美国得克萨斯大学Karen Uhlenbeck教授获得美国数学会2007年度"Steele 重大贡献奖"。Uhlenbeck教授的主要获奖工作是她奠定了规范场数学理论的解析基础。Uhlenbeck教授是当今国际最杰出的女数学家之一。美国艺术与科学院院士，曾获得1983年麦克阿瑟天才奖（MacArthur Prize Fellowship）,美国国家科学奖章。
Uhlenbeck教授在回应Steele奖的陈述中写道：
"接下来的伟大变革是丘成桐一手倡导发起并勉力开创的，他将分析的技巧引入到拓扑，微分几何与代数几何问题中。丘成桐年轻时（的胆识）令人敬佩。"
Citation Karen Uhlenbeck
The 2007 Steele Prize for a Seminal Contribution to Mathematical Research is awarded to Karen Uhlenbeck for her foundational contributions in analytic aspects of mathematical gauge theory. These results appeared in the two papers:
"Removable singularities in Yang-Mills fields," Comm. Math. Phys. 83 (1982), 11-29; and
"Connections with Lp bounds on curvature," Comm. Math. Phys. 83 (1982),31-42.
Connections are local objects in differential geometry, just as functions are local. But there are two crucial differences. First, connections admit automorphisms, called gauge transformations. Thus there are severaldifferent local representations of a connection. Second, the basic elliptic equation on functions-the Laplace equation-is linear whereas its counterpart on connections-the Yang-Mills equation-is nonlinear and not even elliptic as it stands. Its nonellipticity is tied up with the existence of automorphisms. One of Uhlenbeck's fundamental results proves the existence of good local representatives for connections, called Coulomb gauges. The Yang-Mills equations become elliptic when restricted to Coulomb gauges, and so Uhlenbeck deduces many basic theorems: smoothness of solutions, compactness of solutionswith bounds on the curvature, etc. Uhlenbeck also proves that solutionsto the Yang-Mills equations defined on a punctured ball with suitable boundedness on the curvature extend over the puncture. (Compare the mucheasier Riemann removable singularities theorem in complex analysis.) These theorems and the techniques Uhlenbeck introduced to prove them are the analytic foundation underlying the many applications of gauge theory to geometry and topology. The most immediate was Donaldson's work in the1980's on smooth structures on 4-manifolds through invariants of the anti-self-dual equations, a system of first order partial differential equations closely related to the Yang-Mills equations. Recall that Donaldsonproved the existence of topological 4-manifolds which admit no smooth structure and topological 4-manifolds which admit inequivalent smooth structures. These equations have also advanced the theory of stable vectorbundles in algebraic geometry. The analysis of various dimensional reductions of the anti-self-dual equations, the monopole and vortex equations, and other closely related equations of gauge theory begins with Uhlenbeck's theorems. More recently, these gauge theoretic ideas have yieldednew insights in symplectic and contact geometry.
Biographical Note
Karen K. Uhlenbeck spent her early years in New Jersey, after which sheattended the University of Michigan. She received her PhD in 1968 underthe direction of Richard Palais at Brandeis University. She has held posts at MIT, Berkeley, the University of Illinois in both Champaign-Urbanaand Chicago, and the University of Chicago. Since 1988 she has held theSid W. Richardson Foundation Regents Chair in Mathematics at the University of Texas in Austin.
Karen Uhlenbeck is a Fellow of the National Academy of Sciences and theAmerican Academy of Arts and Sciences. She received a MacArthur Prize Fellowship in 1983, the Commonwealth Award for Science and Technology in 1995, and the Presidential Medal of Science in 2000. Uhlenbeck is a co-founder of the IAS/Park City Mathematics Institute and the program for Women and Mathematics in Princeton.
Response from Karen Uhlenbeck
I thank the American Mathematical Society, its members, and the Steele Prize Committee for the honor and the award of the Steele Prize.
This honor confirms what I have been suspecting for quite some time. I am becoming an old mathematician, if I am not already there. It gives mecause to look back at my research and teaching. All in all, I have foundgreat delight and pleasure in the pursuit of mathematics. Along the wayI 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.
My mathematical career has intersected some exciting mathematical changes. My thesis, written under Richard Palais, was written in the thick ofthe days of "Global Analysis", a period in which the tools and methods of differential topology were applied to analysis problems. This fell into disfavor, but it must be admitted that these ideas are today taken asa matter of course as part of the subject of analysis. During my days asan analyst, I wrote a paper on the regularity of elliptic systems, which I still think is the hardest paper I ever wrote.
The next revolution was single-handedly sponsored and spearheaded by S.-
T. Yau, who introduced techniques of analysis into the problems of topology, differential geometry and algebraic geometry. Mind you, S.-T. Yau was quite something in his younger days! I am quite proud of the paper Iwrote with Jonathan Sachs on minimal spheres. Next we come to the introduction of gauge theory into topology, where I did the work which is cited in the award. I had started work on the analysis of gauge theory afterhearing a lecture by Michael Atiyah on gauge theory at the University of Chicago and was fully prepared to understand the thesis of his studentSimon Donaldson, which used the two papers cited in this award. The work of Donaldson and Cliff Taubes, whom I met when he was still a graduatestudent, was the start of a new era in four-manifold topology. Finally,
due to what was now an addiction to intellectual excitement, I tried tofollow the influence of physics on geometry which is associated with thename of Ed Witten. My work in integrable systems grew out of this connection with physics. This part of my career was not entirely successful.The more physics I learned, the less algebraic geometry I seemed to know.
Given that I started my academic career in the late sixties at the University of California, Berkeley, during the Vietnam War, where protests and tear gas were commonplace, it must be said that I rarely found mathematics and the academic life boring.The accomplishments of which I am mostproud are not exactly mathematical theorems. One does mathematics because one has to, and if it is appreciated, all the better! However, encouraged by my young and enthusiastic colleague Dan Freed, I became involvedin educational issues. We were among the founders of the IAS/Park CityMathematics Institute. The original intent was to bring mathematics researchers, students and high school teachers together. This is now an ongoing institution with a yearly summer school overseen by the Institute for Advanced Study at Princeton. The Women and Mathematics Program at IASis an outgrowth of the Park City Institute. Founded by my collaborator Chuu-Lian Terng and I, the original purpose was to encourage and preparemore women to take part in the Park City Summer School. It has now grownto a self-sufficient two-week yearly program sponsored by IAS. I watchwith real delight the emergence of our graduates into prominence in themathematics community.
Another outcome of this involvement with education is our Saturday Morning Math Group at the University of Texas. We started this in conjunctionwith the beginnings of Park City. It is now an ongoing program which our graduate students organize for local high school students. It is oftencited and much boasted of by our university. Finally, I would like to boast further of my department at the University of Texas. During the years that I have held an endowed chair in this department, we have becomeone of the leading departments of mathematics, admittedly below the topranked, but still quite respectable.
Certainly this is due mostly to my colleagues, but I take a little credit. Our primary benefactor is also due some praise. We used to "thank Peter" after a particularly enjoyable colloquium talk and dinner, and I doagain now.
Starting from my days in Berkeley, the issue of women has never been farfrom my thoughts. I have undergone wide swings of feeling and opinion on the matter. I remain quite disappointed at the number 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.
I want to end by thanking my thesis advisor, Richard Palais; my two present collaborators, Chuu-Lian Terng and Andrea Nahmod; my longtime friendand supporter, S.-T. Yau; my colleagues, particularly Dan Freed and Lorenzo Sadun; as well as all my collaborators, PhD students and assistants. My husband, Bob Williams, is due a share in this award.
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