PDE/Analysis Seminar

The Kato square root problem for divergence form elliptic operators is the equivalence statement $\norm{\sqrt{- \mathrm{div} \br{A \nabla}}u} \simeq \norm{\nabla u}$, where  $A$ is a complex matrix-valued function.

We shall discuss the concept of sparse dominations of singular integral operators, in particular for those operators whose kernel do not satisfy any regularity estimate.

A coarse embedding between metric spaces is, intuitively speaking, a map which preserves, in a weak sense, the geometry at large distances.

We look at some unsolved proplems.

We will discuss the Laplacian flow for closed $G_2$ structures.

In the first part of this talk we give a brief survey of the Hardy and Rellich inequalities.