Browsing by Author "Witherspoon, Sarah"

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Cohomology of Noncommutative Rings: Structure and Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/545; National Science Foundation
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.
Homological Techniques for Noncommutative Algebras and Tensor Categories
Mathematics; https://hdl.handle.net/20.500.14641/545; National Science Foundation
Many objects in nature, such as flowers, crystals, and molecules, exhibit symmetry. Symmetry can be described mathematically through motions, for example, a rotation after which the object appears the same. Such motions collectively form what is called a symmetry group. In quantum physics, this classical notion of symmetry group is no longer enough to capture all symmetries and is replaced by a notion more suited to quantum phenomena, termed quantum symmetry groups, or by related more general mathematical structures. This project carries out basic research in quantum symmetry, which is important in current applications of mathematics such as quantum computing and quantum information science. The project aims to develop and apply techniques for breaking down instances of quantum symmetry into smaller components, facilitating understanding. This work will expand the techniques available for answering questions about quantum symmetry that arise in applications. The project involves training of students through research involvement. Quantum groups, noncommutative algebras, and tensor categories are the settings for quantum symmetry studied in this project, which uses homological techniques to shed light on their structure. The research involves several directions. The investigator will study the cohomology of Hopf algebras and tensor categories, specifically to investigate a conjecture that cohomology of finite tensor categories is always finitely generated. The investigator aims to establish results on the structure of support varieties for finite tensor categories, harnessing homological techniques combined with geometry to obtain information such as a tensor product property and criteria for wild representation type. She plans to establish a direct connection between two different techniques that were developed recently, the homotopy lifting method and loops in exact categories, calling on techniques used in connection with A-infinity structures. She will also continue to develop improved techniques for understanding the Lie structure of the Hochschild cohomology of twisted tensor product algebras, particularly skew group algebras.