Algebraic Combinatorics for Computational Dynamics and Stochastics

ERC project summary: A surprising convergence of certain areas of mathematical sciences is currently taking place. Research fields including numerical integration of dynamical systems, rough paths theory and regularity structures for stochastic partial differential equations, combinatorial techniques in control theory and other seemingly very separate areas of mathematics all share common foundations where algebraic structures on trees and their underlying combinatorial algebras are central. The ACCORDS project is developing differential geometric, combinatorial and computational aspects of techniques originating from B-series and Lie-Butcher theory in computational dynamics and from rough paths and regularity structures in stochastic differential equations. Through advancement of the underlying mathematical theories, creation of open source software, publication of research papers and organizing international conferences, the project has as a main goal to provide a common mathematical and computational framework unifying these diverse fields of mathematics. The new theoretical and computational techniques will be applied to problems in computational dynamics, stochastic differential equations, control theory, integrable systems, differential- and symplectic geometry, geometric mechanics and calculus of variations. We are at a pivotal point in these developments, where establishing a common mathematical framework supported by powerful computational techniques and software implementations can lead to far-reaching results and advancement of diverse mathematical disciplines. The PI of this project founded Lie?Butcher theory, generalizing classical B-series to manifolds and Lie groups. In recent research this is substantially developed with co-workers. With their track record in organizing inter-disciplinary conferences and workshops, this research group is in a unique position to lead this research and communicate results between different research areas.