Abstract: One can construct an A-model analogue of the Gauss-Manin connection using genus 0 Gromov-Witten invariants, which also make perfect sense over fields of positive characteristics and p-adic fields. I will explain how arithmetic gadgets including p-curvature, Fontaine-Laffaile modules, and over-convergent Frobenii (conjecturally) arise in this context, and discuss certain extensions in the q-difference setting based on quasimap K-theory. This is based on joint works with Lee, Pomerleano, and Seidel.

Abstract: A recent manuscript written by Claude under the direction of Levent Alpoge constructs a 1-parameter family of compact complex threefolds diffeomorphic to S^6. I will describe the main geometric ideas behind the construction and outline the proof. If time permits, I will remark on possible future research directions.

Abstract: McMullen constructed automorphisms of complex K3 surfaces with positive entropy—and thus chaotic dynamics on a large set—that nonetheless have a Siegel disk: an open region on which the automorphism is holomorphically conjugate to an irrational rotation. Strikingly, such an automorphism forces the K3 surface to be non-projective, so McMullen's examples cannot be written down using explicit equations. Instead they are obtained indirectly via the global Torelli theorem.

In this talk we construct a p-adic analogue: a positive-entropy automorphism with a Siegel disk on a non-projective rigid-analytic K3 surface over a p-adic field. Our construction produces the surface directly, without the Torelli theorem, which makes it considerably more flexible.