Bubbling and Regularity Issues in Geometric Non-linear Analysis

Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear effect. One observes a strict splitting in space between an almost linear regime and a dominantly non-linear regime which has two major characteristics : it requires a quantized amount of energy and arises along rectifiable objects of special geometric interest (geodesics, minimal surfaces, J-holomorphic curves, special lagrangian manifolds, mean-curvature flows...). 2000 Mathematics Subject Classification: 35D10, 35J20, 35J60, 49Q20, 58E15, 58E20.