- Series
- Student Algebraic Geometry Seminar
- Time
- Friday, February 5, 2021 - 9:00am for 1 hour (actually 50 minutes)
- Location
- SAGS Microsoft Teams
- Speaker
- Tim Duff – Georgia Tech – tduff3@gatech.edu – https://timduff35.github.io/timduff35/
- Organizer
- Marc Härkönen
Homotopy continuation methods are numerical methods for solving polynomial systems of equations in many unknowns. These methods assume a set of start solutions to some start system. The start system is deformed into a system of interest (the target system), and the associated solution paths are approximated by numerical integration (predictor/corrector) schemes.
The most classical homotopy method is the so-called total-degree homotopy. The number of start solutions is given by Bézout's theorem. When the target system has more structure than start system, many paths will diverge, This behavior may be understood by working with solutions in a compact projective space.
In joint work with Telen, Walker, and Yahl, we describe a generalization of the total degree homotopy which aims to track fewer paths by working in a compact toric variety analagous to projective space. This allows for a homotopy that may more closely mirror the structure of the target system. I will explain what this is all about and, time-permitting, touch on a few twists we discovered in this more general setting. The talk will be accessible to a general mathematical audience -- I won't assume any knowledge of algebraic geometry.