Foundations of transcendental methods in numerical algebraic geometry – 10000 DIGITS

Modeling nonlinear constraints by polynomial equations and inequalities raises fundamental theoretical issues, many of which have been answered by algebraic geometry. As of applications, nonlinearity is also a formidable computational challenge. What is the probability that a given spacecraft collides with a debris? How many smooth rational curves of degree 4 lie on a given quartic surface? These are questions with an underlying algebraic model and longing for computational answers.

Based on recent proof-of-concept works, I propose new foundational methods in numerical nonlinear algebra, motivated by the need for reliability and applicability. The joint development of theoretical aspects, algorithms and software implementations will turn these proof-of-concepts into breakthroughs.

Concretely, I will develop a theory of transcendental continuation for the numerical computation of a wide range of multiple integrals, based on a striking combination of algebraic geometry, symbolic algorithms and numerical ODE solvers. This would enable the computation of many integrals (e.g. volume of semialgebraic sets, or periods of complex varieties) with rigorous error bounds and high precision, more than thousands of digits. Building upon transcendental continuation, I propose to design algorithms to compute certain algebraic invariants of complex varieties related to algebraic cycles and Hodge classes, far beyond the current reach of symbolic methods. This surprising application is backed by a recent success on Picard group computation.

Applications include algebraic geometry, with the development of computational tools to experiment on concrete examples and build databases that document the largest possible range of behavior. Besides, the computation of multiple integrals with rigorous error bounds applies and science and engineering, in areas where trusting the result is more important than performance, such as spacecraft collision probability estimation.