We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions, and its several drawbacks, leading to the notion of AMLE (Absolutely Minimizing Lipschitz Extension). We then present a rigorous and detailed analysis of the equivalence between being absolutely minimizing Lipschitz and enjoying comparison with cones. Finally, we explore some consequences of this geometric notion, chiefly the derivation of a Harnack inequality.
José Miguel Urbano is a Professor of Mathematics at the University of Coimbra (Portugal). He holds a Ph.D. from the University of Lisbon and did a postdoc at Northwestern University in Chicago. He is the author of the book The Method of Intrinsic Scaling and of over 50 scientific papers in the area of Nonlinear Partial Differential Equations. He has supervised four Ph.D. students and ten postdoctoral fellows. He was a member of the Scientific Council for the Exact Sciences and Engineering of the Portuguese Science Foundation (FCT) and has served as an evaluator of grants and research projects for the EU (Marie-Curie Fellowships), ERC (Starting Grants), the Academy of Finland, the Latvian Council of Science and FCT. He has taught short courses at IMPA (Rio de Janeiro, Brazil), the University of Florence (Italy), Aalto University (Finland), the Federal University of Ceará (Fortaleza, Brazil), KAUST (Saudi Arabia) and Seoul National University (South Korea). He is a Corresponding Academician of the Sciences Class (Mathematics Section) of the Lisbon Academy of Sciences.