Mathematical Sciences: Topological Applications of Algebraic Cycles

9401533研究人员将继续探索代数簇上代数圈群的拓扑性质。他发现这是一个应用于拓扑和代数几何问题的矿藏。代数圈被看作是从代数簇范畴到CW-复形范畴中的群对象范畴的拓扑群函子。这个函子将主纤维与闭包涵体联系起来，它们的同伦群为他希望更深入地理解的代数簇提供了一个同调理论。建立了任意簇的交集理论，推广和简化了已有的拟投射簇理论。在纯拓扑学方面，研究者的研究利用代数环建立了等变无限循环空间，并由此得到的广义上同调理论与其他几个理论相关联。作为这种新旧理论相互作用的结果，他已经回答了关于全陈类映射的长期猜想，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结合在一起来说明所涉及的系数系统、相关的转移映射等。研究人员研究的一个基本方面和指导原则是它的统一性，在这种特性下，不同的数学领域发挥作用，并产生他的代数技术的多方面应用。因此，他获得了一种非常可取的思想经济，在这个意义上，对各种看似不同的问题采取简洁和统一的方法，在很短的时间内就能对每个问题产生深刻的结果。他首先赋予了代数几何中的基本对象--代数圈--一种被证明具有显著性质的拓扑。然后他意识到，这些圈不仅为代数簇(对代数几何学家具有基本重要性)提供了新的不变量，而且它们还与正在进行的稳定同伦理论的主流研究密切相关，稳定同伦理论是代数拓扑学的一个主要主题。当他探索涉及他的代数循环的结构时，其他自然结构在这种背景下出现。因此，从等变的观点研究圈也提供了对有限群的表示理论和群上同调的各个方面的洞察，增强了他的统一方法的有用性。***

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. ***