10:00am - 12:00pmNumerical methods in algebraic geometry

Chair(s): Jose Israel Rodriguez (UW Madison, United States of America), Paul Breiding (MPI MiS)

This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods.

(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)

Minimal problems in multiview 3D reconstruction via homotopy continuation

__Anton Leykin__

Georgia Tech

This will be a short survey of a class of problems in computer vision, for which it is plausible to construct efficient solvers based on polynomial homotopy continuation. For some of these problems alternative solvers do not exist at the moment.

The problems we consider concern relative camera pose recovery from points and lines in more than 2 views. In addition to classical point correspondences and line correspondences, we use incidence correspondences, which result from points lying on lines. We show how to build a solver based on a parameter homotopy coming from this framework.

(This survey is based on collaboration and discussions with many people involved in the Nonlinear Algebra semester at ICERM in Fall 2018.)

Computing the real CANDECOMP/PARAFAC decomposition of real tensors

__Tsung-Lin Lee__

National Sun Yat-sen University

The real Candecomp/Parafac decomposition (CPD) has many applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Several methods have been provided for computing the CPD such as alternating least squares (ALS), nonlinear least squares (NLS) and unconstrained nonlinear optimization. Those methods may take many iterations to converge and are not guar-anteed to converge to the solution. Recently, homotopy continuation techniques have been applied in computing tensor decomposition. In this talk, the real CPD of a real unbalanced tensor will be considered.

Computing transcendental invariants of hypersurfaces via homotopy

__Emre Sertoz__

Max-Planck-Institute MiS, Leipzig

Deep geometric properties of each projective variety is encoded in a matrix of complex numbers, called its periods. Knowing the periods of a variety, one can often say quite a lot about the type of subvarieties it contains using LLL methods, without resorting to symbolic elimination. However, numerical computation of periods have been previously confined to curves in the plane and to varieties enjoying many symmetries. We will demonstrate how periods of hypersurfaces can be computed using a form of homotopy and how they can be studied to reveal the geometry of the hypersurface.

On the nonlinearity interval in parametric semidefinite optimization

__Tingting Tang__

University of Notre Dame

We consider the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective vector along a fixed direction. We characterize the so-called transition point of the optimal partition where the ranks of a maximally complementary optimal solution suddenly change, and the nonlinearity interval of the optimal partition where the ranks of maximally complementary optimal solutions stay constant. The continuity of the optimal set mapping on the basis of Painleve-Kuratowski set convergence in a nonlinearity interval is investigated. We show that not only the continuity might fail, even the sequence of maximally complementary optimal solutions might jump in the interior of a nonlinearity interval. Finally, we present a procedure stemming from numerical algebraic geometry to efficiently compute nonlinearity intervals.