Leroy P. Steele Prize
The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories.
The prizes have b ...
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. He is currently Professor Emeritus in the Mathematics Department at
Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...
.
Biography
Academic career
Leon Simon, born 6 July 1945, received his BSc from the
University of Adelaide
The University of Adelaide (informally Adelaide University) is a public research university located in Adelaide, South Australia. Established in 1874, it is the third-oldest university in Australia. The university's main campus is located on N ...
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 is a public research university based in Adelaide, South Australia, with a footprint extending across 11 locations in South Australia and the Northern Territory. Founded in 1966, it was named in honour of British navigator ...
as a lecturer, then at
Australian National University
The Australian National University (ANU) is a public research university located in Canberra, the capital of Australia. Its main campus in Acton encompasses seven teaching and research colleges, in addition to several national academies and ...
as a professor, at the
University of Melbourne
The University of Melbourne is a public research university located in Melbourne, Australia. Founded in 1853, it is Australia's second oldest university and the oldest in Victoria. Its main campus is located in Parkville, an inner suburb nor ...
, the
University of Minnesota
The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...
, at
ETH Zurich
(colloquially)
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, image = ETHZ.JPG
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, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ac ...
, 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. Among his doctoral students there is
Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Career
Born in Celina, Ohio, and a 1 ...
Fellow
A fellow is a concept whose exact meaning depends on context.
In learned or professional societies, it refers to a privileged member who is specially elected in recognition of their work and achievements.
Within the context of higher education ...
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 Nevanlinna Prize (to be rename ...
MacTutor History of Mathematics Archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathemati ...
. 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 ...
fellow of the Royal Society
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges 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
The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories.
The prizes have b ...
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
The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories.
The prizes have b ...
, 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 In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : ''U'' → ...
, 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 i ...
s and Lyapunov-Schmidt reduction. The resulting Łojasiewicz−Simon inequalities are of interest in and of themselves and have found many applications in geometric analysis.
Simon's primary applications of his Łojasiewicz−Simon inequalities deal with the uniqueness of tangent cones of minimal surfaces and of tangent maps of
harmonic map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
s, making use of the deep regularity theories of William Allard,
Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Career
Born in Celina, Ohio, and a 1 ...
, and Karen Uhlenbeck. 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
In differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold. First introduced by Richard S. Hamilton, Yamabe flow is for noncompact manifolds, and is the
negative ''L'' ...
. 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
In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is def ...
and in
Hubert Bray Hubert Lewis Bray is a mathematician and differential geometry, differential geometer. He is known for having proved the Riemannian Penrose inequality. He works as professor of mathematics and physics at Duke University.
Early life and education
...
's proof of the
Riemannian Penrose inequality
In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannia ...
. 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 or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
Joel Spruck
Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University ...
. 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's resolution of the
positive mass theorem
The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...
and
Gerhard Huisken
Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huiske ...
'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 surf ...
.
Robert Bartnik
Robert Bartnik (1956-2022) was an Australian mathematician based at Monash University. He is known for his contributions to the rigorous mathematical study of general relativity. He received his bachelor's and master's degrees from Melbourne Univ ...
and Simon considered the problem of prescribing the boundary and mean curvature of a spacelike hypersurface of Minkowski space. 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 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 ass ...
and rectifiability. By combining their results with earlier results of Harold Donnelly and Charles Fefferman, 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
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s, identifying a simple combination of
Simons' formula
In the mathematical field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) is a fundamental equation in the study of minimal submanifolds. It was discovered by James ...
with the stability inequality which produced various curvature estimates. As a consequence, they were able to re-derive some results of
Simons Simons is a surname of Scandinavian origins and a variant of Sigmundsson, a patronymic surname with roots in proto-Germanic ''*segaz'' and ''*mundō'', giving a rough translation of "protection through victory".
Notable people
A
* Alan ...
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
Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great grandchild of Ludwig August Colding.
Biography
He was born in Copenhagen, Denmark, to Torben Holck Colding ...
and
William Minicozzi
William Philip Minicozzi II is an American mathematician. He was born in Bryn Mawr, Pennsylvania, Bryn Mawr, Pennsylvania, in 1967.
Career
Minicozzi graduated from Princeton University in 1990 and received his Ph.D. from Stanford University in 199 ...
, as part of their analysis of singularities 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 surf ...
. 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 geometric measure theory. 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 ...
, 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,
Joel Hass
Joel Hass is an American mathematician and a professor of mathematics and at the University of California, Davis.Peter Scott.Freedman, Michael; Hass, Joel; Scott, Peter. Least area incompressible surfaces in 3-manifolds. Invent. Math. 71 (1983), no. 3, 609–642.