This project lies at the interface of the fields of geometry and probability, and is a follow-up to the GSM DTU that kicked off in 2016. The consortium of PIs has identified the need for this new application in order to expand on the GSM DTU to include new colleagues and their complementary expertise. In particular, this added expertise allows not only for a broader spectrum and wider impact, but also for better and stronger bridges between the subjects encompassed by this proposal. In addition, several of the PhD candidates of the current project are well on their way in their projects. As such, the GPS DTU will allow PIs from the GSM DTU to continue to build on the success of the initial project.Guided by internationally recognized researchers, the DTU will provide a flourishing environment for all participants and will strive to attract high-level students. The training environment will be broad and diverse, allowing candidates to acquire mathematical breadth, but also to have access to transferable skills training. This global approach to the enrichment and development of PhD candidates will be beneficial for their future careers, within or outside of academia. The scientific themes to be treated in the project are broad but all revolve around geometry and probability and their interactions. Geometric and probabilistic methods are of paramount importance in mathematics and their applications. Specific topics of interest include the geometry of low dimensional manifolds and their moduli spaces, probabilistic approximations, stochastic analysis, and arithmetic geometry including the study of elliptic curves.