Loewner theory

Abstract: We have developed a new approach to the powerful Loewner theory (an area in the confluence of complex analysis and differential equations). This approach, that includes both radial and chordal versions of the Loewner equations, still raises many issues of great interest (stochastic variants, relationships with Physics,…) and is one of my priority areas of research today (with recent papers in Crelle’s Journal, Math. Ann., TAMS, Rev. Mat. Iberoam.,… having all of them an elevated number of citations). Apart from the intrinsic interest in finding a more general version of previous keynote works, our contribution has been deeply used to obtain extensions of Loewner theory in several complex variables, an issue that until our results had been developed rather superficially, and new results about quasi-conformal extensions of univalent maps.