CAREER: Realizing Derived Equivalences

职业：实现派生等价

• 批准号: 0134938
• 负责人: Brooke Shipley
• 金额: $ 30.48万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2004-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0134938&HistoricalAwards=false
• 关键词: CAREER; Realizing; Derived; Equivalences

DMS-0134938Brooke E. ShipleyThe research component of this proposal involves several questions which fall into two overall projects. One project is to consider when an equivalence between derived categories is induced by a more richly structured equivalence. The goal here is to develop an obstruction theory for realizing derived equivalences by underlying structured equivalences. In particular, this may have applications to the question associated to Broue's Conjecture in representation theory of which stable equivalences lift to derived equivalences. The second project is to develop homotopy theoretic algebraic models. One piece of this project is to continue work with John Greenlees on algebraic models for rational equivariant cohomology theories. More generally, the investigator proposes to develop algebraic models for various stable homotopy theories. The proposed research projects involve the interplay beween the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. In a project onrealizing derived equivalences, the investigator hopes to use techniques developed in algebraic topology to attack questions which originate in algebra. These techniques include the use of a theory of obstructions, which determines whether certain constructions are possible. In another project the investigator plans on extending existing algebraic models to include structures involving symmetries. The educational component of this proposal also includes several parts. The investigator will be involved in the summer research programs for undergraduates at the University of Chicago and Purdue University. With Lucho Avramov, she will organize a workshop on topics related to derived categories of interest to a broad range of algebraically related fields. With Jim McClure and Guershon Harel, the investigator will develop a new course for mathematics education majors based on Euler's ``Elements of Algebra". The goals of this course are to help mathematical education students master skills in algebra, develop confidence for teaching algebra, and understand the motivation for material contained in a standard abstract algebra course. The investigator will also participate in several aspects of the Women in Science Program at Purdue University.

DMS-0134938Brooke E. Shipley 该提案的研究部分涉及几个属于两个总体项目的问题。 一个项目是考虑派生类别​​之间的等价何时由更丰富的结构等价引起。 这里的目标是发展一种阻碍理论，通过底层的结构化等价来实现派生等价。 特别是，这可能适用于与表示论中的布劳猜想相关的问题，其中稳定等价提升为派生等价。 第二个项目是开发同伦理论代数模型。 该项目的一部分是继续与 John Greenlees 合作研究有理等变上同调理论的代数模型。 更一般地说，研究者建议为各种稳定同伦理论开发代数模型。 拟议的研究项目涉及代数结构和拓扑研究、形状或空间研究之间的相互作用。 代数拓扑学家使用代数结构来描述和简化拓扑现象。 在实现派生等价的项目中，研究人员希望使用代数拓扑中开发的技术来解决源自代数的问题。 这些技术包括使用障碍物理论，该理论确定某些构造是否可行。 在另一个项目中，研究人员计划扩展现有的代数模型以包括涉及对称的结构。 该提案的教育部分还包括几个部分。 该研究人员将参与芝加哥大学和普渡大学本科生的夏季研究项目。 她将与 Lucho Avramov 一起组织一个研讨会，主题涉及广泛的代数相关领域的派生类别。 研究者将与 Jim McClure 和 Guershon Harel 一起，基于欧拉的《代数原理》为数学教育专业的学生开发一门新课程。该课程的目标是帮助数学教育学生掌握代数技能，培养代数教学的信心，并理解标准抽象代数课程中包含的材料的动机。研究者还将参与妇女参与科学计划的多个方面 普渡大学。