Mathematical Sciences: Some Problems in Algebraic and Geometric Topology

9626562彼得森这个项目主要涉及流形及其对称性。流形是局部同胚于欧几里得空间的第二可数Hausdorff拓扑空间。高维流形和低维流形的研究有很大的不同，本课题主要研究高维流形现象。非线性相似问题是关于有限群的两个表示是否可以是同胚的而不是线性同构的问题。这个问题本质上是高维的，因为很容易看出，这种现象不可能发生在低于四维的维度上。这个项目的另一个主要部分是Novikov和Borel猜想。这些猜想与高维流形的分类有关。当然，分类在所有维度上都是有趣的，但在高维度和低维度上的方法是相当不同的。该项目与数学的许多分支都有联系，特别是与分析有关。所用的方法主要来自受控拓扑学，在分析中也有类似的受控拓扑学问题，特别是通过Baum-Connes猜想。虽然高维流形不容易被我们的心灵所接近，但它们在现实生活中每当有许多自由度时就会自然而然地出现，例如，具有许多关节和枢轴的机械连杆的所有可能配置的空间。***

9626562 Pedersen This project is mainly concerned with manifolds and their symmetries. A manifold is a second countable Hausdorff topological space which is locally homeomorphic to euclidean space. There is quite a difference in studying manifolds in high dimensions and in low dimensions, and this project is mainly concerned with high dimensional phenomena. The non-linear similarity problem is the question whether two representations of finite groups can be homeomorphic without being linearly ismorphic. This question is high-dimensional in nature, since it is easy to see that such phenomena cannot occur in dimensions lower than four. The other main part of this project is the Novikov and Borel conjectures. These conjectures are concerned with classification of manifolds in high dimensions. Classification is, of course, interesting in all dimensions, but the methods are quite different in high dimensions and in low dimensions. The project has connections to many branches of mathematics, notably to analysis. The main methods used are from controlled topology, and there are analogues of controlled topological problems in analysis, notably through the Baum-Connes conjectures. While high dimensional manifolds are not readily accessible to our mind's eye, they arise naturally in real-life situations whenever there are numerous degrees of freedom, as, for example, the space of all possible configurations of a mechanical linkage with numerous joints and pivots. ***