballscan be cut, glued and rotated. Thus dimensions 3 and 4 representa shared frontier of the two subjects, though this frontier isapproached from different directions. One of the main goals ofthis conference is to promote cross-fertilization between expertson 4-manifolds and experts on higher categories and quantum fieldtheories.Michael Freedman made groundbreaking contributions to the studyand classification of 4-dimensional spaces (manifolds), which area central topic in the study of topology, and have connectionswith algebra, geometry and physics. Ideas from physics, inparticular quantum field theory, imply that there ought to becertain constructions which describe the topology of4-dimensional spaces, that is, intrinsic properties which do notdepend on the measure of length or angles on the space.Mathematically, these theories are formulated in the algebraiclanguage of category theory and require substantial newdevelopments in that area. The object of the conference is tobring together experts on 3- and 4-dimensional manifolds togetherwith experts in category theory and quantum field theory toexplore the interactions between these topics. Additionalgeometric structures, such as broken Lefschetz fibrations,contact and symplectic structures, smooth structures, and gaugetheories will also be explored at the conference.