Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS173, part 2: Numerical methods in algebraic geometry
Time:
Friday, 12/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F012
30 seats, 57m^2

Presentations
10:00am - 12:00pm

Numerical 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)

 

Numerical Root Finding via Cox Rings

Simon Telen
KU Leuven

In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points on a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We present a numerical linear algebra algorithm for computing the corresponding matrices, and from these matrices a set of homogeneous coordinates of the roots of the system. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables.

 

Numerical computation of monodromy action over R

Margaret Regan
University of Notre Dame

The monodromy group (over the complex numbers) is a geometric invariant that encodes the structure of the solutions for a parameterized family of polynomial systems and can be computed using numerical algebraic geometry. Since a naive extension to the real numbers is very restrictive, this talk will explore a new approach over the real numbers which is computed piece-wise to obtain tiered characteristics of the real solution set. This talk will conclude with an application in kinematics to help highlight the computational method and impact on calibration.

 

Adaptive step size control for homotopy continuation methods

Sascha Timme
TU Berlin

At the heart of homotopy continuation methods lies the numerical tracking of implicitly defined paths by a predictor-corrector scheme. For efficient path tracking the predictor step size must be chosen appropriately. We present a new adaptive step size control which changes the step size based on computational estimates of local geometric information as well as the order of the used predictor method. We also give an update on the Julia package HomotopyContinuation.jl.

 

Numerical homotopies from Khovanskii bases

Elise Walker
Texas A&M

Homotopies are useful numerical methods for solving systems of polynomial equations. I will present such a homotopy method using Khovanskii bases. Finite Khovanskii bases provide a flat degeneration to a toric variety, which consequentially gives a homotopy. The polyhedral homotopy, which is implemented in PHCPack, can be used to solve for points on a general linear slice of this toric variety. These points can then be traced via the Khovanskii homotopy to points on a general linear slice of the original variety. This is joint work with Michael Burr and Frank Sottile.