New frontiers with relative modular categories: Meet our PIMS PDF at the University of Saskatchewan, Matthew Rupert .
By Lisa Sammoh, Communications, and Events Assistant.
Matthew Rupert is a PIMS postdoctoral fellow at the University of Saskatchewan under the supervision of Steven Rayan, Alex Weekes, and Curtis Wendlandt. His research interests are in quantum algebra, category theory, and the Kazhdan-Lusztig correspondence for vertex algebras. He obtained his PhD in pure mathematics at the University of Alberta in 2020 under the supervision of Thomas Creutzig, where he studied unrolled quantum groups and their representation theoretic connections to the singlet vertex operator algebras. After completing his PhD, Matthew was a Visiting Assistant Professor at Utah State University from August 2020 to June 2022.
Tell us about your academic journey.
During my undergraduate studies at the University of Alberta, I began with the physics program, but after taking several mathematics courses, including a summer course in ring theory, I switched to mathematical physics and later pure mathematics. I realized that I really enjoy the formalism of mathematics, particularly algebra. Under the supervision of Thomas Creutzig, who works primarily on vertex algebras, I pursued my master’s studies, where I discovered a preference for quantum groups over vertex algebras. This led me to research unrolled quantum groups, which have interesting applications to vertex algebras for my PhD. I continued this work for my PhD, and completed the degree in November 2020. As a Visiting Assistant Professor at Utah State University (USU) in Logan, Utah, I was fortunate to work with Nathan Geer on a novel class of categories called relative modular categories, which have applications to quantum topology, until July 2022. Currently, I hold a PIMS postdoctoral fellowship at the University of Saskatchewan with cosponsors Curtis Wendlandt, Steven Rayan, and Alex Weekes. Curtis and I knew each other as graduate students at the University of Alberta. He works on quantum groups called Yangians with a focus on their R-matrix presentations. In our discussions, we realized we had some overlapping research interests, and Curtis encouraged me to apply for the PIMS fellowship so we could pursue them.
What field are you in and what is your current research on?
My interests broadly include quantum algebra, representation theory, and category theory. Currently, my research is focused on exploring the connections between relative modular categories, unrolled quantum groups, and vertex operator algebras. Vertex operator algebras (VOAs) are symmetry algebras for two dimensional conformal field theory. Certain VOAs have surprising connections to quantum groups through their representation theory, and there is a particular example called the singlet whose representation theory seems to be equivalent to that of the unrolled quantum groups. Specifically, I aim to prove that the representation theory of unrolled quantum groups is equivalent to that of the singlet vertex algebra. Moreover, I am investigating techniques for constructing new examples of relative modular categories, which can be used to build topological quantum field theories.
What does research and life balance mean to you?
The pursuit of tenure can create pressure to sacrifice personal life for productivity, but my priorities have shifted since becoming a parent. Since my wife and I welcomed our first child last August, nothing is more fulfilling than hearing your child’s first laugh or seeing them learn to do something new. Balancing research and family means maintaining an active research program without neglecting my family life. My wife and I love being outdoors. We hiked in the beautiful Rocky Mountains during my Visiting Assistant Professorship in Logan, Utah. Though it’s been difficult to do outdoor activities with a newborn, we plan to explore the Saskatchewan landscape and visit Prince Albert National Park this summer.
Matthew will be speaking at the PIMS Emergent Research Seminar Series, on April 12, 2023, at 9:30 AM Pacific. Details on his talk and poster, Equivalences of Categories of Modules Over Quantum Groups and Vertex Algebras, can be found here.
