Symmetry, Curvature Reduction and EquivAlence Methods

About this project

<p>&nbsp; The goal of this GRIEG proposal at the interface of geometry, algebra, and PDE is to answer questions of fundamental importance for a variety of geometric structures beyond classical Riemannian geometry. Of particular focus are the broad classes of Cartan and parabolic geometries, which include conformal, projective, CR, and ODE geometry, (2,3,5)-distributions, parabolic contact structures, and many more besides. These structures will be examined along the lines of the central themes of this SCREAM proposal: Symmetry, Curvature Reduction, and EquivAlence Methods. Objectives include: (1) Investigate geometric robots whose configuration spaces support interesting geometric structures, in particular those whose symmetries form a simple Lie algebra. (2) Refine the Cartan reduction algorithm for classifying homogeneous structures and apply it to low-dimensional geometries of broad interest. (3) Examine parabolic geometries enhanced by additional geometric structures. (4) Establish a geometric interpretation of dispersionless integrability for a large class of differential equations. (5) Study parabolic contact structures and notions of contactification. (6) Solve problems of direct relevance to Penrose's Conformal Cyclic Cosmology programme. To tackle these problems, this GRIEG proposal lays the groundwork for a collaborative research effort between primary groups centred in Warsaw, Poland (led by Professor Paweł Nurowski) and Tromso, Norway (led by Professor Boris Kruglikov). The Polish team will consist of six scientists, including two students, which will strengthen the Differential Geometry scene in Warsaw. On the Norwegian side, the proposal will augment, by two postdocs, the already established Geometry and Mathematical Physics group (including Assiciate Professor Dennis The), currently supported by the Trond Mond Foundation / Tromso Research Foundation until 2022.</p>

Bilateral cooperation results

The bilateral collaboration within the project is satisfactory, taking into account not only joint paper but also other forms of collaboration. Work on the project's tasks was carried out in rich international collaboration with leading scientists, mainly physicists. As a result the three application were submited and several further grant applications are currently under elaboration by the project members, in particular two (The, Schneider) to be submitted soon to the Research Council of Norway.

Project partners

• University of Tromsø – The Arctic University of Norway Donor project partner· Norway