Abstract: Conformally covariant differential operators,
including the Yamabe operator, feature heavily in studies of the
scalar and Q-curvatures as well as in (sharp) Sobolev inequalities.
In this talk, I discuss multilinear analogues of these operators,
known as conformally covariant polydifferential operators. In one
direction, I show that these operators are ubiquitous; for example,
there is such an operator associated to each variational scalar
Riemannian invariant. In another direction, I describe a general
approach to classifying local minimizers of a related Sobolev
quotient. This talk is partially based on joint work with Yueh-Ju Lin
and Wei Yuan.