Karen Uhlenbeck: Is First Woman to Win Abel Prize for Mathematics

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Dr. Uhlenbeck helped pioneer geometric analysis, developing techniques now commonly used by many mathematicians.

Karen Uhlenbeck published many of her major papers in her late 30s and received a MacArthur Fellowship in 1983.CreditAndrea Kane/Institute for Advanced Study

For the first time, one of the top prizes in mathematics has been given to a woman.

On Tuesday, the Norwegian Academy of Science and Letters announced it has awarded this year’s Abel Prize — an award modeled on the Nobel Prizes — to Karen Uhlenbeck, an emeritus professor at the University of Texas at Austin. The award cites “the fundamental impact of her work on analysis, geometry and mathematical physics.”

One of Dr. Uhlenbeck’s advances in essence described the complex shapes of soap films not in a bubble bath but in abstract, high-dimensional curved spaces. In later work, she helped put a rigorous mathematical underpinning to techniques widely used by physicists in quantum field theory to describe fundamental interactions between particles and forces.

In the process, she helped pioneer a field known as geometric analysis, and she developed techniques now commonly used by many mathematicians.

“She did things nobody thought about doing,” said Sun-Yung Alice Chang, a mathematician at Princeton University who served on the five-member prize committee, “and after she did, she laid the foundations of a branch of mathematics.”

Dr. Uhlenbeck, who lives in Princeton, N.J., learned that she won the prize on Sunday morning.

“When I came out of church, I noticed that I had a text message from Alice Chang that said, Would I please accept a call from Norway?” Dr. Uhlenbeck said. “When I got home, I called Norway back and they told me.”

Dr. Uhlenbeck, 76, a visiting associate at the Institute for Advanced Study in Princeton, said she had not decided what to do with the $700,000 that accompanies the honor.

There is no Nobel Prize in mathematics, and for decades, the most prestigious awards in math were the Fields Medals, awarded in small batches every four years to the most accomplished mathematicians who are 40 or younger. Maryam Mirzakhani, in 2014, is the only woman to receive a Fields Medal.

The Abel, named after the Norwegian mathematician Niels Hendrik Abel, is set up more like the Nobels. Since 2003, it has been given out annually to highlight important advances in mathematics. The previous 19 laureates — in three years, the prize was split between two mathematicians — were men, including Andrew J. Wiles, who proved Fermat’s last theorem and is now at the University of Oxford; Peter D. Lax of New York University; and John F. Nash Jr., whose life was portrayed in the movie “A Beautiful Mind.”

In her early work, Dr. Uhlenbeck essentially figured out the shape of soap films in higher-dimensional curved spaces. This is an example of what mathematicians call optimization problems, which are often very difficult and can have zero solutions, one solution or many solutions.

“You can ask a question of when you have a soap bubble in this n-dimensional space,” she said. “You don’t know ahead of time what the shapes of those minimal soap bubbles are going to be.”

The universe is often lazy, looking for solutions that take the least amount of energy.

In a flat plane, an example of an optimization problem can be stated trivially: The shortest distance between two points is a straight line. Even on a curved surface, like Earth, the question has an easy answer — an arc known as a great circle.

With soap films and bubbles — two-dimensional surfaces in a three-dimensional space — the problem starts to get more complicated.

To minimize the forces of surface tension, a bubble forms in the shape with the least amount of area to wrap around a given volume — a sphere. When two or more bubbles touch each other or when a soap film forms inside of a twisted metal loop, the shapes become more complicated but still contort to take up the smallest amount of area.

In yet higher dimensions, “The theory becomes dramatically harder, and standard techniques just don’t work,” said Dan Knopf, who worked with Dr. Uhlenbeck at the University of Texas.

Dr. Uhlenbeck showed that the problem was not unsolvable everywhere, although at a finite number of points, the calculations would not converge. Thus, one could get a handle on the answer by handling those troublesome points separately.

“Karen developed some revolutionary techniques,” Dr. Knopf said. “And roughly speaking, she found solutions of an approximate problem and then tried to take limits of these approximate solutions to get actual solutions.”

Dr. Uhlenbeck later worked on what are called gauge theories, used by physicists in quantum field theory to describe interactions of subatomic particles. A gauge theory basically says that how the particles behave should not change depending on how you look at it. That is, the laws of physics should not change if the experiment is moved to the left or rotated.

But the answers sometimes seemed to blow up to infinity. She was able to recast the problem in a way that removed the infinities.

Dr. Uhlenbeck began publishing her major papers in her late 30s. In principle, that would have been early enough that she could have been recognized with a Fields Medal, but her ideas took time to spread.

In 1983, at 41, she received broader recognition with a MacArthur Fellowship, which comes with a bundle of money — $204,000 in Dr. Uhlenbeck’s case.

In 1990, she became the second woman to give one of the highlighted plenary talks at the International Congress of Mathematicians, a quadrennial meeting. At each congress, there are 10 to 20 plenary talks, but for decades, all of the speakers had been men. (Emmy Noether, a prominent German mathematician, was the first woman to give a plenary talk, in 1932.)

“That was almost more unnerving” than being the first woman to receive an Abel, Dr. Uhlenbeck said.

Dr. Uhlenbeck said she recognized that she was a role model for women who followed her in mathematics.

“Looking back now I realize that I was very lucky,” she said. “I was in the forefront of a generation of women who actually could get real jobs in academia.”

But she also noted: “I certainly very much felt I was a woman throughout my career. That is, I never felt like one of the guys.”

To find an influential woman, she looked to television.

“Like many people in my generation,” Dr. Uhlenbeck said, “my role model was Julia Child.”

For the first time, mathematics’ most prestigious prize has been awarded to a woman, Karen Uhlenbeck

Karen Uhlenbeck, a mathematician and professor at the University of Texas, has been awarded this year’s Abel Prize, a mathematics prize modeled after the Nobels. It’s the first time the prize has gone to a woman.

The Abel Prize is awarded by the King of Norway to mathematicians who have greatly influenced their field, and includes a cash award of 6 million Norwegian kroner (about $700,000). The first prize was awarded in 2003.
Uhlenbeck, 76, is known for her work with partial differential equations. However, her decades-long career spans multiple disciplines, including physics, geometry and quantum theory.
“Uhlenbeck’s research has led to revolutionary advances at the intersection of mathematics and physics,” Paul Goldbart, dean of the University of Texas’ College of Natural Sciences said in a release. “Her pioneering insights have applications across a range of fascinating subjects, from string theory, which may help explain the nature of reality, to the geometry of space-time.”
One of Uhlenbeck’s most famous contributions was her theories of predictive mathematics inspired by soap bubbles. The thin, curved surface area of a soap bubble is an example of a “minimal surface,” a surface that forms itself into a shape that takes up the least amount of area. Examining how these surfaces behave can help researchers better understand a wide amount of phenomena across a wide array of scientific studies.
“Her theories have revolutionized our understanding of minimal surfaces, such as those formed by soap bubbles, and more general minimization problems in higher dimensions,” Hans Munthe-Kaas, chair of the Abel Committee said in a release.

U.S. Mathematician Becomes First Woman To Win Abel Prize, ‘Math’s Nobel’

“I find that I am bored with anything I understand,” Karen Uhlenbeck once said – and that sense of curiosity is part of why she won the prestigious Abel Prize, from the Norwegian Academy of Science and Letters.

Uhlenbeck, an influential mathematician who was for decades a professor at the University of Texas at Austin and who has sought to encourage women to study mathematics, has become the first woman to win the Abel Prize — often called the Nobel Prize of math.

Uhlenbeck’s complex and wide-ranging work includes analyzing the “minimal surfaces” of soap bubbles and finding ways to unite geometry and physics through new mathematical approaches. She’s widely respected for her work on esoteric topics, such as partial differential equations and the calculus of variations.

“Uhlenbeck’s research has led to revolutionary advances at the intersection of mathematics and physics,” said Paul Goldbart, a professor of physics who is also the dean of UT’s college of natural sciences. In a statement about Uhlenbeck winning the Abel Prize, he added, “Her pioneering insights have applications across a range of fascinating subjects, from string theory, which may help explain the nature of reality, to the geometry of space-time.”

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The Norwegian academy said it recognized Uhlenbeck “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.”

The Abel Prize includes an award of 6 million Norwegian kroner (around $700,000). Uhlenbeck will formally receive the prize from Norway’s King Harald V, in a ceremony in Oslo on May 21.

“Karen Uhlenbeck is a founder of modern geometric analysis,” said Hans Munthe-Kaas, chair of the academy’s Abel Committee. “Her perspective has pervaded the field and led to some of the most dramatic advances in mathematics over the last 40 years.”

As that statement implies, Uhlenbeck has been a star in theoretical mathematics for decades. She won a MacArthur Fellowship in 1983, after publishing a sequence of influential papers on harmonic mapping and gauge theory — some of which she wrote alone and some in which she collaborated with mathematicians such as Richard Schoen and Jonathan Sacks.

In 1986, Uhlenbeck became the first female mathematician to be elected to the National Academy of Sciences. She was awarded a National Medal of Science in 2000. And the American Mathematical Society awarded her the Steele Prize in 2007, for decades of contributions to research.

As it recognized Uhlenbeck’s work in advancing the understanding of theoretical mathematics, the Norwegian committee also noted her professional impact and her standing as a role model.

“As a child, she loved reading and dreamed of becoming a scientist,” the committee said. “Today, Uhlenbeck is Visiting Senior Research Scholar at Princeton University as well as Visiting Associate at the Institute for Advanced Study (IAS). She is one of the founders of the Park City Mathematics Institute (PCMI) at IAS, which aims to train young researchers and promote mutual understanding of the interests and challenges in mathematics.”

Within IAS, Uhlenbeck co-founded the Women and Mathematics program in 1993, seeking to encourage women’s interest in the field.

“I am aware of the fact that I am a role model for young women in mathematics,” Uhlenbeck said, according to a release from Princeton University, where she has also worked. “It’s hard to be a role model, however, because what you really need to do is show students how imperfect people can be and still succeed. … I may be a wonderful mathematician and famous because of it, but I’m also very human.”

When she accepted the Steele Prize in 2007, Uhlenbeck said it was her work in education, not her mathematical theorems, that gave her the most pride. She also said that changing a culture that doesn’t encourage girls and women to pursue careers in mathematics “is a momentous task in comparison” to her other accomplishments.

“I remain quite disappointed at the numbers of women doing mathematics and in leadership positions,” she said. “This is, to my mind, primarily due to the culture of the mathematical community as well as harsh societal pressures from outside.”

Uhlenbeck worked at the University of Texas at Austin for more than 25 years. She attended the University of Michigan and received her Ph.D. at Brandeis University in 1968.

Uhlenbeck has said the variety of fields she studied — and her knack for applying ideas from one area to explore concepts in another — stemmed from “an addiction to intellectual excitement.”

Here’s how she described her work in 1997:

“Mathematicians look at imaginary spaces constructed by scientists examining other problems. I started out my mathematics career by working on Palais’ modern formulation of a very useful classical theory, the calculus of variations.

I decided Einstein’s general relativity was too hard, but managed to learn a lot about geometry of space-time. I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three manifold topology, learned gauge field theory and then some about applications to four manifolds, and have recently been working on equations with algebraic infinite symmetries. I find that I am bored with anything I understand.”

That’s from Uhlenbeck’s contribution to the book Journeys of Women in Science and Engineering: No Universal Constant, as quoted in the release from Princeton, where Uhlenbeck is currently a visiting scholar.

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