Group Proposal in Topology

9803633五月 这个拓扑学的小组项目涉及拓扑学和代数、代数K理论和几何学的相关领域的广泛主题的研究。 梅在稳定同伦理论和相关领域研究以整体结构为中心的各个领域。 他和他的合作者开辟了稳定的同伦理论严重的点集层次代数研究与他们最近的新方法高度结构化的环和模块理论。 环、模和代数现在可以被定义为在谱的对称monoidal范畴中具有良好行为乘积的对象，允许许多以前不可能的构造和应用。 他和Greenlees开辟了新的相互作用之间的等变和非等变稳定同伦理论与他们的分析泰特上同调和他们的证明完成定理模谱超过MU。 Rothenberg研究几何拓扑中的分析和组合挠不变量。 他和他的合作者探索了经典不变量的推广到等变和非紧的情况，并为具有紧纤维的纤维束构建了torsioninvariants，这些是共同源类，而不仅仅是数值不变量。 温伯格的工作中心几何和分析紧空间的奇异性和非紧流形。 他的外科手术理论分层空间可以直接适用于各种问题，并导致了一个改变的视角对集团行动，无论是在回答老问题和informalizing新的。 这个观点在猜想的层次上与同调流形的理论完全结合，在定理的层次上也有所结合。 此外，对证明这些命题所涉及的内容的详细分析似乎与逻辑、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。 与三个初级教师和十五个当前的研究生在拓扑学在芝加哥，拓扑学计划支持这项赠款是世界上最大的之一。 在1996-98年的三年里，它已经毕业了十个新的博士学位，1998-99年在麻省理工学院（2），密歇根州（2），伊利诺伊州（2），伯克利，纽约市立大学，罗格斯大学和犹他州工作。 学生资助的工作领域广泛的代数和几何拓扑，其中一些工作的接口之间的代数拓扑和代数geometryon一方面和几何拓扑和微分geometryon另一方面。 虽然重点是拓扑学，但这项资助支持的工作影响了数学的许多其他领域。 其中一些也影响了当前的数学物理学工作，在那里，研究人员所研究的拓扑和几何结构有直接的相关性。

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.***