TQFT and Low Dimensional Topology

Gilmer将继续研究在环分环整数上定义的拓扑量子场论。他将继续与Masbaum一起研究相关的映射类群表示作用于这些环及其有限商上的模上。他还将研究Masbaum定义的Torelli群的相关无限维摄动表示。Gilmer将研究结点和其他具有无限循环覆盖的空间的强位移等价类不变量。他将使用这些结构来研究低维拓扑中的问题。Gilmer还将研究他与实投影平面上曲线集合相关联的连杆的不变量。这将导致对实代数曲线拓扑结构的限制。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，拓扑学有大量来自物理学的思想。拓扑量子场论（TQFT）是一个与高能物理、量子计算以及数论等其他数学领域密切相关的拓扑领域。Gilmer将要探索的拓扑量子场论已经被其他人提出用于量子计算。Gilmer还希望使用TQFT作为研究拓扑学经典问题的工具。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 the study of topological quantum field theories defined over rings of cyclotomic integers. He will continue to study with Masbaum the associated mapping class groups representations acting on modules over these rings and also over their finite quotients. He will also study the related infinite dimensional perturbative representation of the Torelli group that Masbaum has defined. Gilmer will study strong shift equivalence class invariants of knots and other spaces which are equipped with an infinite cyclic cover. He will use these structures to study questions in low dimensional topology. Gilmer will also study invariants of links that he has associated to collections of curves in the real projective plane. This should lead to restrictions on topology of real algebraic curves.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 had a large influx of ideas from physics. Topological quantum field theory (TQFT) is an area of topology with intimate connections to high energy physics, quantum computing as well as other areas of mathematics, for instance number theory. The same topological quantum field theories that Gilmer will be exploring have been proposed by others for applications to quantum computing. Gilmer would also like to use TQFT as a tool to study classical questions in topology. Low dimensional topology is important for chemistry and biology as it has implications for the mechanism of DNA, and other molecular configurations.