Group Proposal in Topology

这个拓扑学的小组项目涉及拓扑学以及代数、代数K-理论和几何的相关领域的广泛主题的研究。可围绕全局结构稳定同伦理论及相关领域研究多个领域。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，为严肃的点集水平代数研究打开了稳定同伦理论的大门。环、模和代数现在可以被定义为具有对称么半群谱范畴中行为良好的乘积的对象，从而允许许多以前不可能实现的构造和应用。他和Greenlees通过对Tate上同调的分析和对MU上模谱的一个完备性定理的证明，开辟了等变和非等变稳定同伦理论之间的新的相互作用。Rothenberg研究几何拓扑学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致情况下的推广，并构造了具有紧致纤维的纤维丛的扭转不变量，这些是上同调类，而不仅仅是数值不变量。Weinberger的工作集中在具有奇点的紧致空间和非紧致流形上的几何和分析。他的关于分层空间的外科手术理论可以直接应用于各种问题，并导致了对群体行动的看法的改变，无论是在回答旧问题方面，还是在阐述新问题方面。这个观点在猜想的层面上与同调流形理论是完全结合的，在某种程度上也是在定理的层面上。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、复杂性理论和非对易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。芝加哥大学有三名初级教师和十五名拓扑学研究生，这笔资金支持的拓扑学项目是世界上最大的项目之一。在1996-98的三年里，它已经毕业了10个新的博士，1998-99年间在麻省理工学院(2)、密歇根州(2)、伊利诺伊州(2)、伯克利、纽约大学、罗格斯大学和犹他州工作。获得助学金资助的学生从事代数和几何拓扑学的广泛领域的研究，其中一些学生一方面研究代数拓扑学和代数几何之间的接口，另一方面研究几何拓扑学和微分几何之间的接口。虽然重点是拓扑学，但这笔拨款支持的工作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，在数学物理中，研究人员研究的各种拓扑和几何结构具有直接的相关性。

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.***