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This lecture by Melanie Rupflin explores quantitative stability estimates for the Dirichlet energy, addressing the challenges posed by singularity formation in the analysis of almost critical points or almost minimizers. Discover how energy concentration at multiple scales and points leads to bubble tree formations consisting of harmonic maps that describe both bulk behavior and micro-domain energy concentration. Learn about the fundamental questions in quantitative analysis of this variational problem, including whether almost minimizers/critical points can be considered close to singular minimizers/critical points. The presentation particularly focuses on the challenges of comparing maps defined on domains with different topologies and developing optimal methods to measure and control the distance between them.
