UBC DG-MP-PDE Seminar: Rustum Choksi

In this talk, we explore the simple, yet rich, paradigm of optimal quantization (alternatively, optimal centroidal Voronoi tessellations (CVT)). The 3D Gersho's conjecture may be viewed as a crystallization conjecture and asserts the periodic structure, as the number of generators tends to infinity, of the optimal CVT. In joint work with Xin Yang Lu (Lakehead University), we present certain bounds which, combined with a 2D approach introduced by P. Gruber, reduce the resolution of the 3D Gersho's conjecture to a finite (albeit very large) computation of an explicit convex problem in finitely many variables. We then address some current work for optimal quantization of the 2-sphere.

In the second part of the talk, we address the role of numerical algorithms for finding optimal (or close to optimal) CVTs on the 2D flat torus. For small numbers of generators, this will expose some interesting observations and conjectures about defect vs perfect lattice structures. This is joint work with Ivan Gonzales and JC Nave (McGill University) and with two former undergraduates Dragos Cristian Manta and Jack Tisdell.