PRIN Grant 202244A7YL (2023-2025) “Gradient Flows and Non-Smooth Geometric Structures with Applications to Optimization and Machine Learning"

ABSTRACT

The project stems from a long-standing collaboration between a number of researchers
in the fields of optimal transport of probability measures gradient flows in metric
spaces and the geometry of metric-measure spaces in particular with Riemnannian
Ricci curvature bounded from below and from a flourishing research activity in
optimization and machine learning. Some of these analytical tools have received
increasing interest from the applied community revealing new aspects and promising
research directions. Particular challenging questions concern the limit behaviour of
discrete structures as graphs with their energy counterpart towards continuous models
typically represented by non-smooth metric-measure spaces. In this respect weak
topologies as the ones induced by optimal transport metrics and the asymptotic
behaviour of gradient flows play a crucial role in the analysis and provide a promising
background for more refined results and investigations. The project aims to face these
problems and to combine the current knowledge to develop new geometric analytical
and computational methods in both theoretical and applied directions.