A classical result by McDuff shows that the space of symplectic ball embeddings into many simple symplectic four-manifolds is connected. In this talk, on the other hand, we show that the space of symplectic bi-disk embeddings often has infinitely many connected components, even for simple target spaces like the complex projective plane, or the symplectic ball. This extends earlier results by Gutt-Usher and Dimitroglou-Rizell. The proof uses almost toric fibrations and exotic Lagrangian tori. Furthermore, we will discuss natural quantitative questions arising in this context. This talk is based on joint work in progress with Grigory Mikhalkin and Felix Schlenk.

The Mobius strip; collapsing the equator; exploding a point in the plane; geometric definition of blowups; the secant construction; pull-backs of curves under blowup.

The algebraic definition of blowups; the blowup in affine charts; birational maps; the strict and total transforms of varieties and vector fields.

Axiomatic definition of blowups; Cartier divisors; base change property; persistence properties.

First examples of resolutions; curves and surfaces; effect of blowups on polynomials; singularity invariants; lexicographic orderings.

Choice of centers of blowup; descent in dimension; lexicographic decrease of invariant; transversality; obstructions in positive characteristic; resolution of planar vector fields.