Scale and conformal symmetries play a central role in various areas of theoretical physics, ranging from statistical mechanics and quantum field theory to string theory. A natural question is under what conditions scale invariance implies the larger symmetry of conformal invariance.  In this talk, I will discuss non-perturbative arguments for this enhancement in two and four dimensions, highlighting the assumptions under which they hold. I will first explain how Zamolodchikov's $c$-theorem, under suitable assumptions, can be used to establish the enhancement of scale invariance to local conformal invariance in two dimensions. I will then turn to four dimensions and discuss the argument of Dymarsky, Komargodski, Schwimmer, and Theisen based on dilaton scattering amplitudes and unitarity. I will conclude by commenting on possible generalisations to other dimensions and, time permitting, recent progress on the question.

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.