On March 20, the American mathematician Karen Uhlenbeck became the first woman to win the Abel Prize, considered to be the Nobel Prize’s equivalent in mathematics.
Norway’s King Harald V will bestow Uhlenbeck with the prize in a ceremony on May 21.
The Abel Prize was created in 2003 by the Norwegian Academy of Science and Letters. It is awarded once every year, like the Nobel Prizes are, to people who have made outstanding contributions to mathematics. Apart from a certificate, each winner also receives 6 million Norwegian kroner (Rs 4.83 crore).
In the last 17 years, 19 men have won the prize, of whom 11 worked in the US.
According to Uhlenbeck’s citation, she received the prize this year for:
… her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.
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Uhlenbeck was born in 1942 in Cleveland, Ohio. She chose to become a mathematician after a freshman’s course at the University of Michigan in the early 1960s. She would later say that the fact that mathematics could be a solitary pursuit without significant consequences for her work was also a motivating factor.
As a young academic I worked by myself a lot. In fact, that was one of the attractions of mathematics. I am the eldest of four children and I consider dealing with my siblings the hardest thing I’ve ever done in my life. That had a great impact on my choosing a career – I wanted a career where I didn’t have to work with other people.
– Celebratio Mathematica, 1996
That solitude extended only to her mathematical collaborations. Aware of her status as a role model, Uhlenbeck helped launch a mentoring programme for young women mathematicians at the Institute for Advanced Study, Princeton, in 1991. She also founded the Program for Women and Mathematics at the same institute; it helps make mathematics more inclusive and provides financial support. Mathematics as a field has a particularly more acute problem of gender imbalance than science.
No one ever talks about this phenomenon of who is actually succeeding in the sciences and engineering – foreign-born men and women. I try to talk about this with my students. It’s difficult, however, because you’re not supposed to talk about it. In the large classes of engineering students I teach, I’m seeing a lot more diversity – women, Hispanics, African-Americans. It can be done, not just by white, Anglo men.
– Journeys of Women in Science and Engineering: No Universal Constants, 1997
During the course of her graduate studies at Brandeis University, Massachusetts, two other mathematicians sowed the seeds of Uhlenbeck’s first major contribution: bubbling.
In 1963, Richard Palais, her advisor, and Stephen Smale devised what has since been known as the Palais-Smale compactness condition. Imagine a rubber band is wrapped around the surface of a cylinder. This ‘mapping’ of the band is called harmonic if it has the lowest potential energy in this configuration. Say the cylinder is longer than it is wide. So the rubber band’s harmonic mapping will be to wrap around the cylinder’s diameter instead of around its length.
This is why harmonic maps are also called least-expanding maps: the band is completely in contact with surface of the object with the least amount of stretching.
The lowest possible elastic potential energy corresponding to the harmonic map is called the Dirichlet energy.
It’s evidently easy to discover harmonic maps when we’re working with a three-dimensional rubber band and a three-dimensional object. But when either – or both – of them are higher-dimensional objects, it’s nearly impossible to intuit what the harmonic map might be.
To solve this problem, Palais and Smale came up with a compactness condition: if a parameter satisfies this condition, then reducing the value of this parameter on a map should guide the map towards its harmonic state. For example, if the elastic potential energy of a rubber band satisfies the Palais-Smale compactness condition, then ways to reduce its elastic potential energy will also be the ways by which we take the band to a more harmonic mapping.
However, using this check, mathematicians found that the Dirichlet energy satisfies the compactness condition only for a rubber band of dimension 1 – as if it were a vanishingly thin thread.
In the late 1970s, Uhlenbeck and a postdoc named Jonathan Sacks found that there were alternative energy measures that passed the compactness test on flat (2D) surfaces. So they set out to check if, as these energy measures approached the Dirichlet energy, the corresponding maps approached the harmonic map. They found that this was the case almost everywhere on the flat surface. The exceptions were a finite number of points where the map started to form a bubble instead.
This discovery prompted a new area of study called geometric analysis. Mathematicians have used Uhlenbeck’s and Sacks’s finding to discover bubbles on different kinds of surfaces in different conditions. Physicists working with abstract surfaces, such as in string theory, use it to find ‘bubble singularities’ – called so because the bubble walls off an area of the surface for itself – that might correspond to new physical phenomena.
Even as her mathematical star was on the rise, Uhlenbeck was having a difficult time finding a permanent teaching position at universities because of her husband. Olke Uhlenbeck was a noted biochemist who worked at the University of Illinois, Urbana-Champaign, at the time – a time when universities prevented one partner from being hired in any department if the other was already an employee of the same university.
I was told … that people did not hire women, that women were supposed to go home and have babies. So the places interested in my husband – MIT, Stanford and Princeton – were not interested in hiring me. I remember that I was told that there were nepotism rules and that they could not hire me for this reason, although when I called them on this issue years later they did not remember saying these things. … I hated Urbana-Champaign – I felt out of place mathematically and socially, and it was ugly, bourgeois and flat.
– University of St Andrews, 2009
So she worked at the Massachusetts Institute of Technology, the University of California (Berkeley) and the University of Illinois – first in Urbana-Champaign then in Chicago. She has said she disliked her time in the last institute, and that she felt like a “faculty wife” when she was there. In 1983, she received a MacArthur Prize fellowship and moved to the University of Chicago. Five years later, she moved to the University of Texas, Austin, where she is today a professor emerita.
Coinciding with a period of vocational stability, Uhlenbeck’s earlier work led her to study, and contribute to, the Yang-Mills theory, which physicists use to study the behaviour of fundamental particles, in the early 1980s. Specifically, Uhlenbeck found that bubbling singularities don’t form on isolated points on a four-dimensional surface described by the Yang-Mills equations. This is one of her more famous achievements, called the ‘removal of singularities’ theorem.
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“This technical result was an important step in the understanding of instantons,” said Madhusudhan Raman, a postdoc at the Tata Institute of Fundamental Research, Mumbai, “which in physics account for quantum tunnelling phenomena” in a type of theories called gauge theories.
Simon Donaldson built on top of this work to “make stunning advances in the study of the topology of smooth four-dimensional manifolds,” Raman added. Donaldson would win the Fields Medal in 1986.
In 1984, Uhlenbeck’s research also helped solve an open problem in differential geometry, called the Yamabe problem (after Hidehiko Yamabe).
In 1990, she became the first female plenary speaker at the International Congress of Mathematicians in Kyoto, Japan, in 58 years. The last woman to have spoken there was Emmy Noether in 1932. Curiously, both Noether and Uhlenbeck both worked on the calculus of variations, and proved theorems that physicists consider invaluable.
But while Uhlenbeck did blaze many a trail for women in mathematics, the discipline’s exclusion of women is not something women should have to spend time to overcome. In this regard, Uhlenbeck has made it a little easier for her successors to find, join and stay in mathematics.