Room 70, Ross Building, Jerusalem, Israel
I will review homological mirror symmetry for the torus, which describes Lagrangian Floer theory on T^2 in terms of vector bundles on the Tate elliptic curve --- a version of Lekili and Perutz's works "over Z", where t is the Novikov parameter. Then I will describe a modified form of this story, joint with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on T^2 (an "F-field"). When the fiber of this sheaf of rings is perfectoid of characteristic p, it is possible to specialize to t = 1, and the resulting theory there is described in terms of vector bundles on the equal-characteristic-version of the Fargues-Fontaine curve. That "curve" is a weird geometric object (a one-dimensional scheme, but not a variety) introduced in 2010 in p-adic Hodge theory, for part of the talk I'll share my shaky understanding of the subject.