I present the theory of weighted (Lorentz-)Finsler spaces and
some global results which follow from a lower bound on the weighted (Bakry-Emery) Ricci curvature. This curvature, which we obtain generalizing previous versions, depends on the synthetic dimension $N$ and on the weight.
We show that it is convenient to let the proper time and the notion of completeness depend on a special parameter epsilon. We obtain weighted versions of the Jacobi, Riccati, and Raychaudhuri equations which display it. When the weighted Ricci is bounded from below and the parameter epsilon belongs to a special interval dependent on the synthetic dimension, which we call epsilon-range, curvature retains the standard focalization properties over geodesics as the usual unweigthed theory. From this property we get some global results that are new even for weighted Riemannian manifolds and that unify previous results that indeed dealt separately with the cases $N>n$ or negative $N$. This talk is based on joint work with Yufeng Lu and Shin-ichi Ohta.
The seminar will be held over Zoom. Register in advance for this meeting using the following link: https://utoronto.zoom.us/meeting/register/tJcqf-Cqrj4oH9X0X4db3kOgnEhUVGy5AUwe After registering, you will receive a confirmation email containing information about joining the meeting.