Some Problems in Algebraic and Geometric Topology

这一建议涉及到代数拓扑学在高维流形分类中的应用。PI和他的合作者最近证明了每个有限循环空间都是同伦的，等价于一个紧致的光滑流形，解决了Browder提出的一个40年来的问题，激励了外科理论。Borel，Novikov，Baum-Connes和Farrell-Jones的猜想断言K-理论和L-理论中的集合映射要么是单态，要么是同构。所有这些装配图都可以用受控代数来表示，这是PIU打算继续研究的领域，以获得更大类别的组的结果。流形是局部类似于欧几里德空间的几何对象，但可能是全局弯曲的，如球体。这些空间在数学和物理学中都有出现。一百多年来，拓扑学家一直在研究流形，因为它们掌握着理解空间本质的关键。PI建议扩展现有的代数分类方法，以实现更深层次的几何理解。特别重要的是受控代数方法，它同时捕捉了代数和拓扑学的本质

This proposal concerns the application of Algebraic Topology to the classification of high dimensional manifolds. The PI and collaborators have recently shown that every finite loop space is homotopy equivalent to a compact, smooth manifold, solving a 40 year old problem posed by Browder, motivating Surgery Theory. This still leaves open the question whether all finite H-spaces have this property.The conjectures of Borel, Novikov, Baum-Connes and Farrell-Jones assert that the assembly maps in K- and L-theory are either monomorphisms or isomorphisms. All these assembly maps can be expressed in terms of controlled algebra, an area the PIU intends to continue working on, to obtain results for larger classes of groups.A manifold is a geometrical object which locally is like euclidean space, but may be curved globally such as a sphere. These spaces arise both in mathematics and in Physics. Manifolds have been studied by topologists for over a hundred years as they hold the key to understanding the nature of space. The PI proposes to extend vurrent algebraic classification methods to achieve a deeper geometric understanding. Of particular significance is controlled algebraic methods, which capture the essence of both algebra and topology