On the distribution of the Hodge locus

The lecture will discuss a joint work with Gregorio Baldi and Bruno Klingler.  Given a polarized Z-VHS over a complex, smooth quasi-projective variety S, we describe some properties of the Hodge locus, a countable union of algebraic subvarieties of S where exceptional Hodge tensors appear, by a result of Cattani, Deligne and Kaplan.  We prove the geometric part of the Zilber-Pink conjecture in this context: the maximal atypical part of the Hodge locus of positive period demension arise in a finite number of families. In level at least 3, we show that the typical Hodge is empty and therefore the positive dimensional part of the Hodge locus is algebraic.  For instance the Hodge locus of positive period dimension of the universal family of degree d smooth hypersurfaces in the projective space of dimension n+1 is algebraic.  We also prove that if the typical Hodge locus is not empty, then the Hodge locus is analytically dense in S.  If times permits, we will explain how this result can be combined with the work of Lawrence-Venkatesh in the direction of the Bombieri-Lang conjecture.