# Karen Uhlenbeck

Karen Uhlenbeck
Uhlenbeck in 1982
Born
Karen Keskulla

August 24, 1942 (age 76)
EducationUniversity of Michigan, Ann Arbor (BA)
New York University
Brandeis University (MA, PhD)
Known forCalculus of variations
Geometric analysis
Minimal surfaces
Yang-Mills theory
Spouse(s)Olke C. Uhlenbeck
AwardsMacArthur Fellowship
Noether Lecturer (1988)
National Medal of Science (2000)
Leroy P. Steele Prize (2007)
Abel Prize (2019)
Scientific career
FieldsMathematics
InstitutionsUniversity of Texas, Austin
University of Chicago
University of Illinois, Chicago
University of Illinois, Urbana-Champaign
InfluencesShing-Tung Yau

Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richardson Foundation Regents Chair.[1][2][3] She is currently a visiting associate at the Institute for Advanced Study and a visiting senior research scholar at Princeton University.[4]

Uhlenbeck 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."[5] She is the first woman to win the prize.[6]

## Life and career

Uhlenbeck's first family name, Keskulla, comes from Keskküla and from her grandfather who was Estonian.[7] Uhlenbeck received her B.A. (1964) from the University of Michigan.[1][3] She began her graduate studies at the Courant Institute of Mathematical Sciences at New York University, and married biophysicist Olke C. Uhlenbeck (the son of physicist George Uhlenbeck) in 1965. When her husband went to Harvard, she moved with him and restarted her studies at Brandeis University, where she earned an M.A. (1966) and Ph.D. (1968) under the supervision of Richard Palais.[1][3] Her doctoral dissertation was titled The Calculus of Variations and Global Analysis.[8]

After temporary jobs at the Massachusetts Institute of Technology and University of California, Berkeley, and having difficulty finding a permanent position with her husband because of the "anti-nepotism" rules then in place that prevented hiring both a husband and wife even in distinct departments of a university, she took a faculty position at the University of Illinois at Urbana–Champaign in 1971.[9] However, she disliked Urbana and moved to the University of Illinois at Chicago in 1976. She moved again to the University of Chicago in 1983, and to the University of Texas at Austin as the Sid W. Richardson Foundation Regents Chairholder in 1988.[1][2][3] Uhlenbeck is currently a professor emeritus at the University of Texas at Austin,[10] a visiting associate at the Institute for Advanced Study and a visiting senior research scholar at Princeton University.[4]

### Research

Uhlenbeck is one of the founders of the field of geometric analysis, a discipline that uses differential geometry to study the solutions to differential equations and vice versa.[11] She has also contributed to topological quantum field theory and integrable systems.[1][12]

Together with Jonathan Sacks in the early 1980s, Uhlenbeck established regularity estimates that have found applications to studies of the singularities of harmonic maps and the existence of smooth local solutions to the Yang–Mills–Higgs equations in gauge theory.[EMI][MIC][RSY] In particular, Donaldson describes their joint 1981 paper The existence of minimal immersions of 2-spheres[EMI] as a "landmark paper... which showed that, with a deeper analysis, variational arguments can still be used to give general existence results" for harmonic map equations.[13]

Building on these ideas, Uhlenbeck initiated a systematic study of the moduli theory of minimal surfaces in hyperbolic 3-manifolds (also called minimal submanifold theory) in her 1983 paper, Closed minimal surfaces in hyperbolic 3-manifolds.[14][CMS]

In particular, her work is described by Simon Donaldson in a survey of Yang–Mills geometry as foundational in the analytic aspects of the calculus of variations associated with the Yang–Mills functional.[15] A wider survey of her contributions to the field of calculus of variations was published by Simon Donaldson in the March 2019 issue of Notices of the American Mathematical Society; Donaldson describes the work of Uhlenbeck, along with Shing-Tung Yau, Richard Schoen and several others, as developing a

whole circle of ideas and techniques involving the dimension of singular sets, monotonicity, "small energy" results, tangent cones, etc. [that] has had a wide-ranging impact in many branches of differential geometry over the past few decades and forms the focus of much current research activity.[13]

### Outreach

In 1991, Uhlenbeck co-founded, with Herbert Clemens and Dan Freed, the Park City Mathematics Institute (PCMI) with the mission to "provide an immersive educational and professional development opportunity for several parallel communities from across the larger umbrella of the mathematics profession."[16][4] Uhlenbeck also co-founded the Women and Mathematics Program at the Institute for Advanced Study "with the mission to recruit and retain more women in mathematics."[17][4] British theoretical physicist and author Jim Al-Khalili describes Uhlenbeck as a "role model" for her work in promoting a career in mathematics to young people, particularly women.[18]

### Awards and honors

In March 2019, Uhlenbeck became the first woman to win the Abel Prize,[19] with the award committee citing the decision 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."[5] Hans Munthe-Kaas, who chairs the award committee, stated that "Her theories have revolutionised our understanding of minimal surfaces, such as more general minimisation problems in higher dimensions".[18] Uhlenbeck also won the National Medal of Science in 2000,[1][2][20][21] and the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society in 2007, "for her foundational contributions in analytic aspects of mathematical gauge theory",[1][2] based on her 1982 papers "Removable singularities in Yang–Mills fields"[RSY] and "Connections with bounds on curvature".[CLP]

She became a MacArthur Fellow in 1983[1][2] and a Fellow of the American Academy of Arts and Sciences in 1985.[1][2] She was elected as a member of the National Academy of Sciences in 1986.[1][2][3] She became a Guggenheim Fellow in 2001,[22] an honorary member of the London Mathematical Society in 2008,[1] and a Fellow of the American Mathematical Society in 2012.[23]

She was the Noether Lecturer of the Association for Women in Mathematics in 1988.[12] In 1990, she was a plenary speaker at the International Congress of Mathematicians, as only the second woman (after Emmy Noether) to give such a lecture.[1][2]

Her other awards include the University of Michigan alumna of the year (1984),[3] the Sigma Xi Common Wealth Award for Science and Technology (1995),[3] and honorary doctorates from the University of Illinois at Urbana–Champaign (2000),[1] Ohio State University (2001),[1][24] University of Michigan (2004),[1] Harvard University (2007),[1] and Princeton University (2012).[25]

## Selected publications

### Books

 I4M. Freed, Daniel S.; Uhlenbeck, Karen K. (1984). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. 1. Springer-Verlag, New York. doi:10.1007/978-1-4684-0258-2. ISBN 0-387-96036-8..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em} 2nd ed., 1991. Translated into Russian by Yu. P. Solovyev, Mir, 1988.[26]

### Research articles

 RNL. Uhlenbeck, Karen (1977). "Regularity for a class of non-linear elliptic systems" (PDF). Acta Mathematica. 138 (3–4): 219–240. doi:10.1007/bf02392316. MR 0474389.
 EMI. Sacks, Jonathan; Uhlenbeck, Karen (1981). "The existence of minimal immersions of 2-spheres" (PDF). Annals of Mathematics. Second Series. 113 (1): 1–24. doi:10.2307/1971131. MR 0604040.
 MIC. Sacks, J.; Uhlenbeck, K. (1982). "Minimal immersions of closed Riemann surfaces" (PDF). Transactions of the American Mathematical Society. 271 (2): 639–652. doi:10.2307/1998902. JSTOR 1998902. MR 0654854.
 RSY. Uhlenbeck, Karen K. (1982). "Removable singularities in Yang–Mills fields". Communications in Mathematical Physics. 83 (1): 11–29. doi:10.1007/bf01947068. MR 0648355. Announced in the Bulletin of the American Mathematical Society 1 (3): 579–581, MR0526970
 CLP. Uhlenbeck, Karen K. (1982). "Connections with ${\displaystyle L^{p}}$ bounds on curvature". Communications in Mathematical Physics. 83 (1): 31–42. doi:10.1007/bf01947069. MR 0648356.
 RHM. Schoen, Richard; Uhlenbeck, Karen (1982). "A regularity theory for harmonic maps". Journal of Differential Geometry. 17 (2): 307–335. doi:10.4310/jdg/1214436923. MR 0664498.
 CMS. Uhlenbeck, Karen K. (1983). "Closed minimal surfaces in hyperbolic 3-manifolds". In Bombieri, Enrico. Seminar on Minimal Submanifolds. Annals of Mathematics Studies. 103. Princeton University Press. pp. 147–168. JSTOR j.ctt1b7x7tv.10. MR 0795233.
 EHY. Uhlenbeck, Karen; Yau, Shing-Tung (1986). "On the existence of Hermitian-Yang-Mills connections in stable vector bundles". Communications on Pure and Applied Mathematics (Suppl.: Frontiers of the mathematical sciences, New York, 1985). 39: S257–S293. doi:10.1002/cpa.3160390714. MR 0861491.
 HML. Uhlenbeck, Karen (1989). "Harmonic maps into Lie groups: classical solutions of the chiral model". Journal of Differential Geometry. 30 (1): 1–50. MR 1001271.
 HMY. Uhlenbeck, Karen (1992). "On the connection between harmonic maps and the self-dual Yang-Mills and the sine-Gordon equations". Journal of Geometry and Physics. 8: 283–316. doi:10.1016/0393-0440(92)90053-4. MR 1165884.

## References

1. "Karen Uhlenbeck". Biographies of Women Mathematicians. Agnes Scott College.
2. Katterman, Lee (December 6, 1999). "Michigan Great Karen K. Uhlenbeck: Pioneer in mathematical analysis—and for women mathematicians". The University Record. University of Michigan. Retrieved December 19, 2014.
3. ^ a b c d Garrand, Danielle (March 19, 2019). "A woman just won the prize known as "math's Nobel" — for the first time ever". CBS News. Retrieved March 19, 2019.
4. ^ a b "Citation by the Abel Prize Committee". The Abel Prize. Retrieved March 19, 2019.
5. ^ Chang, Kenneth (March 19, 2019). "Karen Uhlenbeck Is First Woman to Receive Abel Prize in Mathematics – Dr. Uhlenbeck helped pioneer geometric analysis, developing techniques now commonly used by many mathematicians". The New York Times. Retrieved March 19, 2019.
6. ^ Allyn Jackson (2018). "Interview with Karen Uhlenbeck". Celebratio Mathematica.
7. ^
8. ^ Cooke, Roger (2005). The History of Mathematics: A Brief Course (2. ed.). Hoboken, NJ: Wiley-Interscience. p. 76. ISBN 978-0-471-44459-6.
9. ^ "Mathematics Emeritus Faculty". University of Texas at Austin. Retrieved March 19, 2019.
10. ^
11. ^ a b "Karen Uhlenbeck". Profiles of Women in Mathematics: The Emmy Noether Lectures. Association for Women in Mathematics. Retrieved December 19, 2014..
12. ^ a b Donaldson, Simon (2019). "Karen Uhlenbeck and the Calculus of Variations" (PDF). Notices of the American Mathematical Society. 66 (3): 303–313. doi:10.1090/noti1806. Retrieved 20 March 2019.
13. ^ Huang, Zheng; Wang, Biao (2017). "Closed Minimal Surfaces in Cusped Hyperbolic three-manifolds" (PDF). Geometriae Dedicata. 189 (1): 37–57. doi:10.1007/s10711-016-0215-8.
14. ^ Donaldson, Simon K. (2005). "Yang–Mills theory and Geometry" (PDF).
15. ^ "About PCMI: Mission, History, Summer Session". Institute for Advanced Study. Retrieved March 19, 2019.
16. ^ "Women and Mathematics". Institute for Advanced Study. Retrieved March 19, 2019.
17. ^ a b Pallab Ghosh (March 19, 2019). "Bubble maths researcher wins top award". BBC. Retrieved March 19, 2019.
18. ^ "US Mathematician Becomes First Woman To Win Prestigious Abel Prize". NDTV.com. Retrieved 2019-03-20.
19. ^ "The President's National Medal of Science: Recipient Details". National Science Foundation. Retrieved December 13, 2009.
20. ^ "UT Austin mathematics professor wins National Medal of Science". Univ. of Texas. November 13, 2000. Retrieved December 19, 2014..
21. ^ "Three UT Austin professors win prestigious Guggenheim Fellowships". Univ. of Texas. April 23, 2001. Retrieved December 19, 2014..
22. ^ "List of Fellows". American Mathematical Society. Retrieved August 28, 2013..
23. ^ "Honorary Degree – University Awards & Recognition – The Ohio State University". osu.edu. Retrieved March 19, 2019.
24. ^ "Princeton awards six honorary degrees". Princeton University. Retrieved March 8, 2019.
25. ^ Reviews of Instantons and Four-Manifolds: