Karen Uhlenbeck wins the Abel Prize

The Norwegian Academy of Science and Letters has awarded the Abel Prize, a mathematical award often regarded as on a level with the Nobel Prize, to Karen Uhlenbeck of the University of Texas, USA.

The award cited her work in geometric analysis, gauge theory and global analysis, which has application across a broad range of modern mathematics and mathematical physics, including models for particle physics, string theory and general relativity.

Her career began in the mid-1960s, under the advisor Richard Palais. Palais had been exploring some connections between analysis (generalizations of calculus) and topology and geometry (the mathematical theory of shapes and continuous deformations). Palais and noted mathematician Stephen Smale had recently found some new results for harmonic maps, which can be thought of as generalizations of the calculus of variations (techniques for finding maxima and minima according to certain criteria). When the geometric/topological shape of a space is complicated, such as in a very high-dimensional object, determining the range of possible harmonic maps is quite difficult. Palais and Smale devised a condition that guarantees that at least some deformations of these spaces will converge.

In the 1970s, while at the University of Illinois, Urbana-Champaign, Uhlenbeck sought to understand more clearly the conditions under which these harmonic deformation processes would fail to converge. Working together with Jonathan Sacks, she found that these processes do converge at almost all points, but at certain points the maps may develop a “bubble singularity.” Their result has now been applied in numerous areas of mathematics.

Among other things, Uhlenbeck applied her approach to the mathematics behind the standard model of physics. Among other things, she discovered a new coordinate system for which the equations behind the standard model could be studied more easily. Then she proved her remarkable “removable singularity” theorem, which demonstrated that for four-dimensional shapes, the bubbling mentioned earlier cannot occur at isolated points. Thus any finite-energy solution to the Yang-Mills equations behind the standard model that is well-defined in a neighborhood of a point will also extend smoothly to the point.

Uhlenbeck’s results “underpin most subsequent work in the area,” according to Simon Donaldson of Imperial College in London.

Uhlenbeck has broken ground in more than one way. First of all, she is the first woman to receive the Abel award in its seventeen-year history. In 2007, she was the first woman to receive the American Mathematical Society’s Steele Prize. In 1990, she was the second woman to present a plenary lecture at the International Congress of Mathematicians (the first was Emmy Noether, in 1932). Interestingly, she also is a counterexample to the stereotype of the child prodigy that many associate with mathematicians — she didn’t really become very interested in mathematics until her freshman year at the University of Michigan.

Uhlenbeck has long found deep satisfaction and fulfillment in mathematical research. As she wrote in accepting the Leroy P. Steele Prize from the American Mathematical society,

Along the way I have made great friends and worked with a number of creative and interesting people. I have been saved from boredom, dourness, and self-absorption. One cannot ask for more.

Additional details can be read in a very nice Quanta Magazine article by Erica Klarreich, from which part of the above note was summarized.

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