CAREER: Realizing Derived Equivalences

这个提案的研究部分包括几个问题，这些问题分为两个总体项目。一个项目是考虑派生类别之间的等价是由结构更丰富的等价引起的。这里的目标是发展一种障碍理论，通过底层结构化等价来实现派生等价。特别是，这可能适用于与表征理论中的布鲁猜想相关的问题，其中稳定等价提升为派生等价。第二个项目是发展同伦理论代数模型。这个项目的一部分是继续和John Greenlees一起研究有理等变上同调理论的代数模型。更一般地，研究者建议发展各种稳定同伦理论的代数模型。提议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。这些技术包括使用障碍物理论，它决定某些结构是否可能。在另一个项目中，研究者计划扩展现有的代数模型，以包括涉及对称性的结构。这项建议的教育部分也包括几个部分。该研究员将参与芝加哥大学和普渡大学的本科生暑期研究项目。与Lucho Avramov一起，她将组织一个与广泛的代数相关领域的衍生类别相关的主题研讨会。他将与Jim McClure和Guershon Harel合作，以欧拉的“代数原理”为基础，为数学教育专业开发一门新的课程。本课程的目标是帮助数学教育专业的学生掌握代数技能，培养教学代数的信心，并理解标准抽象代数课程中包含的材料的动机。该研究员还将参与普渡大学女性参与科学项目的几个方面。

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.