# NONHOLONOM

Professor Irina Markina has won a grant of the Norwegian Research Council (NFR) in the nomination FRINAT.

## Hovedinnhold

The name of the project is "Analysis and geometry on non-holonomic manifolds with non-degenerate metrics", or shortly "NONHOLONOM". It has been financed with 6,215,000 Norwegian kroner for the period of 3.5 years.

The main objective of the project is the developing of geometry and analysis on non-holonomic manifolds with a special focus on cross-fertilization between theoretical studies and applications in geometric numerical analysis and optimal control problems. Therefore, the project was presented from four groups working in these areas. The main player is the Analysis Group of the Department of Mathematics of the University of Bergen led by professor A. Vasiliev. The collaborators are the Group of Geometric Numerical Analysis from the Department of Mathematics of the University of Bergen, where the main participants are A. Zanna and H. Munthe-Kaas, and the Group of Geometric Numerical Integration and Numerical Optimization from the Department of Mathematical Sciences of the Norwegian University of Science and Technology led by E. Celledoni and B. Owren. Moreover, we have contacts with the Department of Finance and Management Science of the Norwegian School of Economics in the area of symmetry applications to differential and discrete equations, where the main contact person is R. Kozlov. The entire research team consists of 7 senior members and a group of approximately 12 juniors including Ph.D. and post-doc fellows. As one can see, the senior members are scientists whose area of expertise lies in the fields of analysis, geometry, and numerical analysis. The team has more than 230 published scientific papers, 3 books and a number of graduated Master's and Ph.D. students. Moreover, the work within the project is supposed to carry out in close collaboration with a number of internationally recognized leading specialists in analysis, control theory, and robotic systems.

The main objectives of the project are organized into 4 groups. They are:

*Sub-Riemannian manifolds.*Here we study different interesting examples of sub-Riemannian manifolds, geodesic curves on them, and surfaces with some specific properties. We are also interested in properties of sub-Riemannian metrics on manifolds and estimates related to elliptic and parabolic equations on manifolds. A substantial part of our interest lies in the study of the potential theory of subelliptic equations.*Sub-semi-Riemannian manifolds.*These are manifolds endowed with a semi-definite metric. It is an essentially new object the study of which was initiated by the members of the Analysis Group from UiB and any new result in this area would be appreciated. We establish contacts with scientists studying similar objects from the control theory viewpoint. It is interesting to figure out when an invariant metric of this type exists on manifolds with group structures. There are several other open questions in this area.*Infinite-dimensional manifolds with a fiber bundle metric.*Our main interest here is the Lie-Fréchet group of orientation preserving diffeomorphisms of the unit circle and the related manifolds, such as its quotient homogeneous space and its Lie algebra. We are interested not only in the study of these objects from the analytic viewpoint but also in its relation with the stochastic Loewner equation, a hypoelliptic operator underlying this equation, and the space of univalent functions.*Control theory and simulations.*Any kind of problem that can be solved by applying results from the first three groups is here. We invite researchers interested in applications to collaborate in this part of the project.

We plan to open one post-doc position from December 2011, one Ph.D. studentship from January 2012, to invite several researchers for different time periods, and to organize several workshops where we can exchange our knowledge and ideas.