PIMS-SFU Computational Math Seminar: Rustum Choksi

Abstract: Voronoi tessellations give rise to a wealth of analytic, geometric, and computational questions. They are also very useful in modelling.

This talk will consist of three parts.

In the first, I will address the simple, yet rich, question of optimal quantization -- or optimal centroidal Voronoi tessellations (CVT) -- on 3D torus. I will address Gersho's conjecture, a crystallization conjecture which asserts the periodic structure of the optimal CVT, as the number of generators tends to infinity.

In the second part of the talk, I present a new 2D hybrid numerical method for accessing low energy CVTs with tiny basins of attraction.

In the last part of the talk, I will present a new dynamical model for generic crowds in which individual agents are aware of their local Voronoi environment---i.e., neighbouring agents and domain boundary features---and may seek static target locations. Our model incorporates features common to many other ``active matter'' models like collision avoidance, alignment among agents, and homing toward targets. However, it is novel in key respects: the model combines topological and metrical features in a natural manner based upon the local environment of the agent's Voronoi diagram. With only two parameters, it captures a wide range of collective behaviours.

This talk comprises joint works with Xin Yang Lu (Lakehead University) and with Ivan Gonzalez, Jean-Christophe Nave, Jack Tisdell (all at McGill University).