Li-Yau and Harnack estimates for nonlocal diffusion problems

These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on PDEs Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis’', held at Ghent University in October 2025.