Mathematical Sciences: Topology of Arithmetic Groups

9504264康诺利·康诺利将研究拓扑刚性的障碍，如S·卡佩尔的UNIL小组提供的障碍。他将研究将边界与一个末端温和的分层空间相适应的问题。他还将用这些来攻击奎恩刚性猜想(ICM，1986)。他希望采用新的“连续控制”K理论，并用它来创造一种适应这些问题的外科理论。他还在他的项目(规范理论)中加入了4维光滑流形。在这个项目中，康诺利将试图证明：算术群在拓扑上是刚性的。用不那么深奥的术语来说，这意味着自然界的一些基本几何图形(这些几何图形被数学家和物理学家称为局部对称空间)所表现出的极端对称性，最终应该被证明是基于其基本群的纯粹拓扑性质。在高度简单化的术语中，这意味着任何模糊地像圆环(或克莱因瓶)的东西在拓扑上都应该等同于圆环(或克莱因瓶)。拓扑等价的一个例子是圆和椭圆；它们不是欧几里德几何意义上的全等图形，但它们在拓扑上是等价的，因为一个拉伸一下就会产生另一个。***

9504264 Connolly Connolly will be studying obstructions to topological rigidity such as those provided by the group UNIL of S. Cappell. He will be investigating the problem of fitting a boundary onto a stratified space with a tame end. He will also be using these to attack the Quinn Rigidity Conjecture (ICM, 1986). He wishes to take the new "continuous control" K-theory and use it to create a Surgery theory which is adapted to these problems. He is also including 4-dimensional smooth manifolds in his project (Gauge Theory). In this project Connolly will be attempting to show that: Arithmetic Groups are Topologically Rigid. In less epigrammatic terms, this means that the extreme symmetry exhibited by some of the fundamental geometries of nature (these geometries are called Locally Symmetric Spaces by mathematicians and physicists), should eventually be shown to be based on purely topological properties of their fundamental groups. In highly oversimplified terms, it means that anything that is vaguely like a torus (or a Klein bottle) should be topologically equivalent to a torus (or a Klein bottle). An example of a topological equivalence is given by a circle and ellipse; they are not congruent figures in the sense of Euclidean geometry, but they are topologically equivalent, because a little stretching of the one produces the other. ***