09:00 to 10:00 Douglas Arnold (University of Minnesota)Finite Element Exterior Calculus - 1 These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications. INI 1 10:00 to 11:00 Elizabeth Mansfield (University of Kent)Introduction to Lie groups and algebras - 2 I will discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Kurusch Ebrahimi-Fard (Norwegian University of Science and Technology)Why B-series, rooted trees, and free algebras? - 2 "We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes. INI 1 12:30 to 14:00 Lunch at Murray Edwards College 14:00 to 15:00 Charlie Elliott (University of Warwick)PDEs in Complex and Evolving Domains II INI 1 15:00 to 16:00 Christian Lubich (Eberhard Karls Universität Tübingen)Variational Gaussian wave packets revisited The talk reviews Gaussian wave packets that evolve according to the Dirac-Frenkel time-dependent variational principle for the semi-classically scaled Schr\"odinger equation. Old and new results on the approximation to the wave function are given, in particular an $L^2$ error bound that goes back to Hagedorn (1980) in a non-variational setting, and a new error bound for averages of observables with a Weyl symbol, which shows the double approximation order in the semi-classical scaling parameter in comparison with the norm estimate. The variational equations of motion in Hagedorn's parametrization of the Gaussian are presented. They show a perfect quantum-classical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion. A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter. The material presented here for variational Gaussians is part of an Acta Numerica review article on computational methods for quantum dynamics in the semi-classical regime, which is currently in preparation in joint work with Caroline Lasser. INI 1 16:00 to 17:30 Poster session and Afternoon tea