Problems in 3-Dimensional Topology and Gauge Theory and Applications

提案：DMS-9971020 PI：Paul Kirk摘要。 该项目关注三维拓扑中拓扑不变量的计算和应用，重点是分析和微分几何机器的使用。所研究的领域包括Chern-Simons泛函的Hessian谱流和相关的3-流形规范理论不变量，包括SU（3）$ Casson不变量和Atiyah-Patodi-Singer rho不变量。 也要研究twistedAlexander不变量的结，链接，字符串和关系到卡森-戈登不变量和应用的柯克-利文斯顿组合结和链接不变量。 柯克的工作在纯数学的目标是解决和澄清几何问题的三维。一方面，三维空间的设置对我们所有人来说都很熟悉，因为我们生活的宇宙是三维的。但是，在这个领域的几何问题是非常困难的奇怪的原因，在这样低的维度有“没有足够的空间移动。”“这种家庭友好和数学困难之间的紧张关系使这一领域成为数学家研究的一个特别有趣的领域。柯克的方法，该地区是联合收割机传统的三维技术与其他领域的方法，特别是方法，使他们的方式进入数学从物理。

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.