Computational Dynamics and Stochastics on Manifolds

Our research in the years 2020-2021 has significantly advanced our understanding of connection algebras. A connection is the geometric structure of a manifold which defines parallelism and geodesic curves. This type of geometric information is essential in many applications such as information geometry, shape analysis and numerical integration of differential equations. Prior to the CODYSMA project, we found that the geometric structures of invariant connections [Nomizu 1954] has an algebraic parallel. E.g. flat geometries are described by preLie algebras, Klein geometries by postLie algebras and symmetric spaces by Lie Admissible Triple Algebras. A detailed understanding of these structures is essential to do systematic computations of many geometric problems. It has been a major (positive) surprise to realise that we can also describe algebras of general non-invariant connections, e.g. general Riemanninan connections, in terms of postLie algebras. A major goal CODYSMA was to provide a detailed understanding of connection algebras for all invariant connections. General connections seemed to be out of reach, but now it is probable that we will be able to develop a rich theory for the general case. This is work in progress which will have major impact on the project in the next years. In 2022-23 we have seen significant breakthroughs in the understanding of algebras of general connections. This opens for generalisations of B-series methods and related techniques for geometric problems in general geometries. Theories developed within the Codysma project now seems to find applications in the theory of regularity structures for stochastic PDEs, and it is significant interest for this internationally, last under the French GdR meeting on renormalisation in Calais, November 2022, as well as in the program "Signatures for Images" at Centre for Advanced Study (2023-24), Oslo, lead by Kurusch Ebrahimi-Fard and Fabian Harang.

A surprising convergence of certain areas of mathematical sciences is currently taking place. Research fields including numerical integration of dynamical systems, rough path 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. Algebraic structures arising naturally from the geometry of invariant connections on manifolds are at the very heart of these mathematical developments. The CoDySMa project is developing geometric, algebraic and computational techniques through an interaction between Lie-Butcher theory in computational dynamics and algebraic techniques arising from rough paths and regularity structures in stochastic analysis. Through advancement of the underlying mathematical theories, open source software, publication and organisation of international conferences and workshops, the project has as a main goal to provide a common mathematical and computational framework unifying diverse fields of mathematics, and develop new mathematical and computational techniques. 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. The PI of this project founded Lie-Butcher theory, generalising classical B-series to manifolds and Lie groups. In recent research with co-workers, these geometric and algebraic structures are substantially developed and applied to rough paths on manifolds. The Norwegian research group is complemented with leading international experts. Together they form a unique research group with expertise to achieve the project goals.