TQFT and Low Dimensional Topology

TQFT 和低维拓扑

• 批准号: 0604580
• 负责人: Patrick Gilmer
• 金额: $ 16.88万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604580&HistoricalAwards=false
• 关键词: TQFT; Low; Dimensional; Topology

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.

吉尔默将继续研究定义在割圆整数环上的拓扑量子场论。他将继续与Masbaum一起研究作用在这些环上的模以及它们的有限商上的相关映射类群表示。他还将研究Masbaum定义的Torelli群的相关无穷维微扰表示。Gilmer将研究纽结和其他具有无限循环覆盖的空间的强移位等价类不变量。他将使用这些结构来研究低维拓扑中的问题。吉尔默还将研究他与真实射影平面中的曲线集合相关联的链接的不变量。这应该导致对实代数曲线的拓扑的限制。拓扑学是研究内在形状的。它有时被称为“橡皮片”几何学，因为人们研究的物体经常可以被扭曲和拉伸，但不能被撕裂。最近，拓扑学从物理学中涌入了大量的想法。拓扑量子场论(TQFT)是与高能物理、量子计算以及数论等数学领域有着密切联系的拓扑学领域。吉尔默将探索的同样的拓扑量子场理论已经被其他人提出应用于量子计算。吉尔默还想用TQFT作为研究拓扑学中经典问题的工具。低维拓扑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。