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This talk will be held over the IHP Zoom link:

While by a result of McDuff the space of symplectic embeddings of a closed 4-ball into an open 4-ball is connected, the situation for embeddings of cubesÌýC4=D2×D2Ìýis very different. For instance, for the open ballÌýB4Ìýof capacity 1, there exists an explicit decreasing sequenceÌýc1,c2,⋯→1/3Ìýsuch that forÌýc<ckÌýthere are at leastÌýkÌýsymplectic embeddings of the closed cubeÌýC4(c)Ìýof capacityÌýcÌýintoÌýB4Ìýthat are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings ofÌýC4(1/3)ÌýintoÌýB4.

A similar result holds for several other targets, like the open 4-cube, the complex projective plane, the product of two equal 2-spheres, or a monotone product of such manifolds and any closed monotone toric symplectic manifold. The proof uses exotic Lagrangian tori.

This is joint work with Joé Brendel and Grisha Mikhalkin.