Fixed-point-free fusion automorphisms

Abstract

Understanding the symmetries of algebraic objects can help decrease their complexity by several fold, and this reduction in complexity is intensified when there are fewer and fewer fixed-points. Typically one requires the unit or neutral element to be fixed by any symmetry, so a symmetry for which this is the only fixed-point has been called "fixed-point-free". These are the most active of symmetries and have strong implications for the objects they act upon. For example, a finite group with a fixed-point-free automorphism must be solvable; if the automorphism is order 2 then the group is abelian and the symmetry is inversion. In this talk, we will attempt to generalize classical results on fixed-point-free automorphisms of finite groups to fusion rings and categories. Of particular interest is understanding when/if combinatorial symmetries of fusion rings can be lifted to categorical symmetries of fusion categories. This talk will be self-contained, aimed toward a general audience, and will include copious amounts of examples.

Speaker Intro

I grew up in a rural area in northern Michigan and did not consider mathematics as a career until I was in my 20's; I cooked professionally for about 10 years. I've since earned degrees in the states of Michigan, Oregon, and Washington, and done postdoctoral research in Australia, Canada, and the United States. My research will probably stay related to tensor categories and their many applications, but I will always think about whatever problems seem interesting to me in the moment. I am a year-round alpine climber, and an avid wildlife photographer.