In this talk we will give an introduction to maximally hypoelliptic differential operators. This is a class of differential operators generalizing elliptic operators and includes operators like Hormander’s sum of squares. We will present our work where we define a principal symbol and show that maximally hypoellipticity is equivalent to invertibility of our principal symbol generalizing the classical regularity theorem for elliptic operators. 

We will also give a topological index formula for maximally hypoelliptic differential operators using our symbol. Explicit examples of index computations will be included at the end.

This talk is based on joint work with Androulidakis and Yuncken.