Group proposal in topology

DMS-0204615J。这个项目涉及拓扑学、几何学和相关数学领域的广泛课题的研究。可以研究稳定同伦理论、等变代数拓扑等领域中的各种范畴结构和同伦结构。尽管他目前的工作主要是基础性的，但他的学生和合作者等人对此进行了广泛的计算应用。他和他的合作者最近通过结构保持函子证明了所有新的高结构谱范畴都是Quillen等价模型范畴，从而统一了稳定同伦理论的基础。并对等变稳定同伦的基础进行了类比和更深层次的统一。在代数拓扑学中，May一直致力于等变和非等变同伦理论的各种其他项目。他还一直致力于各种根植于代数拓扑学但具有更一般性质的主题。特别是，他最近得到了丰富的弱$n-范畴的新定义，并在高等范畴理论中带头进行了一个非常大规模的统一计划。温伯格研究几何拓扑学和微分几何的各个领域。他对变换群，特别是具有群作用的流形上的外科理论有着长期的兴趣。特别是，他研究了与消除阻碍非等变理论直接推广的“缺口假说”有关的问题。Weinberger对Novikov、Borel和Baum-Connes猜想也有长期的兴趣，他最近在这些猜想上取得了进展。在一个新的方向上，Weinberger与Nabutovsky进行了大规模的合作，涉及到逻辑在黎曼变分问题和模空间的大规模几何中的应用。他最近的一些结果很难用简单的描述来描述。例如，他应用了Browder关于有限H-空间的一个旧定理，得到了关于“社会选择问题”的一个定理。另一方面，在最近与Farb的工作中，他证明了局部对称流形上的“隐藏对称”迫使它是算术的。此外，May，芝加哥大学的代数拓扑组包括4名非终身教职员工和8名研究生。几何小组包括温伯格和其他四名终身教职员工，六名非终身教职员工和十五名研究生。在这些群之间，以及它们与芝加哥的其他群，如几何朗兰兹群和代数几何群之间，有相当大的相互作用。这项由这笔资金资助的研究是芝加哥大学正在进行的一系列研究项目的一部分。

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.