Solving polynomial equations is at the heart of many problems in pure and applied mathematics. Although this is an ancient problem, it is still considered very challenging. Real-world applications require advanced, structure-exploiting symbolic and/or numerical algorithms. The Numerical Nonlinear Algebra research group develops such algorithms, as well as the underlying theory. Next to solving equations, we are interested in closely related problems such as polynomial optimization, elimination of variables, and tensor decomposition.
There is a clear need for efficient symbolic-numeric algorithms to test or formulate conjectures in algebraic geometry. Conversely, numerical nonlinear algebra motivates several fundamental questions. For instance, the complexity of Gröbner-basis-like algorithms is governed by an algebraic invariant called the regularity of a graded ring, and the theory of solving sparse polynomial equations is largely based on toric geometry. Our group identifies such questions and seeks to answer them.
A selected application is in particle physics. More precisely, we investigate the use of (numerical) nonlinear algebra in the study of scattering amplitudes. These are functions encoding probability densities of interactions between fundamental particles in collider experiments. Surprising connections to moduli spaces, polyhedral geometry, ideals in (possibly non-commutative) rings, discriminants, … motivate this effort. Numerical nonlinear algebra plays a key role in the development of positive geometry, which is a new branch of mathematics aspiring to provide the right framework for studying these connections.