The theory of multiple orthogonal polynomials has gained a lot of attention in the past few years, partially due to its connection to several models of mathematical physics, such as random matrices, non-intersecting paths and planar growth, to mention a few.
Motivated towards understanding the zeros of such polynomials in great
generality, in this talk we plan to discuss the notion of vector critical measures for a vector logarithmic energy. We obtain several structural results for vector critical measures, in particular showing that they are intimately connected with quadratic differentials on compact Riemann surfaces. Employing techniques on quadratic differentials, we are able to construct families of such critical measures that should ultimately describe the zero distribution
for certain classes of multiple orthogonal polynomials.

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