Spectral Invariants and Low-Dimensional Topology

PI提出研究谱不变量，特别是广义狄拉克算子的eta不变量在几何拓扑和几何问题中的计算和应用。重点将是使用解析和微分几何机制来研究谱流，Atiyah-Patodi-Singer rho不变量，以及$SU(3)$规范理论的Casson不变量。主要目标是利用狄拉克算子的边值问题理论为谱不变量开发可用的剪切和粘贴机制。PI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。但是这个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”。PI在该领域的方法是将传统的三维技术与其他数学学科（如微分几何和分析）的方法相结合。

DMS-0202148Paul KirkThe PI proposes to study computations and applications ofspectral invariants, inparticular the eta-invariant of generalized Dirac operators, toproblems in geometric topology and geometry. The focus will be touse analytic and differential geometric machinery to study spectralflow, the Atiyah-Patodi-Singer rho-invariant, and the$SU(3)$-gauge theoretic Casson invariant. The main goal is to develop usable cut-and-paste machinery for spectral invariants usingthe theory of boundary-value problems for Dirac operators.The PI'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 universe we live in is 3-dimensional. But geometric problems in this area turn out to be very difficult for the odd reason that in such low dimensions there is "not enough room to move." The PI's approach to the area is to combine traditional 3 dimensional techniques withmethods from other mathematical disciplines such as differential geometry and analysis.