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The Abel Prize Laureate 2019 Brochure The Abel Prize Laureate 2019 Karen Keskulla Uhlenbeck University of Texas at Austin www.abelprize.no Karen Keskulla Uhlenbeck receives 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. Citation The Norwegian Academy of Science and critical points of functionals representing Letters has decided to award the Abel geometric quantities such as energy and Prize for 2019 to volume. For example, minimal surfaces are critical points of the area and harmonic Karen Keskulla Uhlenbeck maps are critical points of the Dirichlet University of Texas at Austin energy. Uhlenbeck’s major contributions include foundational results on minimal for her pioneering achievements in surfaces and harmonic maps, Yang-Mills geometric partial differential equations, theory, and integrable systems. gauge theory and integrable systems, and for the fundamental impact of her work Minimal surfaces and bubbling on analysis, geometry and mathematical analysis physics. An important tool in global analysis, preceding the work of Uhlenbeck, is the Palais—Smale compactness condition. Karen Keskulla Uhlenbeck is a founder This condition, inspired by earlier work of modern Geometric Analysis. Her of Morse, guarantees existence of perspective has permeated the field minimisers of geometric functionals and and led to some of the most dramatic is successful in the case of 1-dimensional advances in mathematics in the last domains, such as closed geodesics. 40 years. Uhlenbeck realised that the condition Geometric analysis is a field of of Palais—Smale fails in the case of mathematics where techniques of surfaces due to topological reasons. The analysis and differential equations are papers of Uhlenbeck, co-authored with interwoven with the study of geometrical Sacks, on the energy functional for maps and topological problems. Specifically, of surfaces into a Riemannian manifold, one studies objects such as curves, have been extremely influential and surfaces, connections and fields which are describe in detail what happens when © In 1987 Karen K. Uhlenbeck moved to the University of Texas at Austin to take up the Sid W. Richardson Foundation Regents’ Chair in mathematics. She would remain at the University of Texas until the end of her working career. Currently, Uhlenbeck is a Visiting Senior Research Scholar at Princeton University as well as a Visiting Associate at the Institute for Advanced Study (IAS). PHOTO: Marsha Miller 5 the Palais-Smale condition is violated. of all subsequent research in the area of A minimising sequence of mappings gauge theory. converges outside a finite set of singular Gauge theory involves an auxiliary points and by using rescaling arguments, vector bundle over a Riemannian they describe the behaviour near the manifold. singularities as bubbles or instantons, The basic objects of study are which are the standard solutions of the connections on this vector bundle. After minimising map from the 2-sphere to the a choice of a trivialisation (gauge), a target manifold. connection can be described by a matrix In higher dimensions, Uhlenbeck valued 1-form. Yang-Mills connections in collaboration with Schoen wrote are critical points of gauge-invariant two foundational papers on minimising functionals. Uhlenbeck addressed and harmonic maps. They gave a profound solved the fundamental question of understanding of singularities of solutions expressing Yang-Mills equations as of non-linear elliptic partial differential an elliptic system, using the so-called equations. The singular set, which in the Coulomb gauge. This was the starting case of surfaces consists only of isolated point for both Uhlenbeck’s celebrated points, is in higher dimensions replaced by compactness theorem for connections a set of co-dimension 3. with curvature bounded in Lp, and for her The methods used in these later results on removable singularities revolutionary papers are now in the for Yang-Mills equations defined on standard toolbox of every geometer punctured 4-dimensional balls. The and analyst. They have been applied removable singularity theory for Yang- with great success in many other partial Mills equations in higher dimensions was differential equations and geometric carried out much later by Gang Tian and contexts. In particular, the bubbling Terence Tao. Uhlenbeck’s compactness phenomenon appears in many works in theorem was crucial in Non-Abelian partial differential equations, in the study Hodge Theory and, in particular, in the of the Yamabe problem, in Gromov’s work proof of the properness of Hitchin’s on pseudoholomorphic curves, and also map and Corlette’s important result on in physical applications of instantons, the existence of equivariant harmonic especially in string theory. mappings. Another major result of Uhlenbeck is Gauge theory and Yang-Mills her joint work with Yau on the existence equations of Hermitian Yang-Mills connections on After hearing a talk by Atiyah in Chicago, stable holomorphic vector bundles over Uhlenbeck became interested in gauge complex n-manifolds, generalising an theory. She pioneered the study of Yang- earlier result of Donaldson on complex Mills equations from a rigorous analytical surfaces. This result of Donaldson- point of view. Her work formed a base Uhlenbeck-Yau links developments 6 in differential geometry and algebraic families. Based on this observation, geometry, and is a foundational result for Uhlenbeck described algebraically applications of heterotic strings to particle harmonic mappings from spheres into physics. Grasmannians relating them to an infinite Uhlenbeck’s ideas laid the analytic dimensional integrable system and foundations for the application of gauge Virasoro actions. This seminal work led to theory to geometry and topology, to the a series of further foundational papers by important work of Taubes on the gluing Uhlenbeck and Chuu-Lian Terng on the of self-dual 4-manifolds, to the ground- subject and the creation of an active and breaking work of Donaldson on gauge fruitful school. theory and 4-dimensional topology, and The impact of Uhlenbeck’s pivotal many other works in this area. The book work goes beyond geometric analysis. written by Uhlenbeck and Dan Freed on A highly influential early article was “Instantons and 4-Manifold Topology” devoted to the study of regularity theory instructed and inspired a generation of of a system of non-linear elliptic differential geometers. She continued to equations, relevant to the study of the work in this area, and in particular had an critical map of higher order energy important result with Lesley Sibner and functionals between Riemannian Robert Sibner on non self-dual solutions manifolds. This work extends previous to the Yang-Mills equations. results by Nash, De Giorgi and Moser on regularity of solutions of single non- Integrable systems and harmonic linear equations to solutions of systems. mappings Karen Uhlenbeck’s pioneering The study of integrable systems has results have had fundamental impact on its roots in 19th century classical contemporary analysis, geometry and mechanics. Using the language of mathematical physics, and her ideas gauge theory, Uhlenbeck and Hitchin and leadership have transformed the realised that harmonic mappings from mathematical landscape as a whole. surfaces to homogeneous spaces come in 1-dimensional parametrised A biography of Karen Keskulla Uhlenbeck By Professor Jim Al-Khalili, Fellow of the Royal Society, University of Surrey In 1990, in Kyoto, Japan, Karen Uhlenbeck Ohio in 1942. Her father, Arnold Keskulla, became only the second woman to give was an engineer and her mother, Carolyn a Plenary Lecture at the International Windeler Keskulla, an artist and school Congress of Mathematicians – ICM – the teacher. The family moved to New Jersey largest and most important gathering of when Karen was in third grade. As a young mathematicians in the world. It is held girl, she was curious about everything. Her every four years, and the first woman parents instilled in her a love of art and to do this was Emmy Noether in 1932. music, and she developed a lifelong love Such a shocking statistic reflects just of the outdoors, regularly roaming how hard it is for many women to achieve the local countryside near her home. the recognition they deserve in a male- Most of all, she loved reading, dominated field. shutting herself away whenever she But by that point in her career, could to devour advanced science Uhlenbeck had already established books, staying up late at night and even herself as one of the world’s preeminent reading secretly in class. She dreamed of mathematicians, having overcome becoming a research scientist, particularly many hurdles, both personally and if it meant avoiding too much interaction professionally. In 2000, she received the with other people; not that she was a shy US National Medal of Science. Yet for child, but rather because she enjoyed the many, the recognition of her achievements peace and solitude of her own company. should have been far greater, for her work The last thing she wanted to do was has led to some of the most important to follow in her mother’s footsteps and advances in mathematics in the last end up teaching – an attitude that would 40 years. change dramatically later in life. Karen Keskulla Uhlenbeck, the eldest Uhlenbeck’s love affair with mathe- of four children, was born in Cleveland, matics developed only after she had 8 © Karen Uhlenbeck giving a talk at the Institute for Advanced Study. PHOTO: Andrea Kane 9 started at university. Having been inspired where she studied general relativity and in high school by the writings of great the geometry of space-time – topics physicists such as Fred Hoyle and George that would shape her future research Gamow, she enrolled at the University work. Although a pure mathematician, of Michigan, initially planning to major in Uhlenbeck has drawn inspiration for her physics. However, she soon discovered work from theoretical physics and, in that the intellectual challenge of pure return, she has had a major influence in mathematics was what really excited her.
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