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In topology and geometry, we l having theorems that say things like:

"object on a space $X$ = object on a cover $Y$ + morphisms + compatibility"

These are called descent theorems. For example, in algebraic geometry, quasicoherent sheaves satisfy Zariski descent (where $Y$ is any open cover of $X$). Topology, too, has descent for sheaves over open covers. But there, an even funnier descent question exists: sheaves on a manifold $X$ have a notion of singular support---a closed subset of $T*X$ that projects down to the usual support---and so we can ask, is it possible for a descent theorem to hold for a closed cover of the singular support (but not changing the $X$!)? Very strange... but it is exactly this version of descent that matches Zariski descent under so-called toric mirror symmetry! In this talk, I would like to explore the idea of descent through examples, introduce a monadic formulation of descent, and illustrate "singular support descent" through the coolest theorem I know: toric mirror symmetry for $\mathbb{P}^1$ [FLTZ 2011]. Time permitting, I will also indicate a result with Peng Zhou on when such descent holds more generally.

Pizza at 11:50am