Richard Schoen

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Richard Schoen
Richard Schoen in 1976
Born	(1950-10-23) October 23, 1950 (age 70) Fort Recovery, Ohio[1]
Nationality	American
Alma mater	Stanford University
Known for	Differentiable sphere theorem Schoen–Yau conjecture Solution of positive mass conjecture
Spouse(s)	Doris Fischer-Colbrie
Awards	Bôcher Memorial Prize (1989) Wolf Prize (2017) Heinz Hopf Prize (2017) Lobachevsky Prize (2017)[2] Rolf Schock Prize (2017)[3]
Scientific career
Fields	Mathematics
Institutions	Stanford University University of California, Berkeley University of California, Irvine
Doctoral advisor	Leon Simon Shing-Tung Yau
Doctoral students	Hubert Bray José F. Escobar Ailana Fraser Chikako Mese William Minicozzi André Neves

Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry.

Born in Celina, Ohio, and a 1968 graduate of Fort Recovery High School, he received his B.S. from the University of Dayton in mathematics. He then received his PhD in 1977 from Stanford University and is currently an Excellence in Teaching Chair at the University of California, Irvine. His surname is pronounced "Shane," perhaps as a reflection of the regional dialect spoken by some of his German ancestors.

Schoen is a 1983 MacArthur Fellow.

Contributions[]

Schoen has investigated the use of analytic techniques in global differential geometry. In 1979, together with his former doctoral supervisor, Shing-Tung Yau, he proved the fundamental positive energy theorem in general relativity. In 1983, he was awarded a MacArthur Fellowship, and in 1984, he obtained a complete solution to the Yamabe problem on compact manifolds. This work combined new techniques with ideas developed in earlier work with Yau, and partial results by Thierry Aubin and Neil Trudinger. The resulting theorem asserts that any Riemannian metric on a closed manifold may be conformally rescaled (that is, multiplied by a suitable positive function) so as to produce a metric of constant scalar curvature. In 2007, Simon Brendle and Richard Schoen proved the differentiable sphere theorem, a fundamental result in the study of manifolds of positive sectional curvature. He has also made fundamental contributions to the regularity theory of minimal surfaces and harmonic maps.

His students include Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves.^[4]

Awards and honors[]

For his work on the Yamabe problem, Schoen was awarded the Bôcher Memorial Prize in 1989. He joined the American Academy of Arts and Sciences in 1988 and the National Academy of Sciences in 1991, and won a Guggenheim Fellowship in 1996. In 2012 he became a fellow of the American Mathematical Society.^[5] In 2015, he was elected Vice President of the American Mathematical Society.^[6] He received the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman.^[7] In the same year, he was awarded the by Kazan Federal University.^[8]

Selected publications[]

• Schoen, Richard M.; Simon, Leon; Yau, Shing-Tung (1975), "Curvature estimates for minimal hypersurfaces", Acta Mathematica, 134 (3–4): 275–288, doi:10.1007/bf02392104, MR 0423263
• Schoen, Richard M.; Yau, Shing-Tung (1979), "On the proof of the positive mass conjecture in general relativity", Communications in Mathematical Physics, 65 (1): 45–76, Bibcode:1979CMaPh..65...45S, doi:10.1007/bf01940959, MR 0526976, S2CID 54217085
• Fischer-Colbrie, Doris; Schoen, Richard M. (1980), "The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature", Communications on Pure and Applied Mathematics, 33 (2): 199–211, doi:10.1002/cpa.3160330206, MR 0562550
• Schoen, Richard M.; Yau, Shing-Tung (1981), "Proof of the positive mass theorem. II", Communications in Mathematical Physics, 79 (2): 231–260, Bibcode:1981CMaPh..79..231S, doi:10.1007/bf01942062, MR 0612249, S2CID 59473203
• Schoen, Richard M.; Uhlenbeck, Karen (1982), "A regularity theory for harmonic maps", Journal of Differential Geometry, 17 (2): 307–335, doi:10.4310/jdg/1214436923, MR 0664498
• Schoen, Richard M. (1984), "Conformal deformation of a Riemannian metric to constant scalar curvature", Journal of Differential Geometry, 20 (2): 479–495, doi:10.4310/jdg/1214439291, MR 0788292
• Gromov, Mikhael; Schoen, Richard M. (1992), "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one", Institut des Hautes Études Scientifiques. Publications Mathématiques, 76: 165–246, doi:10.1007/bf02699433, MR 1215595, S2CID 118023776
• Schoen, Richard M.; Wolfson, Jon (2001), "Minimizing area among Lagrangian surfaces: the mapping problem", Journal of Differential Geometry, 58 (1): 1–86, arXiv:math/0008244, doi:10.4310/jdg/1090348282, MR 1895348, S2CID 13995369
• Brendle, Simon; Schoen, Richard M. (2009), "Manifolds with 1/4-pinched curvature are space forms", Journal of the AMS, 22 (1): 287–307, arXiv:0705.0766, Bibcode:2009JAMS...22..287B, doi:10.1090/s0894-0347-08-00613-9, MR 2449060, S2CID 2901565

See also[]

• Almgren–Pitts min-max theory
• Harmonic map
• Sphere theorem

References[]

External links[]

• Personal web site
• O'Connor, John J.; Robertson, Edmund F., "Richard Schoen", MacTutor History of Mathematics archive, University of St Andrews
• Richard Schoen at the Mathematics Genealogy Project
• Sormani, Christina (August 2018). "The Mathematics of Richard Schoen" (PDF). Notices of the American Mathematical Society. 65 (11): 1349–1376. doi:10.1090/noti1749.