报告题目：Knots, links, spatial graphs and their representation varieties

报告人：谢羿，北京大学北京国际数学研究中心

时间：2022年10月21日（周五）15:30-16:30

腾讯会议：647-4448-8711 

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摘要：The fundamental groups can strongly reflect the geometric and topological properties of 3-manifolds. One approach to understand the fundamental groups is to study their representations into linear groups such as SU(2) or SO(3). In the past, the existence of non-abelian SU(2) representations has been proved for the fundamental groups of  many different classes of 3-manifolds, including the complement of any non-trivial knot. Most of these results are obtained using techniques from gauge theory. More surprisingly, these techniques have been generalized by Kronheimer and Mrowka to the situation of spatial graphs (graphs embedded in the space). It is known that the four color theorem is equivalent to the existence of certain representations of planar cubic graphs (viewed as spatial graphs) into the Klein 4-group, which is a subgroup of SO(3). Therefore Kronheimer and Mrowka’s theory provides a potential way to obtain a computer-free proof of the four color theorem. In this talk, I will give an introduction to their theory and review various results on representations of the fundamental groups of the complements of knots, links and spatial graphs. Part of this talk is joint work with Boyu Zhang.  

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