# Prefaces and a citation for Richard Schoen

Richard M Schoen was sixty years old on 23 October 2010 and the volume

*Surveys in Geometric Analysis and Relativity*(International Press, 2011) edited by Hubert L Bray and William P Minicozzi II (eds.) consisting of 23 survey articles was dedicated "to Richard Schoen in honour of his sixtieth birthday." We give below the Preface of this volume. We also give the citation delivered on the occasion of Schoen being awarded an Hon DSc by the University of Warwick, England, on Tuesday, 14 July 2015. Finally we give the Preface by Hubert L Bray and William P Minicozzi II to the article 'The Mathematics of Richard Schoen',*Notices of the American Mathematical Society***65**(11) (2018), 1349-1376, communicated by Christina Sormani.**1. Preface to Surveys in Geometric Analysis and Relativity.**

This volume of 23 survey articles is dedicated to Richard M Schoen on the occasion of his 60th birthday in recognition of his many important contributions as a leading researcher in geometric analysis and general relativity. We also thank him for the equally important roles he has played as mentor, colleague, collaborator, and friend.

Rick Schoen was born on October 23, 1950 in Celina, Ohio. In 1972 he graduated summa cum laude from the University of Dayton and received an NSF Graduate Fellowship. In March 1977, Rick received his Ph.D. from Stanford University under the direction of Leon Simon and Shing-Tung Yau, and soon after received a Sloan Postdoctoral Fellowship. His early work was on minimal surfaces and harmonic maps. By the time that Rick received his Ph.D., he had already proven major results, including his 1975 curvature estimates paper with Simon and Yau.

In the late 1970's, Schoen and Yau developed new tools to study the topological implications of positive scalar curvature. This work grew out of their study of stable minimal surfaces, eventually leading to their proof of the positive mass theorem in 1979. All together, their work was impressive for the way it connected neighbouring fields, first using analysis to understand geometry, and then using geometry to understand physics.

In the early 1980's, Rick published a number of fundamental papers on minimal surfaces and harmonic maps. His work on minimal surfaces includes an influential Bernstein theorem for stable minimal surfaces with Doris Fischer-Colbrie. Rick met his future wife Doris in Berkeley, where Doris received her Ph.D. in 1978. They have two children, Alan and Lucy, seen in the photographs in this book, both of whom graduated from Stanford.

Other works from the early 1980's include an extremely useful curvature estimate for stable surfaces, a uniqueness theorem for the catenoid, and a partial regularity theory for stable hypersurfaces in high dimensions with Leon Simon. In 1982, Rick and Karen Uhlenbeck proved the partial regularity of energy minimising harmonic maps. In 1983, Rick was awarded the very prestigious MacArthur Prize Fellowship.

Rick is also very well known for his celebrated solution to the remaining cases of the Yamabe problem in 1984, this time using a theorem from physics, namely the positive mass theorem, to solve a famous problem in geometry. The resulting fundamental theorem in geometry, that every smooth Riemannian metric on a closed manifold admits a conformal metric of constant scalar curvature, had been open since the 60's. This work was cited in 1989 when Rick received the Bôcher prize of the American Mathematical Society. His work on scalar curvature at this time set the direction for the field for the next 25 years.

Rick was elected to the American Academy of Arts and Sciences in 1988 and the National Academy of Sciences in 1991. He has been a Fellow of the American Association for the Advancement of Science since 1995 and won a Guggenheim Fellowship in 1996.

Starting around 1990, Rick began two major programs. The first was to develop a theory of harmonic maps with singular targets, starting with a joint paper with Mikhail Gromov where they used harmonic maps to establish $p$-adic superrigidity for lattices in groups of rank one. In a series of papers, Rick and Nick Korevaar laid the foundations for a general theory of mappings to NPC spaces, established the basic existence and regularity results, and applied their theory to settle problems in a number of areas of mathematics. The second big program was a variational theory of Lagrangian submanifolds, including the existence and regularity theory, done in a series of papers with Jon Wolfson.

Over the last decade, Rick has continued to make major contributions to geometric analysis and general relativity. Among other results in general relativity, Rick has made fundamental contributions to the constraint equations (with Corvino and others) which dictate the range of possible initial conditions for a spacetime and proved theorems on the topology of higher dimensional black holes (with Galloway). In geometric analysis, he has several important results with Simon Brendle on Ricci flow, including the proof of the differentiable sphere theorem, as well as a compactness theorem for the Yamabe equation with Marcus Khuri and Fernando Marques.

Rick has written 2 books and roughly 80 papers and has solved an impressively wide variety of major problems and conjectures. He has supervised 35 students and counting, and he has hosted many postdocs. Even with his great success, Rick is still one of the hardest working people in mathematics, giving us all the distinct impression that he must love it. His impact on mathematics, both in terms of his ideas and the example he sets, continues to be tremendous.

We would like to thank all of the authors for their contributions, the publishers Lizhen Ji and Liping Wang for their help, as well as Jaigyoung Choe, Michael Eichmair, John Rawnsley, Peter Topping, and Doris Fischer-Colbrie for contributing photographs. We hope you enjoy reading the beautiful survey articles included in this volume as much as we have enjoyed helping to put it all together.

Hubert L Bray and William P Minicozzi II

April 20, 2011

**2. Citation for Schoen's Hon DSc at the University of Warwick.**

This oration was written by Dr Mario Micallef, Mathematics Institute, University of Warwick, and delivered by Professor Chris Hughes, Department of Politics and International Studies.

To Professor Richard Schoen: Hon DSc (3:00 pm ceremony on Tuesday, 14 July 2015).

Richard Melvin Schoen, or Rick, as he is more popularly known, is a distinguished mathematician, an inspiring teacher, and above all, a great person.

Rick works in what has become known as geometric analysis, which involves the study of curved spaces through partial differential equations. Mathematically, Einstein's general theory of relativity consists of a set of partial differential equations which describe how the distribution and movement of matter and energy in the universe determine its curvature. In a joint work with Yau, Rick has proved the positive mass theorem, an achievement that was a watershed for the mathematical development of general relativity. Rick was awarded the Bôcher prize for his solution of the Yamabe problem; only 34 mathematicians have received this prize from the American Mathematical Society over the past 94 years. More recently, he has solved, together with Brendle, the differentiable sphere theorem which challenged differential geometers for the preceding 60 years. He has made fundamental contributions to harmonic map theory (relevant to the mathematical description of liquid crystals) and to the study of surfaces of least area (which model soap films).

But Rick is much more than a problem solver. He is one of those rare mathematicians who has a deep vision for the subject. Yau, a Fields medallist (mathematical equivalent of Nobel prize winner), describes Rick as "the major engine for the great success and development of geometric analysis in the last forty years."

Rick is a gifted teacher. In his lectures, he presents complex ideas and calculations with ease, making them all seem natural and accessible. I was very fortunate to be his first PhD student at the Courant Institute in New York and, since then, he has had 37 more students and more than 110 descendants, three of whom are here today. More are on the way. Rick is amazingly creative, a quality that is rarely attributed to scientists and mathematicians. When you discuss a mathematical problem with Rick, he always makes constructive suggestions on how to sharpen an idea or replace it by a more fruitful approach. Through this generosity with his mathematical insight and wisdom, he attracts a constant stream of visitors and he has been an invaluable mentor to several postdoctoral researchers. He has created a school of geometric analysts, a world-wide community that includes preeminent scholars who hold leading positions at some of the world's finest universities.

Rick is a beacon of widening participation: he grew up on a farm in Fort Recovery, Ohio. He was the tenth child in a family of thirteen! He knows well the rigours of farming life, waking up early in the morning to help with farm duties before going to school. He did his undergraduate studies at his local University of Dayton. There, he excelled in Mathematics and he won a National Science Foundation Fellowship to pursue his PhD studies at Stanford University under the direction of S.-T. Yau and Leon Simon. Since then, Rick has had a highly successful mathematical career. He has held positions at the Courant Institute, UC Berkeley, and UC San Diego before returning to Stanford as Professor in 1987. He spent a year as a distinguished Visiting Professor at the Institute for Advanced Study in Princeton and he served as chair of the Stanford mathematics department for three years. He is currently the Anne T. and Robert M. Bass Professor of Humanities and Sciences. He also holds a distinguished professorship at UC Irvine. This success has been achieved through very hard work. Rick never tells his students to work hard but, seeing him working in his office most of the day, including weekends, we all got the message of the effort and perseverance required to succeed. However, he also finds time for sport. He was a keen baseball player and he currently also enjoys playing tennis. Rick once asked me to join him running; there was no way I could keep up with him and I gave up in less than a minute!

Rick is a connoisseur of mathematics. His mathematical vision and his discernment of fine mathematics is admired and appreciated by everyone. No surprise then, that Rick serves on many committees. He has served on the Board of Governors of the Institute for Mathematics and its Applications and on the Fachbeirat of the Max Planck Institute for Gravitational Physics in Potsdam-Golm, Germany. Currently, he is Scientific Adviser to a big project in geometric analysis here at Warwick.

Rick's prominence in mathematics has been recognised in many ways. He is a member of the American Academy of Arts and Sciences and of the National Academy of Sciences. He has been the recipient of prestigious fellowships. The International Congress of Mathematics is held once every four years - in the same year as the World Cup, but not as widely televised - and it is always a great honour for a mathematician to be invited to speak at a Congress. Rick has been a speaker at three International Congresses, including twice as a plenary speaker.

The Schoen mathematical family will be celebrating Rick's 65th birthday with two conferences. One will start tomorrow at Imperial College and it will provide Rick's students with an opportunity to thank him for his generosity, his guidance and his unfailing support. The other will be held here at Warwick next week. This will honour Rick for his deep and permanent impact on mathematics; Rick's contributions, like those of Gauss, Riemann and Einstein - pioneers of differential geometry and general relativity - will still be taught centuries from now.

Provost, it is an honour and a great personal pleasure for me, on behalf of the Senate, to present to you for admission to the degree of Doctor of Science, honoris causa, Professor Richard Melvin Schoen.

**3. Preface by Hubert L Bray and William P Minicozzi II to 'The Mathematics of Richard Schoen'.**

For more than forty years Richard Schoen has been a leading figure in geometric analysis, connecting ideas between analysis, geometry, topology, and physics in fascinating and unexpected ways. In 2017 Richard Schoen was awarded the Wolf Prize for these fundamental contributions and for his "understanding of the interconnectedness of partial differential equations and differential geometry." In this article we survey some of his many fundamental ideas.

Rick Schoen was born in 1950 in Celina, Ohio. He was the tenth in a family of thirteen children growing up on a farm. 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.

In 1972 Schoen graduated summa cum laude from the University of Dayton and received an NSF Graduate Fellowship. In March 1977 Rick received his PhD from Stanford University under the direction of Leon Simon and Shing-Tung Yau and soon after received a Sloan Postdoctoral Fellowship. His early work was on minimal surfaces and harmonic maps. By the time Schoen received his PhD, he had already proven major results, including his 1975 curvature estimates paper with Simon and Yau.

In the late 1970s Schoen and Yau developed new tools to study the topological implications of positive scalar curvature. This work grew out of their study of stable minimal surfaces, eventually leading to their proof of the positive mass theorem in 1979. Altogether, their work was impressive for the way it connected neighbouring fields, first using analysis to understand geometry and then using geometry to understand physics.

In the early 1980s, Schoen published a number of fundamental papers on minimal surfaces and harmonic maps. His work on minimal surfaces includes an influential Bernstein theorem for stable minimal surfaces with Doris Fischer-Colbrie. Schoen met his future wife, Fischer-Colbrie, in Berkeley, where she received her PhD in 1978. They have two children, Alan and Lucy, both of whom graduated from Stanford.

Other works from the early 1980s include an extremely useful curvature estimate for stable surfaces, a uniqueness theorem for the catenoid, and a partial regularity theory for stable hypersurfaces in high dimensions with Simon. In 1982, Schoen and Karen Uhlenbeck proved the partial regularity of energy-minimising harmonic maps. In 1983 Schoen was awarded the very prestigious MacArthur Prize Fellowship.

Schoen is also very well known for his celebrated solution to the remaining cases of the Yamabe problem in 1984, this time using a theorem from physics, namely the positive mass theorem, to solve a famous problem in geometry. The resulting fundamental theorem in geometry, that every smooth Riemannian metric on a closed manifold admits a conformal metric of constant scalar curvature, had been open since the 1960s. This work was cited in 1989 when Schoen received the Bôcher Prize of the American Mathematical Society. His work on scalar curvature at this time set the direction for the field for the next twenty-five years.

Schoen was elected to the American Academy of Arts and Sciences in 1988 and the National Academy of Sciences in 1991. He has been a Fellow of the American Association for the Advancement of Science since 1995 and won a Guggenheim Fellowship in 1996. Rick was elected Vice President of the AMS in 2015. He was awarded the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman. In 2017 he was also awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize, and the Rolf Schock Prize, to mention only a few of his awards.

Starting around 1990 Schoen began two major programmes. The first was to develop a theory of harmonic maps with singular targets, starting with a joint paper with Mikhail Gromov, where they used harmonic maps to establish $p$-adic superrigidity for lattices in groups of rank one. In a series of papers, Schoen and Nick Korevaar laid the foundations for a general theory of mappings to NPC spaces, established the basic existence and regularity results, and applied their theory to settle problems in a number of areas of mathematics. The second big program was a variational theory of Lagrangian submanifolds, including the existence and regularity theory, done in a series of papers with Jon Wolfson.

Over the last decade, Schoen has continued to make major contributions to geometric analysis and general relativity. Among other results in general relativity, Schoen has made fundamental contributions to the constraint equations (with Corvino and others), which dictate the range of possible initial conditions for a spacetime, proved theorems on the topology of higher-dimensional black holes (with Galloway), and proved the positive mass theorem in dimensions greater than seven (with Yau). In geometric analysis he has several important results with Simon Brendle on Ricci flow, including the proof of the differentiable sphere theorem, as well as a compactness theorem for the Yamabe equation with Marcus Khuri and Fernando Codá Marques.

Schoen has written two books and roughly eighty papers and has solved an impressively wide variety of major problems and conjectures. He has supervised around forty students and counting and has hosted many postdocs. Even with his great success, Schoen is still one of the hardest working people in mathematics, giving us all the distinct impression that he must love it. His impact on mathematics, both in terms of his ideas and the example he sets, continues to be tremendous.

Last Updated December 2023