**Venue:** Seminar Room "-1" – Via Sommarive, 14 Povo - Trento**Hour**: 3.00 p.m.

**Mattia Vedovato***PhD**in Mathematics*

**Abstract:**

Regularity properties for minimizing harmonic maps between Riemannian manifolds have been known since the classical work of Schoen and Uhlenbeck (1982); here an estimate on the Hausdorff dimension of the singular set S(u) of a map u is given: dim(S(u))≤n-3, where n is the dimension of the domain manifold. In this seminar, we are looking deeper into some more recent quantitative results, which describe precisely the structure of S(u). As a starting point, we are recalling the main tools introduced in the work of Cheeger and Naber (2013), which allow to extend the previous estimate to the Minkowski dimension of S(u): in particular, we are stratifying the singular set according to how close the map u is, at any point of S(u), to be invariant with respect to an affine plane. New techniques are then used to improve the aforesaid estimate: following the work of Naber and Valtorta (2017), we obtain an upper bound on the (n-3)-dimensional Minkowski content of S(u). Furthermore, by the use of a suitable version of the Reifenberg Theorem, we manage to show that the above defined singular strata are rectifiable, thus gaining very powerful information on the structure of the singular set.

**Supervisor: **

Francesco Serra Cassano