Ring Spectra, DGAs and Derived Equivalences

环谱、DGA 和导出的等价物

• 批准号: 0706877
• 负责人: Brooke Shipley
• 金额: $ 27.37万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706877&HistoricalAwards=false
• 关键词: Ring; Spectra; DGAs; Derived; Equivalences

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.

Abstractaward：DMS-0706877原理研究者：Brooke E. Shipleythe Pi建议使用同型理论中的方法研究代数派生的等价。 具体而言，这里提出的主要项目是考虑何时可以通过基本的结构等值来实现衍生典型之间的等价。 PI和Dugger表明，差分级代数（DGA）的某些衍生等效性只能通过“拓扑等效”（涉及tiltingspectra）来掩盖，这些差异是通过在较宽的环形光谱上考虑DGA而产生的。 该项目的一个目标是生产简单的不变，以检测DGAS的拓扑等效性。另一个目标是扩展对拓扑等效性DG类别的研究。 在与Dugger进行的正在进行的联合合作中，PI ISALSO调查了两个衍生的等效性的外来例子，没有潜在的结构性等效性。拟议的研究项目涉及代数结构和拓扑结构的相互作用，即塑形空间的研究。代数拓扑师使用代数结构来描述并简化拓扑现象。在一个跨越派生的等价的项目中，研究人员希望在代数拓扑中开发的Usetechniques，以攻击起源于代数的问题。 PI还计划组织会议，这将有助于培训研究生以及迫切结果。 此外，PI将继续与几个促进妇女在科学领域的妇女合作的组织进行参与。