Geometric Integration is an interdisciplinary area of research, which is applying modern abstract geometrical ideas to the numerical solution of differential equations. Situated at the point of intersection between pure and applied mathematics, computer science and mathematical physics, it is an activity, which has in recent years combined ideas from these different fields in a remarkable way and turned them into tools of computational mathematics. Research in Geometric Integration has several goals.
• Geometrical structures are fundamental in the understanding of physical phenomena. In many simulations, it is crucial to develop numerical solution techniques that exactly preserve important underlying geometrical structures. Such geometrically exact algorithms have applications in a wide range of different areas such as structural mechanics, robotics and control theory, molecular dynamics, simulation of particle lattices, celestial mechanics and general relativity. The last years have brought a wide range of different new techniques, but there is still a lot of work to be done in this direction, both pursuing new ideas, and also refining recent ideas and turning them into efficient algorithms and computer programs.
• Object orientation is a fundamental tool in the construction of large software systems involving discrete mathematical structures. It is an important goal to understand and overcome the theoretical and practical difficulties lying in the generalization of these techniques to areas of mathematics involving continuous mathematical structures and differential equations.
• Through the construction of software, abstract mathematical ideas become more concrete and available to applied mathematicians. Thus a focus on computations and software is contributing to bridging the gap between pure and applied mathematics.

The year has been a great inspiration to the participants, and the project has been an important event within the applied mathematics and scientific computing communities in Norway. The advances achieved point in various directions. The most important scientific contributions from this year will be easier to pinpoint in a few years’ time. A summary of some directions pursued is given here:
• Advances of basic Lie group integrators (LGI) and their analysis. This is a class of algorithms that have been studied by our research community for 7–8 years. During the GI special year, advances were made in the analysis and optimization of several of the basic algorithms.
• Advances towards applications of LGI on PDEs. Most of our earlier studies were directed towards Ordinary Differential Equations. The advances towards applying the methods on Partial Differential Equations might be one of the most important developments during the year. We are now seeing these kinds of methods entering into more large-scale computing problems in engineering and computational physics.
• Improved understanding of non-commutative B-series. B-series is a fundamental tool for analysing time integration schemes, which have surprising connections to renormalization theory in physics and optimal control theory. While the classical theory of B-series is a commutative theory, the structure underlying LGI is a non-commutative theory. The algebraic structure of non- commutative B-series became well understood during the year in Oslo.
• Advances in symplectic integration and other algorithms in computational mechanics. Several research reports pushed the knowledge of structure preserving integration of Hamiltonian systems further.
• Advances in linear algebra algorithms. Structure preserving algorithms in Linear Algebra is an area of active research, and work was done in several directions, such as optimizing the number of commutators, computations of matrix exponentials and fast computable coordinates on matrix Lie groups.