Ring Spectra, DGAs and Derived Equivalences

[摘要]获奖：dms -0706877首席研究员：Brooke E. shipleypi提出了用同伦理论的方法研究代数衍生等价。具体来说，这里提出的主要项目是考虑派生类别之间的等价何时可以通过潜在的结构丰富的等价来实现。PI和Dugger已经表明，微分梯度代数（DGAs）的某些导数等价只能通过“拓扑等价”（涉及倾斜谱）来实现，这种“拓扑等价”是在更广泛的环谱背景下考虑DGAs而产生的。该项目的一个目标是产生用于检测DGAs拓扑等价的简单不变量。另一个目标是将拓扑等价的研究扩展到dg范畴。在与Dugger正在进行的联合工作中，PI也在研究一个奇异的例子，即两个衍生的等效dgas没有潜在的结构等效。提出的研究项目涉及代数结构和拓扑研究之间的相互作用，形状或空间的研究。代数拓扑学家用代数结构来描述和简化拓扑现象。在一个实现衍生等价的项目中，研究者希望使用代数拓扑中开发的技术来解决起源于代数的问题。PI还计划组织会议，这将有助于培训研究生以及传播结果。此外，该计划将继续与一些促进妇女参与科学的组织合作。

AbstractAward: DMS-0706877Principal Investigator: Brooke E. ShipleyThe PI proposes to study algebraic derived equivalences usingmethods from homotopy theory. Specifically, the main projectproposed here is to consider when an equivalence between derivedcategories can be realized by an underlying richly structuredequivalence. The PI and Dugger have shown that certain derivedequivalences of differential graded algebras (DGAs) can only berealized by "topological equivalences" (involving tiltingspectra) which arise by considering DGAs in the broader contextof ring spectra. One goal of this project is to produce simpleinvariants for detecting topological equivalences of DGAs.Another goal is to extend the study of topological equivalencesto DG-categories. In on-going joint work with Dugger, the PI isalso investigating an exotic example of two derived equivalentDGAs with no underlying structured equivalence.The proposed research projects involve the interplay beween thestudy of algebraic structures and topology, the study of shapesor spaces. Algebraic topologists use algebraic structures todescribe and simplify topological phenomena. In a project onrealizing derived equivalences, the investigator hopes to usetechniques developed in algebraic topology to attack questionswhich originate in algebra. The PI also plans to organizeconferences which will help to train graduate students as well asdisseminate results. In addition, the PI will continue to beinvolved with several organizations which promote theparticipation of women in science.