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Description

Ever since the seminal work of Bethuel-Brezis-Helein(BBH) on a simplified 2d Ginzburg-Landau system rose to prominence in the ’80s, it has inspired a very active field in nonlinear analysis and mathematical physics that connects the theory of harmonic maps, energy renormalization and hard-analysis concentration estimates. However, up to this date, the beautiful description of vortex configurations achieved in the 2d setting is yet to find an exact analog in higher dimensions; this is due to a weaker characterization of vortex defects, the delicate interplay between the geometry of these codimension 2 objects and their mutual interaction, in dimensions greater than or equal to 3. In this talk I will present a framework that allows to capture an ”effective interaction energy” of nearly parallel vortex filaments in certain 3d settings: this provides a next order or ”renormalization” of the energy of clustering-filaments configurations together with a more accurate description of the vortex region in the spirit of the BBH study in 2d. This is joint work with Robert Jerrard.