TQFT and Low Dimensional Topology

吉尔默将继续研究积分拓扑量子理论。与格雷戈尔Masbaum，他将研究表示映射类组，可以使用积分拓扑量子理论定义。 他还将研究Masbaum定义的Torelli群的普遍表示。Gilmer计划计算节点和其他空间的强移位等价类不变量，这些不变量可以使用TQFT定义。 他希望进一步计算强转移等价不变量，以揭示其拓扑意义。他使用拓扑量子理论不变量来寻找纤维结成为丝带结的障碍。 他想看看是否还有其他障碍。 总的来说，他将尝试使用量子拓扑作为低维拓扑的工具。 与查尔斯利文斯顿，他希望更好地了解哪些不可定向的表面在半4空间可以有边界的一个给定的结在3空间。吉尔默还计划用4维拓扑学来研究真实的射影平面上的真实的代数曲线的拓扑。拓扑学是研究内在形状的。它有时被称为“橡胶片”几何，因为人们研究的物体经常可以扭曲和拉伸，但不能撕裂。最近拓扑学经历了大量涌入的想法从物理学。拓扑量子场论是拓扑学中最新和最令人兴奋的领域之一，与高能物理、量子计算以及其他数学领域（例如数论）有着密切的联系。吉尔默将使用拓扑量子理论作为研究低维拓扑的工具。低维拓扑结构对化学和生物学很重要，因为它对DNA和其他分子构型的机制有影响。

Gilmer will continue studying integral topological quantum theories. With Gregor Masbaum, he will study representations of mapping class groups that can be defined using integral topological quantum theories. He also will study a universal representation of the Torelli group that Masbaum has defined. Gilmer plans to compute strong shift equivalence class invariants of knots and other spaces, which can be defined using TQFT. He wishes to calculate further strong shift equivalnce invariants to uncover their topological meaning. He has used topological quantum theory invariants to find obstructions to fibered knots being ribbon knots. He wants to see if there are further obstructions. In general, he will try to use quantum topology as a tool in low-dimensional topology. Together with Charles Livingston, he wishes to understand better which non-orientable surfaces in half 4-space can have boundary a given knot in 3-space. Gilmer also plans to use 4-dimensional topology to study the topology of real algebraic curves in the real projective plane.Topology is the study of intrinsic shape. It is sometimes called "rubber sheet" geometry as frequently the objects one studies can be twisted and stretched but not torn. Recently topology has experienced a large influx of ideas from physics. Topological quantum field theory is one of the most current and exciting areas of topology with intimate connections to high energy physics, quantum computing as well as other areas of mathematics, for instance number theory. Gilmer will use topological quantum theory as a tool to study low dimensional topology. Low dimensional topology is important for chemistry and biology as it has implications for the mechanism of DNA, and other molecular configurations.