Spectral Invariants and Low-Dimensional Topology

Pi建议研究谱不变量的计算和应用，特别是广义Dirac算子的ETA不变量在几何拓扑和几何问题中的应用。重点将利用解析和微分几何机制来研究谱流、Atiyah-Patodi-Singer Rho不变量和$SU(3)$规范理论Casson不变量。主要目的是利用狄拉克算子的边值问题理论，开发有用的谱不变量的剪切和粘贴机制。PI在纯数学中的工作是以解决和澄清三维几何问题为目标的。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。但这一领域的几何问题被证明是非常困难的，因为奇怪的原因是，在如此低的维度中，“没有足够的移动空间”。PI的方法是将传统的三维技术与其他数学学科的方法相结合，如微分几何和分析。

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.