by Professor Terry Lyons, (University of Oxford)

Abstract: Complex streams of evolving data are better understood by their effects on nonlinear systems that by their values at times. The question of which nonlinear systems would seem to be context dependent, but it is not. Core to rough path theory is a simple universal nonlinear system that captures all the information needed to predict any response to any nonlinear system. This idealised mathematical feature set is known as the signature of the stream. Its abstract simplicity opens the possibilities for understanding and working with streams in the same context free way that calculators work with numbers. Signature-based techniques offer simple to apply universal numerical methods that are robust to irregular data and efficient at representing the order of events and complex oscillatory data. Specific software can be developed and then applied across many contexts. Signatures underpin prize winning contributions in recognizing Chinese handwriting, in detecting sepsis, and in generating financial data, and most recently in the ability to score streams as outliers against a corpus of normal streams. This principled outlier technology has emerged as a powerful unifying technique; it identifies radio frequency interference in astronomical data and brain injury from MEG data. The underpinning theoretical contributions span a range from abstract algebra and non-commutative analysis to questions of organisation of efficient numerical calculation. See www.datasig.ac.uk/. New hyperbolic partial differential equations have been developed that compute the “signature kernel” trick without ever having to introduce signatures. Neural controlled differential equations can directly harness approaches such as the log ode method and consume the control as a rough path. The current step is the rough transformer. For this RoughPy needs to be on the GPU.