Ring Spectra, DGAs and Derived Equivalences

摘要奖：DMS-0706877主要研究人员：布鲁克·E·希普莱PI建议使用同伦理论的方法来研究代数导出的等价性。具体地说，这里提出的主要项目是考虑衍生范畴之间的等价何时可以通过潜在的丰富的结构等价来实现。PI和Dugger证明了微分分次代数(DGA)的某些导数等价只能通过“拓扑等价”(涉及倾斜谱)来实现，这种等价是在更广泛的环谱范围内考虑DGA而产生的。这个项目的一个目标是产生简单的不变量来检测DGA的拓扑等价，另一个目标是将拓扑等价的研究扩展到DG-范畴。在与Dugger正在进行的合作中，PI还研究了两个派生的等价DGA的奇异例子，这些DGA没有潜在的结构等价。建议的研究项目涉及代数结构和拓扑学的相互作用，Shapeor空间的研究。代数拓扑学家使用代数结构来描述和简化拓扑现象。在一个实现派生等价的项目中，研究者希望利用代数拓扑学中发展起来的技术来解决源于代数的问题。国际和平研究所还计划组织会议，帮助培训研究生并传播成果。此外，该协会还将继续参与促进妇女参与科学的几个组织。

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.