Symplectically knotted cubes

Symplectically knotted cubes

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Felix Schlenk, University of Neuchâtel

This talk will be held over the IHP Zoom link:

https://us02web.zoom.us/j/86503014755?pwd=Q2tkQ2gwNXVqNU5RT3crWHo2SmJxUT09

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 cubes C4=D2×D2 is very different. For instance, for the open ball B4 of capacity 1, there exists an explicit decreasing sequence c1,c2,⋯→1/3 such that for c<ck there are at least k symplectic embeddings of the closed cube C4(c) of capacity c into B4 that are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings of C4(1/3) into B4.

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.