Leon Melvyn Simon

Quick Info

Born
6 July 1945

Summary
Leon Simon is an Australian mathematician who made contributions to geometric measure theory, harmonic maps and partial differential equations.

Biography

Leon Simon studied at the University of Adelaide where he was awarded a B.Sc. in 1967. He then continued to undertake research at the University of Adelaide with James H Michael as his research supervisor and was awarded his Ph.D. in 1971 for his thesis Interior Gradient Bounds for Non-Uniformly Elliptic Equations. During his years as a doctoral student, he was employed from 1968 to 1971 as a Tutor in Mathematics by the University.

Following the award of his Ph.D., Simon was appointed as a Lecturer in Mathematics at Flinders University of South Australia for the academic year 1972-73. He then travelled to the United States where he was first Visiting Assistant Professor at Stanford University in 1973, then appointed as Assistant Professor at Stanford holding this position until 1976. He was a Sloan Fellow during 1974-75. Papers which Simon published over this period were: (with his Ph.D. supervisor James H Michael) Sobolev and mean-value inequalities on generalized submanifolds of $\mathbb{R}^{n}$(1973); Global estimates of Hölder continuity for a class of divergence-form elliptic equations (1974); (with Richard M Schoen and Shing-Tung Yau) Curvature estimates for minimal hypersurfaces (1975); Interior gradient bounds for non-uniformly elliptic equations (1976); and Remarks on curvature estimates for minimal hypersurfaces (1976). Let us note that Simon's co-author Richard M Schoen was his first doctoral student and was working on his Ph.D. under Simon's supervision at the time the above paper was written.

In 1976 Simon returned to Australia being a Visiting Professor at the University of Adelaide during the academic year 1976-77, then was appointed Associate Professor at the University of Minnesota where he worked for the year 1977-78 before again returning to Australia when appointed as Professor of Mathematics at the University of Melbourne. He held this chair from 1978 to 1981when he moved to the Australian National University as Professor of Mathematics. He held this post until 1986. In 1983 he was honoured with election as a fellow of the Australian Academy of Science and in the same year he became the third recipient of the Australian Mathematical Society Medal. This Medal is:-
... awarded to a member of the Society under the age of 40 years for distinguished research in the mathematical sciences. A significant portion of the research work should have been carried out in Australia.
Simon published Lectures on geometric measure theory in 1983. In a review, J S Joel writes:-
Many years ago H Federer's book on geometric measure theory appeared and immediately became a reference for those working in and interested in the field. Its very size and comprehensiveness lent it more the nature of a reference book than a textbook. In the meantime there has also been considerable progress in the field, including deeper regularity theorems and a closer connection with differential-geometric and analytic objects. The book under review, which the author says is a preliminary version of a more complete book he hopes to write, is both an introduction to geometric measure theory and related variational problems and to the regularity theory. The exposition is clear throughout ...
In 1986 Simon was appointed Professor in the Mathematics Department at Stanford University. In 1994 he was awarded the Bôcher Memorial Prize by the American Mathematical Society. The citation gives a good overview of Simon's important contributions [1]:-
The 1994 Bôcher Prize is awarded to Leon Simon for his profound contributions towards understanding the structure of singular sets for solutions of variational problems.

Powerful methods were developed in the 1960s to establish the partial regularity of minima and critical points of the Plateau problem and later extended to other variational problems such as the harmonic mapping problem. These results left open basic questions about the structure of the set of singularities exhibited by the solutions of such variational problems.

In a series of papers over the past ten years, Simon has developed methods for analysing this structure. This development began with his 1983 paper "Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems." The first stage of his work on general singular sets is principally described in "Cylindrical tangent cones and the singular set of minimal submanifolds" (1993), and the remaining work appears in his paper "Rectifiability of the singular set of energy minimizing maps" (1995). This latter paper establishes rectifiability for singular sets of energy minimising maps into an arbitrary compact real analytic target manifold.
In 1996 Simon published Theorems on regularity and singularity of energy minimizing maps which was based on lectures which he gave while holding a visiting position at Eidgenössische Technische Hochschule Zürich. Nathan Smale gives the following overview as part of his review of the book:-
The volume under review provides a detailed exposition of the more fundamental results on the regularity theory of energy minimizing maps between Riemanmnian manifolds. After a preliminary section, the book covers the main partial regularity results, due to Schoen and Uhlenbeck, then a fairly comprehensive discussion of minimizing tangent maps including a somewhat simplified proof of the author's celebrated theorem on the uniqueness of tangent maps, and finally an account of recent results on the rectifiability of the singular set. Despite the depth of the results, the book is completely self-contained (assuming only a rudimentary knowledge of real analysis, such as one would find in a first year graduate course), and gets rapidly to the heart of the subject, providing detailed, concise proofs of the theorems.
We have mentioned above some of the honours which Simon has received. Let us finally mention that he was elected a fellow of the American Academy of Arts and Sciences in 1994.

References (show)

1. Leon Simon receives 1994 Bôcher Memorial Prize, Notices Amer. Math. Soc. 41 (2) (1994), 99-100.