Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps

Explore a comprehensive lecture on the Rectifiable-Reifenberg theorem and its applications to the regularity of stationary and minimizing harmonic maps. Delve into the groundbreaking paper by Aaron Naber and Daniele Valtorta, published in Annals of Mathematics 2017. Examine the evolution of techniques for bounding singularities in harmonic maps, from Federer's dimension reduction argument to the quantitative stratification introduced by Cheeger and Naber. Discover how careful analysis of these quantitative aspects leads to sharp rectifiability results for singular sets of minimizing harmonic maps, including uniform Minkowski bounds and improved integrability for the harmonic map itself. Gain insights into the adaptability of these techniques to various scenarios, with particular focus on minimal surfaces and Q-valued maps. This 49-minute talk, presented by Daniele Valtorta at BIMSA for #ICBS2024, offers a deep dive into the fundamental ideas and far-reaching implications of this mathematical breakthrough.