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

Geometric transformations in octrees using shears

Existent algorithms to perform geometric transformations on octrees can be classified in two families: inverse transformation and address computation ones. Those in the inverse transformation family essentially resample the target octree from the source one, and are able to cope with all the affine transformations. Those in the address computation family only deal with translations, but are commonly accepted as faster than the former ones for they do no intersection tests, but directly calculate the transformed address of each black node in the source tree. This work introduces a new translation algorithm that shows to perform better than previous one when very small displacements are involved. This property is particularly useful in applications such as simulation, robotics or computer animation.Postprint (published version