📋 Tender Overview
Foundations Of Transcendental Methods In Computational Nonlinear Algebra
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.
Based On Recent Proof-Of-Concept Works, I Propose New Foundational Methods In Computational 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, I Propose Applications To Diophantine Approximations, Feynman Integrals, And Optimization.
Funded Under
European Research Council (Erc)
Total Cost
€ 1 393 312,50
Eu Contribution
€ 1 393 312,50
Coordinated By
Institut National De Recherche En Informatique Et Automatique
France
Start Date
1 April 2022
End Date
31 March 2027
Objective
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.
Based on recent proof-of-concept works, I propose new foundational methods in computational 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 var
