Group Proposal in Topology

这个拓扑学的小组项目涉及拓扑学和代数、代数k理论和几何相关领域的广泛主题的研究。May主要研究全局结构、稳定同伦理论及相关领域。他和他的合作者用他们最近对高结构环和模理论的新方法，将稳定同伦理论开辟为严肃的点集水平代数研究。环、模和代数现在可以被定义为在对称的单谱范畴中具有良好行为的产物的对象，允许许多以前不可能的结构和应用。他和Greenlees通过分析Tate上同调和证明MU上模谱的补全定理，开辟了等变稳定同伦理论与非等变稳定同伦理论之间的新相互作用。Rothenberg研究几何拓扑中的解析和组合扭转不变量。他和他的合作者探索了经典不变量在等变和非紧化情况下的推广，并构造了具有紧纤维的纤维束的扭转不变量，这些是上同调类而不仅仅是数值不变量。Weinberger的工作中心几何及其对紧致空间奇异点和非紧致流形的分析。他关于分层空间的外科理论可以直接应用于各种问题，并在回答老问题和形成新问题方面改变了对群体行为的看法。这一视角与同调流形理论在猜想的层面上，以及在某种程度上的定理层面上完全结合。此外，对证明这些猜想所涉及的内容的详细分析似乎与逻辑学、复杂性理论和非交换几何有着深刻的联系。该项目的一个主要重点是研究生教育。芝加哥拓扑学有三名初级教师和十五名在读研究生，这项资助支持的拓扑学项目是世界上最大的项目之一。在1996- 1998年的三年间，它有10名新博士毕业，1998-99年在麻省理工学院（2名）、密歇根大学（2名）、伊利诺斯州（2名）、伯克利大学、纽约市立大学、罗格斯大学和犹他大学提供了工作岗位。获得资助的学生在代数和几何拓扑的各个领域进行工作，其中一些是关于代数拓扑和代数几何之间的界面，以及几何拓扑和微分几何之间的界面。虽然研究的重点是拓扑学，但这项资助支持的工作也影响到数学的许多其他领域。其中一些还影响了当前的数学物理工作，研究人员所研究的拓扑和几何结构的种类与之直接相关

9803633May This group project in topology deals with research in a wide rangeof topics in topology and related areas of algebra, algebraic K-theory,and geometry. May studies various areas centering on global structuresin stable homotopy theory and related fields. He and his collaboratorshave opened up stable homotopy theory to serious point-set levelalgebraic study with their recent new approach to highly structured ringand module theory. Rings, modules, and algebras can now be defined asobjects with well-behaved products in a symmetric monoidal category ofspectra, allowing many constructions and applications that were notpossible previously. He and Greenlees have opened up new interactionsbetween equivariant and non-equivariant stable homotopy theory withtheir analysis of Tate cohomology and their proof of a completiontheorem for module spectra over MU. Rothenberg studies analytic andcombinatorial torsion invariants in geometric topology. He and hiscollaborators have explored generalizations of the classical invariantsto equivariant and non-compact situations and have constructed torsioninvariants for fiber bundles with compact fibers, these being cohomologyclasses rather than just numerical invariants. Weinberger's work centerson geometry and analysis on compact spaces with singularities and onnoncompact manifolds. His surgery theory on stratified spaces can beapplied directly to various problems and has led to a changed perspectiveon group actions, both in terms of answering old questions and informulating new ones. The perspective is completely integrated with thetheory of homology manifolds at the level of conjecture, and somewhatat the level of theorem. Furthermore, detailed analysis of what wouldbe involved in proving such conjectures seems to have deep connectionswith logic, complexity theory, and non-commutative geometry. A major emphasis of the project is graduate education. With threejunior faculty and fifteen current graduate students in topology atChicago, the topology program supported by this grant is one of theworld's largest. In the three years 1996-98, it has graduated ten newPhD's, with 1998-99 jobs at MIT (2), Michigan (2), Illinois (2), Berkeley,CUNY, Rutgers, and Utah. Students supported on the grant work in a widevariety of areas of algebraic and geometric topology, and some of themwork on the interfaces between algebraic topology and algebraic geometryon the one hand and between geometric topology and differential geometryon the other. Although focused on topology, the work supported by thisgrant impinges on many other areas of mathematics. Some of it alsoimpinges on current work in mathematical physics, where the kinds oftopological and geometric structures studied by the investigators havedirect relevance.***