Talk page
Title:
Packings and numbers in symplectic geometry
Speaker:
Abstract:
Gromov detected that symplectic diffeomorphisms are much more special than volume preserving transformations. Indeed, he famously proved that the ball $B^{2n}(1)$ of radius 1 does not symplectically embed into the cylinder $B^2(r) x R^{2n-2}$ (of infinite volume!) if $r<1$. This established ``symplectic rigidity''. The question was then to better understand symplectic rigidity. Is symplectic rigidity a basic and simple phenomenon, or are there refinements of Gromov's theorem? To this end, Gromov himself, and then McDuff, Polterovich, and Biran, considered packing problems, asking for how much of the volume of a symplectic manifold (of finite volume) can be symplectically filled by k equal balls. Already for 4-dimensional balls and cubes as targets, interesting sequences of numbers appear. These numbers are better understood by looking at two 1-parametric problems, namely the problems of maximal symplectic embeddings of a 4-dimensional ellipsoid E(1,a) into a 4-ball or a 4-cube P(A,A) = D(A) x D(A) (where D(A) is the 2-disc of area A). The graphs describing the answer to these two problems starts with infinite staircases, that are given in terms of Fibonacci respectively Pell numbers. If the target is a polydisc P(A,bA) (with b an integer \ge 2), then this rich structure disappears, and only finitely many steps remain in the graph. However, for increasing b, the staircase has square root of 2b many steps, so that at large b an infinite staircase reappears. The latter results were obtained in joint work with Dusa McDuff and David Frenkel, and by Dorothee Müller and David Frenkel.
Link:
Workshop: