Geometric Analysis is by nature a field at the intersection of different branches of Mathematics. As such, it gathers the interest of specialists with a diverse set of expertise. Our group reflects this fact, as it is formed by colleagues with backgrounds ranging from analysis on non-smooth spaces to geometric flows, from Lorentzian geometry to optimal control problems.
These complementary visions share a common research interest: the desire of studying singular geometric structures as a mean to better understand more regular ones. Here `structure’, `singular’, `regular’ are purposely vague terms, as our interests span broad areas of knowledge: for instance, we study metric measure spaces carrying weak curvature bounds as a tool to understand degeneration of smooth Riemannian metrics; as another example, geometric evolutions tend to develop singularities, that we study with the intent of capturing the geometry of evolving shapes.
More concretely, defining goals of our proposal are:
1) To build a unified theory of synthetic description of curvature bounds. We aim to cover and encompass both the existing ones about lower sectional and Ricci bounds, to explore the currently little-known setting of non-negative scalar curvature and to pursue a systematic study of the Lorentzian setting. In this latter framework we intend to cover both Lorentz-Finsler and Lorentz-metric structures.
2) To deepen the understanding of geometric evolutions and their relation with underlying curvature assumptions. A first example of question we want to address is the interplay between the level-set flow of p-harmonic functions, the torsion problem with Dirichlet boundary data and the arrival time equation for the Mean Curvature Flow: here the level set flow should act as a sort of suitable interpolation between the other two problems. A second type of question comes from the study of monotonicity formulas. Here, on one side we intend to sharpen the understanding of existing ones to obtain new geometric and functional inequalities, possibly in a quantitative form, while on the other we intend to investigate the presence of new monotone quantities for extrinsic curvature flows. We emphasize that this sort of problems are interdisciplinary by nature and as such will require the joint efforts of our group to be tackled. As they are extremely general and hard, we do not expect to fully solve them, but still we believe we will be able to address some relevant portions. More in general, such hard goals will be used as strong and concrete drivers of our research efforts in the years to come.
Finally, one of the by-products of the current project is to contribute to building a larger and stronger Italian community of Geometric Analysis, with diverse and complementary views on the topic.
