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Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He 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, 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."
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 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.
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, 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.
- "NAS Membership Directory: Federer, Herbert". National Academy of Sciences. Retrieved 15 June 2010.
- Parks, H. (2012) Remembering Herbert Federer (1920–2010), NAMS 59(5), 622-631.
- Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801.
- Federer, H. (1978), "Colloquium lectures on geometric measure theory", Bulletin of the American Mathematical Society 84 (3): 291–338, doi:10.1090/S0002-9904-1978-14462-0, MR 0467473, Zbl 0392.49021.
- Morgan, Frank (2009), Geometric measure theory: A beginner's guide (4th ed.), San Diego, CA: Academic Press, pp. viii+249, ISBN 0-12-506851-4, MR 1775760, Zbl 1179.49050.