Richard Melvin Schoen

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23 October 1950
Celina, Ohio, USA

Richard Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He has been awarded many major mathematics prizes including the Bôcher Memorial Prize, the Wolf Prize and the Rolf Schock Prize.


Richard Schoen is known to his friends and colleagues as Rick Schoen. His parents were Arnold Peter Schoen (1904-1986) and Rosemary Heitkamp (1911-2012). Arnold Schoen (son of Jacob Schoen and Elizabeth Knapke, born on 18 March 1904 in Mount Sterling, Ohio) was a farmer in Allen, Darke County, Ohio. On 8 November 1933 he married Rosemary Heitkamp (known as Rose). Rose Heitkamp, a daughter of Bernard John Heitkamp and Catherine Gehle, was born on 8 November 1911 in Saint Henry, Mercer County, Ohio. Arnold and Rose Schoen had thirteen children of which Richard (the subject of this biography) was the tenth. His older siblings were: Mary C Schoen; Virginia E Schoen; Eileen M Schoen; Janice A Schoen; Harold L Schoen; Patricia A Schoen; James A Schoen; George E Schoen; and Linda C Schoen. Rick [7]:-
... knows well the rigours of farming life, waking up early in the morning to help with farm duties before going to school.
Hubert L Bray and William P Minicozzi II write in [36]:-
He enjoyed farm work and has described driving a tractor to plough the fields as "great for thinking." His mother encouraged the children in their schooling, and his father was always inventing things. His older brothers, Hal and Jim, were both mathematics majors and inspired him to study mathematics.
More from this article is at THIS LINK.

Richard attended Sharpsburg Elementary School in Fort Recovery, then went to Fort Recovery High School. He graduated from the High School in 1968 and, later that year, began his studies at the University of Dayton, in Dayton, Ohio. This was the local university for his home town of Fort Recovery being less than 80 km to the south east. He graduated from Dayton with a B.S. in 1972, receiving the University of Dayton's Distinguished Alumnus Award and the Special Achievement Award, and then, in the same year, he entered Stanford University to undertake graduate studies. He was awarded a National Science Foundation Graduate Fellowship to fund his studies from 1972 to 1975. He explained how he came to have two thesis advisors in an article he wrote in tribute to Shing-Tung Yau [41]:-
I first met Shing-Tung Yau in 1973 when I was a second year Ph.D. student at Stanford University and he was a newly arrived faculty member. We became mathematically involved through a reading course I was doing with Leon Simon on minimal hypersurfaces. This led to a three-way joint work on properties of stable minimal hypersurfaces. I continued to work with both Yau and Leon while I was a student and was officially their joint Ph.D. student. I spent several hours a day working with (mostly learning from) Yau when I was a student. He was interested in anything geometric, and he had ideas for approaching a vast range of problems. This was an incredible opportunity for me, and it gave me a great start on my research career. We wrote two more joint papers while I was a student.
These two joint papers were Curvature estimates for minimal hypersurfaces (1975), and Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature (1976). Schoen considered that he was very lucky to have the opportunity to work with Yau in these early days [41]:-
I have vivid memories of Yau from the early times: his tremendous dedication to his work (he was in his office day and night including weekends), his amazing breadth of knowledge and technique, his openness and generosity with his time.
Schoen left Stanford in 1976 to take up the position of Instructor in Mathematics at the University of California, Berkeley. He was awarded his Ph.D. from Stanford University in 1977 for his thesis Existence and Regularity Theorems for some Geometric Variational Problems. Soon Yau also arrived in Berkeley and their collaboration continued [41]:-
I left Stanford in 1976 to take up a two year instructorship in Berkeley. Yau came to Berkeley during my second year, and we continued our collaboration. It was in Berkeley that we began our work on scalar curvature and the positive mass theorem.
Yau was equally appreciative of having Schoen as a student, collaborator and friend [40]:-
I've collaborated closely, in particular, with Rick Schoen for about forty-five years and have done some of my best work with him. Although he started out as my student, I'm sure I've learned as much from him as he has from me. I truly value his friendship.
It was in Berkeley that Schoen met Doris Helga Fischer-Colbrie. She had been born in Vienna, Austria, on 12 January 1949 and had been educated at the University of California, Berkeley, receiving a BA in 1971 and a Master's degree two years later. She had been a Teaching Assistant at Berkeley while working for her doctorate advised by Blaine Lawson but, when Schoen arrived in Berkeley, she was a Research Assistant. She was awarded a Ph.D. in 1978 for her thesis Minimal Varieties: Theorems on Global Comportment and Local Existence. Later, on 29 October 1983, Richard Schoen and Doris Fischer-Colbrie were married; they had two children Alan (born 29 February 1984) and Lucy (born 25 April 1988). Before they married, they wrote a joint paper The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature which appeared in print in 1980. In it they studied minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. In the same year of 1980, Doris published the single-authored paper Some rigidity theorems for minimal submanifolds of the sphere.

Schoen spent two years as an Instructor at Berkeley, then in 1978 he took up an appointment as an Assistant professor at New York University. He spent two years in this post and during these years he made major breakthroughs with the work that he and Shing-Tung Yau had begun at Berkeley [41]:-
We expanded this work substantially over the next few years, and I remember wonderful times working together at Stanford during the summers of 1978 and 1979. During the 1979-80 academic year Yau organised a special year at the Institute for Advanced Study. This was another formative period in my career since there was so much going on in a wide variety of directions. I learned a lot and did some work that I am still proud of.
As indicated by this quote, Schoen was a Visiting Member at the Institute for Advanced Study, Princeton, during 1979-80, a visit that was funded by a Sloan Postdoctoral Fellowship. Justin Corvino and Daniel Pollack describe the advances made by Schoen at this time [4]:-
[A] watershed for the mathematical development of relativity is the celebrated work of Rick Schoen and S-T Yau from the late seventies on the 'Positive Mass Theorem'. Not only did their work employ serious tools of geometric analysis, including partial differential equations and geometric measure theory, to resolve a question motivated by gravitational physics, but they also established a link between the positivity of the mass of an isolated gravitational system and the relationship between positive scalar curvature and topology, a topic of interest to a broad range of mathematicians. In the early eighties, Schoen brought the Positive Mass Theorem to bear on the resolution of the famous Yamabe problem, providing more evidence to support the development of the mathematical theory of the constraint equations, and inspiring many others to do so.
In 1980 Schoen returned to the University of California, Berkeley, when he was appointed as a Professor. He spent eight years at the University of California, the first four being at Berkeley and the remaining three, 1984-87, being at San Diego. During these seven years, Schoen was invited to address the International Congress of Mathematicians twice. He give a 45 minute invited lecture at the Congress held in Warsaw in 1983, then gave one of the plenary lectures at the Congress held in Berkeley in August 1986. At the Berkeley Congress he gave the lecture New Developments in the Theory of Geometric Partial Differential Equations. Dennis DeTurck describes the content of Schoen's lecture [42]:-
The author surveys recent work on nonlinear elliptic partial differential equations which arise from geometric sources, concentrating especially on the Yamabe problem and the theory of harmonic mappings. For the former, an outline is given of the recent solution of Yamabe's conjecture (that every metric on a compact manifold is pointwise conformally equivalent to one with constant scalar curvature), including the use of the positive mass theorem and a discussion of regularity of weak solutions of Yamabe's equation.
The solution of the Yamabe problem on compact manifolds, which Schoen discussed in this lecture, is one of his greatest achievements. He solved this problem in 1984. He had been awarded a MacArthur Fellowship in August 1983 (two awards went to mathematicians with Karen Uhlenbeck receiving one at the same time) and he held this Fellowship until 1988. He had returned to Stanford University in 1987 and continued to work at Stanford as the Anne T and Robert M Bass Professor of Humanities and Sciences. Honours came rapidly: he was elected to the American Academy of Arts and Sciences in 1988; and he was awarded the Bôcher Memorial Prize by the American Mathematical Society in 1989 [2]:-
... for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a Riemannian metric to constant scalar curvature".
Further honours acknowledged his many achievements: he was elected to the National Academy of Sciences in 1991, he became a fellow of American Association for Advancement of Science in 1995, and in the following year he was awarded a Guggenheim Fellowship. The Stanford University News described his work in the following terms when they announced his election to the National Academy of Sciences on 30 April 1991 [10]:-
Schoen, 40, continues his research in differential geometry, nonlinear partial differential equations and the calculus of variations. He constructs and analyses geometric objects that optimise certain physical or geometric energies. For example, he has developed new ways to understand surfaces of least area spanning a curve in three-dimensional space - the mathematical model for soap films. Schoen's ideas have been applied to a wide range of mathematical problems, from general relativity to questions about rigidity for lattice subgroups of algebraic groups.
We must not give the impression that Schoen's major research contributions stopped in the 1990s. Far from it and, just to give one example of a later highly significant work, let us note his achievement in 2007 when, in collaboration with Simon Brendle, he proved the differentiable sphere theorem. This is a fundamental result in the theory of manifolds with positive sectional curvature.

In addition to the visiting positions which we mentioned above, Schoen was a Visiting Member of the Institute for Advanced Study, Princeton, in the Spring of 1984, a Visiting Professor at the Courant Institute, New York University in academic year 1989-90, Distinguished Visiting Professor at the Institute for Advanced Study, Princeton, in academic year 1992-93, a Visiting Professor at Harvard University in the autumn of 1999, and Eilenberg Chair at Columbia University in the autumn of 2009.

Schoen has published two important books, both in collaboration with Shing-Tung Yau, which were based on lecture courses. In 1994 they published Lectures on differential geometry. We give the first and last paragraphs from a review by Man Chun Leung [9]:-
As the authors note in their introduction, the book under review was written for the lecture series given at Princeton University in 1983 and at the University of California, San Diego, in 1984 and 1985. The book contains significant results in differential geometry and global analysis; many of them are the works of the authors. The main topics are differential equations on a manifold and the relation between curvature and topology of a Riemannian manifold. There are nine chapters in the book, with the last three chapters more like appendices, which focus on problems concerning different areas of differential geometry.
The book under review is very well written. Readers will find comprehensive and detailed discussions of many significant results in geometric analysis. The book is both useful as a reference book for researchers and as a course book for graduate students. With details of proofs and background materials presented in a concise and delightful way, the book provides access to some of the most exciting areas in differential geometry.
The second book is Lectures on harmonic maps (1997). We give a short quote from a detailed review by John C Wood [39]:-
This is a very useful contribution to the literature on harmonic maps. It is not an elementary textbook on harmonic maps ... It is rather a collection of some of the most important topics in, and applications of, harmonic maps, skewed towards the authors' interests. ... this is a book that everyone interested in harmonic maps will benefit from reading.
Other important contributions to mathematics by Schoen include his editorial work. He served on the editorial boards of: the Journal of Differential Geometry, Communications in Analysis and Geometry, Communications in Partial Differential Equations, Calculus of Variations and Partial Differential Equations, and Communications in Contemporary Mathematics. He also does important work for the American Mathematical Society serving on various committees. For example he served on the Committee to Select the Winner of the E H Moore Research Article Prize, the Committee for National Awards and Public Representation, and the Committee to award Steele Prizes.

Few lecturers receive such praise from their undergraduate students as Schoen does. We give just one short quote from many [34]:-
Best Professor I've had so far at Stanford. His lectures are very lucid and compliment the text.
We mentioned above that Schoen lectured twice to the International Congress of Mathematicians in the 1980s. He was also an invited plenary speaker at the International Congress of Mathematicians held in Hyderabad in 2010 when he gave the lecture Riemannian manifolds of positive curvature.

The book [3], Surveys in Geometric Analysis and Relativity, contains 23 survey articles and is dedicated to Richard Schoen on the occasion of his 60th birthday. You can read the Preface to this volume at THIS LINK.

Schoen continued to be awarded prizes and given honours for his outstanding contributions. He was elected a Fellow of the American Mathematical Society in 2012, then he received the H&S Dean's Teaching Awards 2014-15 from Stanford University [25]:-
The Dean's Award for Distinguished Teaching recognises the efforts of exceptional teachers in the School of Humanities and Sciences and is given for excellence in graduate education, achievements in teaching, and first years of teaching at Stanford.
In 2015 he received an Honorary Doctor of Science from the University of Warwick, England [38]:-
Dr Richard Schoen received his PhD from Stanford University in 1977. He held positions at the Courant Institute, UC Berkeley, and UC San Diego before returning to Stanford as Professor in 1987. He is currently the Anne T and Robert M Bass Professor of Humanities and Sciences. He was chair of the Stanford mathematics department from 2001 until 2004. ...
Schoen received the honorary D.Sc. from the University of Warwick on 14 July 2015. An oration was written by Mario Micallef, University of Warwick Mathematics Institute, and delivered by Chris Hughes of the Department of Politics and International Studies. For the oration, see THIS LINK.

The year 2017 was a remarkable one for the awards that Schoen received. In that year he was awarded: the Wolf Prize; the Heinz Hopf Prize; the Lobachevsky Medal and Prize; and the Rolf Schock Prize. The Wolf Prize was awarded jointly to Richard Schoen and Charles Fefferman [21]:-
...for their striking contributions to analysis and geometry.
The citation for Richard Schoen reads [21]:-
Richard Schoen is awarded the Wolf Prize for being a pioneer and a driving force in geometric analysis. His work on the regularity of harmonic maps and minimal surfaces had a lasting impact on the field. His solution of the Yamabe problem is based on the discovery of a deep connection to general relativity. Through his work on geometric analysis, Schoen has contributed greatly to our understanding of the interrelation between partial differential equations and differential geometry. Many of the techniques he developed continue to influence the advance of non-linear analysis.
The 2017 Heinz Hopf Prize was awarded to Schoen [13]:-
... for his outstanding and foundational contributions to differential geometry and geometric analysis.
He was presented with the Prize at the award ceremony in ETH Zurich on 30 October 2017. He delivered two Heinz Hopf Lectures, one on the day of the ceremony and one on the following day.

The winner of the Lobachevsky Medal and Prize, Professor Richard Schoen [24]:-
... is a specialist in differential geometry. He proved the positive energy theorem in general relativity, obtained a complete solution to the Yamabe problem on compact manifolds, and has contributed to the regularity theory of minimal surfaces and harmonic maps.
The award ceremony was held on 1 December 2017, the official celebration of Nikolai Lobachevsky's 225th birthday.

He received the 2017 Rolf Schock Prize in Mathematics for [43]:-
... ground-breaking work in differential geometry and geometric analysis including the proof of the Yamabe conjecture, the positive mass conjecture, and the differentiable sphere theorem.
The award ceremony was held at the Royal Swedish Academy of Sciences on 14 November 2017.

We end this biography by quoting from [4] written by two mathematicians who received their doctorates with Schoen as advisor:-
The mathematical influence of Richard M Schoen can be measured in many ways. His research has fundamentally shaped geometric analysis, and his results form many cornerstones within geometry, partial differential equations and general relativity. Evidence of his influence includes the large number of his students who continue to work in these areas. As two of these students, the authors of this contribution are exceedingly grateful for Rick's mathematical insight and generosity.

References (show)

  1. Arnold Peter Schoen, Sidney Daily News (Sidney, Ohio) (7 April 1986).
  2. Bôcher Memorial Prize 1989: Richard M Schoen, American Mathematical Society.
  3. H L Bray and W P Minicozzi II (eds.), Surveys in Geometric Analysis and Relativity (International Press, 2011).
  4. J Corvino and D Pollack, Scalar Curvature and the Einstein Constraint Equations, in H L Bray and W P Minicozzi II (eds.), Surveys in Geometric Analysis and Relativity (International Press, 2011), 145-188.
  5. Fefferman and Schoen Awarded 2017 Wolf Prize in Mathematics, Notices of the American Mathematical Society 64 (5) (2017), 496-497.
  6. Fellows Database, Alfred P Sloan Foundation.
  7. C Hughes, Oration: Professor Richard Schoen: Hon DSc, University of Warwick (2015).
  8. ICM Plenary and Invited Speakers: Schoen, International Mathematical Union.
  9. M C Leung, Review: Lectures on differential geometry, by R Schoen and S T Yau, Mathematical Reviews MR1333601 (97d:53001).
  10. National Academy of Sciences elects three Stanford scholars, Stanford University News Service (7 May 1991).
  11. PIMS Marsden Memorial Lecture: Richard Schoen, American Mathematical Society (2016).
  12. PIMS Marsden Memorial Lecture: Richard Schoen, Pacific Institute for the Mathematical Sciences (19 July 2016).
  13. Professor Richard Schoen, Laureate 2017: Heinz Hopf Prize, Department of Mathematics, ETH Zurich.
  14. Professor Richard Schoen, Department of Mathematics, Imperial College London.
  15. Professor Richard M Schoen joins as Chair of the 2021 HLMA Scientific Committee, Hang Lung Mathematics Awards (2021).
  16. Professor Richard Schoen, Simons Laufer Mathematical Sciences Institute.
  17. Richard Schoen, Institute for Advanced Study.
  18. Richard Schoen, Clay Mathematics Institute (2013).
  19. Richard Schoen, Rolf Schock Prize - Mathematics 2017, The Royal Swedish Academy of Sciences (2017).
  20. Richard Schoen, Alchetron (11 September 2022).
  21. Richard Schoen, Wolf Prize 2017, Wolf Foundation.
  22. Richard Schoen, Stanford Profiles, Stanford University.
  23. Richard Schoen, Distinguished Professor, Excellence in Teaching Chair, Department of Mathematics, University of California Irvine (2018).
  24. Richard Schoen Announced as the Winner of the 2017 Lobachevsky Medal and Prize, Kazan University (30 August 2017).
  25. Richard Schoen, H&S Dean's Teaching Awards 2014-15, School of Humanities and Sciences, Stanford University.
  26. Richard M Schoen, MacArthur Foundation (1 August 1983).
  27. Richard M Schoen, National Academy of Sciences (1991).
  28. Richard M Schoen, John Simon Guggenheim Memorial Foundation (1996).
  29. Richard Melvin Schoen, American Academy of Arts and Sciences (September 2023).
  30. Rolf Schock Prizes 2017 awarded to four epoch-makers" Press release, The Royal Swedish Academy of Sciences (2017).
  31. Rose Mary Heitkamp Schoen, Find a Grave.*1gplukb*_gcl_au*NzQxNTk0OTkzLjE2OTY4OTA1Nzk.*_ga*MTIzNTE1ODQwMC4xNjU2NjI2MjIw*_ga_4QT8FMEX30*ZTQ4NDdlNzItMWYxZS00MmE4LWEzYWUtODA0YjNiYmY2ZDdlLjI3MS4xLjE2OTcxNDUwMDAuMi4wLjA.*_ga_6R6RRSB9ZD*ZTQ4NDdlNzItMWYxZS00MmE4LWEzYWUtODA0YjNiYmY2ZDdlLjUzLjEuMTY5NzE0NTAwMC4wLjAuMA..
  32. R M Schoen, The theory and applications of harmonic mappings between Riemannian manifolds (American Mathematical Society, Providence, RI, 1992).
  33. R M Schoen, New developments in the theory of geometric partial differential equations (American Mathematical Society, Providence, RI, 1992).
  34. R M Schoen and S-T Yau, Lectures on Harmonic Maps (International Press, Boston, 2013).
  35. R M Schoen and S-T Yau, Lectures on Differential Geometry (International Press, Boston, 2010).
  36. C Sormani (ed.), The Mathematics of Richard Schoen, Notices of the American Mathematical Society 65 (11) (2018), 1349-1376.
  37. Vice President Past Members, American Mathematical Society (2023).
  38. Warwick to honour Nobel Laureates, leading film director, journalist, both CBI & TUC heads, Indonesian author, travel guru and British Library Chief: Dr Richard Schoen Hon DSc, University of Warwick (13 July 2013).
  39. J C Wood, Review: Lectures on harmonic maps, by R Schoen and S T Yau, Mathematical Reviews MR1474501 (98i:58072).
  40. S-T Yau, The shape of a life (Yale University Press, 2019).
  41. Prof Yau from the eyes of colleagues, friends and students, Centre of Mathematical Sciences, Zhejiang University (22 February 2006).
  42. D M deTurck, Review: New developments in the theory of geometric partial differential equations, by Richard Schoen, Mathematical Reviews MR0934219 (89g:58217).
  43. Professor Richard Schoen awarded the 2017 Rolf Schock Prize, University of California Irvine (15 March 2017).
  44. L Gravé-Lazi, Wolf Prize to be awarded to eight laureates from US, UK and Switzerland, The Jerusalem Post (3 January 2017).

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Written by J J O'Connor and E F Robertson
Last Update December 2023