凯伦·乌伦贝克Karen Uhlenbeck

凯伦·乌伦贝克的父亲弗拉基米尔·阿诺尔德 Keskulla是一名工程师，她的母亲Carolyn Windeler Keskulla是一名艺术家。她在乡下长大，是四个孩子中的老大。许多数学家从小就知道数学将是他们的生命，但乌伦贝克并非如此。小时候她对书籍感兴趣，这导致她对科学产生了兴趣。她写道[2]：-

小时候我读了很多书，什么都读。我会去图书馆，然后熬夜读书。我常常在学校课桌下读书。……我们住在乡下，所以没有太多事情可做。我特别喜欢读科学方面的书。大约十二岁时，父亲开始把弗雷德·霍伊尔的天体物理学书籍带回家。我觉得它们非常有趣。我还记得一本叫《一、二、三……无穷大》的平装小书，作者是乔治·伽莫夫，我记得理解那个非常复杂的论证——存在两种不同的无穷——时的兴奋。

乌伦贝克进入密歇根大学，打算学习物理，但学习令人兴奋的数学课程，加上发现物理实验不是她的强项，使她转到了数学。她于1964年获得数学学士学位。

从密歇根大学毕业后，乌伦贝克在纽约的理查·科朗特研究所继续深造。然而此时她结了婚，并决定跟随丈夫去哈佛。她进入布兰迪斯大学，于1966年获得硕士学位。她留在布兰迪斯，在Richard Palais的指导下攻读博士学位，并于1968年获得博士学位。

她的第一个职位是1968-69年在麻省理工学院的一年期职位。然后是另一个临时职位，这次是两年期，1989-71年在加利福尼亚大学乔治·伯克利担任讲师。她描述了自己寻找永久职位的过程[2]：-

在麻省理工学院一年和伯克利两年之后找工作时，我被告知人们不雇用女性，女性应该回家生孩子。所以对我丈夫感兴趣的地方——麻省理工学院、斯坦福和普林斯顿——对雇用我不感兴趣。我记得有人告诉我存在裙带关系规定，因此不能雇用我，尽管多年后我就此问题打电话给他们时，他们不记得说过这些话……我最终去了伊利诺伊大学厄巴纳-香槟分校，因为他们雇用了我，我丈夫也一起来了。回想起来，我意识到他多么慷慨，因为他本可以去麻省理工学院、斯坦福或普林斯顿。我讨厌厄巴纳-香槟——我在数学上和社会上都感到格格不入，而且那里丑陋、庸俗而平淡。

1971年至1976年在厄巴纳-香槟分校任教后，她转到伊利诺伊大学芝加哥分校，在那里被提升为正教授。此时她：-

……与丘成桐成为朋友，我认为正是他慷慨地使我最终且决定性地成为了一名数学家。

1983年，她获得了麦克阿瑟奖 fellowship，并转到芝加哥大学任教授。1988年，乌伦贝克被任命为德克萨斯大学奥斯汀分校教授，同时她还担任该校的Sid W Richardson基金会数学讲席。

乌伦贝克是偏微分方程领域的顶尖专家，她这样描述自己的数学兴趣[2]：-

我研究偏微分方程，这些方程最初源于描述电磁学等现象的需要，但经历了一个世纪的变迁，现在它们以更加技术化的方式被用来考察空间的形状。数学家们研究由科学家们在探讨其他问题时构造的想象空间。我的数学生涯始于研究Palais对一个非常有用的经典理论——变分法——的现代表述。我认为阿尔伯特·爱因斯坦的广义相对论太难，但设法学到了很多关于时空几何的知识。我在偏微分方程方面做了一些非常技术性的工作，对激波问题做了一次不成功的尝试，在尺度不变变分问题中工作过，对三维流形 拓扑学做了一次糟糕的尝试，学习了规范场论以及一些关于其在四维流形上的应用，最近一直在研究具有代数无穷对称性的$n$方程。

乌伦贝克的工作提供了分析工具，将瞬子用作有效的几何工具。在[1]中，西蒙·唐纳森回忆了关于瞬子应用的工作，这项工作使他在1986年获得了菲尔兹奖。他描述了“冒泡”现象，说：-

事实上，乌伦贝克大约在那时[1982年]发表的论文，基本上包含了将这一图景建立在坚实基础上所需的全部分析。这些论文没有明确讨论“冒泡”——也许这些论证被认为对专家来说是显然的，通过与Sacks和乌伦贝克在调和映射情形中的工作类比即可看出。

1988年，乌伦贝克在美国数学会百年庆典上就Instantons and Their Relatives作了演讲。爱德华·威滕在研讨会上接着作了关于Geometry and quantum field theory的报告，他说：-

在我之前的演讲中，乌伦贝克描述了一些纯数学的发展，这些发展至少大致可以归入这一领域。她描述了通过研究非线性偏微分方程组而在几何学中取得的进展。除其他内容外，她概述了西蒙·唐纳森关于四维流形几何、瞬子——即某些非线性偏微分方程组，自对偶杨-米尔斯方程的解——的工作的某些方面，这些方程最初是由物理学家在量子场论的背景下引入的。

两年后，即1990年，爱德华·威滕因其在拓扑量子场论方面的工作而获得了约翰·查尔斯·菲尔兹奖章。在京都举行的同一次国际数学家大会上，乌伦贝克是全会报告人。

在乌伦贝克因其工作而获得的众多荣誉中，尤其应当提到的是，她于1985年当选为美国艺术与科学院的成员，并于次年当选为国家科学院的成员。

她还担任许多期刊的编委；迄今为止的完整列表为Journal of 微分几何（1979-81）、Illinois Journal of Mathematics（1980-86）、Communications in Partial Differential Equations（1983- ）、Journal of the American Mathematical Society（1986-91）、Ergebnisse der Mathematik（1987-90）、Journal of Differential Geometry（1988-91）、Journal of Mathematical Physics（1989- ）、Houston Journal of Mathematics（1991- ）、Journal of 纽结理论（1991- ）、Calculus of Variations and Partial Differential Equations（1991- ）、Communications in Analysis and Geometry（1992- ）。

在她获得的奖章和奖项中，我们首先提到总统国家科学奖章，该奖章于2000年12月1日星期五在华盛顿特区国家建筑博物馆举行的颁奖典礼上授予她：-

表彰她对整体几何学的许多开创性贡献，这些贡献带来了数学物理和偏微分方程理论的进展。她的研究成就与她在数学培训和教育方面的领导力和热情投入相得益彰。

她于2000年获得伊利诺伊大学厄巴纳-香槟分校的名誉博士学位，2001年获得俄亥俄大学的名誉博士学位，2004年获得密歇根大学的名誉博士学位，2007年获得哈佛大学的名誉博士学位。2007年1月6日星期六，她在路易斯安那州新奥尔良举行的联合数学会议上获得了美国数学会的Leroy P 凯瑟琳·斯蒂尔奖。该奖项的颁发情况见[5]：-

……因她在数学规范场论分析方面的奠基性贡献，见论文⟦E1⟧（1982）和⟦E2⟧（1982）。

她在获奖答辞的开头说：——

我感谢美国数学会、其成员以及斯蒂尔奖委员会授予我斯蒂尔奖这一荣誉。这一荣誉证实了我一段时间以来的猜测。我正在成为一名老数学家，如果我还不是的话。它让我有理由回顾我的研究和教学。总而言之，我在追求数学的过程中找到了极大的喜悦和乐趣。一路走来，我结交了很好的朋友，并与许多富有创造力和有趣的人共事。我得以免于无聊、阴郁和自我专注。一个人不能要求更多了。

在她的答辞后面，她说：——

从我在伯克利的日子开始，女性问题就从未远离我的思绪。在这个问题上，我经历了感受和意见的大幅摇摆。我对从事数学工作和担任领导职位的女性人数仍然相当失望。在我看来，这主要是由于数学界的文化以及来自外部的严酷社会压力。与我提到的其他微小成就相比，改变文化是一项艰巨的任务。

2008年，乌伦贝克当选为伦敦数学会荣誉成员。引文开头写道：——

乌伦贝克是享有最高国际声望的杰出数学家，专攻微分几何、非线性偏微分方程和数学物理。乌伦贝克教授是美国最杰出的数学家之一，或许也是我们时代最杰出的女数学家。同时，乌伦贝克在教育领域的努力，尤其是她作为帕克城-IAS数学研究所创始人的角色，为数学界增添了活力。乌伦贝克教授对女数学家的指导堪称传奇，既有正式的（她在IAS共同创办了年度女性数学项目），也有非正式的。

乌伦贝克于2012年当选为美国数学会会士。同样在2012年，她获得了普林斯顿大学荣誉博士学位。2019年3月，她成为首位获得尼尔斯·阿贝尔奖的女性：-

... 因她在几何偏微分方程、规范理论和可积系统方面的开创性成就，以及她的工作对分析、几何和数学物理的根本性影响。

完整的引文[4]相当专业，但我们将其提供给那些具备必要背景知识的人：-

乌伦贝克是现代几何分析的创始人之一。她的视角渗透了整个领域，并引领了过去40年数学中一些最重大的进展。

几何分析是数学的一个领域，其中分析和微分方程的技术与几何和拓扑问题的研究交织在一起。具体来说，人们研究诸如曲线、曲面、联络和场等对象，它们是代表能量和体积等几何量的泛函的临界点。例如，极小曲面是面积的临界点，调和映射是约翰·彼得·古斯塔夫·勒热纳·狄利克雷能量的临界点。乌伦贝克的主要贡献包括关于极小曲面和调和映射、Yang-Mills理论和可积系统的奠基性结果。

在全局分析中，一个重要的工具，先于乌伦贝克的工作，是Palais-斯蒂芬·斯梅尔紧致性条件。这个条件受莫尔斯早期工作的启发，保证了几何泛函极小值的存在性，并在1维域的情况下成功，例如闭测地线。

乌伦贝克意识到，由于拓扑原因，Palais-斯蒂芬·斯梅尔条件在曲面的情况下失败。乌伦贝克与Sacks合著的关于曲面映射到黎曼流形的能量泛函的论文极具影响力，并详细描述了当Palais-斯蒂芬·斯梅尔条件被违反时会发生什么。一个极小化映射序列在有限个奇点之外收敛，通过使用重缩放论证，他们将奇点附近的行为描述为气泡或瞬子，这些是从2球面到目标流形的极小化映射的标准解。

在更高维度，乌伦贝克与理查德·肖恩合作写了两篇关于极小化调和映射的奠基性论文。他们对非线性椭圆偏微分方程解的奇点给出了深刻的理解。奇点集，在曲面的情况下仅由孤立点组成，在更高维度被一个余维3的集所取代。

这些革命性论文中使用的方法如今已成为每位几何学家和分析学家的标准工具箱的一部分。它们已被成功应用于许多其他偏微分方程和几何情境中。特别地，泡状现象出现在偏微分方程的许多工作中，出现在山边英彦问题的研究中，出现在米哈伊尔·格罗莫夫关于伪全纯曲线的工作中，也出现在瞬子的物理应用中，尤其是在弦论中。

在芝加哥听了迈克尔·阿蒂亚的一次演讲后，乌伦贝克对规范理论产生了兴趣。她从严格的 analytical 角度开创了杨-米尔斯方程的研究。她的工作构成了规范理论领域所有后续研究的基础。

规范理论涉及黎曼流形上的一个辅助向量丛。基本研究对象是该向量丛上的联络。在选定一个平凡化（规范）之后，联络可以用一个矩阵值1-形式来描述。杨-米尔斯联络是规范不变泛函的临界点。乌伦贝克提出并解决了将杨-米尔斯方程表示为椭圆系统这一基本问题，使用的是所谓的夏尔·奥古斯丁·德·库仑规范。这既是乌伦贝克关于曲率在$L^{p}$中有界的联络的著名紧致性定理的起点，也是她后来关于定义在去心4维球上的杨-米尔斯方程的可去奇点结果的开端。高维杨-米尔斯方程的可去奇点理论很久以后才由Gang Tian和陶哲轩完成。乌伦贝克的紧致性定理在非阿贝尔威廉·瓦兰斯·道格拉斯·霍奇理论中至关重要，特别是在Hitchin映射的固有性证明和Corlette关于等变调和映射存在性的重要结果中。

乌伦贝克的另一项重要成果是她与丘成桐合作的关于复n维流形上稳定全纯向量丛上埃尔米特杨-米尔斯联络存在性的工作，推广了西蒙·唐纳森早先关于复曲面的结果。这一西蒙·唐纳森-乌伦贝克-丘成桐的结果联系了微分几何与代数几何的发展，并且是异弦理论应用于粒子物理的基础性结果。

乌伦贝克的思想为规范理论应用于几何与拓扑奠定了分析基础，为Taubes关于自对偶4维流形粘合的重要工作、为西蒙·唐纳森关于规范理论与4维拓扑的开创性工作以及该领域的许多其他工作奠定了基础。乌伦贝克与Dan Freed合著的《瞬子与4维流形》一书教导并激励了一代微分几何学家。她继续在这一领域工作，特别是与Lesley Sibner和Robert Sibner一起得到了关于杨-米尔斯方程非自对偶解的重要结果。

可积系统的研究根源于19世纪经典力学。利用规范理论的语言，乌伦贝克和Hitchin意识到从曲面到齐性空间的调和映射以1维参数族的形式出现。基于这一观察，乌伦贝克用代数方法描述了从球面到格拉斯曼流形的调和映射，将它们与一个无穷维可积系统和Virasoro作用联系起来。这一开创性工作引发了乌伦贝克和Chuu-Lian Terng关于该主题的一系列进一步基础性论文，并创建了一个活跃而富有成果的学派。

乌伦贝克的关键性工作的影响超越了几何分析。一篇极具影响力的早期文章致力于研究非线性椭圆方程组正则性理论，这与黎曼流形间高阶能量泛函的临界映射研究相关。这项工作将约翰·福布斯·纳什、恩尼奥·德乔吉和尤尔根·莫泽尔关于单个非线性方程解的正则性的先前结果推广到了方程组的解。

乌伦贝克的开创性成果对当代分析、几何和数学物理产生了根本性影响，她的思想和领导力改变了整个数学格局。

Association for Women in Mathematics将她列入2020年度AWM会士名单，以表彰[3]：-

……她对现代几何分析的开创性和深远贡献；尽管女性进入该领域时面临相当大的挑战，她仍成为我们时代最伟大的数学家之一；她利用自己应对这些挑战的经验，为未来世代的女性创建并维持应对这些挑战的项目。表彰她一生打破障碍；以及她是第一位获得尼尔斯·阿贝尔奖的女性。

她于2023年当选为皇家学会会士。

Karen Uhlenbeck's father Arnold Keskulla, was an engineer and her mother Carolyn Windeler Keskulla, was an artist. She grew up in the country, the eldest of four children. Many mathematicians know from early age that mathematics will be their life but this was not so with Karen Uhlenbeck. As a child she was interested in books and this led her to an interest in science. She writes [2]:-

As a child I read a lot, and I read everything. I'd go to the library and then stay up all night reading. I used to read under the desk in school. ... we lived in the country so there wasn't a whole lot to do. I was particularly interested in reading about science. I was about twelve years old when my father began bringing home Fred Hoyle's books on astrophysics. I found them very interesting. I also remember a little paperback book called "One, Two, Three, (and, in?) Infinity" by George Gamow, and I remember the excitement of understanding this very sophisticated argument that there were two different kinds of infinities.

Uhlenbeck entered the University of Michigan with the intention of studying physics but a combination of studying exciting mathematics courses and finding that physics practicals were not a strong point lead her to change to mathematics. She was awarded a B.S. in mathematics in 1964.

After graduating from the University of Michigan, Uhlenbeck continued her studies at the Courant Institute in New York. However at this time she married and decided to follow her husband when he went to Harvard. She entered Brandeis University and was awarded a Master's Degree in 1966. She remained at Brandeis to study for her doctorate under Richard Palais' supervision, and was awarded a Ph.D. in 1968.

Her first appointment was a one year post in 1968-69 at Massachusetts Institute of Technology. Then another temporary post, this time a two year one as a lecturer at the University of California, Berkeley during 1989-71. She describes her search for a permanent position [2]:-

I was told, when looking for jobs after my year at MIT and two years at Berkeley, that people did not hire women, that women were supposed to go home and have babies. So the places interested in my husband - MIT, Stanford, and Princeton - were not interested in hiring me. I remember that I was told that there were nepotism rules and that they could not hire me for this reason, although when I called them on this issue years later they did not remember saying these things ... I ended up at the University of Illinois, Champaign-Urbana because they hired me, and my husband came along. In retrospect I realized how remarkably generous he was because he could have been at MIT, Stanford, or Princeton. I hated Champaign-Urbana - I felt out of place mathematically and socially, and it was ugly, bourgeois and flat.

After being on the faculty at Urbana-Champaign from 1971 to 1976, she moved to the University of Illinois at Chicago where she was promoted to full professor. At this time she:-

... became friends with S T Yau, whom I credit with generously establishing my finally and definitively as a mathematician.

In 1983 she was awarded a MacArthur Prize Fellowship and moved to a professorship at the University of Chicago. In 1988 Uhlenbeck was appointed Professor in the University of Texas at Austin where she also holds the Sid W Richardson Foundation Regents Chair in Mathematics.

Uhlenbeck is a leading expert on partial differential equations and describes her mathematical interests as follows [2]:-

I work on partial differential equations which were originally derived from the need to describe things like electromagnetism, but have undergone a century of change in which they are used in a much more technical fashion to look at the shapes of space. Mathematicians look at imaginary spaces constructed by scientists examining other problems. I started out my mathematics career by working on Palais' modern formulation of a very useful classical theory, the calculus of variations. I decided Einstein's general relativity was too hard, but managed to learn a lot about geometry of space time. I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three dimensional manifold topology, learned gauge field theory and then some about applications to four dimensional manifolds, and have recently been working $n$ equations with algebraic infinite symmetries.

Uhlenbeck's work provided analytic tools to use instantons as an effective geometric tool. In [1] Simon Donaldson reminisces about the work on the applications of instantons that led him to receive a Fields Medal in 1986. He describes the "bubbling" phenomenon saying:-

In fact the papers of Uhlenbeck which appeared about that time [in 1982] contained essentially all the analysis required to put this picture on a firm footing. The papers do not discuss "bubbling" explicitly - perhaps the arguments were supposed to be obvious to experts by analogy with the work of Sacks and Uhlenbeck in the harmonic maps case.

In 1988 Uhlenbeck lectured on Instantons and Their Relatives at the Centennial Celebration of the American Mathematical Society. Witten, who gave the next talk on Geometry and quantum field theory at the symposium said:-

In the talk just before mine, Karen Uhlenbeck described some purely mathematical developments that at least roughly might be classified in this area. She described advances in geometry that have been achieved through the study of systems of nonlinear partial differential equations. Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.

Two years later, in 1990, Witten received a Fields Medal for his work on topological quantum field theories. At the same International Congress of Mathematicians in Kyoto, Karen Uhlenbeck was a Plenary Speaker.

Among the many honours that Uhlenbeck has received for her work one should mention in particular that she was elected a Member of the American Academy of Arts and Science in 1985 and a Member of the National Academy of Sciences the following year.

She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ).

Among the medals and prizes she has received we mention first the President's National Medal of Science which was awarded to her at an awards ceremony at the National Building Museum, Washington, DC, on Friday, 1 December 2000:-

For her many pioneering contributions to global geometry that resulted in advances in mathematical physics and the theory of partial differential equations. Her research accomplishments are matched by her leadership and passionate involvement in mathematics training and education.

She was awarded honorary doctorates by the University of Illinois, Champaign in 2000, by the University of Ohio in 2001, by the University of Michigan in 2004, and by Harvard University in 2007. On Saturday 6 January 2007 she received the American Mathematical Society's Leroy P Steele Prize at the Joint Mathematics Meeting in New Orleans, Louisiana. The award was made [5]:-

... for her foundational contributions in analytic aspects of mathematical gauge theory in the papers "Removable singularities in Yang-Mills fields" (1982) and "Connections with bounds on curvature" (1982).

Her reply to the Award began:-

I thank the American Mathematical Society, its members and the Steele Prize committee for the honour and the award of the Steele Prize. This honour confirms what I have been suspecting for quite some time. I am becoming an old mathematician, if I am not already there. It gives me cause to look back at my research and teaching. All in all, I have found great delight and pleasure in the pursuit of mathematics. Along the way I have made great friends and worked with a number of creative and interesting people. I have been saved from boredom, dourness and self-absorption. One cannot ask for more.

Later in her response she said:-

Starting from my days in Berkeley, the issue of women has never been far from my thoughts. I have undergone wide swings of feeling and opinion on the matter. I remain quite disappointed at the numbers of women doing mathematics and in leadership positions. This is, to my mind, primarily due to the culture of the mathematical community as well as harsh societal pressures from outside. Changing the culture is a momentous task in comparison to the other minor accomplishments I have mentioned.

In 2008 Uhlenbeck was elected an honorary member of the London Mathematical Society. The citation begins:-

Karen Uhlenbeck is a distinguished mathematician of the highest international stature, specialising in differential geometry, non-linear partial differential equations and mathematical physics. Professor Uhlenbeck is one of the United States' most eminent mathematicians, and perhaps the most distinguished woman mathematician of our time. At the same time, Uhlenbeck's efforts across the educational spectrum, especially her role as a founder of the Park City-IAS Mathematical Institute, have added vitality to the mathematical scene. Professor Uhlenbeck's mentoring is legendary, both formal (she co-founded the annual Women in Mathematics programme at the IAS) and informal, of women mathematicians.

Karen Uhlenbeck was elected a Fellow of the American Mathematical Society in 2012. Also in 2012 she was awarded an honorary doctorate by Princeton University. In March 2019 she became the first woman to receive the 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.

The full citation [4] is rather technical, but we give it for those who have the necessary background knowledge:-

Karen Keskulla Uhlenbeck is a founder of modern Geometric Analysis. Her perspective has permeated the field and led to some of the most dramatic advances in mathematics in the last 40 years.

Geometric analysis is a field of mathematics where techniques of analysis and differential equations are interwoven with the study of geometrical and topological problems. Specifically, one studies objects such as curves, surfaces, connections and fields which are critical points of functionals representing geometric quantities such as energy and volume. For example, minimal surfaces are critical points of the area and harmonic maps are critical points of the Dirichlet energy. Uhlenbeck's major contributions include foundational results on minimal surfaces and harmonic maps, Yang-Mills theory, and integrable systems.

An important tool in global analysis, preceding the work of Uhlenbeck, is the Palais-Smale compactness condition. This condition, inspired by earlier work of Morse, guarantees existence of minimisers of geometric functionals and is successful in the case of 1-dimensional domains, such as closed geodesics.

Uhlenbeck realised that the condition of Palais-Smale fails in the case of surfaces due to topological reasons. The papers of Uhlenbeck, co-authored with Sacks, on the energy functional for maps of surfaces into a Riemannian manifold, have been extremely influential and describe in detail what happens when the Palais-Smale condition is violated. A minimising sequence of mappings converges outside a finite set of singular points and by using rescaling arguments, they describe the behaviour near the singularities as bubbles or instantons, which are the standard solutions of the minimising map from the 2-sphere to the target manifold.

In higher dimensions, Uhlenbeck in collaboration with Schoen wrote two foundational papers on minimising harmonic maps. They gave a profound understanding of singularities of solutions of non-linear elliptic partial differential equations. The singular set, which in the case of surfaces consists only of isolated points, is in higher dimensions replaced by a set of codimension 3.

The methods used in these revolutionary papers are now in the standard toolbox of every geometer and analyst. They have been applied with great success in many other partial differential equations and geometric contexts. In particular, the bubbling phenomenon appears in many works in partial differential equations, in the study of the Yamabe problem, in Gromov's work on pseudoholomorphic curves, and also in physical applications of instantons, especially in string theory.

After hearing a talk by Atiyah in Chicago, Uhlenbeck became interested in gauge theory. She pioneered the study of Yang-Mills equations from a rigorous analytical point of view. Her work formed a base of all subsequent research in the area of gauge theory.

Gauge theory involves an auxiliary vector bundle over a Riemannian manifold. The basic objects of study are connections on this vector bundle. After a choice of a trivialisation (gauge), a connection can be described by a matrix valued 1-form. Yang-Mills connections are critical points of gauge-invariant functionals. Uhlenbeck addressed and solved the fundamental question of expressing Yang-Mills equations as an elliptic system, using the so-called Coulomb gauge. This was the starting point for both Uhlenbeck's celebrated compactness theorem for connections with curvature bounded in $L^{p}$, and for her later results on removable singularities for Yang-Mills equations defined on punctured 4-dimensional balls. The removable singularity theory for Yang-Mills equations in higher dimensions was carried out much later by Gang Tian and Terence Tao. Uhlenbeck's compactness theorem was crucial in Non-Abelian Hodge Theory and, in particular, in the proof of the properness of Hitchin's map and Corlette's important result on the existence of equivariant harmonic mappings.

Another major result of Uhlenbeck is her joint work with Yau on the existence of Hermitian Yang-Mills connections on stable holomorphic vector bundles over complex n-manifolds, generalising an earlier result of Donaldson on complex surfaces. This result of Donaldson-Uhlenbeck-Yau links developments in differential geometry and algebraic geometry, and is a foundational result for applications of heterotic strings to particle physics.

Uhlenbeck's ideas laid the analytic foundations for the application of gauge theory to geometry and topology, to the important work of Taubes on the gluing of self-dual 4-manifolds, to the ground-breaking work of Donaldson on gauge theory and 4-dimensional topology, and many other works in this area. The book written by Uhlenbeck and Dan Freed on "Instantons and 4-Manifolds" instructed and inspired a generation of differential geometers. She continued to work in this area, and in particular had an important result with Lesley Sibner and Robert Sibner on non self-dual solutions to the Yang-Mills equations.

The study of integrable systems has its roots in 19th century classical mechanics. Using the language of gauge theory, Uhlenbeck and Hitchin realised that harmonic mappings from surfaces to homogeneous spaces come in 1-dimensional parametrised families. Based on this observation, Uhlenbeck described algebraically harmonic mappings from spheres into Grasmannians relating them to an infinite dimensional integrable system and Virasoro actions. This seminal work led to a series of further foundational papers by Uhlenbeck and Chuu-Lian Terng on the subject and the creation of an active and fruitful school.

The impact of Uhlenbeck's pivotal work goes beyond geometric analysis. A highly influential early article was devoted to the study of regularity theory of a system of non-linear elliptic equations, relevant to the study of the critical map of higher order energy functionals between Riemannian manifolds. This work extends previous results by Nash, De Giorgi and Jürgen Moser on regularity of solutions of single non-linear equations to solutions of systems.

Karen Uhlenbeck's pioneering results have had fundamental impact on contemporary analysis, geometry and mathematical physics, and her ideas and leadership have transformed the mathematical landscape as a whole.

The Association for Women in Mathematics included her in the 2020 class of AWM Fellows for [3]:-

... her ground-breaking and profound contributions to modern geometric analysis; for establishing a career as one of the greatest mathematicians of our time, despite the considerable challenges facing women when she entered the field; for using her experiences navigating these challenges to create and sustain programs to address them for future generations of women. For a lifetime of breaking barriers; and for being the first woman to win the Abel Prize.

She was elected a Fellow of the Royal Society in 2023.