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.

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.