中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

综合报告

报告人： 李振坤 博士（Stanford University）

题  目： Instanton Floer homology and Dehn surgery on knots

时  间：2022.09.09（星期五），09:00-10:00

地  点：腾讯会议：607-926-119

摘  要：Instanton homology was introduced by Floer in 1980s. It is an infinite dimensional Morse theory built for 3-manifolds or the knots inside them. Instanton Floer homology encodes strong topological information (for example, it detects the genus and fiberednessof knots), and is closely related to the SU(2) representations of the fundamental groups of related 3-manifolds (which, for example, leads to the approval of the Property P conjecture). However, instanton homology is built on a set of partial differentialequations and is almost impossible to be computed directly via solving the equations. These make the computation of instanton Floer homology a crucial task. In this talk, I will present my recent work, largely joint with Fan Ye, on establishing some basicstructural properties for instanton homology, and proving a surgery formula that computes the instanton Floer homology of Dehn surgeries on knots. I will also discuss the applications of these works in proving some new knot detection results, in studying therepresentations of the fundamental groups of 3-manifolds, and in computing the instanton Floer homology of some important families of 3-manifolds and knots.