About abstract spaces and precise values
Can you imagine a space that has more than three dimensions, that is curved, and to top it all off is irregular? What sounds a little like science-fiction has been reality in mathematical research for some time.
In its work, the working group “Stochastic Analysis” at the Hausdorff Center examines and describes these spaces in great detail. For this purpose, the team uses methods from differential calculus.
If you want to be able to imagine the spaces that are treated here, you could, for example, think of a donut. Even, spacetime is curved according to Einstein’s theory of relativity. Mathematicians analyze for example, how the volume of such an abstract space could be calculated accurately and how high the natural frequency of such a space is. For a long time already, it is known that a lot of these properties only depend on two factors: the dimension of a space and its curvature. In this context scientists distinguish between positive and negative curvature, i. e. between the curvature of a sphere and a saddle.
For a lot of theories, these curved multidimensional spaces are still a far too simple model. Because even in the most beautiful space there can be irregularities. For instance, an orange is not a perfect sphere, because of the mark of the branch on which it hung and the many dimples in its peel which disturb the perfection of its geometry. In mathematics, one calls such irregularities “singularities. And singularities follow different rules. Here, the typical definition of curvature loses its meaning. The dimension can also vary, and it does not have to be an integer. Therefore, such spaces cannot be described by means of the usual differential calculus.
To describe the geometry of these spaces more in detail, however, the scientists had to combine different, apparently very distant research areas. The concept of “optimal transportation” was important in this combination. That is what mathematicians call the question of how a certain mass distribution can be transferred most efficiently into another one. In 2005, HCM coordinator Karl-Theodor Sturm and his colleagues had already discovered that there is a link between the curvature of a space and the optimal transportation. The researchers were able to derive information about the curvature and the dimension of a space from the transport behaviour. Since then, these properties can also be calculated for spaces with singularities.
Karl-Theodor Sturm and Matthias Erbar from the Hausdorff Center, together with their colleague Kazumasa Kuwada from Tokyo, succeeded in improving this theory significantly. In their work they showed for the first time that, even in such spaces, it is possible to work with methods from differential calculus if one utilizes curvature and dimension cleverly. Until this work, the mathematicians had not managed to use the information about the dimension which they had derived from the transport behaviour to describe spaces. Hence, the new discovery fills in an important gap in the theory.
With their research, the German-Japanese team has laid the foundation for a better understanding of abstract spaces. The first scientists have already used the new theory for their own work and gained further insights. We can look forward to many exciting discoveries in this research area in the future.
MATTHIAS ERBAR, KAZUMASA KUWADA, KARL-THEODOR STURM (2014): On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces. Inventiones mathematicae, 1-79.