Thursday, December 1, 2022, 2:30 – 3:30pm

In computational topology and geometry, theoretical guarantees for algorithms often take the following form: Start with a finite sample of points from a subspace of \mathbb{R}^n. If the sample is "dense enough" with respect to the subspace, then the algorithm outputs a quantity of interest for the subspace, for example its Betti numbers. The quantities associated to a subspace which determine how dense of a sample is necessary are well studied by computational geometers: the reach, local feature size, and weak feature size of a subspace. Rather than a set of points, an algebraic space X is specified by a system of polynomial equations. To apply the above methods in a theoretically sound way, one must use the space's defining equations both to compute its feature sizes and subsequently a dense point sample from the space.

In this talk, I will discuss new theory and algorithms to compute feature sizes of algebraic manifolds using numerical algebraic geometry methods. The corresponding theory investigates the differential critical point/value theory of a particular class of optimization problems, namely the constrained optimization distance-to-X function d_X:\mathbb{R}^n \to \mathbb{R}, from an algebraic geometry perspective.

This is joint with Sandra Di Rocco, David Eklund, Oliver Gävert, and Jonathan Hauenstein.

Zoom Participation. See announcement.

Event Type: Seminars
Room Number: Virtual Presentation - ET
Building: Remote Access - Zoom
Speaker's Name: PARKER EDWARDS
Speaker Website: sites.nd.edu…
Speaker's Professional Title: Robert and Sara Lumpkins Postdoctoral Research Associate Department of Applied and Computational Mathematics and Statistics, University of Notre Dame
Talk Title: Feature Sizes and Bottlenecks for Algebraic Manifolds
For More Information: anezhad@andrew.cmu.edu
Affiliations: Computer Science Department (CSD)
Organization(s): Department of Mathematical Sciences
Event Website Title: Event Website
Event Website URL: www.cmu.edu…