In Memoriam

We were very much saddened by the death on May 10, 2024 of Jim Simons, a Founding Member of the Association for Mathematical Research. Simons was famous for the brilliance of his mathematics, his breakthroughs in quantitative finance, and the generosity of his philanthropy.

The first of Simons’ two revolutionary contributions to mathematics involved minimal varieties. His paper, “Minimal varieties in Riemannian Manifolds”, [Ann. of Math., 1968], contained the derivation of the fundamental Simons equation which he used, together with basic results of Federer-Fleming, Di Georgi and Almgren in Geometric Measure Theory, to prove the interior regularity for globally volume-minimizing hypersurfaces up to ambient dimension 7. He also established the Bernstein conjecture, which says that a minimal graph in $R^n$ which is defined over all of $R^{n-1}$ is a hyperplane, up to ambient dimension n=8. In this same paper Simons showed that the cone on the Clifford torus, $S^3\times S^3\subset S^7$, is a local volume-minimizer, and he conjectured that this cone minimizes globally. It would therefore be a counterexample to interior regularity in ambient dimension 8. Not long afterwards, this was confirmed in work of E. Bombieri, E. Di Giorgi and E. Guisti, who also showed that the Bernstein Conjecture is false in all higher dimensions. This work of Simons made a fundamental change in one of the main fields of mathematical research.

The second of Simons’ revolutionary contributions was the discovery of Chern-Simons forms whose theory he developed with S.S. Chern in the paper, “Characteristic forms and Geometric Invariants” [Ann. of Math, 1974]. Chern-Simons forms are canonical locally defined differential forms on the total space of a principal $G$-bundle with connection. The exterior derivative of a Chern-Simons form is the pull back to the total space of a corresponding characteristic Chern-Weil form on the base. Chern-Simons forms have many mathematical applications, for example, to subtle conformal non-immersion theorems in Riemannian geometry. They are also deeply related to the Atiyah-Patodi-Singer index theorem. Subsequent work by Simons and J. Cheeger, and later by Simons and D.Sullivan, developed the theory of differential characters. These new mathematical objects are certain functions on cycles which take values in $R/Z$. Their definition permitted a refinement of Chern-Simons theory in which the Chern-Simons forms on the total space of the bundle are replaced by differential characters on the base space. In this context, differential characters constitute the first example of differential cohomology theories, defined much later by M. Hopkins and I. Singer.

Remarkably, not long after its discovery, Chern-Simons theory began to pervade theoretical physics, including quantum field theory and condensed matter theory, a trend which continues up until the present.

In 1976, for his work on minimal varieties and on Chern-Simons invariants, Simons shared the Veblen Prize of the American Mathematical Society with W. Thurston. He was elected to the National Academy of Sciences in 2014.

Another of Jim Simons’ fundamental contributions to science came from his Seminar with Frank Yang in the early 1970’s. Their discovery, that the field of bundle-connections in differential geometry and theory of Yang-Mills fields in particle physics were exactly the same, engendered a long and ongoing, close interaction between mathematicians and physicists, resulting in a new world of basic results in both fields.

The Simons Foundation, founded by Jim and his wife Marilyn, has provided extraordinary support for mathematics and related fields. It “established the Simons Center for Geometry and Physics (at Stony Brook), and has supported programs and institutions as varied as Math for America, the African Mathematics Project, graduate and postdoctoral fellowships in math and computer science, challenge grants to SUNY Stony Brook and MSRI (now the Simons Laufer Mathematical Sciences Institute), Brookhaven National Laboratory, and the endowing of several university chairs.” (https://celebratio.org/Simons_J/cover/183/). More recently, the Simons Foundation has supported many additional mathematics institutions and activities, such as Oberwolfach and arxiv.org, and established the Simons Institute for the Theory of Computing at UC Berkeley.

Jim Simons will be greatly missed.