Mathematical Sciences: Topological Applications of Quantum 3-Manifold Invariants

9626818 Kania-Bartoszynska本项目属于一般低维拓扑领域。更具体地说，它讨论了拓扑量子场论在经典三维流形拓扑中的应用。研究人员将使用量子不变量来构造三维流形在三球面上嵌入的障碍，并构造用于检测透镜空间和塞弗特纤维空间的不变量。此外，量子不变量将被用来制定三维流形中对称性存在的准则。低维拓扑主要涉及对三维和四维空间的分类，其中两个空间被视为等价的，如果一个空间可以变形到另一个空间而不被撕裂-这被称为同胚变换。这种被称为拓扑分类的分类基本上是在三维和四维之外的维度上进行的。事实证明，这两个维度提供了各种进一步的技术困难，其中一些是当前非常感兴趣的话题，因为它们与现代物理学的深层思想相联系。

9626818 Kania-Bartoszynska This project belongs in the general area of low-dimensional topology. More specifically, it deals with applications of topological quantum field theory to classical three-manifold topology. The investigator will use quantum invariants to construct obstructions to the embedding of three-manifolds in the three-sphere and to construct invariants for the detection of lens spaces and Seifert fiber spaces. Additionally, quantum invariants are to be used to formulate criteria for the existence of symmetries in three-manifolds. Low-dimensional topology is mainly concerned with classifying spaces of dimensions three and four, where two spaces are treated as equivalent if one space can be deformed onto the other without tearing - this is called a homeomorphic transformation. Such a classification, known as the topological classification, has been essentially carried out in dimensions other than three and four. It turns out that these two dimensions provide various further technical difficulties, some of which are topics of much current interest as they interface with deep ideas of modern physics.