Mathias Braun - Research

Nonsmooth Lorentzian Geometry

I am currently working on a promising optimal transport approach to Einstein's theory of gravity. In this work, I have obtained results about the timelike curvature-dimension condition – briefly TCD condition – for nonsmooth measured Lorentzian spaces, especially its pathwise interpretation, its local-to-global property, and sharp geometric inequalities. It characterizes the condition "Ricci curvature being bounded from below in timelike directions, and dimension being bounded from above" for smooth spacetimes, hence includes the strong energy condition of Hawking and Penrose. The class of spaces in which the TCD condition makes sense includes spacetime solutions to the Einstein equation with low regularity metric. In this work with Matteo Calisti, we have shown spacetimes with C¹ metrics – the borderline regularity for local existence results for the Einstein equation in four dimensions – and timelike lower bounds for their distributional Ricci tensor to be covered by the synthetic Lorentzian setting. The latter should thus lead to new understandings of (super-)solutions to the Einstein equation. With Shin-ichi Ohta, we have also covered Finsler spacetimes here.

Dirichlet Spaces with Singular Ricci Bounds

I am also active in the positive signature case, namely Dirichlet spaces with singular Ricci bounds. These singularities can arise e.g. from spatial interior or boundary irregularities, or from simple unboundedness of the "Ricci tensor". In this work with Karen Habermann and Karl-Theodor Sturm, we have characterized variable Ricci bounds in the Lagrangian and Eulerian picture. Moreover, here I have developed a second order calculus on spaces with Ricci curvature bounded from below by a signed measure in the extended Kato class. This setting includes a vast class of curvature singularities which are still controlled by the heat kernel. The analysis of such spaces might lead to applications to Ricci flow, where curvature never stays bounded by flowing through singularities.

Stochastic Tensor Calculus

A side project I am curious about is a nonsmooth analogue of Eells–Elworthy–Malliavin's construction of diffusion processes on Riemannian manifolds. A key difficulty to date is that this approach is highly extrinsic. The prototypical setting should be the above mentioned Dirichlet spaces with lower Ricci bounds, in which both Brownian motion and an intrinsic tensor calculus make sense. A successful treatment might lead to stochastic derivative and representation formulas for heat semigroups on functions, 1-forms, vector fields, etc. à la Bismut–Elworthy–Li and Feynman–Kac. In turn, this entails pathwise insights into the behavior of the aforementioned analytic objects.