Introduction to quantitative De Giorgi methods

The theory of De Giorgi (1958) and Nash (1959) solves Hilbert's 19th problem and constitutes a major advance in the analysis of PDEs in the 20th century. This theory concerns the H\"older regularity of solutions to elliptic and parabolic equations with non-regular coefficients, and it was extended by Moser (1960) to include the Harnack inequality. This course reviews the classical De Giorgi method in the elliptic and parabolic cases and introduces its recent extension to hypoelliptic equations which appear naturally in kinetic theory. The simplest case is the Kolmogorov equation with a rough diffusion coefficients matrix in the kinetic variable. We present compactness arguments but emphasize the recently developed quantitative methods based on the construction of trajectories. These lecture notes are self-contained and can be used as a general introduction to the topic.