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Home , Abel Prize, Dan Freed, Karen Uhlenbeck, Lennart Carleson, Lesley Sibner, Louis Nirenberg

The 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, and integrable systems, and for the fundamental impact of her work on analysis, and mathematical .

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 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 . Her of Morse, guarantees existence of perspective has permeated the minimisers of geometric functionals and and led to some of the most dramatic is successful in the case of 1-dimensional advances in 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 , 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 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 , 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 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 . Uhlenbeck’s compactness phenomenon appears in many works in theorem was crucial in Non-Abelian partial differential equations, in the study and, in particular, in the of the , 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 . 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 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 and foundations for the application of gauge Virasoro actions. This seminal work led to theory to geometry and , 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 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 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 ’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 , 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 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 – ICM – the teacher. The family moved to 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 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 , 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. shaping it by developing ideas with It also meant she didn’t have to do any lab a wide range of different applications. work, which she disliked. For example, physicists had predicted Graduating in 1964, she married her the existence of mathematical objects biophysicist boyfriend Olke Uhlenbeck called instantons, which describe the a year later and decided to embark on behaviour of surfaces in four-dimensional postgraduate study. Already well aware space-time. Uhlenbeck became one of the predominantly male and often of the world’s leading experts in this misogynistic culture in academia, she field. The classic textbook, Instantons avoided applying to prestigious schools and 4-Manifolds, which she co-wrote in such as Harvard, where Olke was heading 1984 with Dan Freed, inspired a whole for his PhD and where competition to generation of mathematicians. succeed was likely to be fierce. Instead, In 1971, she became an assistant she enrolled at professor at the University of Illinois at where she received a generous graduate Urbana-Champaign where she felt isolated fellowship from the National Science and undervalued. So, five years later she Foundation. There, she completed her left for the University of Illinois at Chicago. PhD in mathematics, working on the Here, there were other female professors, ; a technique that who offered advice and support, as well involves the study of how small changes as other mathematicians who took her in one quantity can help us find the work more seriously. In 1983, she took maximum or minimum value of another up a full professorship at the University quantity – like finding the shortest of Chicago, establishing herself as one distance between two points. You might of the preeminent mathematicians of think this would be a straight line, but her generation. Her interests included it is not always so straightforward. For nonlinear partial differential equations, example, if you have to drive through differential geometry, gauge theory, a busy city, the quickest route is not topological quantum field theory and necessarily the shortest. Needless, to say, integrable systems. In 1987, she moved Uhlenbeck’s contribution to the field was to the University of Texas at Austin to take somewhat more complicated than this! up the Sid W. Richardson Foundation After a brief teaching period at Regents’ Chair in mathematics. There, she MIT, she moved to Berkeley, California, broadened her understanding of physics

10 by studying with winning science. She has come a long way from physicist . She would the young girl who wished to be alone. remain at the University of Texas until the For a while, she struggled to come to end of her working career. terms with her own success, but now Uhlenbeck’s most noted work says she appreciates it as a privilege. focused on gauge theories. Her papers She has stated that she is aware of analysed the Yang-Mills equations in four being a role model, for young female dimensions, laying some of the analytical mathematicians in particular, but that groundwork for many of the most exciting “it’s hard, because what you really need ideas in modern physics, from the to do is show students how imperfect Standard Model of particle physics to the people can be and still succeed. Everyone search for a theory of quantum gravity. knows that if people are smart, funny, Her papers also inspired mathematicians pretty, or well-dressed they will succeed. Cliff Taubes and , paving But it’s also possible to succeed with the way for the work that won Donaldson all of your imperfections. I may be a the in 1986. wonderful mathematician and famous Uhlenbeck, now back in New Jersey, because of it, but I’m also very human.” remains a staunch advocate for greater Karen Uhlenbeck is certainly a remarkable gender diversity in mathematics and in human.

Karen K. Uhlenbeck will meet schoolchildren for mathematical games in Archimedes’ Labyrinth in the Botanical Garden at the during the 2019 Abel Week. The labyrinth is designed by Hans Munthe-Kaas, chair of the Abel Committee. Photo: Avinor/Daniel Volle - Pandora Film CC BY-SA 3.0

12 A glimpse of the Laureate’s work “I’m Forever Blowing Bubbles”

Arne B. Sletsjøe, ass. professor, Department of Mathematics,

I’m forever blowing bubbles, Soap bubbles are beautiful objects, Pretty bubbles in the air. perfectly shaped and with a marvellous They fly so high, play of colours, due to interference of Nearly reach the sky, light reflecting off the front and back Then like my dreams, surfaces of the soap film. Soap bubbles They fade and die. are beautiful objects in a mathematical setting as well, as they constitute Jaan Kenbrovin examples of minimal surfaces. When the enclosed volume of air inside the bubble is fixed, the soap film will minimize the wall tension, pulling the bubble into the shape of the least surface enclosing a fixed volume, known for centuries to be a perfect sphere. If we instead of blowing the bubble, dip a heavily deformed wire loop into a soap bubble solution, the soap film will form a disc with its boundary given by the wire loop and of minimal area. Unlike the sphere-shaped bubble, this film has equal pressure on each side, hence it is a surface with zero mean curvature, i.e. the average curvature along all directions is zero. Even if the soap film almost instantly is able to form a , computing the shape of the surface analytically is a rather complicated task.

13 Among curves connecting two points of minimal surfaces, Uhlenbeck wanted to in space, we can always find a shortest understand what happens when Condition path. The analogous statement is not true C is violated. In a paper co-authored for surfaces when considering their area. with Jonathan Sacks, they describe in The problem is that in order to reduce the detail the situation where you cannot area of a surface, a consequence could rely on the conclusion of Condition C. be that the surface is shrunk to a curve, They construct a sequence of mappings which of course does not count as a of a sphere into the target space which minimal surface. An example of this is the satisfies Condition C, but in such a way minimal tubular surface connecting two that their limit does not. Outside of a finite parallel circles. If the distance between the set of singular points everything works circles is small compared to their radius, well, but near the singularities the so- the minimal surface looks like a slightly called bubbling phenomenon appears. concave cylinder. When pulling the circles The area of the limit surface is strictly apart the cylinder will shrink in the region less than the limit of the areas of the between the circles, forming a surface surfaces in the sequence. The difference known as a catenoid. At a certain point, is concentrated in a finite set of isolated the middle part of the curved cylinder will points, being the limit of “bubbles” in collapse along the line connecting the the sequence of surfaces. The idea and centres of the two parallel circles. When the methods of this revolutionary paper pulling the circles further apart there is no has since it was published become a tubular minimal surface connecting them. successful mathematical tool. In particular, the bubbling phenomenon has had Mapping spaces great influence as a method for solving In 1968 Karen Uhlenbeck received her problems in various parts of mathematics. Ph.D. from Brandies with the thesis “The Calculus of Variations and Global Footprints of gauge Analysis”. Her supervisor was Richard Karen Uhlenbeck also left her footprints Palais, who a few years earlier and in the field calledgauge theory. Gauge together with , had theory is a mathematical theory introduced the so-called Palais-Smale introduced by Hermann Weyl in 1918, Condition C. This condition gives a which originated in theoretical physics and criterion for the existence of minimizers Einstein´s theory of general relativity. A for functionals on mapping spaces. key idea in Einstein´s work is that laws of “Minimizers for functionals on mapping physics should be the same in all frames spaces” is a more general phrase than of reference. This is also the general idea “find the surface of least area”, but of a gauge theory, to find connections that Condition C can also be applied to the compare measurements taken at different minimal surface problem, but then it fails. points in a space and look for quantities Motivated by the general non-existence that do not change. The physical

14 interpretation was brought further by Yang which both originate from a wish to and Mills in the fifties, in what is now understand nature. When mathematicians called the Yang-Mills equations. To reveal get interested in such problems, the theory the secrets of theoretical physics you have propagates into theoretical constructions to work in a (at least) four-dimensional far beyond the tangible objects of nature. space, three spatial coordinates and one But even if the mathematical theory seems time-coordinate. A physical law should to be soaring, scientists often benefit from be the same wherever you are located in the generalized theory. Karen Uhlenbecks space-time, i.e. independent of choice of mathematical achievements constitute frame of reference. important examples of such processes.

Minimal surfaces (I’m Forever Blowing Bubbles is an Karen Uhlenbeck attacked this problem American song from the Broadway from the mathematical point of view. musical The Passing Show of 1918. It She pioneered the study of Yang-Mills was released in 1918, the same year as equations in a rigorous analytical way. Hermann Weyl introduced the notion Her work formed a base of all subsequent of a gauge. In addition to the obvious research in the area of gauge theory. Her connection to minimal surfaces, the analysis of the Yang-Mills equations in four lyrics may have some associations to dimensions together with C. H. Taubes, mathematical research in general. The also laid the ground for the theories of song is also adapted as the club anthem Simon Donaldson, who later was awarded of West Ham United, a London-based the Fields Medal in 1986 for his work on football club.) the topology of four-manifolds. Minimal surfaces and gauge theory are two separate fields of mathematics

15 Karl Johans gate, the main street of Oslo during the Abel week. photo: Thomas Brun/NTB

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About the Abel Prize

The Abel Prize is an international award Developing Countries, and the Bernt for outstanding scientific work in the field Michael Holmboe Memorial Prize for of mathematics, including mathematical excellence in teaching mathematics in aspects of computer science, mathe­ . In addition, national mathematical matical physics, probability, numerical contests, and various other projects and analysis, scientific computing, statistics, activities are supported in order to and also applications of mathe­matics in stimulate interest in mathematics the sciences. among children and youth. The Abel Prize has been awarded At the Heidelberg Laureate Forum since 2003 by the Norwegian Academy in Germany young mathematicians get of Science and Letters. The choice of the opportunity to meet winners of the laureates is based on the recommen­ Abel Prize. dations from the Abel Committee. The prize carries a cash award of 6 million Call for nominations 2020 NOK (about 650,000 Euro or about The Norwegian Academy of Science and 730,000 USD). Letters hereby calls for nominations for The prize is named after the the Abel Prize 2020, and invites you (or exceptional Norwegian mathematician your society or institution) to nominate (1802–1829). According candidate(s). Nominations are confidential to the statutes of the Abel Prize the and a nomination should not be made objective is both to award the annual Abel known to the nominee. Prize, and to contribute towards raising the status of mathematics in society and Deadline for nominations for the Abel Prize stimulating the interest of children and 2020 is September 15, 2019. young people in mathematics. Among initiatives supported are the Please consult www.abelprize.no for Abel Symposium, the International more information. Mathematical Union’s Commission for

18 © DNVA / Snøhetta

19 The laureate wall in The Abel-room at the Norwegian Academy of Science and Letters. photo: Eirik Furu Baardsen

2018 Robert P. Langlands “for his visionary program connecting to ” 2017 “for his pivotal role in the development of the mathematical theory of wavelets.” 2016 Sir Andrew J. Wiles “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.” 2015 John Forbes Nash, Jr. and “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.” 2014 Yakov G. Sinai “for his fundamental contributions to dynamical systems, , and mathematical physics.” 2013 “for seminal contributions to and for their transformative impact on number theory, representation theory, and related fields.” 2012 Endre Szemerédi “for his fundamental contributions to dis­crete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.” 2011 “for pioneering discoveries in topology, geometry and .”

2010 John Torrence Tate “for his vast and lasting impact on the theory of numbers.”

2009 Mikhail Leonidovich Gromov “for his revolutionary contributions to geometry.” 2008 John Griggs Thompson and “for their profound achievements in algebra and in particular for shaping modern .” 2007 Srinivasa S. R. Varadhan “for his fundamental contributions to and in particular for creating a unified theory of large deviations.” 2006 “for his profound and seminal contributions to and the theory of smooth dynamical systems.” 2005 Peter D. Lax “for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.” 2004 Sir Michael Francis Atiyah and Isadore M. Singer “for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstand­ing role in building new bridges between mathematics and theoretical physics.” 2003 Jean-Pierre Serre “for playing a key role in shaping the modern form of many parts of mathemat­ics, including topology, algebraic geometry and number theory.”

22 After hearing a talk by 2004 Abel laureate, the late Sir , Karen Uhlenbeck became interested in gauge theory. photo: Didier Vandenbosch

23 Abel Banquet at Akershus Castle Programme Iselin Nybø, Minister of Research and Higher Education, hosts the Norwegian government’s Abel Week 2019 banquet at Akershus Castle in honour of the Abel Laureate (by invitation only) May 20 front

— May 22 page Holmboe Prize Award Ceremony

— photo Jan Tore Sanner, Minister of Education The Abel Lectures

and Integration, presents the Bernt Michael : © Marsha Miller The Laureate will give the Abel Prize lecture Holmboe Memorial Prize for teachers at the University of Oslo, Georg Sverdrups Hus, of mathematics at Oslo Cathedral School Aud. 1. This will be followed by other lectures with topics related to the prize winner’s work Wreath-laying at the Abel Monument by the Abel Prize Laureate in the Palace Park The Abel Party at the Norwegian Academy of Science Dinner in honour of the Abel Laureate and Letters (by invitation only) at the Norwegian Academy of Science and Letters (by invitation only) May 23 — May 21 Abel Prize Lecture and mathematical games — The Abel Laureate gives a lecture at the Abel Prize Award Ceremony University of Bergen. Schoolchildren will invited His Majesty King Harald V presents the to play mathematical games with the prize Abel Prize to the Laureate in the University winner in Archimedes Labyrinth in Bergen Aula, Oslo, Norway Botanical Garden Reception and interview with Register online at: www.abelprize.no from the Abel Laureate mid-April or contact [email protected], at Det Norske Teatret, Oslo, Norway facebook.com/Abelprize

Press contact: For other information: Marina Tofting Håkon Sandbakken [email protected] [email protected] + 47 938 66 312 + 47 454 03 077 Anne-Marie Astad [email protected] + 47 415 67 406