This course is associated with the Thematic Einstein Semester (TES) Varieties, Polyhedra, Computation organised within the framework of the Berlin Mathematics Research Center MATH+ and supported by the Einstein Foundation Berlin.
Solving systems of polynomial equations over the real or complex numbers is a basic, ubiquitous, and highly relevant mathematical task. Over the past two decades, there has been dramatic progress in our ability to practically solve polynomial systems and explore their solution sets.
This course will center around a specific approach to solving sets of polynomial equations: numerical homotopy continuation. Numerical methods are particularly appealing for “large” problems due to their speed in computations. However, numerical computations only yields approximate solutions, the output is not exact. This is why, traditionally, homotopy continuation has been considered as a branch of applied mathematics. For instance, homotopy continuation is a popular tool in computing the solutions of kinematic problems in engineering.
In this course, Students will learn to solve their own systems with the Julia package HomotopyContinuation.jl. Next to solving problems from application, they will also learn how output from numerical homotopy continuation can be used in rigorous mathematical proofs. For instance, an instance of the famous 3264 conics tangent to five given conics was computed using HomotopyContinuation.jl, and all its tangential conics where proven to be real.
Students work on projects involving the solution of polynomial systems. They present their programs and solutions in conjunction with the Milestones Conference of the TES in February.