Log-concave density estimation

The class of log-concave densities on $\mathbb{R}^d$ is a very natural infinite-dimensional generalisation of the class of Gaussian densities.  I will show that it also allows the statistician to have the best of both the parametric and nonparametric worlds, in that one can obtain a fully automatic density estimator in the class (via maximum likelihood), with no tuning parameters to choose.  I'll discuss its computation, methodological consequences and theoretical properties, and in particular very recent results on minimax rates of convergence and adaptation.