Mathematician Karen Uhlenbeck, Bubbles and Bubbling

Karen Keskulla Uhlenbeck, linocut detail, 9.25" x 12.5", variable edition on Japanese kozo paper with chine collé with suminagashi marbling, by Ele Willoughby, 2021

My latest #sciart (or, maybe #mathart) portrait is of mathematician Karen Uhlenbeck. Though a lot of her work is pertinent to physics, this was still a challenge to work out how to tell the story of her mathematics. A founder of modern geometric analysis, she won the 2019 Abel Prize 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." Partial differential equations are pretty central to doing physics, and a sort of mathematics I really enjoy, but apart from that, I know what all the words mean in that previous sentence, but things like gauge theory and analysis aren't part of the math I've studied or used in my own research. As we move forward in time, depicting contemporary (or near contemporary) mathematics is increasingly challenging because it's beyond my ken.

I looked up several of her important papers, seeking geometrical diagrams as a way to tell the story of her research pictorially. Though she describes herself as a visual, geometric (and messy!) thinker, I didn't find anything but a lot of algebra (which I would have had to work hard to follow, let alone use to speak to a wide audience, or painstakingly carve in reverse!) But I found an "in" for depicting minimal surfaces and her methods for removing singularities, through Fields Medalist Sir Simon Donaldson's review of her work, including a schematic of "bubbling" and also some of the great science/math communication and articles about her and her work, which were published when she won the Abel Prize.

Born in Clevland to an engineer father Arnold Keskulla and teacher and artist mother Carolyn Windeler Keskulla, her family moved to New Jersey when she was a child. She spent her childhood enjoying nature, playing football and secretly reading science texts, imagining a job where she could avoid people. She started a degree in physics but switched to math at the University of Michigan. In the wake of the Soviet Union's launch of Sputnik the US Congress passed the National Defense Education Act, to encourage more participation in STEM including encouraging women and minorities in those fields. The university added an honours math course in response, which showed her there was more to math than she ever knew and it changed her life. “The structure, elegance and beauty of mathematics struck me immediately, and I lost my heart to it,” she wrote. She switched majors. She chose the Courant Institute of Mathematical Sciences at New York University, for graduate school, avoiding applying to more prestigious schools where she feared the competitive male math culture wouldn't suit her. She married biophysicist Olke C. Uhlenbeck, the son of European intellectuals; his father is well-known physicist George Uhlenbeck. She was quite influenced by her new in-laws. Her parents valued ideas but put an emphasis on earning an income, but her in-laws valued intellectual things and encouraged her pursuit of mathematics. Her husband moved to Harvard and she opted to switch to graduate school at Brandeis University, where she earned her PhD in 1968 with advisor Richard Palais. Palais was exploring the space between analysis (a generalization of calculus) and topology (the study of shapes) and Uhlenbeck enjoyed this in-between world. With Smale, Palais had made advances in the study of harmonic maps, which came out of the calculus of variations and the search for shapes in equilibrium with physical measurements. Uhlenbeck's research built on this.

She found temporary jobs at MIT and Berkeley but was hampered by the classic "two-body problem" for academic spouses (where an couple cannot find two professorships in the same place) and anti-nepotism rules - though she later noted that the institutions which claimed they were precluded from hiring both her and her husband, even in different departments, did not officially have any such anti-nepotism rules on the books. Schools like MIT, Stanford and Princeton which courted her husband, simply declined to hire a woman in math. Her husband generously declined to work at institutions where she could not.

Eventually they both got faculty positions at the University of Illinois at Urbana–Champaign in 1971. But she could not stand life in Urbana, where she was often dismissed as an "academic wife" rather than a professor in her own right. She lacked professional support, did not enjoy teaching, felt undervalued and struggled to find the right research topic for her. But, while there, she met post-doc Jonathan Sacks and they began working whether certain problems converge to harmonic maps; eventually they discovered a sort of "awkward point" or singularity called a bubble where systems cannot converge on a harmonic map. The mathematical concept of a minimal surface minimizes area (or surface tension or energy) the way a soap bubble does in the natural world. They showed there were a finite number of these singularities and that when you rescale a map near them you have "bubbling" as in the diagram in my print. This trick of rescaling was one of her and Sacks' most important contributions.

 She moved to the University of Illinois at Chicago in 1976, separating from her husband. In 1983, she moved to the University of Chicago, where there were other women math professors and peers who took her seriously. She received a Sloan Fellowship and finally knew she could support herself and her research. She learned the value of a professional support network and realized that isolating herself was hurting her and her career. Her work with Sacks was foundational to the development of the new mathematical field of geometrical analysis. This bubbling phenomenon was found in many different sorts of mathematical physics. She took her intuition and techniques honed working on minimal surfaces and began ground-breaking work in gauge theory and the nature of exotic n-spaces, a means of connecting observations at different places that comes up in the quantum mechanical world of fundamental forces, building on the work of mathematical physicists like Weyl, Yang and Mills. With C.H. Taubes, she analyzed Yang-Mills equations in four dimensions; she proved her removable singularities theorem, showing that bubbling cannot occur around isolated points, so if a finite-energy solution exists to the Yang-Mills equations near a point, the solution will extend smoothly to the point. Simon Donaldson's extension of this research won him the prestigious Fields Medal for mathematics in 1986. Uhlenberck won a MacArthur Prize Fellowship or "genius grant" (1983-1988) which made her feel, "obliged to become more ambitious.”

She married mathematician Robert F. Williams, then in 1988, moved to the University of Austin as the Sid W. Richardson Foundation Regents Chairholder and won the Noether Lecture award from the Association for Women in Mathematics. In 1990, she was only the second woman in math to give the plenary lecture at the International Congress of Mathematicians in Kyoto, after none other than mathematical giant Emmy Noether. She founded the Park City Mathematics institute and with Chuu-Lian Terng she co-founded the IAS Women and Mathematics (WAM) program to recruit and retain more women in math, encouraging collaboration and mentorship at all stages of their careers, a complete turn-around from the isolation she sought at the beginning of her career and her avoidance of her fellow women in math because “It was self-evident that you wouldn't get ahead in mathematics if you hung around with women.” Together, they've created a network of nearly 1500 women mathematicians. She realized that one needs role models - not examples of perfection, but people who make errors but pick themselves back up. When asked if she had a role model, she cited beloved chef and TV presenter Julia Child! "She had these fantastic television programs, and she was a real person. She could pick the turkey up off the floor and serve it," Uhlenbeck said. She founded the Distinguished Women in Mathematics Lecture Series at the University of Texas at Austin. Uhlenbeck won the National Medal of Science in 2001 and the 2007 Steele Prize for a Seminal Contribution to Mathematical Research from the American Mathematical Society. Granted yearly by the King of Norway, rather like a math equivalent to the Nobel, the Abel Prize includes a $700,000 award, and recognizes a mathematician’s entire career. Uhlenbeck became the first (and so far only) woman to win the Abel Prize in 2019.  She donated half the funds to organizations encouraging underrepresented (that is, traditionally excluded) minorities in mathematics. She further donated to environmental organizations and the National Academy of Science. She is now an emeritus professor at Austin, as well as a visiting associate at the Institute for Advanced Study and senior research scholar at Princeton University.

Like so many of my subjects, I've found she also loves art and began drawing more than a decade ago, mostly landscapes. She notes, “I discovered the fascinating fact that the problem of scale occurs both in mathematics and in drawing.” In drawing you want to capture the full forest as well as the details of individual plants, just as you want to capture all scales in math. “The hardest part with both is fitting the two scales together. You need the right tools.” Uhlenbeck has provided the world of mathematics and physics with so many of just the right tools.

References

Donaldson, S.. “Karen Uhlenbeck and the Calculus of Variations.” Notices of the American Mathematical Society 66 (2019): 1. DOI:10.1090/noti1806

Sioban Roberts, "In Bubbles, Karen Uhlenbeck Sees a Mathematical Universe," The New York Times, April 8, 2019

Karen Uhlenbeck, Wikipedia, accessed July 2021.

Rachel Crowell, "Karen Uhlenbeck Becomes First Woman To Win Abel Prize For Mathematics," Forbes, March 20, 2019

Dale DeBakcsy, How Mathematical Lone Wolf Karen Uhlenbeck Found Her Pack, Women You Should Know, November 29, 2017.

Karen Uhlenbeck, Karen Uhlenbeck on Being the First Woman to Receive the Abel Prize, remarks made at reception at the Institute for Advanced Study on March 19, 2019.

Karen Uhlenbeck, Coming to grips with success: a profile of Karen Uhlenbeck, Celebratio Mathematica, 1996

Jessica Atlee, Crossing Fields: Karen Uhlenbeck’s pioneering work marries math with physics, Symmetry Magazine, August 15, 2019.

Erica Karreich, Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize, Quanta Magazine, March 19, 2019

Marianne Freiberger, The Abel Prize 2019, Plus magazine, University of Cambridge, March 19, 2019.