Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

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 $C^4 = D^2 \times D^2$ is very different. For instance, for the open ball $B^4$ of capacity 1, there exists an explicit decreasing sequence $c_1, c_2, \dotsc \to 1/3$ such that for $c < c_k$ there are at least $k$ symplectic embeddings of the closed cube $C^4(c)$ of capacity $c$ into $B^4$ that are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings of $C^4(1/3)$ into $B^4$.

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.