On some geometry-grounded problems involving PDEs, Dynamics, and discretization

I use the same title for different talks, changing the selection of topics and the level of detailedness for each topic:) This is a survey talk based on joint projects with D. Chen, S. Ivanov, late Ya. Kurylev, M. Lassas, J. Lu, and, if time permits, with L. Polterovich, B. Kleiner at al. As the main themes, I plan Inverse Problems (Including boundary rigidity and the Inverse Gel’fand’s Transform), presence of positive metric entropy in nearly integrable systems, and discretization in PDEs and metric geometry. In an unlikely event some time is left, a bit of combinatorial group theory and Riemannian geometry. My intention is to try to give an accessible talk, sweeping technical details under the rug. Questions: welcome at any point.