Karen Uhlenbeck, specialist in geometry and analysis, wins the Abel Award

Karen Uhlenbeck, whose pioneering work has launched an entire field of mathematics, received the Abel Award at a ceremony held today in Oslo. The award, created by the Norwegian Academy of Sciences and Letters in 2003, is widely regarded as the mathematical equivalent of a Nobel Prize.

Uhlenbeck's work has "resulted in some of the most dramatic advances in mathematics over the past 40 years," says the award.

His research "inspired a generation of mathematicians," said François Labourie of the Université de Côte d'Azur in France. "She wanders and finds new things that no one has found yet."

Uhlenbeck, born in Cleveland in 1942, was a voracious reader in her childhood, but she was not really interested in mathematics until she enrolled in the first year mathematics class at the University of Michigan. "The structure, the elegance and the beauty of mathematics immediately struck me and I lost all my heart," she writes in the book. Mathematicians: an external vision of the inner world.

Mathematical research had another element that he liked at the time: it's something you can work on your own, if you wish. At the beginning of her life, she said in 1997, "I thought that everything about people was kind of a horrible profession."

In the mid-1960s, Uhlenbeck graduated from Brandeis University, where she chose Richard Palais as a consultant. Palais was exploring what was then mostly unexplored territory between analysis (a generalization of computation) and topology and geometry (which study the structure of forms). "I've been attracted to this – the area between things," said Uhlenbeck in an interview last year in Celebratio Mathematica. "It was like jumping from a bridge where we did not know what was going to happen."

Palais and mathematician Stephen Smale (who then won a Fields Medal for his topology research) had just made a breakthrough on "harmonic maps" that would be the springboard for some of Uhlenbeck's most important results. The study of harmonic maps can be related to a centuries-old mathematical domain called calculus of variations, which looks for shapes in equilibrium with some natural physical measurements, such as energy, length or surface. For example, one of the oldest and most famous problems in the calculation of variations is the problem of the "brachistochrone", posed by Johann Bernoulli in 1696, which asks the curve in which a ball will roll the fastest One point to another.

To understand what a card means to be harmonic, imagine a compact rubber shape, such as a rubber band or rubber sphere. Next, choose a particular way of locating this shape in a given space (such as an infinite space in three dimensions or a donut in three dimensions). This positioning of the form is called the harmonic map if, roughly, it puts the form in equilibrium, which means that the rubber does not sink into a different configuration which has a less elastic potential energy (what mathematicians call energy Dirichlet).

When the space in which you trace the rubbery shape is a complicated object with holes (such as a donut surface or its larger dimension equivalents), a variety of harmonic cards may emerge. For example, if you wrap an elastic band around the center hole of a donut's surface, it can not shrink to a point without leaving the surface of the donut. Instead, it will shrink to the shortest route around the hole.

When the rubbery shape or the target space is a complicated object, perhaps of great size, determining the range of possibilities offered by harmonic maps can be tricky because we can not simply build a physical model and then see what what does rubber do. Intuitively, we could try to construct a harmonic map starting with any map, then looking for ways to deform it, little by little, to bring it closer to balance. But it is not always clear if such a process will eventually converge and reach equilibrium.

This question makes sense for other measures similar to energy as Dirichlet's energy. Palace and Smale have put in place a condition relating to energy measures that, once satisfied, ensures the convergence of at least a part of these deformation processes. The Palais-Smale condition was perfect for the one-dimensional case of harmonic maps, where we map an elastic band (infinitely narrow) in a compact space like a donut sphere or surface. But when the rubbery shape has a dimension greater than one, for example a larger surface or object, the Dirichlet energy does not always satisfy the Palace and Smale condition, which means that the process of progressive deformation a map to reduce the number The Dirichlet's energy can sometimes fail to converge to a harmonic map.

In the mid-1970s, while he was a professor at the University of Illinois, Urbana-Champaign, Uhlenbeck strove to understand how that failure of convergence might look like . Her five years at Urbana-Champaign were not particularly happy – she and her husband were both teachers – and she felt as though she was perceived primarily as a "teacher's wife" – but she met a fellow postdoctoral nominee Jonathan Sacks. . Together, they explored a series of different energy-like measurements on two-dimensional surfaces, each satisfying the Palais-Smale condition and approximating Dirichlet's energy. In each of these alternative energy measurements, the Palais-Smale condition ensures that there is a map to minimize energy consumption. While the energy in question is getting closer and closer to the Dirichlet energy, Uhlenbeck and Sacks have asked, do these maps converge towards a harmonic map?

The answer, they showed in the late 1970s and early 1980s, is "almost". At almost every point on the surface, these maps converge to a harmonic map. But at a finite set of points on the surface, maps can begin to form a kind of very specific singularity called bubble, where there is no way to make sense of a map.

To imagine a singularity of bubble, imagine that you chew gum and you blow a bubble, then you gradually pull on it more and more in your mouth, while maintaining the bubble to the same size. The gum forming the bubble will be diluted and diluted, but the bubble will remain viable throughout this process (at least in an idealized environment in which the gum will be infinitely extensible). But at the end of this process, the bubble will crumble, since you have basically pulled all the chewing gum out of your mouth.

In the same way, as Sacks and Uhlenbeck show, maps that minimize alternative forms of energy converge to a harmonic map almost everywhere, but begin to form bubbles near a handful of points on the surface. . As the energy approaches Dirichlet 's energy, these bubbles will be constructed from smaller and smaller plates of the surface. At the end of the process, when we have reached the actual Dirichlet energy, the map will want to create an entire bubble from a single point. We will have a singularity. Since the compact space in which the surface is being mapped – a sphere-like surface or a donut – only has a finite number of holes around which bubbles can to form, there is only a finite number of these bubbling singularities.