{\bf Abstract:} Obstruction flatness of a strongly pseudoconvex hypersurface $\Sigma$ in a complex manifold refers to the property that any (local) K\"ahler-Einstein metric on the pseudoconvex side of $\Sigma$, complete up to $\Sigma$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $\Sigma$. In general, $u$ has only a finite degree of smoothness up to $\Sigma$.
In this talk, we are interested in obstruction flatness of hypersurfaces $\Sigma$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over a complex manifold $M$, whose dual line bundle induces a K\"ahler metric $g$ on $M$. The main result we will discuss can be summarized as follows: If $(M,g)$ has constant Ricci eigenvalues, then $S(L)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then the corresponding disk bundle admits a complete K\"ahler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of $S(L)$ in terms of the K\"ahler geometry of $(M,g)$ in some special cases.
The talk is based on a recent joint paper with Ebenfelt and Xu.

Suppose we are observing a sample of independent random vectors, knowing that the original distribution was contaminated, so that a fraction of observations came from a different distribution. How to estimate the covariance matrix of the original distribution in this case? In this talk, we discuss an estimator of the covariance matrix that achieves the optimal dimension-free rate of convergence under two standard notions of data contamination: We allow the adversary to corrupt a fraction of the sample arbitrarily, while the distribution of the remaining data points only satisfies a certain (rather weak) moment equivalence assumption. Despite requiring the existence of only a few moments, our estimator achieves the same tail estimates as if the underlying distribution were Gaussian. Based on a joint work with Pedro Abdalla.

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