Herbert Federer (July 23, 1920 – April 21, 2010) was 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 ...
. He is one of the creators of
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
, at the meeting point of
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 ...
and
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
.
[Parks, H. (2012]
''Remembering Herbert Federer (1920–2010)''
NAMS 59(5), 622-631.
Career
Federer was born July 23, 1920, in
Vienna
en, Viennese
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,
Austria
Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
. After emigrating to the US in 1938, he studied mathematics and physics 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 ...
, earning the Ph.D. as a student of
Anthony Morse
Anthony Perry Morse (21 August 1911 – 6 March 1984) 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. Kelle ...
in 1944. He then spent virtually his entire career as a member of the
Brown University
Brown University is a private research university in Providence, Rhode Island. Brown is the seventh-oldest institution of higher education in the United States, founded in 1764 as the College in the English Colony of Rhode Island and Providenc ...
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
The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. By 31 December 2021, it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a ty ...
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 Justin Almgren Jr. (July 3, 1933 – February 5, 1997) was an American mathematician working in geometric measure theory. He was born in Birmingham, Alabama.
Almgren received a Guggenheim Fellowship in 1974. Between 1963 and 1992 he wa ...
(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
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
. In 1987, he and his Brown colleague
Wendell Fleming
Wendell Helms Fleming (born March 7, 1928) is an American mathematician, specializing in geometrical analysis and stochastic differential equations.
Fleming received in 1951 his PhD under Laurence Chisholm Young at the University of Wisconsin ...
won the American Mathematical Society's
Steele Prize
The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories.
The prizes have ...
"for their pioneering work in ''Normal and Integral currents''."
Mathematical work
In the 1940s and 1950s, Federer made many contributions at the technical interface of geometry and measure theory. Particular themes included surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the classical analysis of surfaces. A particularly noteworthy early accomplishment (improving earlier work of
Abram Besicovitch
Abram Samoilovitch Besicovitch (or Besikovitch) (russian: link=no, Абра́м Само́йлович Безико́вич; 23 January 1891 – 2 November 1970) was a Russian mathematician, who worked mainly in England. He was born in Berdyansk ...
) was the characterization of ''purely unrectifiable sets'' as those which "vanish" under almost all projections. Federer also made noteworthy contributions to the study of
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orient ...
in low regularity. The theory of
capacity with modified exponents was developed by Federer and William Ziemer. In his first published paper, written with his Ph.D. advisor
Anthony Morse
Anthony Perry Morse (21 August 1911 – 6 March 1984) 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. Kelle ...
, Federer proved the
Federer–Morse theorem
In mathematics, the Federer–Morse theorem, introduced by , states that if ''f'' is a surjective continuous map from a compact metric space ''X'' to a compact metric space ''Y'', then there is a Borel subset ''Z'' of ''X'' such that ''f'' re ...
which states that any continuous surjection between
compact metric spaces can be restricted to a
Borel subset
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named a ...
so as to become an injection, without changing the image.
One of Federer's best-known papers, ''Curvature Measures'', was published in 1959. The intention is to establish measure-theoretic formulations of second-order analysis in differential geometry, particularly
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
. The
Steiner formula In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an of convex bodies in space. This number depends on the size and shape of the bodies and on their relative orientation to each ...
formed a fundamental precedent for Federer's work; it established that the volume of a neighborhood of a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
in
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 ...
is given by a polynomial. If the boundary of the convex set is a smooth submanifold, then the coefficients of the Steiner formula are defined by its curvature. Federer's work was aimed towards developing a general formulation of this result. The class of subsets that he identified are those of ''positive reach'', subsuming both the class of convex sets and the class of smooth submanifolds. He proved the Steiner formula for this class, identifying generalized
quermassintegral In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an of convex bodies in space. This number depends on the size and shape of the bodies and on their relative orientation to each ...
s (called ''curvature measures'' by Federer) as the coefficients. In the same paper, Federer proved the
coarea formula, which has become a standard textbook result in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
.
Federer's second landmark paper, ''Normal and Integral Currents'', was co-authored with
Wendell Fleming
Wendell Helms Fleming (born March 7, 1928) is an American mathematician, specializing in geometrical analysis and stochastic differential equations.
Fleming received in 1951 his PhD under Laurence Chisholm Young at the University of Wisconsin ...
. In their work, they showed that
Plateau'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 experimented with soap films. The problem is ...
for
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 that ...
s can be solved in the class of
integral currents, which may be viewed as generalized submanifolds. Moreover, they identified new results on the
isoperimetric problem
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 by ...
and its relation to the
Sobolev embedding theorem
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...
. Their paper inaugurated a new and fruitful period of research on a large class of geometric variational problems, and especially minimal surfaces.
In 1969, Federer published his book ''Geometric Measure Theory'', which is among the most widely cited books in mathematics. It is a comprehensive work beginning with a detailed account of
multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
and
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
. The main body of the work is devoted to a study of rectifiability and the
theory of currents. The book ends with applications to the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. Federer's book is considered an authoritative text on this material, and included a number of new results in addition to much material from past research of Federer and others. Much of his book's discussion of currents and their applications are limited to integral coefficients. He later developed the basic theory in the setting of real coefficients.
A particular result detailed in Federer's book is that area-minimizing
minimal hypersurfaces of Euclidean space are smooth in low dimensions. Around the same time,
Enrico Bombieri
Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathema ...
,
Ennio De Giorgi, and
Enrico Giusti
Enrico Giusti (born Priverno, 1940), is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He has ...
proved that a minimal hypercone in eight-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 ...
, first identified by
James Simons
James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hedge fund manager, and philanthropist. He is the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his ...
, is area-minimizing. As such, it is direct to construct area-minimizing minimal hypersurfaces of Euclidean space which have ''singular sets'' of codimension seven. In 1970, Federer proved that this codimension is optimal: all such singular sets have codimension of at least seven. His ''dimension reduction'' argument for this purpose has become a standard part of the literature on geometric measure theory and
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
.
[Leon Simon. Lectures on Geometric Measure Theory.] Later, Federer also found a new proof of the result of Bombieri–De Giorgi–Giusti.
Major publications
Federer was the author of around thirty research papers, along with his famous textbook ''Geometric Measure Theory''.
References
External links
Federer's page at BrownWendell H. Fleming and William P. Ziemer, "Herbert Federer", Biographical Memoirs of the National Academy of Sciences (2014)*
{{DEFAULTSORT:Federer, Herbert
1920 births
2010 deaths
20th-century American mathematicians
21st-century American mathematicians
Mathematicians from Vienna
Members of the United States National Academy of Sciences
Geometers
Brown University faculty
University of California, Berkeley alumni
Austrian emigrants to the United States