The Abel Symposia volume at hand contains a collection of high-quality articles written by the world's leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory.

In recent years we have witnessed a remarkable convergence between individual mathematical disciplines that approach deterministic and stochastic dynamical systems from mathematical analysis, computational mathematics and control theoretical perspectives. Breakthrough developments in these fields now provide a common mathematical framework for attacking many different problems related to differential geometry, analysis and algorithms for stochastic and deterministic dynamics. In the Abel Symposium 2016, which took place from August 16-19 in Rosendal near Bergen, leading researchers in the fields of deterministic and stochastic differential equations, control theory, numerical analysis, algebra and random processes presented and discussed the current state of the art in these diverse fields.

The current Abel Symposia volume may serve as a point of departure for exploring these related but diverse fields of research, as well as an indicator of important current and future developments in modern mathematics.

High-quality articles written by leading experts, including several ICM speakers, the 2014 Fields medalist Martin Hairer and the 2018 Henri Poincaré Prize laureate Percy Deift Presents developments in the field of deterministic and stochastic dynamical systems from the combined points of view of mathematical analysis, computational mathematics and control theory Unique and new emphasis on the fruitful interplay between analytic problems and tools from algebraic combinatorics

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Facilitated Exclusion Process: Jinho Baik et al.- Stochastic Functional Differential Equations and Sensitivity to their Initial Path: D. R. Baños et al.- Grassmannian Flows and Applications to Nonlinear Partial Differential Equations: Margaret Beck et al.- Gog and Magog Triangles: Philippe Biane.- The Clebsch Representation in Optimal Control and Low Rank Integrable Systems: Anthony M. Bloch et al.- The Geometry of Characters of Hopf Algebras: Geir Bogfjellmo et al.- Shape Analysis on Homogeneous Spaces: a Generalised SRVT Framework: E. Celledoni et al.- Universality in Numerical Computation with Random Data. Case Studies, Analytical Results and Some Speculations: Percy Deift et al.- BSDEs with Default Jump: Roxana Dumitrescu et al.- The Faà di Bruno Hopf Algebra for Multivariable Feedback Recursions in the Center Problem for Higher Order Abel Equations: Kurusch Ebrahimi-Fard et al.- Continuous-Time Autoregressive Moving-Average Processes in Hilbert Space: Fred Espen Benth et al.- Pre- and Post-Lie Algebras: The Algebro-Geometry View: Gunnar Fløystad et al.- Extension of the Product of a Post-Lie Algebra and Application to the SISO Feedback Transformation Group: Loïc Foissy.- Infinite Dimensional Rough Dynamics: M. Gubinelli.- Heavy Tailed Random Matrices: How They Differ from the GOE, and Open Problems: Alice Guionnet.- An Analyst's Take on the BPHZ Theorem: Martin Hairer.- Parabolic Anderson Model with Rough Dependence in Space: Yaozhong Hu et al.- Perturbation of Conservation Laws and Averaging on Manifolds: Xue-Mei Li.- Free Probability, Random Matrices, and Representations of Non-Commutative Rational Functions: Tobias Mai et al.- A Review on Comodule-Bialgebras: Dominique Manchon.- Renormalization: a Quasi-Shuffle Approach: Frédéric Menous et al.- Hopf Algebra Techniques to Handle Dynamical Systems and Numerical Integrators: A. Murua et al.- Quantitative Limit Theorems for Local Functionals of Arithmetic Random Waves: Giovanni Peccati et al.- Combinatorics on Words and the Theory of Markoff: Christophe Reutenauer.- An Algebraic Approach to Integration of Geometric Rough Paths: Danyu Yang.