Numerical Certification: Bridging Computation and Proof
ABSTRACT
Modern numerical methods, such as homotopy continuation, can efficiently solve systems of nonlinear equations, yet they often lack rigorous guarantees. This talk addresses a central challenge: how can we elevate numerical computation to the level of rigorous proof?
Focusing on recent work in certified homotopy tracking, we show how interval arithmetic can be used to rigorously validate solution paths and prevent numerical failures. After a brief overview of numerical algebraic geometry, we present the key ideas of certified tracking and its implementation. Finally, we highlight several emerging directions, including certified Galois group computation and certified approximation of algebraic varieties, pointing toward a broader program of turning computation into proof.
