Karen Uhlenbeck

Karen Uhlenbeck is an American mathematician whose work on geometric analysis and gauge theory shaped modern differential geometry. She received the Abel Prize in 2019.

Geometric analysis: The use of analytic methods, especially partial differential equations, to study geometric objects. This field frames Uhlenbeck’s research on geometric structures governed by nonlinear equations.
Uhlenbeck compactness theorem: A compactness result for sequences of connections with bounded curvature energy, after suitable gauge transformations. This theorem supplies a central method for extracting limits from bounded-energy gauge fields.
Abel Prize: An international mathematics prize awarded annually by the Norwegian Academy of Science and Letters. Uhlenbeck received the 2019 prize for her pioneering contributions to geometric partial differential equations, gauge theory, and integrable systems.
Moduli space: A space whose points represent equivalence classes of mathematical objects with specified structure. Gauge-theoretic compactness results help analyze spaces of solutions and their limiting configurations.
Gauge theory: A framework for describing physical fields through symmetries that vary across space and time. Uhlenbeck developed analytic foundations for gauge theories used in geometry and mathematical physics.
Coulomb gauge: A gauge condition in which a connection satisfies a divergence-free equation. Uhlenbeck’s gauge-fixing theorem places connections in this condition under suitable smallness assumptions.
Robert F. Kennedy Visiting Professor: A visiting professorship at Harvard University focused on distinguished scholars and public engagement. Uhlenbeck held this position during her career at the Institute for Advanced Study.
Instanton: A finite-action solution of the Yang–Mills equations in four-dimensional Euclidean space. Instantons are a key setting for the gauge fields and concentration phenomena studied by Uhlenbeck.
Differential geometry: The study of smooth spaces using calculus, including their curvature and geometric structures. Her contributions advanced the analysis of geometric structures on smooth manifolds.
Gauge fixing: The choice of a representative from among mathematically equivalent descriptions related by gauge transformations. Her analytic results show when a well-controlled representative can be chosen.

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