Synthetic timelike lower Ricci curvature bounds

This project, carried out by Clemens Sämann in collaboration with the project host Professor Robert McCann of the University of Toronto, is about mathematical problems in Einstein`s theory of gravitation General Relativity the fundamental theory of space and time. The problem it addresses concerns the language in which General Relativity is formulated differential geometry and physically relevant models that might not fit into that language. It aims at considerably extending the range of applicability of the theory, a goal that promises significant advances both in the underlying mathematical theory and in concrete applications to physical models of the universe. As a geometric theory General Relativity was developed for spaces that behave nicely from a mathematical perspective. Every object involved is assumed to be described by functions of a very regular character, meaning that for example functions are not allowed to have jumps. However, in reality, models of the universe as a whole of more localized situations such as stars indeed do exhibit such jumps. This obviously leads to problems when using the classical theory of General Relativity. Moreover, the central idea of General Relativity is that the matter (or, equivalently, energy) content of the universe determines its shape, and conversely, that the curvature of space determines the movement of matter. The central object describing the shape of space is curvature, a highly complex entity that depends strongly on the language of differential geometry. However, in other fields of mathematics outside of General Relativity, there are successful approaches to describing curvature by other means and thereby addressing the problem of non-niceness. The main goal of the project is to develop a complete theory that can handle such physically relevant situations as described above and to obtain a notion of curvature independent of assumptions concerning the niceness of the underlying objects. To this end, completely new methods developed recently by the applicant and Michael Kunzinger permit a mathematical treatment of such problems and a recent breakthrough result of McCann relates geometric properties of space and time to more robust notions. The innovative idea of the proposed project is that these notions can be cast into the framework of Kunzinger and Sämann and this synthesis will result in a geometrically robust theory of gravitation that simultaneously sheds new light on and considerably extends the geometry of General Relativity. To summarize, this project aims at significantly contributing to the understanding of the universe and other models in General Relativity by developing a robust theory of gravitation outside the classical framework. To this end the expertise and experience of Robert McCann and the knowledge of the applicant will perfectly complement each other to achieve the goals of the project.

This project, carried out by Clemens Sämann in collaboration with the project host Professor Robert McCann of the University of Toronto, was about mathematical problems in Einstein's theory of gravitation - General Relativity - the fundamental theory of space and time. The problem it addressed concerns the language in which General Relativity is formulated - differential geometry - and physically relevant models that might not fit into that language. Its main aim was to considerably extend the range of applicability of the theory, a goal that promises significant advances both in the underlying mathematical theory and in concrete applications to physical models of the universe. As a geometric theory General Relativity was developed for spaces that behave "nicely" from a mathematical perspective. Every object involved is assumed to be described by functions of a very regular character, meaning that for example functions are not allowed to have "jumps". However, in reality, models of the universe as a whole of more localized situations such as stars indeed do exhibit such "jumps". This obviously leads to problems when using the classical theory of General Relativity. Moreover, the central idea of General Relativity is that the matter (or, equivalently, energy) content of the universe determines its shape, and conversely, that the curvature of space determines the movement of matter. The central object describing the shape of space is curvature, a highly complex entity that depends strongly on the language of differential geometry. However, in other fields of mathematics outside of General Relativity, there are successful approaches to describing curvature by other means and thereby addressing the problem of "non-niceness". The main goal of the project was to develop a complete theory that can handle such physically relevant situations as described above and to obtain a notion of curvature independent of assumptions concerning the "niceness" of the underlying objects. To this end, completely new methods developed recently by the applicant and Michael Kunzinger permit a mathematical treatment of such problems and a recent breakthrough result of McCann relates geometric properties of space and time to more robust notions. The innovative idea of the proposed project was that these notions can be cast into the framework of Kunzinger and Sämann and this synthesis will result in a geometrically robust theory of gravitation that simultaneously sheds new light on and considerably extends the geometry of General Relativity. To summarize, the aims of the project was to significantly contribute to the understanding of the universe and other models in General Relativity by developing a robust theory of gravitation outside the classical framework.