Herbert Federer
Encyclopedia
Herbert Federer was an American mathematician
. He is one of the creators of geometric measure theory
, at the meeting point of differential geometry and mathematical analysis
.
, Austria
. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley
, earning the Ph.D. as a student of Anthony Morse
in 1944. He then spent virtually his entire career as a member of the Brown University
Mathematics Department, where he eventually retired with the title of Professor Emeritus.
Federer wrote more than thirty research papers in addition to his book Geometric measure theory. The Mathematics Genealogy Project
assigns him nine Ph.D. students and well over a hundred subsequent descendants. His most productive students include the late Frederick J. Almgren, Jr.
(1933–1997 ) a professor at Princeton for 35 years, and his last student, Robert Hardt, now at Rice University.
Federer was a member of the National Academy of Sciences
. In 1987, he and his Brown colleague Wendell Fleming won the American Mathematical Society's Steele Prize "for their pioneering work in Normal and Integral currents."
— the problem of finding a (k+1)-dimensional least-area surface spanning a given k-dimensional boundary cycle in n-dimensional Euclidean space. Their solution inaugurated a new and fruitful period of research on a large class of geometric variational problems — especially minimal surfaces — via what came to be known as Geometric Measure Theory.
had proven this for 1-dimensional sets in the plane, but Federer's generalization, valid for subsets of arbitrary dimension in any Euclidean space, was a major technical accomplishment, and later played a key role in Normal and Integral Currents.
In 1958, Federer wrote Curvature Measures, a paper that took some early steps toward understanding second-order properties of surfaces lacking the differentiability properties typically assumed in order to discuss curvature. He also developed and named what he called the coarea formula
in that paper. That formula has become a standard analytical tool.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He is one of the creators of geometric measure theory
Geometric measure theory
In mathematics, geometric measure theory is the study of the geometric properties of the measures of sets , including such things as arc lengths and areas. It uses measure theory to generalize differential geometry to surfaces with mild singularities called rectifiable sets...
, at the meeting point of differential geometry and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
.
Career
Federer was born July 23, 1920, in ViennaVienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
, Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, earning the Ph.D. as a student of Anthony Morse
Anthony Morse
Anthony Perry Morse was an American mathematician who worked in both analysis, especially measure theory, and in the foundations of mathematics. He is best known as the co-creator, together with John L. Kelley, of Morse–Kelley set theory. This theory first appeared in print in General...
in 1944. He then spent virtually his entire career as a member of the Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
Mathematics Department, where he eventually retired with the title of Professor Emeritus.
Federer wrote more than thirty research papers in addition to his book Geometric measure theory. The Mathematics Genealogy Project
Mathematics Genealogy Project
The Mathematics Genealogy Project is a web-based database for the academic genealogy of mathematicians. As of September, 2010, it contained information on approximately 145,000 mathematical scientists who contribute to "research-level mathematics"...
assigns him nine Ph.D. students and well over a hundred subsequent descendants. His most productive students include the late Frederick J. Almgren, Jr.
Frederick J. Almgren, Jr.
Frederick Justin Almgren, Jr. was a mathematician working in geometric measure theory...
(1933–1997 ) a professor at Princeton for 35 years, and his last student, Robert Hardt, now at Rice University.
Federer was a member of the National Academy of Sciences
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
. In 1987, he and his Brown colleague Wendell Fleming won the American Mathematical Society's Steele Prize "for their pioneering work in Normal and Integral currents."
Normal and integral currents
Federer's mathematical work separates thematically into the periods before and after his watershed 1960 paper Normal and integral currents, co-authored with Fleming. That paper provided the first satisfactory general solution to Plateau's problemPlateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who was interested in soap films. The problem is considered part of the calculus of variations...
— the problem of finding a (k+1)-dimensional least-area surface spanning a given k-dimensional boundary cycle in n-dimensional Euclidean space. Their solution inaugurated a new and fruitful period of research on a large class of geometric variational problems — especially minimal surfaces — via what came to be known as Geometric Measure Theory.
Earlier work
During the 15 years or so years prior to that paper, Federer worked at the technical interface of geometry and measure theory. He focused particularly on surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the analysis of surfaces. His 1947 paper on the rectifiable subsets of n-space characterized purely unrectifiable sets by their "invisibility" under almost all projections. A. S. BesicovitchAbram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch was a Russian mathematician, who worked mainly in England. He was born in Berdyansk on the Sea of Azov to a Karaite family.-Life and career:...
had proven this for 1-dimensional sets in the plane, but Federer's generalization, valid for subsets of arbitrary dimension in any Euclidean space, was a major technical accomplishment, and later played a key role in Normal and Integral Currents.
In 1958, Federer wrote Curvature Measures, a paper that took some early steps toward understanding second-order properties of surfaces lacking the differentiability properties typically assumed in order to discuss curvature. He also developed and named what he called the coarea formula
Coarea formula
In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of the integral of the level sets of another function...
in that paper. That formula has become a standard analytical tool.