Frederick Almgren

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Frederick Justin Almgren junior (born July 3, 1933 in Birmingham , Alabama , † February 5, 1997 in Princeton , New Jersey ) was an American mathematician who dealt with topology, calculus of variations, differential geometry and minimal surfaces.

Life

Almgren completed an engineering degree from Princeton University in 1955 and was then a fighter jet pilot in the US Navy. From 1958 he studied mathematics at Brown University with Herbert Federer , who founded the geometric dimension theory with Wendell Fleming around this time . In 1962 he did his doctorate with Federer (The homotopy groups of the integral cycle groups, published in: Topology, Vol. 1, 1962, pp. 257–299) and was then an instructor at Princeton University, 1963 to 1965 at the Institute for Advanced Study ( as well as 1974/75, 1981/82, 1985, 1989 and 1992), from 1965 Assistant Professor, from 1968 Associate Professor and from 1972 Professor in Princeton (most recently as Henry Burchard Fine Professor). In 1970 he was an exchange scientist at the Steklow Institute in Saint Petersburg . He was diagnosed with bone cancer in 1996 and died a year later of complications from pneumonia following a bone marrow transplant.

He was a Sloan Research Fellow from 1968 to 1970 and a Guggenheim Fellow from 1974 to 1975. He was a member of the American Association for the Advancement of Science . He was editor and one of the founders of the journal Experimental Mathematics. In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki ( Minimal surfaces: tangent cones, singularities and topological types ) and in 1970 in Nice ( Geometric measure theory and elliptic variational problems ).

Almgren was married twice. From his first marriage he had a son and a daughter who also became mathematicians. In 1973 he was married to the mathematician Jean Taylor (professor at Rutgers University ), who did her doctorate with him. He had a daughter with her.

plant

Almgren was known for his work on geometric analysis, especially the study of minimal surfaces.

In 1966 he succeeded in solving the amber problem for four dimensions (see the article Ennio de Giorgi ).

In the 1960s he initiated the study of "varifolds" as a generalization of the currents describing orientable surfaces in the geometric dimension theory to the non-orientable case (later expanded by WK Allard).

In the 1970s he investigated minimal problems that were closer to the problem of the shape of the surfaces of soap bubbles (with edges) than the classic plateau problem (by Tibor Radó and Jesse Douglas ) or the formulation by Wendell and Fleming (mass minimizing integral currents) . In the case of the classic plateau problem, for example, some unphysical assumptions such as self-penetration of the surfaces are allowed. Almgren obtained proof of existence and regularity, which Jean Taylor further developed. They showed, for example, that their soap bubble model reproduced the experimental observation that three surfaces meet in a line and four meet in an isolated point.

From 1974 he worked on the proof that the dimension of the singular sets of mass-minimizing d-dimensional hypersurfaces have at most dimension d − 2 and the d-dimensional measure zero. His original proof, which he worked on for 10 years, was 1,700 pages long. It wasn't published until 2000 (edited by Jean Taylor and Vladimir Scheffer ).

Almgren also worked on computer simulations of minimal surfaces and made a film about them.

In the 1980s he worked with Jean Taylor, among others, on evolution equations of the dynamics of surfaces in differential geometry, based on the ideas of the "mean curvature flow" by Ken Brakke (1975). In particular, they were interested in applications in modeling crystal growth.

Fonts

  • Plateau's Problem: an invitation to varifold geometry , Benjamin, 1966, new edition AMS 2001
  • with Jean Taylor: The geometry of Soap Films and Soap Bubbles , Scientific American July 1976
  • Selected Works , AMS 1999 (editor Jean Taylor)
  • The theory of varifolds , Lectures Notes, Princeton 1965
  • Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints , Memoirs AMS 1976

literature

Web links

References

  1. ↑ Frederick Almgren in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. ^ Almgren, Some interior regularity theorems for minimal surfaces and an extension of the Bernstein's theorem, Ann. of Math., Vol. 85, 1966, pp. 277-292
  3. ↑ "Mass" in the sense of the geometric theory of measure, it corresponds to the area measure counted with a multiplicity of possible overlays
  4. ↑ Vladimir Scheffer, Jean E. Taylor (Eds.): Almgren's big regularity paper: Q-valued functions minimizing Dirichlet's Integral and the regularity of area-minimizing rectifiable currents up to codimension 2 , World Scientific, 2000, 955 pages
  5. ↑ Fred Almgren, Jean E. Taylor, Lihe Wang: Curvature driven flows: a variational approach , SIAM Journal on Control and Optimization 31, 1993, pp. 387-438
personal data
SURNAME Almgren, Frederick
ALTERNATIVE NAMES Almgren, Frederick Justin junior (full name)
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH July 3, 1933
PLACE OF BIRTH Birmingham , Alabama
DATE OF DEATH February 5, 1997
Place of death Princeton