Group proposal in topology

拓扑中的组提议

• 批准号: 0204615
• 负责人: J May
• 金额: $ 53.05万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204615&HistoricalAwards=false
• 关键词: Group; proposal; topology

DMS-0204615J. Peter MayThis project deals with research in a wide range of topics in topology, geometry, and related areas of mathematics. May studies a variety of categorical and homotopical structures in stable homotopy theory, equivariant algebraic topology, and other fields. Although his current work is primarily foundational, his students and collaborators, among others, make extensive calculational applications of it. He and his collaborators have recently unified the foundations of stable homotopy theory by proving that all of the new highly structured categories of spectra are Quillen equivalent model categories, via structure-preserving functors. The analogous, and deeper, unification of the foundations of equivariant stable homotopy has also been carried out. Within algebraic topology, May has been working on various other projects in equivariant and non-equivariant homotopy theory. He has also been working on various topics that have roots in algebraic topology but are of a more general nature. In particular, he has recently obtained a new definition of enriched weak $n$-categories and has spearheaded a very large scale unification project in higher category theory. Weinberger studies various areas of geometric topology and differential geometry. He has a longstanding interest in transformation groups, especially surgery theory on manifolds with group actions. In particular, he studies problems concerned with removing the ``gap hypothesis'' that obstructs the direct generalization of the nonequivariant theory. Weinberger also has a longstanding interest in the Novikov, Borel, and Baum-Connes conjectures, and he has made recent progress on them. In a new direction, Weinberger has been engaged in a large scale collaborationwith Nabutovsky that concerns applications of logic to Riemannian variational problems and to the large scale geometry of moduli spaces. Some of his recent results defy easy characterization. For one example, he has applied an old theorem of Browder about finite H-spaces to obtain a theorem about the "social choice problem". For another, in recent work with Farb he has shown that "hidden symmetries"on a locally symmetric manifold force it to be arithmetic.Besides May, the algebraic topology group at Chicago includes four nontenured faculty and eight graduate students. The geometry group includes Weinberger and four other tenured faculty, six nontenured faculty, and fifteen graduate students. There is considerable interactionbetween these groups, and between them and other groups at Chicago, such as the geometric Langlands group and the algebraic geometry group. The research funded by this grant is part of a web of research projects in progress at the University of Chicago.

DMS-0204615J。Peter May这个项目涉及拓扑学，几何学和数学相关领域的广泛主题的研究。梅在稳定同伦理论、等变代数拓扑和其他领域研究了各种范畴和同伦结构。虽然他目前的工作主要是基础性的，但他的学生和合作者，以及其他人，都将其广泛地应用于计算。他和他的合作者最近通过证明所有新的高度结构化的谱范畴都是奎伦等价模型范畴，通过结构保持函子，统一了稳定同伦理论的基础。类似的，更深层次的，统一的基础等变稳定同伦也进行了。在代数拓扑，五月一直致力于各种其他项目的等变和非等变同伦理论。他还一直致力于各种议题，有根源的代数拓扑结构，但更一般的性质。特别是，他最近获得了一个新的定义，丰富的弱$n$-类别，并率先在一个非常大规模的统一项目在更高的范畴理论。温伯格研究几何拓扑学和微分几何的各个领域。他有一个长期的兴趣在转换组，特别是外科理论流形与组行动。特别是，他研究有关消除“间隙假说”，阻碍了直接推广的非等变理论的问题。温伯格也对诺维科夫、博雷尔和鲍姆-康纳斯理论有着长期的兴趣，最近他在这些理论上取得了进展。在一个新的方向，温伯格一直从事大规模的合作与Nabutovsky的关注应用逻辑黎曼变分问题和大规模几何的模空间。他最近的一些成果难以简单描述。例如，他应用了Browder关于有限H-空间的一个老定理来获得关于“社会选择问题”的一个定理。另一方面，在最近的工作与法布，他已经表明，“隐藏的对称性“的局部对称流形的力量，它是算术。除了五月，代数拓扑组在芝加哥包括四个非终身教职员工和八名研究生。几何组包括温伯格和其他四个终身教职员工，六个非终身教职员工，和15名研究生。有相当大的interactionbetween这些团体，他们之间和其他团体在芝加哥，如几何朗兰兹组和代数几何组。这项研究是芝加哥大学正在进行的一系列研究项目的一部分。