Problems in 3-Dimensional Topology and Gauge Theory and Applications

提案：DMS-9971020PI: Paul kirk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chen - simons泛函的hessian的谱流和相关的3流形规范理论不变量，包括$SU(3)$ Casson不变量和atiyah - patodi - singer不变量。还将研究结、环和弦的扭曲亚历山大不变量以及与casson - gordon不变量的关系，以及柯克-利文斯顿组合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，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。但这一领域的几何问题变得非常困难，原因很奇怪，在如此低的维度中，“没有足够的移动空间”。模糊熟悉和数学困难之间的紧张关系使这个领域成为数学家研究的一个特别有趣的领域。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，特别是那些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.