We present some results on sub-Riemannian geodesics in stratified groups recently obtained in collaboration with E. Le Donne, G.P. Leonardi, and R. Monti. One of the main open questions in the field is the regularity of length minimizers, that is not trivial due to the presence of the so called abnormal curves. We provide a characterization of abnormal curves in stratified groups showing that they are contained in certain algebraic varieties; this poses several questions ranging from analysis to algebraic geometry. Some applications will be discussed.
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