Mathematical Sciences: Algebraic Cycles and the Homotopy Theory of Groups

9400235弗里德兰德·弗里德兰德计划使用代数几何和代数拓扑学的技术，继续他对代数圈的研究。Friedlander等人的程序很有希望，因为它引入了一个新的视角，并引入了代数拓扑学的技术。几个具体的研究途径似乎已经成熟，可以进一步探索：代数圈和同调上的拓扑过滤，Lawson同调和态上同调之间的对偶性，代数圈同调伪装下的Motivic复形，以及代数圈空间中的陈类。除了对代数圈的研究外，Friedlander还将继续努力研究无穷小代数群的上同调所隐含的几何。Priddy计划继续他的计划，研究群的分类空间的同伦型和相关的构造。同伦理论中的许多重要问题都与有限李群或紧李群的空间分类有关。近年来，拓扑学和群论，特别是有限群的上同调和模表示理论之间也有了有趣而有力的联系。随着西格尔和沙利文猜想的解决，这一领域在最近几年迅速发展到了回答基本问题的地步。其中最重要的可能是确定在素数p处完成的分类空间的稳定同伦类型和不稳定同伦类型与其基础群的p-局部结构之间的确切关系。代数几何是利用几何技术研究多项式方程(即代数簇)的解集的学科。对代数几何问题的部分回答导致了从计算机科学的复杂性理论到几何拓扑到数论等领域的进步。弗里德兰德打算借用代数拓扑学中的方法以及现代代数几何技术来研究代数簇。拓扑学的使用涉及对连续变化的结构族的研究，这些结构族传统上被认为是通过其他方法来考虑的。人们希望，这些新技术将为深入了解代数几何中长期存在的问题提供帮助。代数拓扑学是用代数技巧研究几何对象的一门学科。令人兴奋的新发展导致了群论的进步，使用了代数拓扑学，从而颠倒了通常信息流的方向。群是发生在所有科学中的基本对称性，包括涉及代码和物理结构的领域。Priddy希望这种新的方法将使人们更好地理解这些领域之间的关系。***

9400235 Friedlander Friedlander plans to continue his investigation of algebraic cycles, using techniques from algebraic geometry and algebraic topology. The program of Friedlander and others holds promise in that it introduces a new perspective and imports techniques of algebraic topology. Several specific avenues of research appear ripe for further exploration: topological filtrations on algebraic cycles and homology, duality between Lawson homology and morphic cohomology, motivic complexes in the guise of algebraic cycle homology, and Chern classes in the context of algebraic cycle spaces. In addition to this study of algebraic cycles, Friedlander will continue his efforts to study the geometry implicit in the cohomology of infinitesimal algebraic groups. Priddy plans to continue his program to study the homotopy type of classifying spaces of groups and related constructions. Many of the most important questions in homotopy theory are related to classifying spaces of finite or compact Lie groups. Recently there have also developed interesting and powerful connections between topology and group theory, especially the cohomology and modular representation theory of finite groups. With the solution of the Segal and Sullivan Conjectures, this area has developed rapidly in recent years to the point where fundamental questions are being answered. Perhaps the most important of these is to determine the exact relationship between the stable and unstable homotopy types of a classifying space, completed at a prime p, and the p-local structure of its underlying group. Algebraic geometry is the study of solution sets of polynomial equations (i.e., algebraic varieties) using geometric techniques. Partial answers to questions in algebraic geometry have led to progress in fields ranging from complexity theory of computer science to geometric topology to number theory. Friedlander intends to study algebraic varieties, using methods borrowed from algebraic topology, as well as modern techniques of algebraic geometry. The use of topology involves the study of continuously varying families of structures, which have traditionally been considered by other means. The hope is that these new techniques will offer insight into deep and long-standing problems of algebraic geometry. Algebraic topology is the study of geometric objects by means of algebraic techniques. Exciting new developments have led to advances in group theory, using algebraic topology, thus reversing the direction of the usual flow of information. Groups are the fundamental symmetries occurring in all sciences, including areas involving codes, and structures in physics. Priddy hopes that this new approach will lead to a better understanding of the relationship between these fields. ***