CAREER: Realizing Derived Equivalences

职业：实现派生等价

• 批准号: 0417206
• 负责人: Brooke Shipley
• 金额: --
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0417206&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. SHIPLELETHTHE THENE THEN THEN THENE THEN THUS THENE COMENTER涉及几个问题，这些问题属于两个整体项目。 一个项目是考虑何时由更丰富的等效性引起派生类别之间的等价性。 这里的目的是通过基本的结构性等价来开发一种实现衍生等效的阻塞理论。 特别是，这可能在代表理论中与布鲁伊的猜想相关的问题有应用，在该理论中，稳定的等效率提高了派生的等价。 第二个项目是开发同型理论代数模型。 该项目的一部分是继续与约翰·格林利斯（John Greenlees）合作，以建立代数模型，以进行理性的同学理论。 更一般而言，研究人员建议开发各种稳定同型理论的代数模型。 拟议的研究项目涉及代数结构和拓扑的研究，形状或空间的研究。 代数拓扑师使用代数结构来描述和简化拓扑现象。 在实现派生的等价的项目中，研究人员希望使用代数拓扑中开发的技术来攻击起源于代数的问题。 这些技术包括使用障碍理论，这决定了某些结构是否可能。 在另一个项目中，研究者计划将现有代数模型扩展到包括涉及对称性的结构。 该提案的教育部分还包括几个部分。 研究人员将在芝加哥大学和普渡大学的本科生夏季研究计划中参与。 与卢乔·阿夫拉莫夫（Lucho Avramov）一起，她将组织一个与广泛相关的代数相关领域相关类别的主题的讲习班。 借助Jim McClure和Guershon Harel，研究者将根据Euler的``代数的要素''开发一门新课程，以开发一门新课程，以实现数学教育的目标，以帮助代数的数学教育学生掌握技能，在代数方面发展信心，并了解妇女在标准的Alge AlgeBra中的材料，并在进行绘画方面进行了研究，该方面是摘要的，该方面是绘制的，该方面是绘制的。普渡大学。