Algorithmic Symplectic Packing

Abstract

In this article we explore a symplectic packing problem where the targets and domains are 2n-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to ${\mathbb{Z}}^n$, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Miller Maley, Mastrangeli, and Traynor showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to k = 12 simplices, along with lower bounds for higher values of k. We present a modified algorithmic approach that allows us to determine the k-simplex packing widths for up to k = 13 simplices in dimension four and up to k = 8 simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.

Keywords:

• Symplectic packings
• symplectic capacities
• combinatorial optimization
• mixed integer linear programming
• semidefinite programming

Acknowledgments

We gratefully acknowledge the advice of Hansjörg Geiges and Felix Schlenk.

Notes

1 At this point we can’t refrain from telling that a colored version of the picture in Figure 6 is the logo of the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” of the German Science Foundation that has supported our research. When it was chosen in 2016, we had no idea of its rare feature, we just found it visually pleasing.

Additional information

Funding

This research is part of a project in the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics,” funded by the DFG.