Herbert Federer

Herbert Federer, an American mathematician, is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.

Federer was born July 23, 1920, in Vienna. 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 A.P. Morse in 1944. He then spent virtually his entire career as a member of the Brown University Mathematics Department, where he currently holds the title of Professor Emeritus.

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 Brown colleague Wendell Fleming. That paper provided the first satisfactory general solution to the so-called Plateau problem--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.

Federer is a member of the National Academy of Sciences. In 1987, he and Fleming won the American Mathematical Society's Steele Prize "for their pioneering work in "Normal and Integral currents"."

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. Besicovitch 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.

Geometric measure theory

Federer is perhaps best known for his treatise "Geometric measure theory" ("GMT"), published in 1969. Intended as both a text and a reference work, the book is unusually complete, general and authoritative: Its nearly 600 pages cover a substantial amount of linear and multilinear algebra, give a profound treatment of measure theory, integration and differentiation, and then move on to rectifiability, theory of currents, and finally, variational applications. Nevertheless, the book's unique style exhibits a rare and artistic economy that still inspires admiration, respect--and exasperation. A more accessible introduction may be found in F. Morgan's book listed below.

Federer's wrote more than thirty research papers in addition to the book GMT. 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.

References

* citation
last = Federer
first = Herbert
authorlink = Herbert Federer
title = Geometric measure theory
publisher = Springer-Verlag New York Inc.
location = New York
year = 1969
pages = xiv+676
isbn = 978-3540606567
id= MathSciNet|id=0257325
ISBN status = May be invalid - please double check
unused_data = |series Die Grundlehren der mathematischen Wissenschaften, Band 153
*citation|first=H. |last=Federer|title=Colloquium lectures on geometric measure theory|journal= Bull. Amer. Math. Soc. |volume= 84 |issue= 3 |year=1978|pages= 291–338
url=http://www.ams.org/bull/1978-84-03/S0002-9904-1978-14462-0/|doi=10.1090/S0002-9904-1978-14462-0
* citation
last = Morgan
first = Frank
title = Geometric measure theory: A beginner's guide
edition = Third edition
publisher = Academic Press Inc.
location = San Diego, CA
year = 2000
pages = x+226
isbn = 0-12-506851-4
id=MathSciNet|id=1775760
ISBN status = May be invalid - please double check

External links

* [http://www.math.brown.edu/faculty/federer.html Federer's page at Brown]
*