Apollonian circle packings -- more precisely, the collection of curvatures in such a packing -- are a particularly pretty example of the orbit of thin Kleinian group.  For many such packings, the collection consists entirely of integers.  Such orbits are conjectured to follow a local-global principle in the sense that, except for a congruence obstruction modulo 24, all sufficiently large integers are expected to appear.  I will describe some of what I've learned and participated in concerning proving the positive density of curvatures in an Apollonian circle packing, and, more recently, in a prime component of such a packing.  The methods and players include quadratic forms, the circle method, expander graphs and strong approximation.

Given a smooth projective variety X and a smooth divisor D \subset X, one can study the enumerative geometry of counting curves in X with tangency conditions along D. There are (at least) two theories associated to it: relative Gromov--Witten invariants of (X,D) and orbifold Gromov--Witten invariants of the r-th root stack X_{D,r}. For a sufficiently large r, Abramovich--Cadman--Wise proved that genus zero relative invariants are equal to genus zero orbifold invariants of root stacks (with a counterexample in genus 1). I will explain the precise relation in higher genus. If time permits, I will also talk about further results on structures of relative Gromov--Witten theory and mirror symmetry. This is based on joint works with Hisan-Hua Tseng, Honglu Fan and Longting Wu.

After sketching the idea of an infinite-dimensional category --- more precisely an (∞,n)-category for n between 0 and ∞ --- I'll explain how one can do rigorous work with infinite-dimensional categories in a "model-independent" fashion, at least in the cases n = 0 or n = 1. The key idea is to reason inside an ∞-cosmos, an axiomatization of the "universe" in which ∞-categories live as objects. This is joint work with Dominic Verity.

Some claim that "(∞,1)-category theory is just like 1-category theory except you put an `∞' in front of everything," but there is at least one case where this optimism is unwarranted: it's easy to state and prove the Yoneda lemma for ordinary categories but extremely complicated to even define the Yoneda embedding for an (∞,1)-category. In this talk we'll introduce left fibrations and cocartesian fibrations, which can be used respectively to model (homotopy coherent) functors of (∞,1)-categories valued in (∞,0)- or (∞,1)-categories. We then describe the construction of another homotopy coherent functor, the comprehension functor associated to a fixed cocartesian fibration, which "unstraightens" a homotopy coherent diagram into cocartesian fibration. We conclude by explaining how a special case of this construction provides the sought-for Yoneda embedding. This is joint work with Dominic Verity.