Low-Dimensional Topology and Fundamental Groups

PI将以基群对低维流形几何性质的影响为中心研究低维拓扑中的问题。本文将采取三个一般的研究方向：具有规定基本群的辛光滑4-流形的渐近地理问题，SU(3) Casson不变量的手术性质，以及与流形基本群的酉表示相关的Atiyah-Patodi-Singer不变量的同调性质。拓扑学的主题是研究在变形或拉伸下不变的空间性质。基本组是一个简单的定义，但非常强大的数学对象，测量圆如何在不同的空间中变形。最具挑战性的研究空间是“低维流形”，即那些具有3维或4维的空间，就像我们生活的宇宙一样。PI建议使用基本群及其与空间几何和分析的关系来研究这些问题。

The PI will study problems in low-dimensional topology centered around the impact of the fundamental group on the geometric properties of a low dimensional manifold. Three general directions of investigation will be taken: an asymptotic geography problem for symplectic and smooth 4-manifolds with prescribed fundamental groups, surgery properties of the SU(3) Casson invariant, and the homotopy properties of the Atiyah-Patodi-Singer invariant associated to a unitary representation of the fundamental group of a manifold.The subject of topology is the study of properties of space that are unchanged under deformation or stretching. The fundamental group is a simple to define but very powerful mathematical object that measured how circles can be deformed in different spaces. Themost challenging spaces to study are the "low dimensional manifolds", i.e. those which have dimension 3 or 4, like the universe we live in. The PI proposes to study these using the fundamental group and its relation to the geometry and analysis of of a space.