Cohomology of Noncommutative Rings: Structure and Applications

表示论是数学的一个分支，它用代数方法研究对称性和运动，例如，通过将信息编码为数字数组或矩阵。它出现在许多科学调查中，例如，在关于宇宙形状的问题中，化学结构的对称性，以及量子计算。上同调是一种工具，它可以将表示理论的信息分解成更小、更容易理解的成分。PI在上同调和表示论方面的研究旨在回答许多数学和科学环境中出现的有关基本结构的难题。她的研究计划影响了许多其他数学家，特别是她指导的学生和博士后。她领导的研究团队都在她的大学和国际上，她共同组织会议在她的研究领域，她正在写一本书在高级研究生水平。这个项目涉及几个相互关联的问题的结构Hochschild上同调和Hopf代数上同调，并在表示论和代数变形理论的应用。结合环的Hochschild上同调有一个Lie结构，它是一个重要的工具，但很难管理。其中一些困难最近通过PI和其他人的工作克服了，他们开发了新的技术来理解任意分辨率的Lie结构。这为理解Hochschild上同调的结构打开了更大的潜在进展之门，例如，与其他描述建立联系，例如通过扩张空间上的上导子和循环。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有限生成猜想在Hopf代数上同调。PI将使用各种技术来证明一些重要的Hopf代数类的猜想，包括扭曲张量积的分辨率和Anick分辨率。PI将致力于相关的支持品种，以理解有关这些Hopf代数和相关类别的表示问题，使用她正在开发的技术与博士后处理张量类别的模块类别。

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.