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I will discuss recent results on minimal hypersurfaces with cylindrical tangent cones of the form $C \times \mathbf R$, where $C$ is a minimal quadratic cone, such as the Simons cone over $\mathbf S^3 \times \mathbf S^3$. I will talk about a uniqueness result for such tangent cones in a certain non-integrable situation, as well as a precise description of such minimal hypersurfaces near the singular set under a symmetry assumption.
http://cvgmt.sns.it/seminar/872/