Understanding the nature of singularities which arise in minimal submanifolds has a rich history. In the 1960's it was established that codimension one area minimisers are smoothly embedded away from a (generally unavoidable) set of codimension 7, referred to as the singular set. Nowadays, due to the work of N. Wickramasekera, we know that this regularity result is actually a consequence of just stability (i.e., non-negative second variation of area) as well as the absence of a certain singularity type know as a classical singularity (which includes, for example, crossings of smoothly embedded submanifolds). One can then ask if the singular set obeys more refined estimates than just a dimension bound. The work of Naber-Valtorta provides Minkowski content bounds for codimension one area minimisers, and work of A. Song shows that one can achieve measure bounds for codimension one minimal hypersurfaces which have finite index and have sufficiently small singular set a priori. In this talk, I will discuss recent work, joint with Nicolau S. Aiex and Sean McCurdy, where we improve on these results, proving a Minkowski content bound on the singular set in the finite index setting when one just rules out classical singularities. In particular, we are able to control the measure of the entire tubular neighbourhood of the singular set.

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