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Online Seminar: Stratifications in algebra and topology May 11, 2020 (10:00 AM PDT - 11:30 AM PDT)
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Location: MSRI: Online/Virtual
Speaker(s) Aaron Mazel-Gee (California Institute of Technology)
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Stratifications In Algebra And Topology


Zoom link.


I will present a new theory of stratified noncommutative geometry, which gives a means of decomposing and reconstructing (higher) categories.

I'll motivate this theory by describing how it interacts with ordinary stratified geometry in the dual contexts of (i) quasicoherent sheaves on a scheme and (ii) constructible sheaves on a stratified topological space (e.g. a singular manifold). This leads to an adelic reconstruction theorem for quasicoherent sheaves, which generalizes the classical arithmetic fracture square for abelian groups as well as chromatic reconstruction for spectra.

I'll also explain an enhancement that accounts for (e.g. ∅, braided, or symmetric) monoidal structures. Using various fundamental operations on stratifications such as pullback, pushforward, and refinement, this gives a means of studying tensor-triangulated categories that is compatible with -- but not bound to -- the theory of Balmer spectra.

If time permits, I'll indicate how this theory gives a novel method of computing equivariant cohomology.

This represents joint work with David Ayala and Nick Rozenblyum.

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Stratifications In Algebra And Topology

H.264 Video 25056_28451_8343_Stratifications_in_Algebra_and_Topology.mp4