Robert Osserman
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Robert "Bob" Osserman (December 19, 1926 – November 30, 2011) was an American mathematician who worked in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. He is specially remembered for his work on the theory of
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s. Raised in
Bronx The Bronx () is a borough of New York City, coextensive with Bronx County, in the state of New York. It is south of Westchester County; north and east of the New York City borough of Manhattan, across the Harlem River; and north of the New ...
, he went to
Bronx High School of Science The Bronx High School of Science, commonly called Bronx Science, is a public specialized high school in The Bronx in New York City. It is operated by the New York City Department of Education. Admission to Bronx Science involves passing the Sp ...
(diploma, 1942) and
New York University New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then- Secretary of the Treasury Albert Gallatin. In 1832, ...
. He earned a Ph.D. in 1955 from
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
with the thesis ''Contributions to the Problem of Type'' (on
Riemann surfaces In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
) supervised by
Lars Ahlfors Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Background Ahlfors was born in Helsinki, Finland. His mother, Si ...
. He joined Stanford University in 1955. He joined the
Mathematical Sciences Research Institute The Simons Laufer Mathematical Sciences Institute (SLMath), formerly the Mathematical Sciences Research Institute (MSRI), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, Califo ...
in 1990. He worked on geometric function theory, differential geometry, the two integrated in a theory of minimal surfaces,
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
, and other issues in the areas of
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, geometry,
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an ...
and complex function theory. Osserman was the head of mathematics at
Office of Naval Research The Office of Naval Research (ONR) is an organization within the United States Department of the Navy responsible for the science and technology programs of the U.S. Navy and Marine Corps. Established by Congress in 1946, its mission is to pla ...
, a
Fulbright Lecturer The Fulbright Program, including the Fulbright–Hays Program, is one of several United States Cultural Exchange Programs with the goal of improving intercultural relations, cultural diplomacy, and intercultural competence between the people ...
at the
University of Paris The University of Paris (french: link=no, Université de Paris), Metonymy, metonymically known as the Sorbonne (), was the leading university in Paris, France, active from 1150 to 1970, with the exception between 1793 and 1806 under the French Revo ...
and
Guggenheim Fellow 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 ...
at the
University of Warwick , mottoeng = Mind moves matter , established = , type = Public research university , endowment = £7.0 million (2021) , budget = £698.2 million (202 ...
. He edited numerous books and promoted mathematics, such as in interviews with celebrities
Steve Martin Stephen Glenn Martin (born August 14, 1945) is an American actor, comedian, writer, producer, and musician. He has won five Grammy Awards, a Primetime Emmy Award, and was awarded an Honorary Academy Award in 2013. Additionally, he was nominat ...
and
Alan Alda Alan Alda (; born Alphonso Joseph D'Abruzzo; January 28, 1936) is an American actor, screenwriter, and director. A six-time Emmy Award and Golden Globe Award winner, he is best known for playing Captain Benjamin "Hawkeye" Pierce in the war come ...
. He was an
invited speaker at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...
(ICM) of 1978 in
Helsinki Helsinki ( or ; ; sv, Helsingfors, ) is the Capital city, capital, primate city, primate, and List of cities and towns in Finland, most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of U ...
. He received the Lester R. Ford Award (1980) of the Mathematical Association of America for his popular science writings. H. Blaine Lawson,
David Allen Hoffman David Allen Hoffman is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University. In 1985, together with William Meeks, he proved that Costa's surface was embedded. He is a fellow ...
and
Michael Gage Michael Gage may refer to: * Michael Gage (mathematician) * Michael Gage (politician) {{hndis, Gage, Michael ...
were Ph.D. students of his. Robert Osserman died on Wednesday, November 30, 2011 at his home.


Mathematical contributions


The Keller–Osserman problem

Osserman's most widely cited research article, published in 1957, dealt with the
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
:\Delta u=f(u). He showed that fast growth and monotonicity of is incompatible with the existence of global solutions. As a particular instance of his more general result: Osserman's method was to construct special solutions of the PDE which would facilitate application of the
maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
. In particular, he showed that for any real number there exists a rotationally symmetric solution on some ball which takes the value at the center and diverges to infinity near the boundary. The maximum principle shows, by the monotonicity of , that a hypothetical global solution would satisfy for any and any , which is impossible. The same problem was independently considered by
Joseph Keller Joseph Bishop Keller (July 31, 1923 – September 7, 2016) was an American mathematician who specialized in applied mathematics. He was best known for his work on the "geometrical theory of diffraction" (GTD). Early life and education Born i ...
, who was drawn to it for applications in electrohydrodynamics. Osserman's motivation was from differential geometry, with the observation that the
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 ...
of the Riemannian metric on the plane is given by :-e^\Big(\frac+\frac\Big). An application of Osserman's non-existence theorem then shows: By a different maximum principle-based method,
Shiu-Yuen Cheng Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
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 ...
generalized the Keller–Osserman non-existence result, in part by a generalization to the setting of a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
. This was, in turn, an important piece of one of their resolutions of the Calabi–Jörgens problem on rigidity of affine hyperspheres with nonnegative mean curvature.


Non-existence for the minimal surface system in higher codimension

In collaboration with his former student H. Blaine Lawson, Osserman studied the
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
problem in the case that the codimension is larger than one. They considered the case of a graphical minimal submanifold of euclidean space. Their conclusion was that most of the analytical properties which hold in the codimension-one case fail to extend. Solutions to the boundary value problem may exist and fail to be unique, or in other situations may simply fail to exist. Such submanifolds (given as graphs) might not even solve the Plateau problem, as they automatically must in the case of graphical hypersurfaces of Euclidean space. Their results pointed to the deep analytical difficulty of general elliptic systems and of the minimal submanifold problem in particular. Many of these issues have still failed to be fully understood, despite their great significance in the theory of
calibrated geometry In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (''M'',''g'') of dimension ''n'' equipped with a differential ''p''-form ''φ'' (for some 0 ≤ ''p'' ≤ ''n'') which is a calibration, meaning ...
and the Strominger–Yau–Zaslow conjecture.


Books

*''Two-Dimensional Calculus'' ( Harcourt, Brace & World, 1968; Krieger, 1977; Dover Publications, Inc, 2011) ; ; *''A Survey of Minimal Surfaces'' (1969, 1986) *''Poetry of the Universe: A Mathematical Exploration of the Cosmos'' (
Random House Random House is an American book publisher and the largest general-interest paperback publisher in the world. The company has several independently managed subsidiaries around the world. It is part of Penguin Random House, which is owned by Ger ...
, 1995)


Awards

*
John Simon Guggenheim Memorial Foundation The John Simon Guggenheim Memorial Foundation was founded in 1925 by Olga and Simon Guggenheim in memory of their son, who died on April 26, 1922. The organization awards Guggenheim Fellowships to professionals who have demonstrated exceptional ...
fellow (1976) * Lester R. Ford Award (1980) *2003
Joint Policy Board for Mathematics The Joint Policy Board for Mathematics (JPBM) consists of the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. The Board has nea ...
Communications Award.


Topics named after Robert Osserman

* Chern–Osserman inequality * Osserman conjecture in Riemannian geometry * Osserman manifolds *
Osserman's theorem In mathematics, Nirenberg's conjecture, now Osserman's theorem, states that if a neighborhood of the sphere is omitted by the Gauss map of a complete minimal surface, then the surface in question is a plane. It was proved by Robert Osserman Rober ...


Selected research papers

* * * * * *


References

{{DEFAULTSORT:Osserman, Robert 1926 births 2011 deaths 20th-century American mathematicians 21st-century American mathematicians New York University alumni Harvard University alumni Stanford University Department of Mathematics faculty Differential geometers American textbook writers