Cohomology of Noncommutative Rings: Structure and Applications

表象理论是数学的一个分支，它以代数的方式研究对称性和运动，例如，通过对数字数组或矩阵等信息进行编码。它出现在许多科学研究中，例如，在关于宇宙的形状、化学结构的对称性和量子计算的问题中。上同源是一种工具，它将表征理论信息分解成更小、更容易理解的组成部分。PI在上同调和表示理论方面的研究旨在回答在许多数学和科学环境中出现的关于基本结构的难题。她的研究项目影响了许多其他数学家，特别是她指导的学生和博士后。她领导着她所在大学和国际上的研究团队，她在她的研究领域联合组织会议，她正在写一本高级研究生水平的书。这个项目涉及几个相互关联的问题，关于Hochschild上同调和Hopf代数上同调的结构，以及在表示理论和代数形变理论中的应用。结合环的Hochschild上同调具有Lie结构，这是一个重要的工具，但很难管理。其中一些困难最近通过PI和其他人的工作被克服了，他们开发了新的技术来理解任意分辨率的谎言结构。这打开了在理解Hochschild上同调的结构方面取得更多潜在进展的大门，例如，通过扩张空间上的上导子和环与其他描述建立联系。PI还将研究变形理论中的相关问题。这个项目的另一部分涉及Hopf代数上同调中的一个有限生成猜想。PI将使用各种技巧证明一些重要的Hopf代数的猜想，包括扭张量积的分解和Anick分解。PI将致力于相关的支持变体，以理解关于这些Hopf代数和相关类别的表示的问题，使用她与博士后开发的技术来处理张量类别上的模块类别。

Representation theory is a branch of mathematics that studies symmetry and motion algebraically, for example, by encoding such information as arrays of numbers, or matrices. It arises in many scientific inquiries, for example, in questions about the shape of the universe, symmetry of chemical structures, and in quantum computing. Cohomology is a tool that pulls apart representation-theoretic information into smaller, more easily understandable components. The PI's research in cohomology and in representation theory aims at answering hard questions about fundamental structures arising in many mathematical and scientific settings. Her research program impacts that of many other mathematicians, particularly the students and postdocs that she mentors. She leads research teams both at her university and internationally, she co-organizes conferences in her research area, and she is writing a book at an advanced graduate level.This project concerns several inter-related problems on the structure of Hochschild cohomology and Hopf algebra cohomology, and on applications in representation theory and algebraic deformation theory. Hochschild cohomology of an associative ring has a Lie structure that is an important tool and yet is difficult to manage. Some of this difficulty was recently overcome through work of the PI and others in developing new techniques for understanding the Lie structure in terms of arbitrary resolutions. This opens the door to much more potential progress in understanding the structure of Hochschild cohomology, for example, to making connections to other descriptions such as by coderivations and loops on extension spaces. The PI will also work on related questions in deformation theory. Another part of this project concerns a finite generation conjecture in Hopf algebra cohomology. The PI will prove the conjecture for some important classes of Hopf algebras using a variety of techniques, including resolutions for twisted tensor products and the Anick resolution. The PI will work on related support varieties for understanding questions about representations of these Hopf algebras and related categories, using techniques she is developing with a postdoc for handling module categories over tensor categories.