Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps.
Early life and education Schoen was born in Celina, Ohio, on October 23, 1950. In 1968, he graduated from 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 with Leon Simon and Shing-Tung Yau as advisors.
Career After faculty positions at the Courant Institute, NYU, University of California, Berkeley, and University of California, San Diego, he was Professor at Stanford University from 1987 to 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. His surname is pronounced "Shane." Schoen received an NSF Graduate Research Fellowship in 1972 and a Sloan Research Fellowship 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, Schoen was awarded the Bôcher Memorial Prize in 1989. In 1988, he was elected to the American Academy of Arts and Sciences and to the National Academy of Sciences in 1991, became Fellow of the American Association for the Advancement of Science in 1995, and won a Guggenheim Fellowship in 1996. In 2012 he became a Fellow of the American Mathematical Society. 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. He was awarded an Honorary Doctor of Science from the University of Warwick in 2015. He received the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman. In the same year, he was awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize by Kazan Federal University, and the Rolf Schock Prize. He has had over 44 doctoral students, including Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves. Schoen has investigated the use of analytic techniques in global differential geometry, 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 used Yau's earlier Liouville theorems to extend the rigidity phenomena found earlier by James Eells 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 made a foundational contribution to the regularity theory of energy-minimizing harmonic maps. The techniques they developed, making extensive use of monotonicity formulas, have been very influential in the field of geometric analysis 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 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 spaces rather than Riemannian manifolds, 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.
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