We present new regularity results for the singular sets of Q-valued harmonic functions. This work is a collaboration with Camillo de Lellis, Andrea Marchese and Emanuele Spadaro (see arXiv: 1612.01813).
For a Q-valued Dirichlet minimizing harmonic functions, we show that its singular set is m-2 rectifiable. Moreover, we also prove uniform Minkowski bounds for the set of Q-points. The proof is based on a Reifenberg-type theorem obtained in collaboration with Aaron Naber (see arXiv:1504.02043), and the technique is very versatile in nature and can be used to tackle many different problems in GMT. For example, this techinque can be adapted to study the singularities of thin obstacle problems and liquid crystals (see arXiv:1703.00678 - arXiv:1706.02734 ).