Limits of Voronoi and Delaunay Cells

Abstract: Voronoi diagrams of finite point sets partition space into regions. Each region contains all points which are nearest to one point in the finite point set. Voronoi diagrams (and their generalizations and variations) have been an object of interest for hundreds of years by mathematicians spanning many fields, and they have numerous applications across the sciences. Recently, Cifuentes, Ranestad, Sturmfels, and Weinstein defined Voronoi cells of varieties, in which the finite point set is replaced by a real algebraic variety. Each point y on the variety has a cell of points in the ambient space corresponding to those points which are closer to y than any other point on the variety. In this talk, we present the limiting behavior of Voronoi diagrams of finite sets, where the finite sets are sampled from the variety and the sample size increases. In this setting, we observe that many interesting features of the variety can be seen in a Voronoi Diagram, including its medial axis, curvatures, normals, reach, and singularities.

commutative algebraalgebraic geometrynumber theory

Audience: learners

Experimental Talks in Algebraic Geometry

Series comments: Description: Exploring ways to virtually communicate math

The recent shift from in-person to online talks provides us with an opportunity to explore new and interesting ways to communicate and learn things related to algebraic geometry. The goal of this seminar is to provide a space for speakers and audience members to do this exploration.

Each speaker will be given 50 minutes to create and lead an environment where topics related to algebraic geometry can be communicated and learned. How the speaker uses their time and creates such an environment is entirely up to them. They may try mixing pre-recorded and live aspects of a talk, they might try leading group work, who knows, it is entirely up to the speaker. The only request made of speakers is that they think critically about new ways to best use their time to promote the communication and learning of math. After the 50 minutes, a short group discussion will be held discussing how this form of online math communication went, and how it might be built upon. Everyone participating is asked to keep in mind that this is a place for experimentation.

This event is being organized by Juliette Bruce in association with the Algebraic Geometry Syndicate (AGS) discord server. Updates about the talks will be posted both here and on the AGS discord server. If you would like to join the AGS discord server please email juliette.bruce 'at' math.wisc.edu.