Karen Uhlenbeck: Geometry, Physics, and the Abel Prize

A Curious Child in New Jersey

Karen Keskulla Uhlenbeck was born on 24 August 1942 in Cleveland, Ohio. Her father, Arnold Edward Keskulla, was an engineer, and her mother, Carolyn Windeler Keskulla, was an artist. The family moved to New Jersey, where Karen grew up with a love of the outdoors and a voracious appetite for reading.

As a child, Uhlenbeck was drawn to science and mathematics, but she has often said that she initially imagined herself becoming a researcher in physics or biology rather than mathematics. She was an avid reader who devoured every book she could find.

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Education

Uhlenbeck entered the University of Michigan as an undergraduate, intending to study physics. She was drawn to mathematics by its elegance and precision, and she graduated with a bachelor's degree in mathematics in 1964.

She began graduate school at the Courant Institute of Mathematical Sciences at New York University but transferred to Brandeis University, where she completed her PhD in 1968 under Richard Palais. Her thesis was on the calculus of variations — specifically, on variational problems in Riemannian geometry.

Finding a permanent position proved difficult. As one of very few women in mathematics in the late 1960s, Uhlenbeck faced significant institutional barriers. She has spoken openly about the challenges:

After positions at MIT, the University of California at Berkeley, and the University of Illinois at Urbana-Champaign, she moved to the University of Chicago in 1983 and then to the University of Texas at Austin in 1988, where she held the Sid W. Richardson Foundation Regents Chair in Mathematics.

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Geometric Analysis and the Calculus of Variations

Uhlenbeck's mathematical work lies at the intersection of geometry, analysis, and mathematical physics. She is one of the founders of geometric analysis — the use of partial differential equations and variational methods to study geometric problems.

Minimal Surfaces and Harmonic Maps

Uhlenbeck made fundamental contributions to the study of harmonic maps — maps $u: (M, g) \to (N, h)$ between Riemannian manifolds that are critical points of the energy functional:

$E(u) = \frac{1}{2} \int_M |du|^2 \, dV_g$

The Euler–Lagrange equation for this functional is the harmonic map equation:

$\tau(u) = \operatorname{trace}_g \nabla du = 0$

where $\tau(u)$ is the tension field. Harmonic maps generalize geodesics, harmonic functions, and minimal surfaces.

In a series of influential papers in the 1980s, Uhlenbeck (together with Jonathan Sacks) studied harmonic maps from surfaces. They proved:

This bubbling phenomenon — the formation of "bubbles" of concentrated energy — became a central concept in geometric analysis and has analogues throughout PDE theory and mathematical physics.

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Gauge Theory: The Yang–Mills Equations

Uhlenbeck's most celebrated work concerns gauge theory, the mathematical framework underlying the Standard Model of particle physics. A gauge field is a connection on a principal bundle, and the Yang–Mills equations are the Euler–Lagrange equations for the Yang–Mills energy:

$\mathcal{YM}(A) = \int_M |F_A|^2 \, dV_g$

where $A$ is a connection on a principal $G$-bundle over a Riemannian manifold $M$, and $F_A$ is its curvature. The Yang–Mills equations are:

$d_A^* F_A = 0$

Uhlenbeck Compactness

In her groundbreaking 1982 papers, Uhlenbeck proved two fundamental results on the analysis of Yang–Mills connections:

The key technical tool is Uhlenbeck's gauge-fixing lemma (also called the Coulomb gauge theorem): on a sufficiently small ball, any connection with small enough curvature can be put into Coulomb gauge ($d^* A = 0$) by a gauge transformation. In Coulomb gauge, the Yang–Mills equations become an elliptic system, and standard PDE techniques apply.

These results were essential for Simon Donaldson's revolutionary work on the topology of 4-manifolds, for which he received the Fields Medal in 1986. Donaldson has repeatedly acknowledged that his work would not have been possible without Uhlenbeck's foundational analysis.

Removability of Singularities

Uhlenbeck proved that isolated singularities of Yang–Mills connections in dimension 4 are removable: if $A$ is a finite-energy Yang–Mills connection on $B^4 \setminus \{0\}$, then $A$ extends to a smooth connection on all of $B^4$. This is analogous to classical removable singularity theorems in complex analysis.

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Self-Dual and Anti-Self-Dual Connections

In dimension 4, the curvature $F_A$ decomposes into self-dual and anti-self-dual parts:

$F_A = F_A^+ + F_A^-$

Instantons are connections with $F_A^+ = 0$ (anti-self-dual) or $F_A^- = 0$ (self-dual). They are absolute minimizers of the Yang–Mills energy within their topological class, since:

$\mathcal{YM}(A) = \|F_A^+\|^2 + \|F_A^-\|^2 \geq \left| \|F_A^+\|^2 - \|F_A^-\|^2 \right| = 8\pi^2 |k|$

where $k$ is the second Chern number. Uhlenbeck's compactness and removability theorems provided the analytical foundation for the moduli theory of instantons.

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Integrable Systems and Loop Groups

In later work, Uhlenbeck studied connections between gauge theory and integrable systems. In her influential 1989 paper "Harmonic maps into Lie groups," she showed that harmonic maps from surfaces into Lie groups can be studied using loop group techniques, connecting gauge theory to the theory of completely integrable systems.

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The Abel Prize

In 2019, Karen Uhlenbeck was awarded the Abel Prize — the first woman to receive this honor. The citation recognized 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."

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Mentorship and Advocacy

Throughout her career, Uhlenbeck has been a passionate advocate for women and underrepresented groups in mathematics. She co-founded the Park City Mathematics Institute (PCMI) and the Women and Mathematics program at the Institute for Advanced Study in Princeton, where she currently holds a visiting position.

She has mentored numerous students and junior mathematicians and has spoken publicly about the challenges of being a woman in mathematics:

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Awards and Honors

• MacArthur Fellowship (1983)
• National Medal of Science (2000)
• Leroy P. Steele Prize for Seminal Contribution to Research (2007) — from the AMS
• Abel Prize (2019)
• Member of the National Academy of Sciences and the American Philosophical Society

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Legacy

Karen Uhlenbeck's work fundamentally shaped geometric analysis and mathematical gauge theory. Her compactness and regularity theorems for Yang–Mills connections provided the analytical backbone for an entire field — from Donaldson's invariants of 4-manifolds to the Seiberg–Witten theory and beyond. Her contributions to harmonic maps and the calculus of variations continue to influence active research.

As a pioneer for women in mathematics, her impact extends well beyond her theorems.

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References

• Uhlenbeck, K., "Removable singularities in Yang–Mills fields," Communications in Mathematical Physics, 83(1), 1982.
• Uhlenbeck, K., "Connections with $L^p$ bounds on curvature," Communications in Mathematical Physics, 83(1), 1982.
• Sacks, J. and Uhlenbeck, K., "The existence of minimal immersions of 2-spheres," Annals of Mathematics, 113(1), 1981.
• Uhlenbeck, K., "Harmonic maps into Lie groups: classical solutions of the chiral model," Journal of Differential Geometry, 30(1), 1989.
• Wikipedia — Karen Uhlenbeck
• Abel Prize 2019 — Citation
• Institute for Advanced Study — Karen Uhlenbeck
• University of Texas — Karen Uhlenbeck