CAREER: Realizing Derived Equivalences

DMS-0134938布鲁克·E·希普本提案的研究部分涉及几个问题，这些问题属于两个总体项目。一个项目是考虑派生类别之间的等价性何时由结构更丰富的等价性诱导。这里的目标是发展一种障碍理论，通过潜在的结构对等来实现派生对等。特别是，这可能适用于与Broue猜想有关的问题，即稳定等价提升到派生等价的表示论。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与John Greenlees合作建立有理等变上同调理论的代数模型。更一般地，研究者建议为各种稳定同伦理论建立代数模型。拟议的研究项目涉及研究代数结构和拓扑学、研究形状或空间之间的相互作用。代数拓扑学家使用代数结构来描述和简化拓扑现象。在一个实现派生等价的项目中，研究人员希望使用代数拓扑学中发展起来的技术来解决源于代数的问题。这些技术包括使用障碍理论，该理论确定某些结构是否可能。在另一个项目中，研究人员计划扩展现有的代数模型，以包括涉及对称性的结构。这项提案的教育部分还包括几个部分。这位调查员将参与芝加哥大学和普渡大学本科生的暑期研究项目。与卢乔·阿夫拉莫夫一起，她将组织一个研讨会，主题与广泛的代数相关领域感兴趣的派生类别有关。该研究人员将与Jim McClure和Guershon Harel一起，以欧拉的《代数原理》为基础，为数学教育专业的学生开发一门新课程。这门课程的目标是帮助数学教育专业的学生掌握代数技能，培养教授代数的信心，并理解标准抽象代数课程中包含的材料的动机。研究人员还将参与普渡大学女性科学计划的几个方面。

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.