Mathematical Sciences: Topological Applications of Quantum 3-Manifold Invariants

9626818 Kania-Bartoszynska这个项目属于低维拓扑的一般领域。更具体地说，它涉及应用拓扑量子场论经典三流形拓扑。研究人员将使用量子不变量来构造三球面中三流形嵌入的障碍物，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标准的对称性存在于三个流形。低维拓扑学主要关注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上而不撕裂-这被称为同胚变换。这种分类，称为拓扑分类，基本上是在三维和四维以外的维度上进行的。事实证明，这两个维度提供了各种进一步的技术困难，其中一些是当前备受关注的话题，因为它们与现代物理学的深刻思想相结合。

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.