Frederick J. Almgren's Books

Selected Works of Frederick J. Almgren, Jr.

By: Frederick J. Almgren

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Selected Works of Frederick J. Almgren, Jr.

By: Frederick J. Almgren

This volume offers a unique collection of some of the work of Frederick J. Almgren, Jr., the man most noted for defining the shape of geometric variational problems and for his role in founding The Geometry Center. Included in the volume are the following: a summary by Sheldon Chang of the famous 1700 page paper on singular sets of area-minimizing $m$-dimensional surfaces in $Rn$, a detailed summary by Brian White of Almgren's contributions to mathematics, his own announcements of several longer papers, important shorter papers, and memorable expository papers. Almgren's enthusiasm for the subject and his ability to locate mathematically beautiful problems that were "ready to be solved" attracted many students who further expanded the subject into new areas. Many of these former students are now known for the clarity of their expositions and for the beauty of the problems that they work on. As Almgren's former graduate student, wife, and colleague, Professor Taylor has compiled an important volume on an extraordinary mathematician. This collection presents a fine comprehensive view of the man's mathematical legacy

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Page Count

638

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821810677

ISBN 13

9780821810675

Almgren's Big Regularity Paper

Q-valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-minimizing Rectifiable Currents Up to Codimension 2

By: Frederick J. Almgren,Vladimir Scheffer,Jean E. Taylor

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Almgren's Big Regularity Paper

Q-valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-minimizing Rectifiable Currents Up to Codimension 2

By: Frederick J. Almgren,Vladimir Scheffer,Jean E. Taylor

Fred Almgren created the excess method for proving regularity theorems in the calculus of variations. His techniques yielded Holder continuity except for a small closed singular set. In the sixties and seventies Almgren refined and generalized his methods. Between 1974 and 1984 he wrote a 1,700-page proof that was his most ambitious exposition of his ground-breaking ideas. Originally, this monograph was available only as a three-volume work of limited circulation. The entire text is faithfully reproduced here. This book gives a complete proof of the interior regularity of an area-minimizing rectifiable current up to Hausdorff codimension 2. The argument uses the theory of Q-valued functions, which is developed in detail. For example, this work shows how first variation estimates from squash and squeeze deformations yield a monotonicity theorem for the normalized frequency of oscillation of a Q-valued function that minimizes a generalized Dirichlet integral. The principal features of the book include an extension theorem analogous to Kirszbraun's theorem and theorems on the approximation in mass of nearly flat mass-minimizing rectifiable currents by graphs and images of Lipschitz Q-valued functions.

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Page Count

976

Publisher

World Scientific

Published Date

ISBN 10

9810241089

ISBN 13

9789810241087

Plateau's Problem

An Invitation to Varifold Geometry

By: Frederick J. Almgren (Jr.)

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Plateau's Problem

An Invitation to Varifold Geometry

By: Frederick J. Almgren (Jr.)

There have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past 25 to 30 years. Many of the researchers who have produced these excellent results were inspired by this little book - or by Fred Almgren himself. The book is indeed a delightful invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behavior of soap films.When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This new tool makes possible much exciting new analysis and many new results. Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. The author's hope in writing this book was to encourage young mathematicians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well.

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American Mathematical Soc.

Published Date

ISBN 10

0821827472

ISBN 13

9780821827475

Geometric Measure Theory and the Calculus of Variations

By: William K. Allard,Frederick J. Almgren

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Geometric Measure Theory and the Calculus of Variations

By: William K. Allard,Frederick J. Almgren

These twenty-six papers survey a cross section of current work in modern geometric measure theory and its applications in the calculus of variations. Presently the field consists of a jumble of new ideas, techniques and intuitive hunches; an exchange of information has been hindered, however, by the characteristic length and complexity of formal research papers in higher-dimensional geometric analysis. This volume provides an easier access to the material, including introductions and summaries of many of the authors' much longer works and a section containing 80 open problems in the field. The papers are aimed at analysts and geometers who may use geometric measure-theoretic techniques, and they require a mathematical sophistication at the level of a second year graduate student. The papers included were presented at the 1984 AMS Summer Research Institute held at Humboldt State University. A major theme of this institute was the introduction and application of multiple-valued function techniques as a basic new tool in geometric analysis, highlighted by Almgren's fundamental paper Deformations and multiple-valued functions. Major new results discussed at the conference included the following: Allard's integrality and regularity theorems for surfaces stationary with respect to general elliptic integrands; Scheffer's first example of a singular solution to the Navier-Stokes equations for a fluid flow with opposing force; and Hutchinson's new definition of the second fundamental form of a general varifold.

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Page Count

484

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821868071

ISBN 13

9780821868072

Book's by Frederick J. Almgren

Selected Works of Frederick J. Almgren, Jr.

Selected Works of Frederick J. Almgren, Jr.

Almgren's Big Regularity Paper

Almgren's Big Regularity Paper

Plateau's Problem

Plateau's Problem

Geometric Measure Theory and the Calculus of Variations

Geometric Measure Theory and the Calculus of Variations

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