Quantitative stratification and critical sets of harmonic functions

Given a harmonic function defined on the unit ball of R^n, we discuss techniques to
obtain effective volume estimates on the tubular neighborhood of its critical sets.


We use a technique recently introduced by proff. Jeff Cheeger and Aaron Naber,
called quantitative stratification technique. It is based on approximate symmetries
of the function u at different scales. Studying how these approximate symmetries
interact with each other, we obtain the effective volume estimates.

These results
are described in a preprint available on arXiv.

We also discuss possible improvements
of the results using a refined quantitative differentiation argument and packing
estimates for semi-algebraic sets.