AMS Short Course

Polynomial systems, homotopy continuation and applications

January 2-3, 2023, Boston, MA

Organized by Timothy Duff, University of Washington, and Margaret Regan, Duke University.

Systems of multivariate polynomial equations are ubiquitous throughout mathematics and neighboring scientific fields. Numerical homotopy continuation methods are a fundamental technique for both solving these polynomial systems and determining more refined information about their structure. A research community has blossomed around the subject, with important work on both basic methods and applications that are ripe for sharing with a general mathematical audience.

Continuation methods are a well-studied subject in numerical analysis. They are especially potent when applied to problems where polynomial equations depending on both certain unknown quantities (variables) and physical measurements (parameters) must be satisfied. By working over the complex numbers, many of these methods enjoy the property of being globally convergent with probability one. For instance, a general parameter continuation theorem (established by Morgan and Sommese in 1989) establishes that all isolated solutions can be computed via tailor-made homotopies which operate in a problem’s natural parameter space. Another useful paradigm (introduced in the 90s by Sommese, Wampler, and Verschelde) is numerical algebraic geometry, allowing these techniques to be extended to problems whose solution sets are not 0-dimensional.  A number of freely available software packages exist: most prominently, Bertini, HomotopyContinuation.jl, HOM4PS, NAG4M2, and PHCPACK. These tools power latest advances in polynomial homotopy continuation, with an array of conferences and research programs worldwide reflecting the latest developments.

The field is also outward-looking: particularly in recent years, where it has stretched the state of the art and provided new insights in far-flung fields including computer vision, enumerative geometry, kinematic design, power engineering, and systems biology. The current level of interest presents a perfect opportunity to offer a short course on the subject, accessible to students and professional mathematicians alike.

Registration will open in mid-August.  Please check this page for updates.

AMS Short Courses

The AMS Short Course is a two-day program held just prior to the annual JMM, the Short Course introduces mathematicians and students to emergent areas of applied mathematics to fuel curiosity, discovery and research. Led by experts, activities in each course address theoretical issues, numerical challenges and practical applications. Read a feature story about the experiences of participants in the 2022 Short Course.