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Clifford H Taubes Awards
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1. Oswald Veblen Prize in Geometry 1991.
1.1. The Oswald Veblen Prize in Geometry.
Oswald Veblen (1880-1960), who served as President of the Society in 1923 and 1924, was well known for his mathematical work in geometry and topology. In 1961, the Trustees of the Society established a fund in memory of Professor Veblen, contributed originally by former students and colleagues, and later doubled by his widow. Since 1964, the fund has been used for the award of the Oswald Veblen Prize in Geometry. Subsequent awards were made at five-year intervals. A total of ten awards have been made: Christos D Papakyriakopolous (1964), Raoul H Bott (1964), Stephen Smale (1966), Morton Brown and Barry Mazur (1966), Robion C Kirby (1971), Dennis P Sullivan (1971), William P Thurston (1976), James Simons (1976), Mikhael Gromov (1981), Shing-Tung Yau (1981), and Michael H Freedman (1986). At present (1991), the award is supplemented from the Steele Prize Fund, bringing the value of the Veblen Prize to $4000, divided equally between this year's recipients.
1.2. The American Mathematical Society 1991 Oswald Veblen Prize in Geometry.
The American Mathematical Society's 1991 Oswald Veblen Prize in Geometry was awarded at the Joint Mathematics Meetings in San Francisco to Andrew J Casson of the University of California at Berkeley and to Clifford H Taubes of Harvard University.
The 1991 Veblen Prize was awarded by the AMS Council on the basis of a recommendation by a selection committee consisting of Edgar H Brown, Jr., Michael H Freedman (chair), and Mikhael Gromov.
1.2. Clifford H Taubes. Citation.
Clifford Henry Taubes is awarded the 1991 Oswald Veblen Prize in Geometry for his foundational work in Yang-Mills theory. Taubes, since the time of his Ph.D. thesis and book on vortices and monopoles (co-authored with Arthur Jaffe), has done as much as any individual to forge emerging physical concepts into powerful mathematical tools.
The harnessing of Yang-Mills theory by mathematicians began with Karen Uhlenbeck's work on the singularities of, and curvature estimates for, the solutions of these equations. From this beginning, Taubes laid a geometric and analytical foundation for the study of the Yang-Mills functional. His initial paper- Self-dual Yang-Mills connections on non-self-dual 4-manifolds (1982) - contained the technical basis for Simon Donaldson's first celebrated non-existence theorem. Similarly, antecedents of Andreas Floer's remarkable "homology theory" occur in Taubes' use of spectral flow to determine the signs in his paper, "Casson's invariant and gauge theories" (1990).
The understanding that solutions to the Yang-Mills equations could be constructed over metrically arbitrary 4-manifolds is due to Taubes. A fundamental tool in the analysis of Yang-Mills fields on manifolds is "neck-pulling," wherein the underlying manifold degenerates. The behaviour of the Yang-Mills field is carefully tracked during the degeneration so that knowledge of the limit yields implications about the original fields. Taubes pioneered this method in his analysis of End periodic 4-manifolds ("Gauge theory on asymptotically periodic 4-manifolds," 1987). Among the topological implications was a proof that there are uncountably many smooth structures on Taubes' most recent work (unpublished) analyses the behaviour of Yang-Mills connections along ends which have nontrivial first homology. Important topological consequences of such connections have recently been discovered by his students Tom Mrowka and Bob Gompf.
Cliff Taubes laid much of the foundation for a remarkable decade of Yang-Mills theory.
1.3. Clifford Taubes. Biographical Sketch.
Clifford Henry Taubes was born on February 21, 1954 in New York City. He received his B.A. from Cornell University (1975) and his M.S. (1978) and Ph.D. (1980) degrees from Harvard University. From 1980 to 1983, he was a junior fellow at Harvard University. After that, he held a position as acting associate professor at the University of California at Berkeley before returning to Harvard as professor of mathematics in 1985. He held a National Science Foundation (NSF) Mathematical Sciences Postdoctoral Fellowship (1984-1987). He is currently a member of the NSF Advisory Committee for the Mathematical Sciences. In 1990, he was elected to the American Academy of Arts and Sciences.
Taubes presented an address at the Special Session on Nonlinear Generalizations of Maxwell's Equations at the AMS Meeting at the University of Massachusetts at Amherst in October 1981; an AMS Invited Address at the Joint Mathematics Meetings in Eugene, Oregon in August 1984; and a 45-minute address at the International Congress of Mathematicians at Berkeley in 1986. He was also an invited speaker at the Symposium on the Mathematical Heritage of Hermann Weyl, held at Duke University in May 1987. His areas of research are differential geometry, nonlinear partial differential equations, and mathematical physics.
2. Élie Cartan Prize 1993.
Oswald Veblen (1880-1960), who served as President of the Society in 1923 and 1924, was well known for his mathematical work in geometry and topology. In 1961, the Trustees of the Society established a fund in memory of Professor Veblen, contributed originally by former students and colleagues, and later doubled by his widow. Since 1964, the fund has been used for the award of the Oswald Veblen Prize in Geometry. Subsequent awards were made at five-year intervals. A total of ten awards have been made: Christos D Papakyriakopolous (1964), Raoul H Bott (1964), Stephen Smale (1966), Morton Brown and Barry Mazur (1966), Robion C Kirby (1971), Dennis P Sullivan (1971), William P Thurston (1976), James Simons (1976), Mikhael Gromov (1981), Shing-Tung Yau (1981), and Michael H Freedman (1986). At present (1991), the award is supplemented from the Steele Prize Fund, bringing the value of the Veblen Prize to $4000, divided equally between this year's recipients.
1.2. The American Mathematical Society 1991 Oswald Veblen Prize in Geometry.
The American Mathematical Society's 1991 Oswald Veblen Prize in Geometry was awarded at the Joint Mathematics Meetings in San Francisco to Andrew J Casson of the University of California at Berkeley and to Clifford H Taubes of Harvard University.
The 1991 Veblen Prize was awarded by the AMS Council on the basis of a recommendation by a selection committee consisting of Edgar H Brown, Jr., Michael H Freedman (chair), and Mikhael Gromov.
1.2. Clifford H Taubes. Citation.
Clifford Henry Taubes is awarded the 1991 Oswald Veblen Prize in Geometry for his foundational work in Yang-Mills theory. Taubes, since the time of his Ph.D. thesis and book on vortices and monopoles (co-authored with Arthur Jaffe), has done as much as any individual to forge emerging physical concepts into powerful mathematical tools.
The harnessing of Yang-Mills theory by mathematicians began with Karen Uhlenbeck's work on the singularities of, and curvature estimates for, the solutions of these equations. From this beginning, Taubes laid a geometric and analytical foundation for the study of the Yang-Mills functional. His initial paper- Self-dual Yang-Mills connections on non-self-dual 4-manifolds (1982) - contained the technical basis for Simon Donaldson's first celebrated non-existence theorem. Similarly, antecedents of Andreas Floer's remarkable "homology theory" occur in Taubes' use of spectral flow to determine the signs in his paper, "Casson's invariant and gauge theories" (1990).
The understanding that solutions to the Yang-Mills equations could be constructed over metrically arbitrary 4-manifolds is due to Taubes. A fundamental tool in the analysis of Yang-Mills fields on manifolds is "neck-pulling," wherein the underlying manifold degenerates. The behaviour of the Yang-Mills field is carefully tracked during the degeneration so that knowledge of the limit yields implications about the original fields. Taubes pioneered this method in his analysis of End periodic 4-manifolds ("Gauge theory on asymptotically periodic 4-manifolds," 1987). Among the topological implications was a proof that there are uncountably many smooth structures on Taubes' most recent work (unpublished) analyses the behaviour of Yang-Mills connections along ends which have nontrivial first homology. Important topological consequences of such connections have recently been discovered by his students Tom Mrowka and Bob Gompf.
Cliff Taubes laid much of the foundation for a remarkable decade of Yang-Mills theory.
1.3. Clifford Taubes. Biographical Sketch.
Clifford Henry Taubes was born on February 21, 1954 in New York City. He received his B.A. from Cornell University (1975) and his M.S. (1978) and Ph.D. (1980) degrees from Harvard University. From 1980 to 1983, he was a junior fellow at Harvard University. After that, he held a position as acting associate professor at the University of California at Berkeley before returning to Harvard as professor of mathematics in 1985. He held a National Science Foundation (NSF) Mathematical Sciences Postdoctoral Fellowship (1984-1987). He is currently a member of the NSF Advisory Committee for the Mathematical Sciences. In 1990, he was elected to the American Academy of Arts and Sciences.
Taubes presented an address at the Special Session on Nonlinear Generalizations of Maxwell's Equations at the AMS Meeting at the University of Massachusetts at Amherst in October 1981; an AMS Invited Address at the Joint Mathematics Meetings in Eugene, Oregon in August 1984; and a 45-minute address at the International Congress of Mathematicians at Berkeley in 1986. He was also an invited speaker at the Symposium on the Mathematical Heritage of Hermann Weyl, held at Duke University in May 1987. His areas of research are differential geometry, nonlinear partial differential equations, and mathematical physics.
2.1. The Élie Cartan Prize.
The Élie Cartan Prize is a triennial prize which was created by the Académie des sciences in 1980 intended to reward, without distinction of nationality, a mathematician aged at most 45 years, having accomplished an important work either by the introduction of new ideas, or by the solution of a difficult problem.
2.2. The 1993 Élie Cartan Prize.
The 1993 Élie Cartan Prize was awarded to Clifford Taubes, professor at Harvard University, for his work on Yang Mills equations and geometry.
3. Clay Research Award 2008.
The Élie Cartan Prize is a triennial prize which was created by the Académie des sciences in 1980 intended to reward, without distinction of nationality, a mathematician aged at most 45 years, having accomplished an important work either by the introduction of new ideas, or by the solution of a difficult problem.
2.2. The 1993 Élie Cartan Prize.
The 1993 Élie Cartan Prize was awarded to Clifford Taubes, professor at Harvard University, for his work on Yang Mills equations and geometry.
3.1. The Clay Research Awards.
The Clay Mathematics Institute was founded in September 1998 by Landon T Clay, a Boston businessman, and his wife, Lavinia D Clay. Its establishment grew out Landon Clay's belief in the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life.
The Clay Research Awards have been made almost every year since 1999. Presented annually at the Clay Research Conference, the awards celebrate the outstanding achievements of the world's most gifted mathematicians. Awardees receive the bronze sculpture "Figureight Knot Complement VII/CMI" by sculptor Helaman Ferguson. Although perhaps less well known outside the mathematical world than the Clay Millennium Prize Problems, the Clay Research Award is widely appreciated within it. The first award in 1999 was made to Andrew Wiles.
3.2. The 2008 Clay Research Awards.
In 2008 two Clay Research Awards were presented, one to Clifford Taubes and one to Claire Voisin. The 2008 Clay Research Award was made to Clifford Taubes for his proof of the Weinstein conjecture in dimension three.
3.2. Citation for Clifford Taubes.
The Weinstein conjecture is a conjecture about the existence of closed orbits for the Reeb vector field on a contact manifold. A contact manifold is an odd-dimensional manifold with a one-form such that wedged with the th exterior power of is everywhere nonzero. In particular, the kernel of is a maximally nonintegrable field of hyperplanes in the tangent bundle. The Reeb vector field generates the kernel of and pairs to one with . Alan Weinstein asked some thirty years ago whether this vector field must, in all cases, have a closed orbit. (The unit sphere in complex -space with the annihilator of the maximal complex subspace of the real tangent space is an example of a contact manifold and contact 1-form. In this case, the orbits of the Reeb vector field generate the circle action whose quotient gives the associated complex projective space.) Note, by contrast, that there exist non-contact vector fields, even on the 3-sphere, with no closed orbits. These are the counter-examples (due to Schweitzer, Harrison and Kuperberg) to the Seifert conjecture. Hofer affirmed the Weinstein conjecture in many 3-dimensional cases, for example the three-sphere and contact structures on any 3-dimensional, reducible manifold. Taubes' affirmative solution of the Weinstein conjecture for any 3-dimensional contact manifold is based on a novel application of the Seiberg-Witten equations to the problem.
4. NAS Award in Mathematics 2008.
The Clay Mathematics Institute was founded in September 1998 by Landon T Clay, a Boston businessman, and his wife, Lavinia D Clay. Its establishment grew out Landon Clay's belief in the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life.
The Clay Research Awards have been made almost every year since 1999. Presented annually at the Clay Research Conference, the awards celebrate the outstanding achievements of the world's most gifted mathematicians. Awardees receive the bronze sculpture "Figureight Knot Complement VII/CMI" by sculptor Helaman Ferguson. Although perhaps less well known outside the mathematical world than the Clay Millennium Prize Problems, the Clay Research Award is widely appreciated within it. The first award in 1999 was made to Andrew Wiles.
3.2. The 2008 Clay Research Awards.
In 2008 two Clay Research Awards were presented, one to Clifford Taubes and one to Claire Voisin. The 2008 Clay Research Award was made to Clifford Taubes for his proof of the Weinstein conjecture in dimension three.
3.2. Citation for Clifford Taubes.
The Weinstein conjecture is a conjecture about the existence of closed orbits for the Reeb vector field on a contact manifold. A contact manifold is an odd-dimensional manifold with a one-form such that wedged with the th exterior power of is everywhere nonzero. In particular, the kernel of is a maximally nonintegrable field of hyperplanes in the tangent bundle. The Reeb vector field generates the kernel of and pairs to one with . Alan Weinstein asked some thirty years ago whether this vector field must, in all cases, have a closed orbit. (The unit sphere in complex -space with the annihilator of the maximal complex subspace of the real tangent space is an example of a contact manifold and contact 1-form. In this case, the orbits of the Reeb vector field generate the circle action whose quotient gives the associated complex projective space.) Note, by contrast, that there exist non-contact vector fields, even on the 3-sphere, with no closed orbits. These are the counter-examples (due to Schweitzer, Harrison and Kuperberg) to the Seifert conjecture. Hofer affirmed the Weinstein conjecture in many 3-dimensional cases, for example the three-sphere and contact structures on any 3-dimensional, reducible manifold. Taubes' affirmative solution of the Weinstein conjecture for any 3-dimensional contact manifold is based on a novel application of the Seiberg-Witten equations to the problem.
4.1. The National Academy of Sciences Award in Mathematics.
The National Academy of Sciences Award in Mathematics was initially established by the American Mathematical Society in commemoration of its Centennial in 1988 as the" National Academy of Sciences Award in Mathematics". At that time, it was funded mainly by gifts to the Society from Morris Yachter and Sydney Gould. A $5,000 award was given every four years for excellence of research in the mathematical sciences published within the past ten years. After making the 2012 award to Michael J Hopkins, the award was suspended. We note that in 2018, the prize was renamed the Maryam Mirzakhani Prize in Mathematics after Maryam Mirzakhani, the first woman to win a Fields Medal, and a major push to increase the endowment was initiated.
4.2. The NAS Award in Mathematics 2008.
The 2008 National Academy of Sciences Award in Mathematics was made to Clifford Taubes, Harvard University, for "groundbreaking work relating to Seiberg-Witten and Gromov-Witten invariants of symplectic 4-manifolds, and his proof of the Weinstein conjecture for all contact 3-manifolds."
4.3. Notices of the American Mathematics Society on Taubes' work.
The Notices asked D Kotschick, Ludwig-Maximilians-Universität München, and T S Mrowka, Massachusetts Institute of Technology, to comment on the work of Taubes. They responded:
By his own account, Cliff Taubes would like to be considered a topologist. Ignoring this wish, most of his colleagues see him as a great geometric analyst, whose work has had a profound impact on geometry, topology and mathematical physics.
Starting out in mathematics with a physics background, Taubes did some of the early foundational work in mathematical gauge theory studying vortices and Bogomolny monopoles and building up a substantial existence theory for the self-dual Yang-Mills, or instanton, equations on four-manifolds. The latter was, of course, crucial in Donaldson's celebrated application of gauge theory to four-dimensional differential topology. Taubes himself proved the existence of uncountably many exotic differentiable structures on ; he reinterpreted Casson's invariant in terms of gauge theory and proved a homotopy approximation theorem for Yang-Mills moduli spaces. Taubes also proved a powerful existence theorem for anti-self-dual conformal structures on four-manifolds.
When Witten proposed the study of the so-called Seiberg-Witten equations in 1994, Cliff Taubes was one of the handful of mathematicians who quickly worked out the basics of the theory and launched it on its meteoric path. From an analyst's point of view, the quasi-linear Seiberg-Witten equations may seem rather mundane, at least when compared to the challenges offered up by the Yang-Mills equations. True to this spirit, Taubes announced at the time that he would never again write a paper more than twenty pages long. Of course, this resolution lasted only for about six months! After that, Taubes wrote a whole series of deep, technical, and very long papers that became known by the slogan 'Seiberg-Witten - Gromov'. These papers establish the most profound results known to this day about the Seiberg-Witten equations, linking their solutions on symplectic four-manifolds to Gromov's pseudo-holomorphic curve theory in a very precise way.
Taubes's work on the Seiberg-Witten equations remains one of the cornerstones underpinning the current very productive symbiosis between symplectic geometry and low-dimensional topology. Nevertheless, it was a shock to many when Taubes knocked off one of the holy grails of symplectic topology last year. The Weinstein conjecture predicts the existence of periodic orbits for the Reeb flows of arbitrary contact forms on closed three-manifolds. Many special cases had been proved by a variety of methods from symplectic geometry and a proof of the full conjecture was one of the ultimate goals of symplectic field theory. However, Taubes's proof follows a rather different line, using gauge theory and deploying a strategy similar to his work on 'Seiberg-Witten - Gromov'. The proof also hinges on a novel estimate for the spectral flow of a family of Dirac-type operators.
It is in the nature of Cliff Taubes's work that his papers are not usually short or easy to read. Rather they are difficult and original, and technically demanding by necessity. His faithful readers take comfort in the knowledge that Cliff is at least as hard on himself as he is on the readers, and they appreciate his very personal style peppered with what they affectionately refer to as 'Cliffisms'.
4.4. Biographical Sketch of Taubes.
Clifford Taubes grew up in Rochester, New York. After his undergraduate education at Cornell University, he received a Ph.D. in physics from Harvard University in 1980. After a Harvard Junior Fellowship, taken in the mathematics department at Harvard, he taught for two years at the University of California, Berkeley. Since 1985 he has been at Harvard, where he is the William Petschek Professor of Mathematics. He received the AMS Veblen Prize (1991) and the Élie Cartan Prize of the French Mathematical Society (1993). He is a member of both the American Academy of Arts and Sciences and of the National Academy of Sciences.
5. Shaw Prize in Mathematics 2009.
The National Academy of Sciences Award in Mathematics was initially established by the American Mathematical Society in commemoration of its Centennial in 1988 as the" National Academy of Sciences Award in Mathematics". At that time, it was funded mainly by gifts to the Society from Morris Yachter and Sydney Gould. A $5,000 award was given every four years for excellence of research in the mathematical sciences published within the past ten years. After making the 2012 award to Michael J Hopkins, the award was suspended. We note that in 2018, the prize was renamed the Maryam Mirzakhani Prize in Mathematics after Maryam Mirzakhani, the first woman to win a Fields Medal, and a major push to increase the endowment was initiated.
4.2. The NAS Award in Mathematics 2008.
The 2008 National Academy of Sciences Award in Mathematics was made to Clifford Taubes, Harvard University, for "groundbreaking work relating to Seiberg-Witten and Gromov-Witten invariants of symplectic 4-manifolds, and his proof of the Weinstein conjecture for all contact 3-manifolds."
4.3. Notices of the American Mathematics Society on Taubes' work.
The Notices asked D Kotschick, Ludwig-Maximilians-Universität München, and T S Mrowka, Massachusetts Institute of Technology, to comment on the work of Taubes. They responded:
By his own account, Cliff Taubes would like to be considered a topologist. Ignoring this wish, most of his colleagues see him as a great geometric analyst, whose work has had a profound impact on geometry, topology and mathematical physics.
Starting out in mathematics with a physics background, Taubes did some of the early foundational work in mathematical gauge theory studying vortices and Bogomolny monopoles and building up a substantial existence theory for the self-dual Yang-Mills, or instanton, equations on four-manifolds. The latter was, of course, crucial in Donaldson's celebrated application of gauge theory to four-dimensional differential topology. Taubes himself proved the existence of uncountably many exotic differentiable structures on ; he reinterpreted Casson's invariant in terms of gauge theory and proved a homotopy approximation theorem for Yang-Mills moduli spaces. Taubes also proved a powerful existence theorem for anti-self-dual conformal structures on four-manifolds.
When Witten proposed the study of the so-called Seiberg-Witten equations in 1994, Cliff Taubes was one of the handful of mathematicians who quickly worked out the basics of the theory and launched it on its meteoric path. From an analyst's point of view, the quasi-linear Seiberg-Witten equations may seem rather mundane, at least when compared to the challenges offered up by the Yang-Mills equations. True to this spirit, Taubes announced at the time that he would never again write a paper more than twenty pages long. Of course, this resolution lasted only for about six months! After that, Taubes wrote a whole series of deep, technical, and very long papers that became known by the slogan 'Seiberg-Witten - Gromov'. These papers establish the most profound results known to this day about the Seiberg-Witten equations, linking their solutions on symplectic four-manifolds to Gromov's pseudo-holomorphic curve theory in a very precise way.
Taubes's work on the Seiberg-Witten equations remains one of the cornerstones underpinning the current very productive symbiosis between symplectic geometry and low-dimensional topology. Nevertheless, it was a shock to many when Taubes knocked off one of the holy grails of symplectic topology last year. The Weinstein conjecture predicts the existence of periodic orbits for the Reeb flows of arbitrary contact forms on closed three-manifolds. Many special cases had been proved by a variety of methods from symplectic geometry and a proof of the full conjecture was one of the ultimate goals of symplectic field theory. However, Taubes's proof follows a rather different line, using gauge theory and deploying a strategy similar to his work on 'Seiberg-Witten - Gromov'. The proof also hinges on a novel estimate for the spectral flow of a family of Dirac-type operators.
It is in the nature of Cliff Taubes's work that his papers are not usually short or easy to read. Rather they are difficult and original, and technically demanding by necessity. His faithful readers take comfort in the knowledge that Cliff is at least as hard on himself as he is on the readers, and they appreciate his very personal style peppered with what they affectionately refer to as 'Cliffisms'.
4.4. Biographical Sketch of Taubes.
Clifford Taubes grew up in Rochester, New York. After his undergraduate education at Cornell University, he received a Ph.D. in physics from Harvard University in 1980. After a Harvard Junior Fellowship, taken in the mathematics department at Harvard, he taught for two years at the University of California, Berkeley. Since 1985 he has been at Harvard, where he is the William Petschek Professor of Mathematics. He received the AMS Veblen Prize (1991) and the Élie Cartan Prize of the French Mathematical Society (1993). He is a member of both the American Academy of Arts and Sciences and of the National Academy of Sciences.
5.1. The Shaw Prizes.
In 2002, under the auspice of Run Run Shaw, a visionary philanthropist, the Shaw Prize Foundation was established. The inaugural Shaw Prize was presented two years later in 2004. The Shaw Prize consists of three annual awards, namely the Prize in Astronomy, the Prize in Life Science and Medicine, and the Prize in Mathematical Sciences. Each of these awards carries the amount of 1 million dollars.
The Shaw Prize honours individuals, regardless of race, nationality, gender, and religious belief, who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The Shaw Prize is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.
The Shaw Prize Certificate presented to each laureate is mounted elegantly on a dark brown leather folder. Within the folder, the left side features a decorative rendering, displaying the Shaw Prize medal in relief, and the motto of the Prize, "for the benefit of humankind", in gold engraving. The decoration comes in three distinct colours, each representing one of the award categories. For mathematics, the colour is red. On the certificate, "The Shaw Prize in Mathematical Sciences", engraved in gold, is displayed prominently. Below is the name of the laureate, meticulously handwritten by a local calligrapher, followed by the citation and the date of the award. Each certificate is signed by the Chair of the Board of Adjudicators and the Chair of the Shaw Prize Council.
5.2. The 2009 Shaw Prize in Mathematics.
The 2009 Shaw Prize in Mathematics was awarded jointly to Simon K Donaldson and Clifford H Taubes for their many brilliant contributions to geometry in 3 and 4 dimensions.
5.3. The contribution of Simon K Donaldson and Clifford H Taubes.
Geometry and Physics have been closely related from the earliest times and the differential calculus of Newton and Leibniz became the common mathematical tool that connected them. The geometry of 2-dimensional surfaces was fully explored by these techniques in the 19th century. It was closely related to algebraic curves and also to the flow of fluids.
Extending our understanding to 3-dimensional space and 4-dimensional space-time has been fundamental for both geometers and physicists in the 20th and 21st centuries. While the calculus is still employed, the problems are now much deeper and totally new phenomena appear.
Simon K Donaldson and Clifford H Taubes are the two geometers who have transformed the whole subject by pioneering techniques and ideas originating in theoretical physics, including quantum theory.
Electromagnetism is governed by the famous differential equations of Clerk Maxwell and these equations were used in the early 20th century by William Hodge as geometric tools. They were particularly useful in the geometry associated with algebraic equations, extending the work of the 19th century mathematician Bernhard Riemann.
The physical forces involved in the atomic nucleus are governed by the Yang-Mills equations which generalise Maxwell's equations but, being non-linear, are much deeper and more difficult. It was these equations which Donaldson used, basing himself on analytical foundations of Taubes, to derive spectacular new results. These opened up an entirely new field where more and more subtle geometric results have been established by Donaldson, Taubes and their students. The inspiration has frequently come from physics, but the methods are those of differential equations.
A key strand of this newly developing theory is the close relation that has been found between solutions of the Yang-Mills equations and the geometry of surfaces embedded in 4 dimensions. A definitive result in this direction is a beautiful theorem of Taubes which essentially identifies certain "quantum invariants" with others of a more classical nature. Many old conjectures have been settled by these new techniques, but many more questions still pose a challenge for the future. Donaldson and Taubes between them have totally changed our geometrical understanding of space and time.
Mathematical Sciences Selection Committee
The Shaw Prize
16 June 2009, Hong Kong
5.4. Essay on the 2009 Shaw Prize.
This essay was based on the speech by Sir Michael Atiyah who was chairman of Mathematical Sciences Selection Committee:
Over the past 30 years, geometry in 3 and 4 dimensions has been totally revolutionised by new ideas emerging from theoretical physics. Old problems have been solved but, more importantly, new vistas have been opened up which will keep mathematicians busy for decades to come.
While the initial spark has come from physics (where it was extensively pursued by Edward Witten), the detailed mathematical development has required the full armoury of non-linear analysis, where deep technical arguments have to be carefully guided by geometric insight and topological considerations.
The two main pioneers who both initiated and developed key aspects of this new field are Simon K Donaldson and Clifford H Taubes. Together with their students, they have established an active school of research which is both wide-ranging, original and deep. Most of the results, including some very recent ones, are due to them.
To set the scene, it is helpful to look back over the previous two centuries. The 19th century was dominated by the geometry of 2-dimensional surfaces, starting with the work of Abel on algebraic functions, and developing into the theory of complex Riemann surfaces. By the beginning of the 20th century, Poincare had introduced topological ideas which were to prove so fruitful, notably in the work of Hodge on higher dimensional algebraic geometry and also in the global analysis of dynamical systems.
In the latter half of the 20th century there was spectacular progress in understanding the topology of higher dimensional manifolds and fairly complete results were obtained in dimensions 5 or greater. The two "low dimensions" of 3 and 4, arguably the most important for the real physical world, presented serious difficulties but these were expected to be surmounted, along established lines, in the near future.
In the 1980's this complacent view was shattered by the impact of new ideas coming from physics. The first breakthrough was made by Simon K Donaldson in his PhD thesis where he used the Yang-Mills equations of -gauge theory to study 4-dimensional smooth (differentiable) manifolds. Specifically, Donaldson studied the moduli (or parameter) space of all -instantons, solutions of the self-dual Yang-Mills equations (which minimise the Yang-Mills functional), and used it as a tool to derive results about the 4-manifold. This instanton moduli space depends on a choice of Riemannian metric on the 4-manifold but Donaldson was able to produce results which were independent of the metric.
There are serious analytical difficulties in carrying out this programme and Donaldson had to rely on the earlier work of Karen Uhlenbeck and Clifford H Taubes. As these new ideas were developed and expanded by Donaldson, Taubes and others, spectacular results came tumbling out. Here is an abbreviated list, which shows the wide and unexpected gulf between topological 4-manifold (where the problems had just been solved by Michael Freedman) and smooth 4-manifold:
One of Taubes' great achievements was to relate Seiberg-Witten invariants to those introduced earlier by Gromov for symplectic manifolds. Such manifolds occur both as phase spaces in classical mechanics and in complex algebraic geometry, through the Kahler metrics inherited from projective space and exploited by Hodge. Although symplectic manifolds need not carry a complex structure, they always carry an almost (i.e., non-integrable) complex structure. Gromov introduced the idea of "pseudo-holomorphic curves" on symplectic manifolds and obtained invariants by suitably counting such curves. Taubes, in a series of long and difficult papers, proved that, for a symplectic 4-manifold, the Seiberg-Witten invariants essentially coincide with the Gromov-Witten invariants (an extension of the Gromov invariants). The key step in the work of Taubes is the construction of a pseudo-holomorphic curve from a solution of the Seiberg-Witten equations. This is fundamental since it connects gauge theory (a theory of potentials and fields) to sub-varieties (curves). Roughly, it represents a kind of non-linear duality.
In fact, extending complex algebraic geometry to symplectic manifolds (of any even dimension) was again pioneered by Donaldson who proved various existence theorems such as the existence of symplectic submanifolds. In the apparently large gap between algebraic geometry and theoretical physics, symplectic manifolds form a natural bridge and the recent results of Donaldson, Taubes and others provide, so to speak, a handrail across the bridge.
All this work in 4 dimensions has an impact on 3 dimensions, especially through the work of Andreas Floer, and Taubes has made many contributions in this direction. His most outstanding result is his very recent proof, in 3 dimensions, of a long-standing conjecture of Alan Weinstein. This asserts the existence of a closed orbit for a Reeb vector field on a contact 3-manifold. Contact 3-manifold arise naturally as level sets of Hamiltonian functions (energy) on a symplectic 4-manifold, and the Weinstein conjecture now asserts the existence of a closed orbit of the Hamiltonian vector field. This latest tour de force of Taubes exhibits his real power as a geometric analyst.
In recent years Donaldson has turned his attention to the hard problem of finding Hermitian metrics of constant scalar curvature on compact complex manifolds. The famous solution by Yau of the Calabi conjecture is an example of such problems. Donaldson has recast the constant scalar curvature problem in terms of moment maps, an idea derived from symplectic geometry which played a key role in gauge theory. This construction of metrics is a much deeper problem, being extremely non-linear but Donaldson has already made incisive progress on the analytical questions involved. This new work of Donaldson represents an exciting new advance which is currently attracting much attention.
This quick summary of the contributions of both Donaldson and Taubes shows how they have transformed our understanding of 3 and 4 dimensions. New ideas from physics, together with deep and delicate analysis in a topological framework, have been the hallmark of their work. They are fully deserving of the Shaw Prize in Mathematical Sciences for 2009.
Mathematical Sciences Selection Committee
The Shaw Prize
7 October 2009, Hong Kong
5.5. Clifford Taubes autobiography.
[Note by EFR. In his autobiography, Taubes confused Alan Weinstein and Erick Weinberg, creating a fictitious physicist called Eric Weinstein. We have corrected his error in the version we give below.]
I graduated from Cornell University with a degree in Physics in 1975, and then spent a year as a graduate student in the Astronomy Department at Princeton University. I found this not to my liking, and was fortunate enough to have met Bill Press, who helped facilitate a transfer to Harvard University where I enrolled as a graduate student in the Physics Department. I worked for a year there under the tutelage of Bill Press and Larry Smarr (then a Junior Fellow at Harvard), but soon moved to the more theoretical side of that department. My first substantive paper on gauge theoretic mathematics was the lucky result of hearing a lecture at Harvard by the physicist Erick Weinberg from Columbia. He posed a problem about the existence of solutions to the so-called vortex equations; these come from the Ginzburg-Landau model for superconducting vortices. I went home and stumbled on a proof that the postulated solutions did indeed exist. This was roughly in 1978, and to my amusement, these same vortex equations have been with me in one form or another for the past thirty years.
I did further work to elucidate the structure of the solutions to these vortex equations, but I also studied a non-abelian analogue that is called the Bogolmony monopole equations. These equations came from a version of the then new Weinberg-Salam theory that unifies the weak and electromagnetic forces. Only one solution to the latter was known to exist at the time. During this time at Harvard, I was influenced most by Raoul Bott. Raoul had his class on differential geometry and topology; I and myriad others were enthralled by his glorious lectures.
I more or less finished writing my PhD thesis in the fall of 1979, and with six months until the June 1980 graduation, was at a bit of a loose end. Raoul Bott suggested that I hang out at the Institute of Advanced Studies to talk with a differential equations specialist by the name of Karen Uhlenbeck. Steve Adler was kind enough to arrange an unofficial sort of stay, and so I headed down to Princeton. Karen had a profound effect on my subsequent view of mathematics. She taught me (and is still teaching me) a tremendous amount, although I am too dim to learn more than a fraction of her wisdom.
It was while visiting the Institute that I tripped over a general existence theorem for the Bogolmony monopoles. This good fortune came late one night in the Princeton library.
Shortly after receiving my PhD, I applied my monopole equation techniques to prove an existence theorem for the instanton equations on 4-dimensional spaces. I was starting a Harvard Junior Fellowship, with an office in Harvard's Mathematics Department. My first theorem required that the ambient space have a positive definite intersection form. I told Raoul about my theorem, and he asked: 'Well, can you tell me some spaces with such intersection forms?' I said 'Gee, I dunno, maybe the 4-sphere and the complex projective plane?' I didn't pursue this question; this was the first of many serious errors of judgement on my part.
My fellow Shaw laureate, Simon Donaldson (then a graduate student at Oxford) had the sense to ask and pursue this intersection form question, and so came to his first great theorem about smooth, 4-dimensional manifolds. I met Simon about a year later while spending the fall at Oxford at the invitation of Sir Michael Atiyah. Here, I truly began my lifelong study of mathematics under Simon's unknowing guidance. I freely admit that I owe Simon a tremendous debt for all that I have learned from him (and am still learning from him) over the subsequent years; and for his kindness to me as well.
The work on the instanton equations and their applications by Simon and then others, opened a vast new vista with regard to the interplay between geometry and topology in three and four dimensions. I was fortunate enough to be in at the beginning, and to have stumbled to more than my fair share of useful theorems.
A second great vista opened with Ed Witten's suggestion in 1994 to use what are called the Seiberg-Witten equations to study questions in four dimensional differential topology. I was again most fortunate to be in at the start of this new revolution; and to have Tom Mrowka and Peter Kronheimer as great friends and compatriots in the subsequent investigations of Witten's proposal. I owe them both a huge and steadily increasing debt as well.
I attribute my success to luck. I have been very, very lucky in life; and also in being influenced by great and wise people. I have mentioned some already; two others who taught me about both life and mathematics are Rob Kirby and S-T Yau.
7 October 2009, Hong Kong
In 2002, under the auspice of Run Run Shaw, a visionary philanthropist, the Shaw Prize Foundation was established. The inaugural Shaw Prize was presented two years later in 2004. The Shaw Prize consists of three annual awards, namely the Prize in Astronomy, the Prize in Life Science and Medicine, and the Prize in Mathematical Sciences. Each of these awards carries the amount of 1 million dollars.
The Shaw Prize honours individuals, regardless of race, nationality, gender, and religious belief, who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The Shaw Prize is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.
The Shaw Prize Certificate presented to each laureate is mounted elegantly on a dark brown leather folder. Within the folder, the left side features a decorative rendering, displaying the Shaw Prize medal in relief, and the motto of the Prize, "for the benefit of humankind", in gold engraving. The decoration comes in three distinct colours, each representing one of the award categories. For mathematics, the colour is red. On the certificate, "The Shaw Prize in Mathematical Sciences", engraved in gold, is displayed prominently. Below is the name of the laureate, meticulously handwritten by a local calligrapher, followed by the citation and the date of the award. Each certificate is signed by the Chair of the Board of Adjudicators and the Chair of the Shaw Prize Council.
5.2. The 2009 Shaw Prize in Mathematics.
The 2009 Shaw Prize in Mathematics was awarded jointly to Simon K Donaldson and Clifford H Taubes for their many brilliant contributions to geometry in 3 and 4 dimensions.
5.3. The contribution of Simon K Donaldson and Clifford H Taubes.
Geometry and Physics have been closely related from the earliest times and the differential calculus of Newton and Leibniz became the common mathematical tool that connected them. The geometry of 2-dimensional surfaces was fully explored by these techniques in the 19th century. It was closely related to algebraic curves and also to the flow of fluids.
Extending our understanding to 3-dimensional space and 4-dimensional space-time has been fundamental for both geometers and physicists in the 20th and 21st centuries. While the calculus is still employed, the problems are now much deeper and totally new phenomena appear.
Simon K Donaldson and Clifford H Taubes are the two geometers who have transformed the whole subject by pioneering techniques and ideas originating in theoretical physics, including quantum theory.
Electromagnetism is governed by the famous differential equations of Clerk Maxwell and these equations were used in the early 20th century by William Hodge as geometric tools. They were particularly useful in the geometry associated with algebraic equations, extending the work of the 19th century mathematician Bernhard Riemann.
The physical forces involved in the atomic nucleus are governed by the Yang-Mills equations which generalise Maxwell's equations but, being non-linear, are much deeper and more difficult. It was these equations which Donaldson used, basing himself on analytical foundations of Taubes, to derive spectacular new results. These opened up an entirely new field where more and more subtle geometric results have been established by Donaldson, Taubes and their students. The inspiration has frequently come from physics, but the methods are those of differential equations.
A key strand of this newly developing theory is the close relation that has been found between solutions of the Yang-Mills equations and the geometry of surfaces embedded in 4 dimensions. A definitive result in this direction is a beautiful theorem of Taubes which essentially identifies certain "quantum invariants" with others of a more classical nature. Many old conjectures have been settled by these new techniques, but many more questions still pose a challenge for the future. Donaldson and Taubes between them have totally changed our geometrical understanding of space and time.
Mathematical Sciences Selection Committee
The Shaw Prize
16 June 2009, Hong Kong
5.4. Essay on the 2009 Shaw Prize.
This essay was based on the speech by Sir Michael Atiyah who was chairman of Mathematical Sciences Selection Committee:
Over the past 30 years, geometry in 3 and 4 dimensions has been totally revolutionised by new ideas emerging from theoretical physics. Old problems have been solved but, more importantly, new vistas have been opened up which will keep mathematicians busy for decades to come.
While the initial spark has come from physics (where it was extensively pursued by Edward Witten), the detailed mathematical development has required the full armoury of non-linear analysis, where deep technical arguments have to be carefully guided by geometric insight and topological considerations.
The two main pioneers who both initiated and developed key aspects of this new field are Simon K Donaldson and Clifford H Taubes. Together with their students, they have established an active school of research which is both wide-ranging, original and deep. Most of the results, including some very recent ones, are due to them.
To set the scene, it is helpful to look back over the previous two centuries. The 19th century was dominated by the geometry of 2-dimensional surfaces, starting with the work of Abel on algebraic functions, and developing into the theory of complex Riemann surfaces. By the beginning of the 20th century, Poincare had introduced topological ideas which were to prove so fruitful, notably in the work of Hodge on higher dimensional algebraic geometry and also in the global analysis of dynamical systems.
In the latter half of the 20th century there was spectacular progress in understanding the topology of higher dimensional manifolds and fairly complete results were obtained in dimensions 5 or greater. The two "low dimensions" of 3 and 4, arguably the most important for the real physical world, presented serious difficulties but these were expected to be surmounted, along established lines, in the near future.
In the 1980's this complacent view was shattered by the impact of new ideas coming from physics. The first breakthrough was made by Simon K Donaldson in his PhD thesis where he used the Yang-Mills equations of -gauge theory to study 4-dimensional smooth (differentiable) manifolds. Specifically, Donaldson studied the moduli (or parameter) space of all -instantons, solutions of the self-dual Yang-Mills equations (which minimise the Yang-Mills functional), and used it as a tool to derive results about the 4-manifold. This instanton moduli space depends on a choice of Riemannian metric on the 4-manifold but Donaldson was able to produce results which were independent of the metric.
There are serious analytical difficulties in carrying out this programme and Donaldson had to rely on the earlier work of Karen Uhlenbeck and Clifford H Taubes. As these new ideas were developed and expanded by Donaldson, Taubes and others, spectacular results came tumbling out. Here is an abbreviated list, which shows the wide and unexpected gulf between topological 4-manifold (where the problems had just been solved by Michael Freedman) and smooth 4-manifold:
- Many compact topological 4-manifold which have no smooth structure.
- Many inequivalent smooth structures on compact 4-manifold.
- Uncountably many inequivalent smooth structures on Euclidean 4-space.
- New invariants of smooth structures.
One of Taubes' great achievements was to relate Seiberg-Witten invariants to those introduced earlier by Gromov for symplectic manifolds. Such manifolds occur both as phase spaces in classical mechanics and in complex algebraic geometry, through the Kahler metrics inherited from projective space and exploited by Hodge. Although symplectic manifolds need not carry a complex structure, they always carry an almost (i.e., non-integrable) complex structure. Gromov introduced the idea of "pseudo-holomorphic curves" on symplectic manifolds and obtained invariants by suitably counting such curves. Taubes, in a series of long and difficult papers, proved that, for a symplectic 4-manifold, the Seiberg-Witten invariants essentially coincide with the Gromov-Witten invariants (an extension of the Gromov invariants). The key step in the work of Taubes is the construction of a pseudo-holomorphic curve from a solution of the Seiberg-Witten equations. This is fundamental since it connects gauge theory (a theory of potentials and fields) to sub-varieties (curves). Roughly, it represents a kind of non-linear duality.
In fact, extending complex algebraic geometry to symplectic manifolds (of any even dimension) was again pioneered by Donaldson who proved various existence theorems such as the existence of symplectic submanifolds. In the apparently large gap between algebraic geometry and theoretical physics, symplectic manifolds form a natural bridge and the recent results of Donaldson, Taubes and others provide, so to speak, a handrail across the bridge.
All this work in 4 dimensions has an impact on 3 dimensions, especially through the work of Andreas Floer, and Taubes has made many contributions in this direction. His most outstanding result is his very recent proof, in 3 dimensions, of a long-standing conjecture of Alan Weinstein. This asserts the existence of a closed orbit for a Reeb vector field on a contact 3-manifold. Contact 3-manifold arise naturally as level sets of Hamiltonian functions (energy) on a symplectic 4-manifold, and the Weinstein conjecture now asserts the existence of a closed orbit of the Hamiltonian vector field. This latest tour de force of Taubes exhibits his real power as a geometric analyst.
In recent years Donaldson has turned his attention to the hard problem of finding Hermitian metrics of constant scalar curvature on compact complex manifolds. The famous solution by Yau of the Calabi conjecture is an example of such problems. Donaldson has recast the constant scalar curvature problem in terms of moment maps, an idea derived from symplectic geometry which played a key role in gauge theory. This construction of metrics is a much deeper problem, being extremely non-linear but Donaldson has already made incisive progress on the analytical questions involved. This new work of Donaldson represents an exciting new advance which is currently attracting much attention.
This quick summary of the contributions of both Donaldson and Taubes shows how they have transformed our understanding of 3 and 4 dimensions. New ideas from physics, together with deep and delicate analysis in a topological framework, have been the hallmark of their work. They are fully deserving of the Shaw Prize in Mathematical Sciences for 2009.
Mathematical Sciences Selection Committee
The Shaw Prize
7 October 2009, Hong Kong
5.5. Clifford Taubes autobiography.
[Note by EFR. In his autobiography, Taubes confused Alan Weinstein and Erick Weinberg, creating a fictitious physicist called Eric Weinstein. We have corrected his error in the version we give below.]
I graduated from Cornell University with a degree in Physics in 1975, and then spent a year as a graduate student in the Astronomy Department at Princeton University. I found this not to my liking, and was fortunate enough to have met Bill Press, who helped facilitate a transfer to Harvard University where I enrolled as a graduate student in the Physics Department. I worked for a year there under the tutelage of Bill Press and Larry Smarr (then a Junior Fellow at Harvard), but soon moved to the more theoretical side of that department. My first substantive paper on gauge theoretic mathematics was the lucky result of hearing a lecture at Harvard by the physicist Erick Weinberg from Columbia. He posed a problem about the existence of solutions to the so-called vortex equations; these come from the Ginzburg-Landau model for superconducting vortices. I went home and stumbled on a proof that the postulated solutions did indeed exist. This was roughly in 1978, and to my amusement, these same vortex equations have been with me in one form or another for the past thirty years.
I did further work to elucidate the structure of the solutions to these vortex equations, but I also studied a non-abelian analogue that is called the Bogolmony monopole equations. These equations came from a version of the then new Weinberg-Salam theory that unifies the weak and electromagnetic forces. Only one solution to the latter was known to exist at the time. During this time at Harvard, I was influenced most by Raoul Bott. Raoul had his class on differential geometry and topology; I and myriad others were enthralled by his glorious lectures.
I more or less finished writing my PhD thesis in the fall of 1979, and with six months until the June 1980 graduation, was at a bit of a loose end. Raoul Bott suggested that I hang out at the Institute of Advanced Studies to talk with a differential equations specialist by the name of Karen Uhlenbeck. Steve Adler was kind enough to arrange an unofficial sort of stay, and so I headed down to Princeton. Karen had a profound effect on my subsequent view of mathematics. She taught me (and is still teaching me) a tremendous amount, although I am too dim to learn more than a fraction of her wisdom.
It was while visiting the Institute that I tripped over a general existence theorem for the Bogolmony monopoles. This good fortune came late one night in the Princeton library.
Shortly after receiving my PhD, I applied my monopole equation techniques to prove an existence theorem for the instanton equations on 4-dimensional spaces. I was starting a Harvard Junior Fellowship, with an office in Harvard's Mathematics Department. My first theorem required that the ambient space have a positive definite intersection form. I told Raoul about my theorem, and he asked: 'Well, can you tell me some spaces with such intersection forms?' I said 'Gee, I dunno, maybe the 4-sphere and the complex projective plane?' I didn't pursue this question; this was the first of many serious errors of judgement on my part.
My fellow Shaw laureate, Simon Donaldson (then a graduate student at Oxford) had the sense to ask and pursue this intersection form question, and so came to his first great theorem about smooth, 4-dimensional manifolds. I met Simon about a year later while spending the fall at Oxford at the invitation of Sir Michael Atiyah. Here, I truly began my lifelong study of mathematics under Simon's unknowing guidance. I freely admit that I owe Simon a tremendous debt for all that I have learned from him (and am still learning from him) over the subsequent years; and for his kindness to me as well.
The work on the instanton equations and their applications by Simon and then others, opened a vast new vista with regard to the interplay between geometry and topology in three and four dimensions. I was fortunate enough to be in at the beginning, and to have stumbled to more than my fair share of useful theorems.
A second great vista opened with Ed Witten's suggestion in 1994 to use what are called the Seiberg-Witten equations to study questions in four dimensional differential topology. I was again most fortunate to be in at the start of this new revolution; and to have Tom Mrowka and Peter Kronheimer as great friends and compatriots in the subsequent investigations of Witten's proposal. I owe them both a huge and steadily increasing debt as well.
I attribute my success to luck. I have been very, very lucky in life; and also in being influenced by great and wise people. I have mentioned some already; two others who taught me about both life and mathematics are Rob Kirby and S-T Yau.
7 October 2009, Hong Kong
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