Symmetry, Curvature Reduction and EquivAlence Methods

Description

  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.

Summary of project results

This was a fundamental research project in pure mathematics, specically in differential geometry. For more than a century this area was dominated by Riemannian geometry, which is a curved version of Euclidean geometry familiar from school. There is an abundance of other geometries based on different notions than distance, namely the geometry of conformal structures (angles), of geodesics (shortest lines), of constraint velocities (nonholonomic mechanics), of differential equations (evolutionary dynamics), etc. They have a collective name of Cartan geometries. Some of these geometries nd applications in alternative theories of gravity and other physical theories, but the majority existed as mathematical abstractions. The goal of the project was to investigate a uniform broad class of Cartan geometries with the new tools developed in mathematics in recent decades. The project was called SCREAM because its central themes are Symmetry, Curvature Reduction, and EquivAlence Methods. Symmetry is fundamental in all natural sciences and is tantamount to niceness of the geometric structure. Curvature is a mathematical counterpart of gravity and it is a mechanism of Reduction of Symmetry. EquivAlence is a way to make Equalor Align seemingly different geometric structures, and we wanted exploit various Methods for this
purpose. 

We sought to implement and refine the techniques of Cartan geometries in order to answer questions of fundamental importance for a variety of geometric structures beyond the classical setting. This concerns computing the symmetry size, establishing existence of solutions to physically motivated equations, recognizing geometries via their invariants, and investigating geometric robots whose con figuration spaces support fascinating geometric structures.

The project description specified: “We expect that as a result of studies of the objectives a number of publications in worlds class mathematical journals will be produced. This will be typically at the end of the time span of each Objective”. This was done in full generality. While some specific objectives of the general research tasks were not processed as was initially projected, we found alternative routes and equally important substitutions, making the progress in all research tasks. This attracted attention of experts and was published in respected international journals.
Furthermore, it was claimed that “The two groups in Warsaw and in Tromsø will run a weekly working seminar in which the problems of the proposal and its partial results will be presented and discussed”. This again was done in full generality. Our bi-weekly meeting got recognition and attracted attention of both senior researchers and young specialists.
Also from the project description: “Every year we plan at least one 1-week visit of the PI’s group to Tromsø and one 1-week visit of the Partner’s group to Warsaw”. This was modified in view of pandemic restrictions, but was compensated by the online meetings and exchange of visits on the later stage of the project development.
It was promised that “the groups will jointly organize two conferences in Poland that will gather experts and the members of the research teams”. We did better, as we also organized a workshop in Nara, Japan, on equivalence method in geometries beyond the parabolic
geometries scheme, and a very important final meeting in Paris, in the last months of the project, focusing on Cartan Geometry. Taking into account that the idea of the project was developed during Abel Symposium 2019 in Ålesund Norway, dedicated to Lie Theory, this
closed the loop of ideas the SCREAM program addressed. Finally, “Members will participate in science festivals, public lectures, lectures for school classes, etc”. This was performed well as projected.

In the proposal we volunteered to have events with a blend of science and society in the program. One of such events, a concert accompanying the SCREAM project mathematics conference “GRIEG meets Chopin”, had a surprising socio-economic impact. After contacting the world-renowned jazz pianist Leszek Możdżer, to ask if he would be willing to give a piano recital for the participants of the conference, we realized that we can help him to build a totally new piano on which the music during the concert would be played. This initiated a `piano project’ within the SCREAM proposal. As a result:

• We constructed a totally new concert piano, with a 10-scale equally tempered octaves, which we called an acoustic decaphonic piano,
• We organized the World Premiere of the Acoustic Decaphonic Piano event, with a concert by Leszek Możdżer
• We were asked by the SAP Renovation Company, a World leader in the piano renovation industry, to help them in building a totally new Polish acoustic concert piano, the first one after the 1989 political changes .
• We now collaborate with the SAP Renovation Company in two projects of creating totally new innovative acoustic concert pianos; together with the SAP company we intend to present our pianos at the Polish Pavilion at the 2025 EXPO exhibition in Osaka, Japan.

Summary of bilateral 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.