Abstract: 

In this talk, we present a new method of proving global pointwise Harnack inequalities for positive solutions of parabolic equations, such as the classical heat equation, porous medium equation, and $p$-heat equation. Our approach is based on a multipoint maximum principle argument, which does not rely on additional estimates such as those by Aronson-Benilan or Esteban-Vazquez. We demonstrate our main techniques by providing a new proof of Li-Yau's celebrated Harnack inequality for positive solutions of the heat equation on $\mathbb{R}^d$. This talk is based on joint work with Ben Andrews and Daniel Hauer.