Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. 

This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek. 

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

Let X be an integral projective variety of degree at least 2 defined over Q, and let B>0 an integer. The dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown, and Salberger, provides a certain uniform upper bound on the number of rational points of height at most B lying on X. 

Shifting to the geometric setting (where X may be defined over C(t)), the collection of C(t)-rational points lying on X of degree at most B naturally has the structure of an algebraic variety, which we denote by X(B). In ongoing work with Tijs Buggenhout and Floris Vermeulen, we uniformly bound the dimension and, when the degree of X is at least 6, the number of irreducible components  of X(B) of largest possible dimension​ analogously to dimension growth bounds. We do this by developing a geometric determinant method, and by using results on rational points on curves over function fields. 

Joint with Tijs Buggenhout and Floris Vermeulen.