Robert Bryant showed that any closed immersed Willmore sphere in Euclidean three-space is the inversion of a complete minimal sphere with embedded planar ends. We proved that the Willmore Morse Index of the closed surface can be computed by using unbounded Area-Jacobi fields of the related minimal surface. As a consequence, we get that all immersed Willmore spheres are unstable except for the round sphere. This talk is based on work with Jonas Hirsch and Rob Kusner.

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Higher Spin Gravities are supposed to be minimalistic extensions of gravity that embed it into a quantum consistent theory. However, such minimality turns out to be in tension with the field theory approach, as well as with the numerous no-go theorems. We report on the recent progress in constructing Higher Spin Gravities and testing quantum effects therein. AdS/CFT relates Higher Spin Gravities to a variety of interesting three-dimensional CFT's, e.g. to the Ising model. These CFT's were conjectured to exhibit a number of remarkable dualities, in particular, the three-dimensional bosonization duality. We show how Higher Spin Gravity can be useful to prove the bosonization duality at least in the large N limit and make new predictions for the correlation functions.

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