Harold William Rosenberg (born 19 February 1941 in

New York City New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...

) is an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who works on

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

. Rosenberg has worked at

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

, at the

Institut des Hautes Études Scientifiques The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics. It is located in Bures-sur-Yvette, just ...

, and at the

University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...

. He currently works at the

IMPA is a recurring fictional character in Nintendo's ''The Legend of Zelda'' series. She is one of the oldest and most frequently recurring characters in the series, having appeared in six titles of ''The Legend of Zelda'' games and several spin-off ...

Brazil Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...

. He earned his Ph.D. at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

in 1963 under the supervision of Stephen P. L. Diliberto. In 2004 he was elected to the

Brazilian Academy of Sciences The Brazilian Academy of Sciences ( pt, italic=yes, Academia Brasileira de Ciências or ''ABC'') is the national academy of Brazil. It is headquartered in the city of Rio de Janeiro and was founded on May 3, 1916. Publications It publishes a lar ...

. His students include Norbert A'Campo, Christian Bonatti, and Michael Herman. In 1993, he studied the hypersurfaces in Euclidean space with a given constant value of an

elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...

of the

shape operator In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...

, known as a ''higher-order mean curvature''. His primary result was to obtain some control of the height of such a surface over a plane containing its boundary. As an application, he was able to derive some rigidity results for complete surfaces with constant higher-order mean curvature. In 2004, he and Uwe Abresch extended the classical ''Hopf differential'', discovered by

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...

in the 1950s, from the setting of surfaces in three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

to the setting of surfaces in products of two-dimensional

space form Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...

s with the real line. They showed that, if the surface has constant mean curvature, then their Hopf differential is holomorphic relative to the natural complex structure on the surface. As an application, they were able to show that any immersed sphere of constant mean curvature must be rotationally symmetric, thereby extending a classical theorem of Alexandrov.

Major publications

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References

Differential geometers Members of the Brazilian Academy of Sciences 1941 births Living people Columbia University faculty UC Berkeley College of Letters and Science alumni Instituto Nacional de Matemática Pura e Aplicada researchers 20th-century American mathematicians 21st-century Brazilian mathematicians Expatriate academics in Brazil {{US-mathematician-stub