Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize

The study of harmonic maps can be traced back to a centuries-old field of mathematics called the calculus of variations, which looks for shapes that are in equilibrium with respect to some natural physical measurement, such as energy, length or area. But when the rubbery shape has dimension greater than one — if it’s a surface, for instance, or some higher-dimensional object — the Dirichlet energy does not always satisfy Palais and Smale’s condition, meaning that the process of gradually deforming a mapping to reduce its Dirichlet energy may sometimes fail to converge to a harmonic map. In a similar way, Sacks and Uhlenbeck showed, the maps that minimize the alternative forms of energy converge to a harmonic map nearly everywhere, but near a handful of points on the surface they start to form bubbles.

Source: www.quantamagazine.org