Nonlinear Diffusion, and Geometric and Functional Inequalities on Smooth Metric Measure spaces

This extended abstract is based on a talk given at the workshop and summer school ``Direct and Inverse Problems with Applications" in Ghent Analysis and PDE Centre in August 2024. It focuses on nonlinear diffusion equations of slow and fast types and their links with some geometric and functional inequalities in the framework of smooth metric measure spaces. The article presents some introduction in a summer school style as well as several new results.