Algebraic Background for Numerical Methods, Control Theory and Renormalization / Dominique Manchon -- From Iterated Integrals and Chronological Calculus to Hopf and Rota-Baxter Algebras / Kurusch Ebrahimi-Fard, Frederic Patras -- Noncommutative Symmetric Functions, Lie Series and Descent Algebras / Jean-Yves Thibon -- From Runge-Kutta Methods to Hopf Algebras of Rooted Trees / Ander Murua -- Combinatorial Algebra in Controllability and Optimal Control / Matthias Kawski -- Algebra is Geometry is Algebra -- Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application / Alexander Schmeding.