Over the past few decades significant advancements have been made in the field of Mathematical Analysis, driven by applications in materials science, image segmentation, shape optimization, data analysis, and machine learning. These developments have led to the emergence of powerful new methods, such as the theory of variational evolution problems, optimal transport, free discontinuity problems, nonlocal/nonlinear diffusions, spectral analysis, mean-field theories, geometric and functional inequalities, non-smooth analysis in metric-measure spaces.

The aim of the school is to offer an overview of some of these recent advancements and to introduce Ph.D. students and young researchers to a range of fascinating problems in this dynamic research area.

A special emphasis will be placed on the following three key aspects:

• functional analytic tools for nonlocal models, in particular fractional Sobolev spaces and fractional diffusions,
• variational methods for gradient flows and evolutionary problems,
• spectral methods and shape optimization.

The target audience primarily includes Ph.D. students, but also advanced Master students and postdocs, in PDEs, Calculus of Variations, Applied Mathematics, Probability, and related fields. More generally, anyone interested in the proposed topics is welcome to attend.