TQFT and Low Dimensional Topology

Gilmer将继续研究积分拓扑量子理论。与Gregor Masbaum一起，他将研究可以用积分拓扑量子理论定义的映射类群的表示。他还将研究马斯鲍姆定义的托雷利群的普遍代表。Gilmer计划计算结点和其他空间的强移位等价类不变量，这些不变量可以用TQFT来定义。他希望进一步计算强位移等价不变量，以揭示它们的拓扑意义。他利用拓扑量子理论不变量找到阻碍纤维结成为带状结的因素。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Charles Livingston一起，他希望更好地理解半4空间中的哪些不可定向曲面可以在3空间中具有给定结的边界。Gilmer还计划使用四维拓扑来研究真实投影平面上的真实代数曲线的拓扑。拓扑学是对内在形状的研究。它有时被称为“橡皮板”几何，因为人们经常研究的对象可以扭曲和拉伸，但不能撕裂。最近，拓扑学经历了大量来自物理学的思想的涌入。拓扑量子场论是当前最令人兴奋的拓扑领域之一，与高能物理、量子计算以及数论等其他数学领域有着密切的联系。Gilmer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都很重要，因为它对DNA和其他分子结构的机制有影响。

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.