Mathematical Sciences: Topology of Arithmetic Groups

小行星9504264 康诺利将研究拓扑刚性的障碍，如S。卡佩尔 他将调查的问题拟合边界到分层空间与驯服结束。 他还将用这些来攻击奎因刚性猜想（ICM，1986）。 他希望采取新的“连续控制”K理论，并使用它来创建一个手术理论，这是适应这些问题。 他还包括4维光滑流形在他的项目（规范理论）。 在这个项目中康诺利将试图表明：算术组是拓扑刚性。 用不那么警句的话来说，这意味着自然界中某些基本几何（这些几何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。 在高度简化的术语中，它意味着任何模糊地像环面（或克莱因瓶）的东西都应该拓扑等价于环面（或克莱因瓶）。 拓扑等价的一个例子是圆和椭圆;它们不是欧几里得几何意义上的全等图形，但它们在拓扑上是等价的，因为一个稍微拉伸就会产生另一个。 ***

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. ***