Are all solutions of a linear partial differential equation smooth? This is one of the most natural questions one can ask for linear partial differential equations. If the answer is yes, then we call the differential equation hypoelliptic. There are many known examples like Cauchy-Riemann equation, Laplace equation.

The main source of hypoelliptic equations comes from elliptic operators. Another very well studied example is Hormander's sum of squares of operators. In this talk, we will generalise these examples by constructing a principal symbol which generalises the classical principal symbol. We show that if the principal symbol is invertible then the operator is hypoelliptic. We thus generalise the main regularity theorem for elliptic differential equations. This answers a conjecture of Helffer and Nourrigat.

This is a joint work with Androulidakis and Yuncken.

Bio: Omar Mohsen is a French mathematician. Mohsen earned his doctorate from Universite Paris Diderot in 2018 under the supervision of Georges Skandalis. He then held a postdoc position in Muenster university for 2 years. Since Septembre 2021, he holds a research/position un Universite Paris Saclay. His research is concerned with applications of C*-algebras in geometry, hypoelliptic differential operators, analysis of pseudodifferential operators.