Karen Keskulla Uhlenbeck

Born: 24 August 1942 in Cleveland, Ohio, USA

Karen Uhlenbeck's father was an engineer and her mother was an artist. She grew up in the country, the eldest of four children. Many mathematicians know from early age that mathematics will be their life but this was not so with Karen Uhlenbeck. As a child she was interested in books and this led her to an interest in science. She writes:-
As a child I read a lot, and I read everything. I'd go to the library and then stay up all night reading. I used to read under the desk in school. ... we lived in the country so there wasn't a whole lot to do. I was particularly interested in reading about science. I was about twelve years old when my father began bringing home Fred Hoyle's books on astrophysics. I found them very interesting. I also remember a little paperback book called "One, Two, Three, (and, in?) Infinity" by George Gamow, and I remember the excitement of understanding this very sophisticated argument that there were two different kinds of infinities.
Uhlenbeck entered the University of Michigan with the intention of studying physics but a combination of studying exciting mathematics courses and finding that physics practicals were not a strong point lead her to change to mathematics. She was awarded a B.S. in mathematics in 1964.

After graduating from the University of Michigan, Uhlenbeck continued her studies at the Courant Institute in New York. However at this time she married and decided to follow her husband when he went to Harvard. She entered Brandeis University and was awarded a Master's Degree in 1966. She remained at Brandeis to study for her doctorate under Richard Palais' supervision, and was awarded a Ph.D. in 1968.

Her first appointment was a one year post in 1968-69 at Massachusetts Institute of Technology. Then another temporary post, this time a two year one as a lecturer at the University of California, Berkeley during 1989-71. She describes her search for a permanent position:-

I was told, when looking for jobs after my year at MIT and two years at Berkeley, 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 ended up at the University of Illinois, Champaign-Urbana because they hired me, and my husband came along. In retrospect I realized how remarkably generous he was because he could have been at MIT, Stanford, or Princeton. I hated Champaign-Urbana - I felt out of place mathematically and socially, and it was ugly, bourgeois and flat.
After being on the faculty at Urbana-Champaign from 1971 to 1976, she moved to the University of Illinois at Chicago where she was promoted to full professor. At this time she:-
... became friends with S T Yau, whom I credit with generously establishing my finally and definitively as a mathematician.
In 1983 she was awarded a MacArthur Prize Fellowship and moved to a professorship at the University of Chicago. In 1988 Uhlenbeck was appointed Professor in the University of Texas at Austin where she also holds the Sid W Richardson Foundation Regents Chair in Mathematics.

Uhlenbeck is a leading expert on partial differential equations and describes her mathematical interests as follows:-

I work on partial differential equations which were originally derived from the need to describe things like electromagnetism, but have undergone a century of change in which they are used in a much more technical fashion to look at the shapes of space. 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 dimensional manifold topology, learned gauge field theory and then some about applications to four dimensional manifolds, and have recently been working n equations with algebraic infinite symmetries.
Uhlenbeck's work provided analytic tools to use instantons as an effective geometric tool. In [1] Simon Donaldson reminisces about the work on the applications of instantons that led him to receive a Fields Medal in 1986. He describes the "bubbling" phenomenon saying:-
In fact the papers of Uhlenbeck which appeared about that time [in 1982] contained essentially all the analysis required to put this picture on a firm footing. The papers do not discuss "bubbling" explicitly - perhaps the arguments were supposed to be obvious to experts by analogy with the work of Sacks and Uhlenbeck in the harmonic maps case.
In 1988 Uhlenbeck lectured on Instantons and Their Relatives at the Centennial Celebration of the American Mathematical Society. Witten, who gave the next talk on Geometry and quantum field theory at the symposium said:-
In the talk just before mine, Karen Uhlenbeck described some purely mathematical developments that at least roughly might be classified in this area. She described advances in geometry that have been achieved through the study of systems of nonlinear partial differential equations. Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
Two years later, in 1990, Witten received a Fields Medal for his work on topological quantum field theories. At the same International Congress of Mathematicians in Kyoto, Karen Uhlenbeck was a Plenary Speaker.

Among the many honours that Uhlenbeck has received for her work one should mention in particular that she was elected a Member of the American Academy of Arts and Science in 1985 and a Member of the National Academy of Sciences the following year.

She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ).

Among the medals and prizes she has received we mention first the President's National Medal of Science which was awarded to her at an awards ceremony at the National Building Museum, Washington, DC, on Friday, 1 December 2000:-

For her many pioneering contributions to global geometry that resulted in advances in mathematical physics and the theory of partial differential equations. Her research accomplishments are matched by her leadership and passionate involvement in mathematics training and education.
She was awarded honorary doctorates by the University of Illinois, Champaign in 2000, by the University of Ohio in 2001, by the University of Michigan in 2004, and by Harvard University in 2007. On Saturday 6 January 2007 she received the American Mathematical Society's Leroy P Steele Prize at the Joint Mathematics Meeting in New Orleans, Louisiana. The award was made:-
... for her foundational contributions in analytic aspects of mathematical gauge theory in the papers "Removable singularities in Yang-Mills fields" (1982) and "Connections with bounds on curvature" (1982).
Her reply to the Award began:-
I thank the American Mathematical Society, its members and the Steele Prize committee for the honour and the award of the Steele Prize. This honour confirms what I have been suspecting for quite some time. I am becoming an old mathematician, if I am not already there. It gives me cause to look back at my research and teaching. All in all, I have found great delight and pleasure in the pursuit of mathematics. 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.
Later in her response she said:-
Starting from my days in Berkeley, the issue of women has never been far from my thoughts. I have undergone wide swings of feeling and opinion on the matter. I remain quite disappointed at the numbers of women doing mathematics and in leadership positions. This is, to my mind, primarily due to the culture of the mathematical community as well as harsh societal pressures from outside. Changing the culture is a momentous task in comparison to the other minor accomplishments I have mentioned.
In 2008 Uhlenbeck was elected an honorary member of the London Mathematical Society. The citation begins:-
Karen Uhlenbeck is a distinguished mathematician of the highest international stature, specialising in differential geometry, non-linear partial differential equations and mathematical physics. Professor Uhlenbeck is one of the United States' most eminent mathematicians, and perhaps the most distinguished woman mathematician of our time. At the same time, Uhlenbeck's efforts across the educational spectrum, especially her role as a founder of the Park City-IAS Mathematical Institute, have added vitality to the mathematical scene. Professor Uhlenbeck's mentoring is legendary, both formal (she co-founded the annual Women in Mathematics programme at the IAS) and informal, of women mathematicians.

Article by: J J O'Connor and E F Robertson

September 2009
MacTutor History of Mathematics