CAREER: Equivariant Homotopy and Algebraic K-Theory

职业：等变同伦和代数 K 理论

• 批准号: 1149408
• 负责人: Teena Gerhardt
• 金额: $ 40.52万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1149408&HistoricalAwards=false
• 关键词: CAREER; Equivariant; Homotopy; Algebraic; Theory

The primary research goal of the proposed project is to use the tools of equivariant stable homotopy theory to study algebraic K-theory and related invariants. Although the definition of algebraic K-theory is not inherently equivariant, the tools of equivariant stable homotopy theory have proven useful for K-theory computations. In particular, one fruitful approach exploits the equivariant structure of topological Hochschild homology (THH) to compute algebraic K-theory. In some cases K-theory computations can be reduced to the computation of equivariant stable homotopy groups of THH, graded by the real representation ring of the circle. Determining which groups need to be computed, computing them, and assembling the groups to recover algebraic K-theory are all important components of this approach. Goals of this project include completing these steps for various specific K-theory computations, as well as defining abstract algebraic objects embodying equivariant structures arising in such computations. Other goals of this research program include describing the structure of higher topological Hochschild homology, and developing and exploring applications of a new equivariant model for THH.Algebraic K-theory is an invariant which can be applied to study basic objects from several fields of mathematics. In particular, algebraic K-theory can be used to study properties of fundamental objects in algebra, called rings. Although higher algebraic K-theory was defined more than 30 years ago, computational progress has been slow. Indeed, even for some very basic rings, the algebraic K-theory is still not known. K-theory computations, however, have important applications to many areas of mathematics. Algebraic K-theory lies in the intersection of algebraic topology, algebraic geometry, and number theory, with applications to motivic homotopy theory, classification of manifolds, special values of L-functions, etc. A goal of this project is to use tools from algebraic topology to not only produce new algebraic K-theory computations, but also to develop the framework and theory to facilitate future computations. This project also includes several educational and mentoring programs centered around the recruitment and retention of women and other underrepresented groups in mathematics. Programs aimed at undergraduate students include the development of a Women in Mathematics course, the creation of a Careers in Science lecture series, and undergraduate research opportunities aimed at early-career undergraduates. For graduate students, career mentoring seminars will be developed both for students at Michigan State University, and more broadly for students and post-docs in the international Algebraic Topology community through a retreat at an upcoming semester-long program. Also included are opportunities for K-12 students from underrepresented groups, as well as a program for female faculty members in science, mathematics, and engineering. Additionally, Gerhardt proposes a research project addressing the question of why many successful female mathematicians choose to leave academic math.

本课题的主要研究目标是利用等变稳定同伦理论的工具研究代数k理论及相关不变量。虽然代数k理论的定义不是固有的等变，但等变稳定同伦理论的工具已被证明对k理论的计算是有用的。特别是利用拓扑Hochschild同调（THH）的等变结构来计算代数k理论。在某些情况下，k理论的计算可以简化为对THH的等变稳定同伦群的计算，这些计算由圆的实表示环来分级。确定需要计算哪些组，计算它们，并将这些组组合起来以恢复代数k理论，这些都是该方法的重要组成部分。该项目的目标包括完成各种特定k理论计算的这些步骤，以及定义包含此类计算中产生的等变结构的抽象代数对象。本研究计划的其他目标包括描述更高拓扑Hochschild同调的结构，以及开发和探索THH的新等变模型的应用。代数k理论是一个不变量，可以应用于研究数学多个领域的基本对象。特别是，代数k理论可以用来研究代数中称为环的基本对象的性质。虽然高等代数k理论在30多年前就被定义了，但计算进展缓慢。事实上，即使对于一些非常基本的环，代数k理论仍然是未知的。然而，k理论计算在许多数学领域都有重要的应用。代数k理论是代数拓扑学、代数几何和数论的交叉学科，在动力同伦理论、流形的分类、l函数的特殊值等方面有广泛的应用。该项目的目标是利用代数拓扑工具不仅产生新的代数k理论计算，而且还开发框架和理论，以促进未来的计算。该项目还包括几个教育和指导项目，重点是招募和保留女性和其他数学领域代表性不足的群体。针对本科生的项目包括开发“女性在数学”课程，创建“科学职业”系列讲座，以及针对早期职业本科生的本科生研究机会。对于研究生，将为密歇根州立大学的学生以及国际代数拓扑社区的学生和博士后开发职业指导研讨会，通过即将到来的学期项目的静修。该项目还包括为来自代表性不足群体的K-12学生提供机会，以及为科学、数学和工程领域的女性教员开设的项目。此外，Gerhardt提出了一个研究项目，解决为什么许多成功的女性数学家选择离开学术数学的问题。