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Founder of geometric analysis honored with Abel Prize

Founder of geometric analysis honored with Abel Prize
Science
The Norwegian Academy of Science and Letters today announced that Karen Uhlenbeck has won the 2019 Abel Prize, a Nobel-level honor in math. Uhlenbeck won for her foundational work in geometric analysis, which combines the technical power of analysisa branch of math that extends and generalizes calculuswith the more conceptual areas of geometry and topology. She is the first woman to receive the prize since the award of 6 million Norwegian kroner (approximately $700,000) was first given in 2003.

Caroline Series, a mathprofessor at the University of Warwick in Coventry, U.K., and president of the London Mathematical Society, says, To see a woman right up there, honored for a lifetime of distinguished work in math, who has made a huge difference to the development of the field in the last 40 yearsthat is hugely important.

Uhlenbeck, 76, spent much of her career at the University of Texas in Austin and is now a visitor at the Institute for Advanced Study (IAS) in Princeton, New Jersey. Her work stands at the heart of several important advances in math, including the revolutionary work in 4D topology by Simon Donaldson of the Simons Center at the State University of New York in Stony Brook. It has also fertilized interactions between math and theoretical physics, including string theory.

An example of the kind of object studied in geometric analysis is a minimal surface. Analogous to a geodesic, a curve that minimizes path length, a minimal surface minimizes area; think of a soap film, a minimal surface that minimizes energy. Analysis focuses on the differential equations governing variations of surface area, whereas geometry and topology focus on the minimal surface representing a solution to the equations. Geometric analysis weaves together both approaches, resulting in new insights.

The field did not exist when Uhlenbeck began graduate school in the mid-1960s, but tantalizing results linking analysis and topology had begun to emerge. In the early 1980s, Uhlenbeck and her collaborators did ground-breaking work in minimal surfaces. They showed how to deal with singular points, that is, points where the minimal surface is no longer smooth or where the solution to the equations is not defined. They proved that there are only finitely many singular points and showed how to study them by expanding them into bubbles. As a technique, bubbling made a deep impact and is now a standard tool.

Born in 1942 to an engineer and an artist, Uhlenbeck is a mountain-loving hiker who learned to surf at the age of 40. As a child she was a voracious reader and was interested in everything, she said in an interview last year with Celebratio.org. I was always tense, wanting to know what was going on and asking questions.

Andrea Kane/Institute for Advanced Study

She initially majored in physics as an undergraduate at the University of Michigan. But her impatience with lab work and a growing love for math led her to switch majors. She nevertheless retained a lifelong passion for physics, and centered much of her research on problems from that field.

In physics, a gauge theory is a kind of field theory, formulated in the language of the geometry of fiber bundles; the simplest example is electromagnetism. One of the most important gauge theories from the 20th century is Yang-Mills theory, which underlies the standard model of elementary particle physics. Uhlenbeck and other mathematicians began to realize that the Yang-Mills equations have deep connections to problems in geometry and topology. By the early 1980s, she laid the analytic foundations for mathematical investigation of the Yang-Mills equations.

Karen is the first person to introduce analytic tools from differential geometry to the study of Yang-Mills equations, says Alice Chang of Princeton University, who serves on the Abel Prize selection committee. Pioneer is the right word for her.

In the early 1990s, Uhlenbeck helped establish the Park City Mathematics Institute in Utah, one of the first vertically integrated programs that brought together schoolteachers, undergraduate majors, graduate students, and researchers. By then Uhlenbeck had risen to the top of her field, but she and other female mathematicians of her age didnt see large numbers of women coming after us, she told Celebratio.org.

Around that time, she took up research in integrable systems, which model certain kinds of physical phenomena such as shallow water waves. She and her collaborator, Chuu-Lian Terng of the University of California, Irvine, founded a mentoring program that was initially held in conjunction with the Park City group and grew into the Women and Mathematics Program at IAS in Princeton. Now in its 26th year, the program brings together about 60 female undergraduates, graduate students, and postdocs for 2 weeks of lectures, panels, and informal interactions.

It takes a person of the stature of Karen to persuade the IAS to host such a program, Chang says. Everywhere I gowhen I give a lecture in Taiwan, or in EuropeI will have women come to me and say that they have participated in the program.

Chang said she is thrilled that Uhlenbeck is getting the Abel Prize and counts her as a personal mentor. But Chang is also careful to point out that the Abel Prize committee stuck strictly to research in choosing the prizewinner. The prize citation does not mention Uhlenbecks mentoring efforts or her role as an inspiration to female mathematicians.

Much has changed in math since Uhlenbeck rose to prominence. When she got her Ph.D., the number of female math professors in the top universities in the United States could be counted on one hand. Today, the numbers are still small, but growing. And in 2014, the Fields Medal, the oldest and most prestigious honor in math and one that is given to mathematicians age 40 or younger, was awarded for the first time to a woman, Maryam Mirzakhani, a mathematician at Stanford University in Palo Alto, California, who died in 2017.

Has the time come for everyone to ignore the gender of major math prizewinners? I dont think weve quite got there yet, Series says. The Abel committee might have chosen somebody else, right? Its significant that she was the one they chose.
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