Richard Schoen
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Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
. He is best known for the resolution of the
Yamabe problem The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: By computing a formula for how the scalar curvatur ...
in 1984.


Career

Born in Celina, Ohio, and a 1968 graduate of Fort Recovery High School, he received his B.S. from the
University of Dayton The University of Dayton (UD) is a private, Catholic research university in Dayton, Ohio. Founded in 1850 by the Society of Mary, it is one of three Marianist universities in the nation and the second-largest private university in Ohio. The univ ...
in mathematics. He then received his PhD in 1977 from
Stanford University Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...
. After faculty positions at the Courant Institute, NYU,
University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
, and
University of California, San Diego The University of California, San Diego (UC San Diego or colloquially, UCSD) is a public university, public Land-grant university, land-grant research university in San Diego, California. Established in 1960 near the pre-existing Scripps Insti ...
, he was Professor at
Stanford University Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...
from 1987–2014, as Bass Professor of Humanities and Sciences since 1992. He is currently Distinguished Professor and Excellence in Teaching Chair at the
University of California, Irvine The University of California, Irvine (UCI or UC Irvine) is a public land-grant research university in Irvine, California. One of the ten campuses of the University of California system, UCI offers 87 undergraduate degrees and 129 graduate and pr ...
. His surname is pronounced "Shane." Schoen received an NSF Graduate Research Fellowship in 1972 and a
Sloan Research Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ...
in 1979. Schoen is a 1983 MacArthur Fellow. He has been invited to speak at the International Congress of Mathematicians (ICM) three times, including twice as a Plenary Speaker. In 1983 he was an Invited Speaker at the ICM in Warsaw, in 1986 he was a Plenary Speaker at the ICM in Berkeley, and in 2010 he was a Plenary Speaker at the ICM in Hyderabad. For his work on the
Yamabe problem The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: By computing a formula for how the scalar curvatur ...
, Schoen was awarded the
Bôcher Memorial Prize The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five year ...
in 1989. He was elected to the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
in 1988 and to the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
in 1991, became
Fellow of the American Association for the Advancement of Science Fellowship of the American Association for the Advancement of Science (FAAAS) is an honor accorded by the American Association for the Advancement of Science (AAAS) to distinguished persons who are members of the Association. Fellows are elected ...
in 1995, and won a
Guggenheim Fellowship Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...
in 1996. In 2012 he became a
Fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
.List of Fellows of the American Mathematical Society
retrieved 2013-07-14.
He received the 2014–15 Dean’s Award for Lifetime Achievements in Teaching from Stanford University. In 2015, he was elected Vice President of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. He was awarded an Honorary Doctor of Science from the
University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands (county), West Midlands and Warwickshire, England. The university was founded i ...
in 2015. He received the
Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...
for 2017, shared with
Charles Fefferman Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He was awarded the Fields Medal in 1978 for his contrib ...
. In the same year, he was awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize by
Kazan Federal University Kazan (Volga region) Federal University (russian: Казанский (Приволжский) федеральный университет, tt-Cyrl, Казан (Идел буе) федераль университеты) is a public research uni ...
, and the
Rolf Schock Prize The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm, Sweden, in 1993 and, since 2005, are awarded every three years. Each recipient current ...
. He has had over 44 doctoral students, including
Hubert Bray Hubert Lewis Bray is a mathematician and differential geometry, differential geometer. He is known for having proved the Riemannian Penrose inequality. He works as professor of mathematics and physics at Duke University. Early life and education ...
,
José F. Escobar José Fernando "Chepe" Escobar (born 20 December 1954, in Manizales, Colombia) was a Colombian mathematician known for his work on differential geometry and partial differential equations. He was professor at Cornell University. He completed hi ...
, Ailana Fraser,
Chikako Mese Chikako Mese is an American mathematician known for her work in differential geometry, geometric analysis and the theory of harmonic maps. She is a professor of mathematics at Johns Hopkins University. Education and career Mese graduated from El ...
,
William Minicozzi William Philip Minicozzi II is an American mathematician. He was born in Bryn Mawr, Pennsylvania, Bryn Mawr, Pennsylvania, in 1967. Career Minicozzi graduated from Princeton University in 1990 and received his Ph.D. from Stanford University in 199 ...
, and
André Neves André da Silva Graça Arroja Neves (born 1975, Lisbon) is a Portuguese mathematician and a professor at the University of Chicago. He joined the faculty of the University of Chicago in 2016. In 2012, jointly with Fernando Codá Marques, he sol ...
.


Mathematical work

Schoen has investigated the use of analytic techniques in global
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, with a number of fundamental contributions to the regularity theory of minimal surfaces and harmonic maps.


Harmonic maps

In 1976, Schoen and
Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
used Yau's earlier Liouville theorems to extend the rigidity phenomena found earlier by
James Eells James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis. Biography Eells studied mathematics at Bowdoin College in Maine and earned his undergraduate degree in 1947. Afte ...
and Joseph Sampson to noncompact settings. By identifying a certain interplay of the Bochner identity for harmonic maps together with the second variation of area formula for minimal hypersurfaces, they also identified some novel conditions on the domain leading to the same conclusion. These rigidity theorems are complemented by their existence theorem for harmonic maps on noncompact domains, as a simple corollary of Richard Hamilton's resolution of the Dirichlet boundary-value problem. As a consequence they found some striking geometric results, such as that certain noncompact manifolds do not admit any complete metrics of nonnegative Ricci curvature. In two papers from the 1980s, Schoen and
Karen Uhlenbeck Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...
made a foundational contribution to the regularity theory of energy-minimizing
harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
s. The techniques they developed, making extensive use of monotonicity formulas, have been very influential in the field of
geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
and have been adapted to a number of other problems. Fundamental conclusions of theirs include compactness theorems for sets of harmonic maps and control over the size of corresponding singular sets.
Leon Simon Leon Melvyn Simon , born in 1945, is a Leroy P. Steele PrizeSee announcemen retrieved 15 September 2017. and Bôcher Memorial Prize, Bôcher Prize-winningSee . mathematician, known for deep contributions to the fields of geometric analysis, g ...
applied such results to obtain a clear picture of the small-scale geometry of energy-minimizing harmonic maps. Later, Mikhael Gromov had the insight that an extension of the theory of harmonic maps, to allow values in
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s rather than
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s, would have a number of significant applications, with analogues of the classical Eells−Sampson rigidity theorem giving novel rigidity theorems for lattices. The intense analytical details of such a theory were worked out by Schoen. Further foundations of this new context for harmonic maps were laid out by Schoen and Nicholas Korevaar.


Minimal surfaces, positive scalar curvature, and the positive mass theorem

In 1979, Schoen and his former doctoral supervisor,
Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
, made a number of highly influential contributions to the study of positive scalar curvature. By an elementary but novel combination of the Gauss equation, the formula for second variation of area, and the Gauss-Bonnet theorem, Schoen and Yau were able to rule out the existence of several types of stable minimal surfaces in three-dimensional manifolds of positive scalar curvature. By contrasting this result with an analytically deep theorem of theirs establishing the existence of such surfaces, they were able to achieve constraints on which manifolds can admit a metric of positive scalar curvature. Schoen and
Doris Fischer-Colbrie Doris Fischer-Colbrie is a ceramic artist and former mathematician. She received her Ph.D. in mathematics in 1978 from University of California at Berkeley, where her advisor was H. Blaine Lawson. Many of her contributions to the theory of minima ...
later undertook a broader study of stable minimal surfaces in 3-dimensional manifolds, using instead an analysis of the stability operator and its spectral properties. An inductive argument based upon the existence of stable minimal hypersurfaces allowed them to extend their results to higher dimensions. Further analytic techniques facilitated the application of topological surgeries on manifolds which admit metrics of positive scalar curvature, showing that the class of such manifolds is topologically rich. Mikhael Gromov and
H. Blaine Lawson Herbert Blaine Lawson, Jr. is a mathematician best known for his work in minimal surfaces, calibrated geometry, and algebraic cycles. He is currently a Distinguished Professor of Mathematics at Stony Brook University. He received his PhD fro ...
obtained similar results by other methods, also undertaking a deeper analysis of topological consequences. By an extension of their techniques to noncompact manifolds, Schoen and Yau were able to settle the important Riemannian case of the
positive mass theorem The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...
in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, which can be viewed as a statement about the geometric behavior near infinity of noncompact manifolds with positive scalar curvature. Like their original results, the argument is based upon contradiction. A more constructive argument, using the theory of harmonic spinors instead of minimal hypersurfaces, was later found by
Edward Witten Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
. Schoen, Yau, and
Leon Simon Leon Melvyn Simon , born in 1945, is a Leroy P. Steele PrizeSee announcemen retrieved 15 September 2017. and Bôcher Memorial Prize, Bôcher Prize-winningSee . mathematician, known for deep contributions to the fields of geometric analysis, g ...
identified a simple combination of the Simons formula with the formula for second variation of area which yields important curvature estimates for stable minimal hypersurfaces of low dimensions. In 1983, Schoen obtained similar estimates in the special case of two-dimensional surfaces, making use of the existence of
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
. Slightly weaker estimates were obtained by Schoen and Simon, although without any dimensional restriction. Fundamental consequences of the Schoen−Simon estimates include compactness theorems for stable minimal hypersurfaces as well as control over the size of "singular sets." In particular, the Schoen−Simon estimates are an important tool in the
Almgren–Pitts min-max theory In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's ...
, which has found a number of applications. The possible presence of singular sets restricts the dimensions in which Schoen and Yau's inductive arguments can be easily carried out. Meanwhile Witten's essential use of spinors restricts his results to topologically special cases. Thus the general case of the positive mass theorem in higher dimensions was left as a major open problem in Schoen and Yau's 1979 work. In 1988, they settled the problem in arbitrary dimension in the special case that the
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
is zero; this has been significant in conformal geometry. In 2017, they released a preprint claiming the general case, in which they deal directly with the singular sets of minimal hypersurfaces.


Yamabe problem and conformal geometry

In 1960,
Hidehiko Yamabe was a Japanese mathematician. Above all, he is famous for discovering that every conformal class on a smooth compact manifold is represented by a Riemannian metric of constant scalar curvature. Other notable contributions include his definitive ...
introduced the "Yamabe functional" on a conformal class of Riemannian metrics and demonstrated that a critical point would have constant
scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
. He made partial progress towards proving that critical points must exist, which was taken further by
Neil Trudinger Neil Sidney Trudinger (born 20 June 1942) is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations. After completing his B.Sc at the University of New England (Australia) ...
and
Thierry Aubin Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry and non-linear partial differential equations. His fundamental contrib ...
. Aubin's work, in particular, settled the cases of high dimension or when there exists a point where the
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
is nonzero. In 1984, Schoen settled the cases left open by Aubin's work, the decisive point of which rescaled the metric by the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
of the Laplace-Beltrami operator. This allowed an application of Schoen and Yau's positive mass theorem to the resulting metric, giving important asymptotic information about the original metric. The works of Yamabe, Trudinger, Aubin, and Schoen together comprise the solution of the
Yamabe problem The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: By computing a formula for how the scalar curvatur ...
, which asserts that there is a metric of constant scalar curvature in every conformal class. In 1989, Schoen was also able to adapt
Karen Uhlenbeck Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...
's bubbling analysis, developed for other geometric-analytic problems, to the setting of constant scalar curvature. The uniqueness of critical points of the Yamabe functional, and more generally the compactness of the set of all critical points, is a subtle question first investigated by Schoen in 1991. Fuller results were later obtained by
Simon Brendle Simon Brendle (born June 1981) is a German mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He ...
, Marcus Khuri,
Fernando Codá Marques Fernando Codá dos Santos Cavalcanti Marques (born 8 October 1979) is a Brazilian mathematician working mainly in geometry, topology, partial differential equations and Morse theory. He is a professor at Princeton University. In 2012, together ...
, and Schoen.


Differentiable sphere theorem

In the 1980s, Richard Hamilton introduced the
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
and proved a number of convergence results, most notably for two- and three-dimensional spaces. Although he and others found partial results in high dimensions, progress was stymied by the difficulty of understanding the complicated
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
.
Simon Brendle Simon Brendle (born June 1981) is a German mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He ...
and Schoen were able to prove that the positivity of Mario Micallef and John Moore's "isotropic curvature" is preserved by the Ricci flow in any dimension, a fact independently proven by Huy Nguyen. Brendle and Schoen were further able to relate their positivity condition to the positivity of
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
and of curvature operator, which allowed them to exploit then-recent algebraic ideas of Christoph Böhm and Burkhard Wilking, thereby obtaining a new convergence theorem for Ricci flow.Böhm, Christoph; Wilking, Burkhard. Manifolds with positive curvature operators are space forms. Ann. of Math. (2) 167 (2008), no. 3, 1079–1097. A special case of their convergence theorem has the
differentiable sphere theorem In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. ...
as a simple corollary, which had been a well-known conjecture in the study of positive sectional curvature for the past fifty years.


Selected publications

* * * * * * * * * * * * * * * * * * * Textbooks * *


See also

*
Almgren–Pitts min-max theory In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's ...
*
Harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
*
Sphere theorem In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. ...


References


External links


Personal web site
* * * {{DEFAULTSORT:Schoen, Richard 1950 births Members of the United States National Academy of Sciences 20th-century American mathematicians 21st-century American mathematicians Differential geometers American relativity theorists Living people MacArthur Fellows Stanford University alumni Stanford University Department of Mathematics faculty Fellows of the American Mathematical Society Fellows of the American Academy of Arts and Sciences Mathematicians from Ohio People from Fort Recovery, Ohio