Mathematical Sciences: Algebraic and Geometric Topology

9605024 Dwyer该项目涉及拓扑学中的几个领域。它的目标之一是将一些经典的指数定理，特别是那些涉及到欧拉算子和签名算子的指数定理表达为同伦理论命题，然后研究这种重新解释所提供的高阶信息。研究人员还研究了四维流形中曲面的几何问题，以及围绕置换群对构形空间同调的作用的代数问题。最后，他们正在研究p紧群（即p紧群）的分类。“同伦李群”)，以及更全面地理解拓扑和数论是如何通过代数k理论的构造混合在一起的。该项目的总体主题是研究几何（拓扑空间的科学，即高维形状）和代数（方程和数字的科学）之间一些引人注目和令人兴奋的联系。这种联系是双向的。例如，决定是否有可能在一个特定的拓扑空间上使用微积分的方法，结果涉及到解决一些相当微妙的方程。另一方面，一个方程的所有解的集合通常可以以一种或另一种方式组合到拓扑空间中。研究人员正在研究几何和代数之间相互作用的一些特殊方面，他们希望采取双重观点将在这两个领域取得重大进展。＊＊＊

9605024 Dwyer The project touches on several areas in topology. One of its goals is to express some classical index theorems, especially those involving the Euler and signature operators, as homotopy theoretic statements, and then to study the higher-order information provided by this reinterpretation. The researchers are also investigating geometric questions about surfaces in four-dimensional manifolds, as well as algebraic issues surrounding the action of permutation groups on the homology of configuration spaces. Finally, they are working towards a classification of p-compact groups (i.e. "homotopy Lie groups") as well as towards a more complete understanding of how topology and number theory are mixed together by the constructions of algebraic K-theory. The general theme of the project is the study of some remarkable and exciting connections between geometry (the science of topological spaces, that is, of higher-dimensional shapes) and algebra (the science of equations and numbers). The connections go both ways. For instance, deciding whether or not it is possible to use the methods of calculus on a particular topological space turns out to involve solving some fairly subtle equations. On the other hand, the collection of all solutions to an equation can often be assembled in one way or another into a topological space. The investigators are looking at some particular aspects of the interplay between geometry and algebra, and they expect that taking a dual point of view will lead to major advances in both areas. ***