10:00am - 12:00pmEuclidean distance geometry and its applications
Chair(s): Kaie Kubjas (Sorbonne Université)
Given a natural number d and a weighted graph G=(V,E), the fundamental problem in Euclidean distance geometry is to determine whether there exists a realization of the graph G in Rd such that distances between pairs of points are equal to the corresponding edge weights. This problem naturally arises in many applications that require recovering locations of objects from the distances between these objects. Usually, measurements of the distances are noisy and there can be missing data. Examples of applications are sensor network localization, molecular conformation, genome reconstruction, robotics and data visualization. Algebraic varieties and semialgebraic sets naturally come up in Euclidean distance geometry, since distances between objects are given by polynomials. Hence questions about uniqueness and finiteness of realizations are often algebraic in nature, whereas realizations are found using semidefinite or nonconvex optimization methods. The goal of this minisymposium is to present theory and applications of Euclidean distance geometry, and connect researchers working in Euclidean distance geometry with applied algebraic geometers.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Isometries in Euclidean, Homogeneous, and Conformal Spaces
Carlile Lavor
University of Campinas, Brazil
It is known, from linear algebra, that isometries in Euclidean Spaces are described by orthogonal transformations up to translations. We will discuss what happens when isometries are considered in Homogeneous and Conformal Spaces.
Auxetic deformations of triply periodic minimal surfaces
Ciprian S. Borcea
Rider University, USA
The notion of one-parameter auxetic deformation, introduced previously for periodic frameworks, can be used in the context of triply periodic minimal surfaces. We exhibit auxetic paths in several new families of triply periodic minimal surfaces of low genus.
Voronoi Cells of Varieties
Maddie Weinstein
University of California, Berkeley, USA
Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.
Critical points of the Hamming and taxicab distance functions
Jonathan Hauenstein
University of Notre Dame, USA
Minimizing the Euclidean distance from a given point to the solution set of a given system of polynomial equations can be accomplished via critical point techniques. This talk will explore extending critical point techniques to minimization with respect to the Hamming distance and taxicab distance. Numerical algebraic geometric techniques are derived for computing a finite set of real points satisfying the polynomial equations which contains a global minimizer. Several examples will be used to demonstrate the new techniques. This is joint work with D. Brake, N. Daleo, and S. Sherman.
