RESEARCH
My research as an algebraist is focused in the overarching field of algebra and involves studying different topics using algebraic techniques. I am specifically interested in representation theory, particularly in using the language of Hopf algebras and quivers to study and classify quantum symmetries and in persistent homology topological data analysis. For more details, I invite you to view my research statement or read a brief summary below.
Research Interests
My area of research is focused in the overarching field of algebra and involves studying different topics using algebraic techniques. I am specifically interested in the representation theory of algebras. This is a subset of abstract algebra that translates problems regarding abstract algebraic structures such as groups, rings, algebras, and modules into linear algebra. This is beneficial as linear algebra is a well studied field.
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Historically, P. Gabriel showed that, for a field, k, the representations of any k-algebra, A, are dictated by a directed graph that is uniquely determined once given some additional conditions. Gabriel called this structure a quiver. Moreover, any quiver, Q, determines a k-algebra, kQ, via information about it's vertices and arrows, called the path algebra of the quiver Q.
Throughout the course of their study, quivers and their path algebras have provided a wealth of information regarding their underlying algebras, associated category of finite-dimensional representations, and more. Over the course of my research career, I have studied two such settings, or applications, for which quivers and their representations provide much insight; Topological Data Analysis (TDA) and quantum symmetries.
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Within the realm of topological data analysis, particularly persistent homology, I was able to show the exact value of the p-presentation distance for 1-dimensional free persistence modules, a measure of stability for persistence modules. Turning my attention to the information that actions of Hopf algebras on path algebras give regarding generalized symmetries, some of which are quantum symmetries, I have been able to parameterize all Hopf actions of Hopf-Ore extensions on path algebras. I hope to continue to move this field forward with my future research.