We establish interval arithmetic as a practical tool for certification in numerical algebraic
geometry. Our software HomotopyContinuation. jl now has a built-in function certify, which …

Numerical nonlinear algebra is concerned with the development of numerical methods to
solve problems in nonlinear algebra. The main computational task is the solution of systems …

In this paper we study circles tangent to conics. We show there are generically $184 $
complex circles tangent to three conics in the plane and we characterize the real …

We introduce the concept of monodromy coordinates for representing solutions to large
polynomial systems. Representing solutions this way provides a time-memory trade-off in a …

A wonderful compactification of an orbit under the action of a semi-simple and simply
connected group is a smooth projective variety containing the orbit as a dense open subset …

Abstract (EN) Physics provides new, tantalizing problems that we solve by developing and
implementing innovative and effective geometric tools in nonlinear algebra. The techniques …

Abstract (EN) Systems of polynomial equations appear both in mathematics, as well as in
many applications in the sciences, economics and engineering. Solving these systems is at …

In this thesis we study certain algebraic varieties from the point of view of birational
geometry. Given a variety we want to describe all its birational models. In general, this is a …