Aaron Mazel-Gee (California Institute of Technology) "Reflection in algebra and topology"
- Monday, September 20, 2021 from 4:10pm to 5:15pm
- https://zoom.us/j/99634752372 Meeting ID: 996 3475 2372 Passcode: 2021_Math
Virtual Math Seminar.
Monday, September 20, 4:10-5:15pm MT
Zoom link, meeting, and password:
Meeting ID: 996 3475 2372
Aaron Mazel-Gee (California Institute of Technology)
Reflection in algebra and topology
Abstract.In this talk, I will discuss a new duality that was recently discovered in joint work with David Ayala and Nick Rozenblyum, which we refer to as reflection.In essence, reflection amounts to two dual methods for reconstructing objects, based on a stratification of the category that they live in. As a basic example, an abelian group can be reconstructed on the one hand in terms of its p-completions and its rationalization, or on the other (reflected) hand in terms of its p-torsion components and its corationalization; and these both come from a certain "closed-open decomposition" of the category of abelian groups.Examples and applications of reflection are abundant, because stratifications are abundant. In algebra, reflection recovers the derived equivalences of quivers coming from BGP reflection functors (hence the terminology "reflection"). In topology, reflection is closely related to Verdier duality, a generalization of Poincaré duality that applies to singular spaces. Moreover, an explicit description of reflection leads to a categorification of the classical Möbius inversion formula, a Fourier inversion theorem for functions on posets.
- Department of Mathematical Sciences