Mathematics Seminar - 10/28/26

Abstract: Many applied problems can be reduced to finding the solutions to a polynomial system of equations.  When such solutions are computed numerically, numerical errors may cause the resulting approximation to have a different topology than the true solution.  In order to elevate the results of such a computation to the level of a theorem, one must use a certified algorithm, that is, an algorithm that also produces a proof of the correctness of its output.  Recently developed algorithms have shown that certified computation is not only theoretically possible, but also practical.  In this talk, I will provide a survey of results on certified algorithms for such topologically correct computations.  In addition, I will discuss problems from both applied algebraic geometry and dynamical systems which rely on certified computations.