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In this article we give an overview of the problem of finding sharp constantsin matrix weighted norm inequalities for singular integrals, the so-calledmatrix $A_2$ conjecture. We begin by reviewing the history of the problem inthe scalar case, including a sketch of the proof of the scalar $A_2$conjecture. We then discuss the original, qualitative results for singularintegrals with matrix weights and the best known quantitative estimates. Wegive an overview of new results by the author and Bownik, who developed atheory of harmonic analysis on convex set-valued functions. This led to theproof the Jones factorization theorem and the Rubio de Francia extrapolationtheorem for matrix weights, two longstanding problems. Rubio de Franciaextrapolation is expected to be a major tool in the proof of the matrix $\A_2$conjecture, and we discuss some ideas which may lead to a complete solution.

Matrix weights, singular integrals, Jones factorization and Rubio de Francia extrapolation