Mathematical Sciences: Topological Applications of Algebraic Cycles

[401533] Lima-Filho研究者将继续探索代数变量上代数环群的拓扑性质。他发现这是拓扑和代数几何问题应用的宝库。代数循环被看作是一个拓扑群函子，从代数变元的范畴到cw -复形范畴中的群对象的范畴。这个函子将主纤维与闭包体联系在一起，它们的同伦群为代数变异提供了一个同伦理论，他希望对此有更深入的了解。本文将建立一个任意变量的交点理论，推广和简化了拟射影变量的交点理论。在纯拓扑方面，研究者利用代数环建立了等变无限环空间，由此得到的广义上同调理论与其他几个理论有联系。作为新旧理论相互作用的结果，他已经回答了关于陈总类图的长期猜想，目前正在计算几个例子，其中代数几何，群表示理论和同伦理论一起阐明了所涉及的系数系统，相关的转移图等等。研究者研究的一个基本方面和指导原则是其统一性，在此基础上，数学的不同领域发挥作用，并产生多方面的代数技术应用。因此，他获得了一种非常理想的思想经济，从某种意义上说，他对各种看似不相干的问题采取了一种简洁统一的方法，在短时间内对每个问题都取得了深刻的结果。他首先赋予代数几何中的基本对象，代数循环，一种拓扑学，这种拓扑学后来被证明具有非凡的性质。然后他意识到这些循环不仅为大的代数变量（对代数几何家至关重要）提供了新的不变量，而且它们也与稳定同伦理论（代数拓扑中的一个主要课题）的主流研究密切相关。当他探索涉及代数循环的结构时，其他自然结构也在这种背景下出现。因此，从等变的角度研究循环也提供了对有限群表示理论和群上同调的各个方面的见解，增强了他的统一方法的实用性。＊＊＊

9401533 Lima-Filho The investigator will continue to explore topological properties of the groups of algebraic cycles on algebraic varieties. He has found this to be a mine of applications to topological and algebraic geometric problems. Algebraic cycles are seen as a topological group functor from the category of algebraic varieties to the category of group objects in the category of CW-complexes. This functor associates principal fibrations to closed inclusions, and their homotopy groups provide a homology theory for algebraic varieties which he wishes to understood more deeply. An intersection theory for arbitrary varieties will be developed, generalizing and simplifying an existing theory for quasiprojective varieties. On the purely topological side, the investigator's research has created equivariant infinite loop spaces using algebraic cycles, and the generalized cohomology theories thus obtained have been related with several other theories. As an outcome of this interplay between new and old theories, he has already answered long-standing conjectures about the total Chern class map and is presently computing several examples where algebraic geometry, group representation theory, and homotopy theory come together to illuminate the coefficient systems involved, the associated transfer maps, and so forth. A fundamental aspect and guiding principle in the investigator's research is its unifying character, under which diverse areas of mathematics come into play and yield multifaceted applications of his algebraic techniques. He thus obtains a highly desirable economy of thought, in the sense that a concise and unified approach to various seemingly disparate questions attains deep results about each in a short period of time. He starts by endowing basic objects in algebraic geometry, the algebraic cycles, with a topology which turns out to have remarkable properties. He then realizes that these cycles provide new invariants not only for alge braic varieties (of fundamental importance to algebraic geometers), but that they also deeply relate with ongoing mainstream research in stable homotopy theory, a major topic in algebraic topology. Other natural structures arise in this setting as he explores constructions involving his algebraic cycles. The study of cycles from an equivariant point of view thus also provides insight into various aspects of the theory of representation of finite groups and group cohomology, enhancing the usefulness of his unified approach. ***