Research Project: Cohomology of Noncommutative Rings: Structure and Applications
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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.
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