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The mathematics of F. J. Almgren, Jr.

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Abstract

Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field. He solved many of them and, in the process, discovered ideas which turned out to be useful for many other problems. This article is a more-or-less chronological survey of Almgren’s mathematical research. (Excerpts from this article appeared in the December 1997 issue of theNotices of the American Mathematical Society.) Almgren was also an outstanding educator, and he supervised the thesis work of nineteen PhD students; the 1997 volume 6 issue of the journalExperimental Mathematics is dedicated to Almgren and contains reminiscences by two of his PhD students and by various colleagues. A general article about Almgren’s life appeared in the October 1997Notices of the American Mathematical Society [MD]. See [T3]for a brief biography.

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Papers of F. J. Almgren, Jr

  1. F. J. Almgren, Jr The homotopy groups of the integral cycle groups,Topology,1, 257–299, (1962).

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  3. F. J. Almgren, Jr Three theorems on manifolds with bounded mean curvature,Bull. Am. Math. Soc.,71, 755–756, (1965).

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  4. F. J. Almgren, Jr Mass continuous cochains are differential forms,Proc. Am. Math. Soc.,16, 1291–1294, (1965).

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  5. F. J. Almgren, JrThe Theory of Varifolds. A variational calculus in the large for the k-dimensional are integrand, Multilithed notes, Princeton University Library, 178, 1965.

  6. F. J. Almgren, JrPlateau’s Problem. An Invitation to Varifold Geometry. Benjamin, W.A., Ed. New York, 1966.

  7. F. J. Almgren, Jr Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem,Ann. Math.,84, 277–292, (1966).

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  8. F. J. Almgren, Jr Existence and regularity of solutions to elliptic calculus of variations problems among surfaces of varying topological type and singularity structure,Bull. Am. Math. Soc,73, 576–680, (1967).

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  9. F. J. Almgren, Jr Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure,Ann. Math.,87, 321–391, (1968).

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  10. F. J. Almgren, Jr Measure theoretic geometry and elliptic variational problems,Bull. Am. Math. Soc,75, 285–304, (1969).

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  11. F. J. Almgren, Jr A maximum principle for elliptic variational problems,J. Functional Anal.,4, 380–389, (1969).

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  12. F. J. Almgren, Jr Measure theoretic geometry and elliptic variational problems,Proceedings of the Symposium on Continuum Mechanics and Related Problems of Analysis, (Tbilisi, 1971), (in Russian) vol.II, 307–324. Izdat. “Mecniereba,” Tbilisi, 1974. f

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  13. F. J. Almgren, Jr Geometric measure theory and elliptic variational problems,Actes du Congrès International des Mathématiciens, (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 813–819, 1971.

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  14. F. J. Almgren, Jr with Allard, W.K. An introduction to regularity theory for parametric elliptic variational problems. Partial differential equations,Proc. Symp. Pure Math.,XXIII, 231–260, 1973;Am. Math. Soc., Providence, RI.

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  15. F. J. Almgren, Jr Geometric variational problems from a measure-theoretic point of view,Global analysis and its applications, (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol.II, Internat. Atomic Energy Agency, Viena, 1–22, 1974.

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  16. F. J. Almgren, Jr Geometric measure theory and elliptic variational problems.Geometric Measure Theory and Minimal Surfaces, (C.I.M.E. Lectures, III Ciclo, Varenna, 1972), Ediziono Cremonese, Rome, 31–117, 1973.

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  17. F. J. Almgren, Jr The structure of limit varifolds associated with minimizing sequences of mappings,Symposia Mathematica,XIV. (Convegno di Teoria Geometrica dell’Integrazione e Varietá Minimali, INDAM, Rome, 1973), Academic Press, London, 1974.

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  18. F. J. Almgren, Jr Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Bull. Am. Math. Soc.,81, 151–154, (1975).

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  19. F. J. Almgren, Jr Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Mem. Am. Math. Soc.,4(165), viii + 199, (1976).

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  20. F. J. Almgren, Jr with Allard, W.K. The structure of stationary one-dimensional varifolds with positive density,Inv. Math.,34, 83–97, (1976).

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  21. F. J. Almgren, Jr with Taylor, J.E. The geometry of soap films and soap bubbles,Sci. Am., 82–93, July (1976).

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  23. F. J. Almgren, Jr with Schoen, R. and Simon, L. Regularity and singularity estimates of hypersurfaces minimizing parametric elliptic variational integrals,Acta Math.,139, 217–265, (1977).

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  24. F. J. Almgren, Jr with Simon, L. Existence of embedded solutions of Plateau’s problem,Annali Scuola Normale Superiore de Pisa (Series IV),VI(3), 447–495, (1979).

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  25. F. J. Almgren, Jr Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal submanifolds and geodesics, Obata, M., Ed.,Proceedings of the Japan-U.S. Seminar on Minimal Submanifolds including Geodesies, Kaigai Publishings, Tokyo, Japan, 1–6, 1978.

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  27. F. J. Almgren, Jr Minimal surfaces: tangent cones, singularities, and topological types.Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Lehto, O. Ed., Acad. Sci. Fennica, Helsinki,2, 767–770, 1980.

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  28. F. J. Almgren, Jr with Allard, W.K. On the radial behavior of minimal surfaces and the uniqueness of their tangent cones,Ann. Math.,113, 215–265, (1981).

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  30. F. J. Almgren, JrMinimal Surfaces. McGraw-Hill Encyclopedia of Science and Technology, 5th ed., McGraw-Hill, New York, 599–600, 1982.

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  31. F. J. Almgren, Jr Minimal surface forms,The Mathematical Intelligencer,4(4), 164–172, (1982).

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  32. F. J. Almgren, Jr Approximation of rectifiable currents by Lipschitz Q-valued functions, Seminar on Minimal Submanifolds,Ann. Math. Studies, Princeton University Press, Princeton, NJ,103, 243–259, (1983).

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  33. F. J. Almgren, Jr Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two,Bull. Am. Math. Soc.,8(2), 327–328, (1983).

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  34. F. J. Almgren, Jr Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, (V. Scheffer and J. Taylor, Eds.),World Scientific, to appear. Currently available electronically at http://www.math.princeton.edu/~scheffer.

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  38. F. J. Almgren, Jr Deformations and multiple valued functions, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984),Proc. Sympos. Pure Math., Am. Math. Soc., Providence, RI,44, 29–130, (1986).

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  39. F. J. Almgren, Jr Applications of multiple valued functions,Geometric Modeling: Algorithms and New Trends, SIAM, Philadelphia, 43–54, 1987.

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  40. F. J. Almgren, Jr Spherical symmetrization,Proceedings of the International Workshop on Integral Functions in the Calculus of Variations (Trieste, 1985), Rend. Circ. Mat. Palermo (2) Suppl., 11–25, 1987.

  41. F. J. Almgren, Jr with Taylor, J.E. Optimal crystal shapes,Variational Methods for Free Surface Interfaces, Concus, P. and Finn, R., Eds., Springer-Verlag, New York, 1–11, 1987.

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  42. F. J. Almgren, Jr with Lieb, E.H. Singularities of energy minimizing maps from the ball to the sphere,Bull. Am. Math. Soc.,17, 304–306, (1987).

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  43. F. J. Almgren, Jr with Browder, W. and Lieb, E.H. Co-area, liquid crystals, and minimal surfaces,Partial Differential Equations, Chern, S.S., Ed., Springer Lecture Notes in Mathematics 1306, Springer-Verlag, New York, 1–22, 1988.

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  44. F. J. Almgren, Jr with Lieb, E.H. Singularities of energy minimizing maps from the ball to the sphere,Ann. Math.,128, 483–530, (1988).

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  46. F. J. Almgren, Jr with Lieb, E.H. Symmetric decreasing rearrangement can be discontinuous,Bull. Am. Math. Soc.,20, 177–180, (1989).

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  48. F. J. Almgren, Jr with Gurtin, M. A mathematical contribution to Gibbs’s analyses of fluid phases in equilibriumPartial Differential Equations and the Calculus of Variations, Progr. Nonlinear Differential Equations Appl., Birkäuser, Boston,1, 9–28, 1989.

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  50. F. J. Almgren, Jr What can geometric measure theory do for several complex variables? Proceedings of the Several Complex Variables Year at the Mittag-Leffler Institute (Stockholm, 1987–1988), Princeton University Press Math. Notes (38), Princeton, NJ, 8–21, 1993.

  51. F. J. Almgren, Jr The geometric calculus of variations and modelling natural phenomena, Statistical thermodynamics and differential geometry of microstructured materials (Minneapolis, MN, 1991),IMA Vol. Math. Appl., Springer-Verlag, New York,51, 1–5, (1993).

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  52. F. J. Almgren, Jr Multi-functions modv, Geometric analysis and computer graphics (Berkeley, CA, 1988), 1–17,Math. Sci. Res. Inst. Publ., Springer-Verlag, New York,17, (1991).

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  53. F. J. Almgren, Jr with Lieb, E.H. The (non)continuity of symmetric decreasing rearrangement, Proceedings of the conference on geometry of solutions to PDE (Cortona, 1988),Symposia Mathematica, Academic Press, Boston, MA, XXX, 1992; also in: Variational methods (Paris, 1988),Progr. Nonlinear Differential Equations Appl, Birkhäuser, Boston, MA, 4,3–16,1990. also in: Differential equations and mathematical physics, (Birmingham, AL, 1990),Math. Sci. Engrg., Academic Press, Boston, MA,186, 183–200, 1992.

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  54. F. J. Almgren, Jr Computing soap films and crystals,Computing Optimal Geometries, video report,Am. Math. Soc., 1991.

  55. F. J. Almgren, Jr with Sullivan, J. Visualization of soap bubble geometries,Leonardo,25, 267–271, (1992).

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  56. F. J. Almgren, Jr with Taylor, J.E. and Wang, L. A variational approach to motion by weighted mean curvature, Computational Crystal Growers Workshop Selected Lectures in Mathematics,Am. Math. Soc., 9–12, (1992).

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  58. F. J. Almgren, Jr Questions and answers about area minimizing surfaces and geometric measure theory, Differential Geometry: partial differential equations on manifolds, (Los Angeles, 1990),Proc. Symposia Pure Math., Am. Math. Soc,51, 29–53, 1992.

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  59. F. J. Almgren, Jr with Taylor, J.E. Flat flow is motion by crystalline curvature for curves with crystalline energies,J. Differential Geom.,42(1), 1–22, (1995).

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  60. F. J. Almgren, Jr with Taylor, J.E. Optimal geometry in equilibrium and growth, Symposium in Honor of Benoit Mandelbrot (Curaçao, 1995),Fractals,3(4), 713–723, (1995).

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  61. F. J. Almgren, Jr Questions and answers about geometric evolution processes and crystal growth,The Gelfand Mathematical Seminars, 1–9, 1993–1995; Gelfand Math. Sem., Birkhäuser, Boston, 1996.

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  62. F. J. Almgren, Jr with Wang, L. Mathematical existence of crystal growth with Gibbs-Thomson curvature effects,J. Geom. Anal, (to appear).

  63. F. J. Almgren, Jr with Rivin, I. The mean curvature integral is invariant under bending, 1–21, Geometry and topology monographs #1, University of Warwick published electronically: www.maths.warwick.ac.uk/gt/main/ml

  64. F. J. Almgren, Jr with Taylor, J. Soap bubble clusters: the Kelvin problem,Forma,11(3), 199–207, (1996).

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  65. F. J. Almgren, JrGlobal Analysis. preprint, (survey/expository).

  66. F. J. Almgren, Jr Isoperimetric inequalities for anisotropic surface energies, unfinished manuscript.

  67. F. J. Almgren, Jr A new look at flat chains modn, unfinished manuscript.

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Correspondence to Brian White.

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White, B. The mathematics of F. J. Almgren, Jr.. J Geom Anal 8, 681–702 (1998). https://doi.org/10.1007/BF02922665

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  • DOI: https://doi.org/10.1007/BF02922665

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