Problems in 3-Dimensional Topology and Gauge Theory and Applications

建议：DMS-9971020PI：Paul KirkABSTRACT。该项目涉及三维拓扑中拓扑不变量的计算和应用，重点是解析和微分几何机械的使用。要研究的领域包括Chern-Simons泛函的Hessian谱流动和相关的三维规范理论不变量，包括$SU(3)$Casson不变量和Atiyah-Patodi-Singer Rho不变量。还将研究纽结、环和弦的两个Alexander不变量及其与Casson-Gordon不变量的关系以及Kirk-Livingston组合纽结和环不变量的应用。柯克在纯数学方面的工作以解决和澄清三维几何问题为目标。一方面，3维的设定是我们所有人都熟悉的，因为我们生活的宇宙是3维的。但这一地区的几何问题是非常困难的，因为奇怪的原因是，在如此低的维度上，没有足够的移动空间。这种模糊性和数学难度之间的紧张关系使这一领域成为数学家特别感兴趣的研究领域。柯克的方法是将传统的三维技术与其他领域的方法结合起来，特别是从物理学进入数学的方法。

Proposal: DMS-9971020PI: Paul KirkABSTRACT. The project concerns the computation and applications oftopological invariants in 3-dimensional topology, with an emphasis onthe use of analytic and differential geometric machinery. Theareas to be studied include the spectral flow of the Hessianof the Chern-Simons functional and related 3-manifold gauge theoryinvariants including the $SU(3)$ Casson invariant andAtiyah-Patodi-Singer rho invariants. Also to be studied are twistedAlexander invariants of knots, links, and strings and relations toCasson-Gordon invariants and also applications of theKirk-Livingston combinatorial knot and link invariants. Kirk's work in pure mathematics has as its goal the solution andclarification of geometric problems in 3 dimensions. The setting of3 dimensions on the one hand is familiar to all of us since the universewe live in is 3-dimensional. But geometric problems in this area turnout to be very difficult for the odd reason that in such low dimensionsthere is "not enough room to move." This tension between a vaguefamiliarity and mathematical difficulty makes this area a particularlyinteresting area for mathematicians to study. Kirk's approach tothe area is to combine traditional 3 dimensional techniques withmethods from other areas, especially methods that have made their wayinto Mathematics from Physics.