Calabi-Yau varieties constitute a wide class which (strictly) includes abelian varieties and K3 surfaces. In this talk, I will discuss  Brantner-Taelman's work on the mixed characteristic deformation theory of Calabi-Yau varieties, and discuss my recent work characterising their "canonical lifts" via $p$-adic Hodge theory. Time permitting, I will also discuss the (in general conjectural) modularity of Calabi-Yau varieties over number fields and its relation to the aforementioned lifts.

In this talk, I will discuss the challenges of solving string theory through the two-dimensional quantum field theory on the string worldsheet. Focusing on the spectrum of superstrings in AdS3, I will explain how conformal field theory techniques yield exact results for special backgrounds, and how methods from integrable models allow us to go beyond these cases.