Toric symplectic ball packing

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.Comment: 17 pages, 6 figure

Symplectic spectral geometry of semiclassical operators

In the past decade there has been a flurry of activity at the intersection of spectral theory and symplectic geometry. In this paper we review recent results on semiclassical spectral theory for commuting Berezin-Toeplitz and h-pseudodifferential operators. The paper emphasizes the interplay between spectral theory of operators (quantum theory) and symplectic geometry of Hamiltonians (classical theory), with an eye towards recent developments on the geometry of finite dimensional integrable systems.Comment: To appear in Bulletin of the Belgian Mathematical Society, 11 page

Computation of the multi-chord distribution of convex and concave polygons

Analytical expressions for the distribution of the length of chords corresponding to the affine invariant measure on the set of chords are given for convex polygons. These analytical expressions are a computational improvement over other expressions published in 2011. The correlation function of convex polygons can be computed from the results obtained in this work, because it is determined by the distribution of chords. An analytical expression for the multi-chord distribution of the length of chords corresponding to the affine invariant measure on the set of chords is found for non convex polygons. In addition we give an algorithm to find this multi-chord distribution which, for many concave polygons, is computationally more efficient than the said analytical expression. The results also apply to non simply connected polygons.Comment: 22 figures, 40 pages, 43 reference