Voronoi Tessellations: Optimal Quantization and Modeling Collective Behavior (Prof. Rustum Choksi)

Title: Voronoi Tessellations: Optimal Quantization and Modeling Collective Behavior

Speaker: Prof. Rustum Choksi, Department of Mathematics and Statistics, McGill University

Abstract:  Given a set of N distinct points (generators) in a domain (a bounded subset of Euclidean space or a compact Riemannian manifold), a Voronoi tessellation is a partition of the domain into N regions (Voronoi cells) with the following property: all points in the interior of the i^th Voronoi cell are closer to the i^th generating point than to any other generator.  Voronoi tessellations give rise to a wealth of analytic, geometric, and computational questions. They are also very useful in mathematical and computational modeling.

This talk will consist of three parts:

• We begin by introducing the basic definitions and geometry of Voronoi tessellations,  centroidal Voronoi tessellations (CVTs), and the notion of optimal quantization.
• We will then address simple, yet rich, questions on optimal quantization on the 2D and 3D torus, and on  the 2-sphere. We will address the geometric nature of the global minimizer (the optimal CVT), presenting a few conjectures and a short discussion on rigorous asymptotic results and their proofs.
• We will then shift gears to address the use Voronoi tessellations in modeling collective behaviors.

Collective behavior in biological systems, in particular the contrast and connection between individual and collective behavior, has fascinated researchers for decades. A well-studied paradigm entails the tendency of groups of individual agents to form flocks, swarms, herds, schools, etc. We will first review some well-known and widely used models for collective behavior.  We will then present a new dynamical model for generic crowds in which individual agents are aware of their local Voronoi environment — i.e., neighboring 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 behaviors. The results of many simulations will be shown.

Rustum Choksi received the PhD degree in mathematics from Brown University, in 1994. He held post-doctoral positions with the Center for Nonlinear Analysis, Carnegie Mellon University and the Courant Institute, New York University. From 1997 to 2010, he was a faculty member with the Department of Mathematics, Simon Fraser University. In 2010, he joined McGill University where he is currently a full professor with the Department of Mathematics and Statistics. His main research interests include interaction of the calculus of variations and partial differential equations with pattern formation.