Leon Melvyn Simon , born in 1945, is a Leroy P. Steele PrizeSee announcemen

retrieved 15 September 2017. and Bôcher Memorial Prize, Bôcher Prize-winningSee . mathematician, known for deep contributions to the fields of

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...

geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfac ...

, and

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s. He is currently Professor Emeritus in the Mathematics Department at

Stanford University Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth ...

Biography

Academic career

Leon Simon, born 6 July 1945, received his BSc from the

University of Adelaide The University of Adelaide is a public university, public research university based in Adelaide, South Australia. Established in 1874, it is the third-oldest university in Australia. Its main campus in the Adelaide city centre includes many Sa ...

in 1967, and his PhD in 1971 from the same institution, under the direction of James H. Michael. His doctoral thesis was titled ''Interior Gradient Bounds for Non-Uniformly Elliptic Equations''. He was employed from 1968 to 1971 as a Tutor in Mathematics by the university. Simon has since held a variety of academic positions. He worked first at

Flinders University Flinders University, established as The Flinders University of South Australia is a public university, public research university based in Adelaide, South Australia, with a footprint extending across a number of locations in South Australia and ...

as a lecturer, then at

Australian National University The Australian National University (ANU) is a public university, public research university and member of the Group of Eight (Australian universities), Group of Eight, located in Canberra, the capital of Australia. Its main campus in Acton, A ...

as a professor, at the

University of Melbourne The University of Melbourne (colloquially known as Melbourne University) is a public university, public research university located in Melbourne, Australia. Founded in 1853, it is Australia's second oldest university and the oldest in the state ...

, the

University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Twin Cities of Minneapolis and Saint ...

, at

ETH Zurich ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ran ...

, and at Stanford. He first came to Stanford in 1973 as Visiting Assistant Professor and was awarded a full professorship in 1986. Simon has more than 100 'mathematical descendants', according to the

Mathematics Genealogy Project The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. it contained information on 300,152 mathematical scientists who contributed to research-level mathematics. For a typical mathematicia ...

. Among his doctoral students there are Richard Schoen, Neshan Wickramasekera and Tatiana Toro.

Honours

In 1983 Simon was awarded the Australian Mathematical Society Medal. In the same year he was elected as a

Fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...

of the

Australian Academy of Science The Australian Academy of Science was founded in 1954 by a group of distinguished Australians, including Australian Fellows of the Royal Society of London. The first president was Sir Mark Oliphant. The academy is modelled after the Royal Soci ...

. He was an invited speaker at the 1983

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

in Warsaw. In 1994, he was awarded the

Bôcher Memorial Prize The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five yea ...

.See his brief biography .See hi
extended biography
at the MacTutor History of Mathematics Archive. The Bôcher Prize is awarded every five years to a groundbreaking author in

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

. In the same year he was also elected a fellow of the American Academy of Arts and Sciences. In May 2003 he was elected a

fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. In 2017 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research.

Research activity

Simon's best known work, for which he was honored with the Leroy P. Steele Prize for Seminal Contribution to Research, deals with the uniqueness of asymptotics of certain nonlinear evolution equations and Euler-Lagrange equations. The main tool is an infinite-dimensional extension and corollary of the Łojasiewicz inequality, using the standard Fredholm theory of

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...

s and Lyapunov-Schmidt reduction. The resulting Łojasiewicz−Simon inequalities are of interest in and of themselves and have found many applications in

. Simon's primary applications of his Łojasiewicz−Simon inequalities deal with the uniqueness of tangent cones of

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s and of tangent maps of harmonic maps, making use of the deep regularity theories of William Allard, Richard Schoen, and

Karen Uhlenbeck Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W ...

. Other authors have made fundamental use of Simon's results, such as Rugang Ye's use for the uniqueness of subsequential limits of Yamabe flow. A simplification and extension of some aspects of Simon's work was later found by Mohamed Ali Jendoubi and others. Simon also made a general study of the Willmore functional for surfaces in general codimension, relating the value of the functional to several geometric quantities. Such geometric estimates have proven to be relevant in a number of other important works, such as in Ernst Kuwert and Reiner Schätzle's analysis of Willmore flow and in Hubert Bray's proof of the Riemannian Penrose inequality. Simon himself was able to apply his analysis to establish the existence of minimizers of the Willmore functional with prescribed topological type. With his thesis advisor James Michael, Simon provided a fundamental

Sobolev inequality In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Re ...

for submanifolds of Euclidean space, the form of which depends only on dimension and on the length of the mean curvature vector. An extension to submanifolds of

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s is due to David Hoffman and Joel Spruck. Due to the geometric dependence of the Michael−Simon and Hoffman−Spruck inequalities, they have been crucial in a number of contexts, including in Schoen and

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

's resolution of the positive mass theorem and Gerhard Huisken's analysis of

mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of sur ...

. Robert Bartnik and Simon considered the problem of prescribing the boundary and

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

of a spacelike hypersurface of

Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...

. They set up the problem as a second-order partial differential equation for a scalar graphing function, giving novel perspective and results for some of the underlying issues previously considered in

Shiu-Yuen Cheng Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...

and Yau's analysis of similar problems. Using approximation by harmonic polynomials, Robert Hardt and Simon studied the zero set of solutions of general second-order elliptic partial differential equations, obtaining information on

Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assi ...

and rectifiability. By combining their results with earlier results of Harold Donnelly and

Charles Fefferman Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He was awarded the Fields Medal in 1978 for his contribu ...

, they obtained asymptotic information on the sizes of the zero sets of the eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold. Schoen, Simon, and Yau studied stable minimal hypersurfaces of

s, identifying a simple combination of Simons' formula with the stability inequality which produced various curvature estimates. As a consequence, they were able to re-derive some results of Simons such as the Bernstein theorem in appropriate dimensions. The Schoen−Simon−Yau estimates were adapted from the setting of minimal surfaces to that of "self-shrinking" surfaces by Tobias Colding and William Minicozzi, as part of their analysis of singularities of

. The stable minimal hypersurface theory itself was taken further by Schoen and Simon six years later, using novel methods to provide geometric estimates without dimensional restriction. As opposed to the earlier purely analytic estimates, Schoen and Simon used the machinery of

. The Schoen−Simon estimates are fundamental for the general

Almgren–Pitts min-max theory In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's m ...

, and consequently for its various applications. William Meeks, Simon, and Yau obtained a number of remarkable results on minimal surfaces and the topology of three-dimensional manifolds, building in large part on earlier works of Meeks and Yau. Some similar results were obtained around the same time by

Michael Freedman Michael Hartley Freedman (born April 21, 1951) is an American mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gen ...

, Joel Hass, and

Peter Scott Sir Peter Markham Scott (14 September 1909 – 29 August 1989) was a British ornithologist, conservation movement, conservationist, painter, naval officer, broadcaster and Sportsperson, sportsman. The only child of Antarctic explorer Robert Fal ...

.Freedman, Michael; Hass, Joel; Scott, Peter. Least area incompressible surfaces in 3-manifolds. Invent. Math. 71 (1983), no. 3, 609–642.

Bibliography

Textbooks. * * * Articles. * * * * * * * * *

References

External links

* {{DEFAULTSORT:Simon, Leon 1945 births Living people 20th-century Australian mathematicians 21st-century Australian mathematicians Fellows of the Royal Society Stanford University Department of Mathematics faculty Fellows of the Australian Academy of Science Fellows of the American Mathematical Society Mathematical analysts University of Adelaide alumni