Anchored symplectic embeddings

Symplectic manifolds exhibit curious behaviour at the interface of rigidity and flexibility. A non-squeezing phenomenon discovered by Gromov in the 1980s was the first manifestation of this. Since then, extensive research has been carried out into when standard symplectic shapes embed inside another -- it turns out that even when volume obstructions vanish, sometimes they cannot. A mysterious connection to Markov numbers, a generalization of the Fibonacci numbers, and an infinite staircase, is exhibited in the study of embeddings of ellipsoids into balls. In other cases ingenious constructions such as folding have been invented to find embeddings. In recent work with Hutchings, Weiler, and Yao, I studied the embedding problem for four-dimensional symplectic shapes in conjunction with the question of existence of symplectic surfaces (called anchors) inside the embedding region, which has revealed some interesting results. In this talk I will survey some of the results and time permitting, discuss the ideas behind the proof and further questions.