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We prove that Patterson–Sullivan measures for relatively Anosov groups are exact dimensional with respect to visual metrics arising from suitable Gromov models. For relatively Morse groups, we further show that the scalar Cartan metric is Gromov hyperbolic, and hence induces a visual metric to which the exact dimensionality result applies. We also prove that the associated Manhattan manifolds are $C^1$-regular, and deduce that the growth indicator is $C^1$ and strictly concave on the interior of the limit cone. This is joint work with Eduardo Reyes.

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.