#### by Rob Kirby

James Harris Simons was born on April 25, 1938, in Newton, Massachusetts. His father was in sales, initially selling films for 20th Century Fox to theaters, then later entering the shoe business with his father-in-law. His mother went to art school and was, according to Jim, “a pretty good painter”,1 though as a stay-at-home mom, she did not exhibit commercially.

Jim attended Newton High School, and at the age of 17 entered MIT, graduating in three years in 1958. Hearing that Chern was going to Berkeley, he decided to go there, too, to be able to work with him. Chern delayed his arrival in Berkeley by a year, however, and Jim ended up writing his PhD thesis [1] under Bert Kostant, finishing in 1961 (but graduating formally in 1962). Jim tells the story of being in a seminar during his second year, and seeing a distinguished Chinese man walk into the seminar room; he asked who that was, and was startled to hear it was Chern. Jim had thought Chern was an abbreviated form of an eastern European name, such as “Chernavski”.

Jim started teaching at MIT in the fall of 1961 as a Moore instructor. His stay there lasted a year, at which point he resigned to go into business with friends who launched a floor tile manufacturing concern in Colombia, South America. This was the first of three occasions when Jim decided to “leave” mathematics, though this initial departure was short-lived. Assembling the manufacturing equipment in Colombia turned out to be a lengthy process, and in the meantime Jim took a summer job calculating and programming Bessel functions for a radar company. It didn’t take long for him to realize that he disliked the work and wished to return to academia.

Jim mentioned his change of heart to Raoul Bott at Harvard, who immediately offered to include him on his upcoming NSF contract for the academic year 1962–63. Not long after, he was offered assistant professorships by Harvard and MIT both. Deciding between the two was a struggle, but Jim had grown up in Boston and was swayed by Harvard’s stature.

Jim spent two years at Harvard, but was restless and decided to leave academia a second time. In 1964 he went to work at the Institute for Defense Analyses (IDA) in Princeton and became a code cracker for four years. The IDA was flexible, allowing him to spend half his time working on their projects, and half on his own math research.

The focus of Jim’s mathematics at this time was minimal varieties.
The Plateau problem asks if a codimension-2
manifold __\( M^{n-2} \)__ in __\( \mathbb{R}^n \)__ can bound a smooth __\( (n-1) \)__-dimensional
manifold of minimal area. The answer was known for __\( n=3 \)__
(Jesse Douglas
received the Fields Medal in 1936 for his solution) and __\( n=4 \)__
[e1].
Previous attempts were made using analysis
and measure theory, but Jim was working from a geometer’s perspective,
and thinking about what it meant to have mean curvature zero. It was known
that any singularities in a minimal surface are cones over minimal
surfaces in the __\( (n-1) \)__-sphere; thus the singularities are
0-dimensional. Jim found a method to deform a cone singularity
to a smooth manifold of smaller area, which worked for __\( n \leq 7 \)__,
thereby solving Plateau’s problem in those dimensions.

The solution also gave a proof (for __\( n\leq 8 \)__) of the generalized Bernstein’s
conjecture which stated that if __\( f:\mathbb{R}^{n-1} \to\mathbb{R} \)__ is a __\( C^{\infty} \)__
function whose graph is a minimal surface in __\( \mathbb{R}^n \)__, then __\( f \)__ is linear.

But the method didn’t hold for the case of __\( n=8 \)__. Jim could show that the cone
on __\( \mathbb{S}^3 \times\mathbb{S}^3 \)__ (embedded smoothly in __\( \mathbb{S}^7 \)__) in the 8-ball
was locally stable and locally minimal. But he did not know how to
show it was globally minimal. He was nonetheless pleased with this body of
results, wrote them up, and circulated the paper, which later appeared
in the *Annals of Mathematics*
[2].

E. Bombieri, E. de Giorgi and E. Giusti [e2] then showed that the cone was indeed a global minimum, and this settled Plateau’s problem and the Bernstein conjecture negatively in higher dimensions. Years later, Bombieri told Jim that after hearing word of Simons’ results, the three Italians had worked for two days straight without sleep to obtain their result.

Shortly after publication of the minimal surfaces work, Jim was fired from the IDA because of his outspoken opposition to the Vietnam War. In September 1968, he accepted an offer from the State University of New York at Stony Brook to become chair of the math department. He recognized this as an opportunity to build the department: Hershel Farkas, Irwin Kra and Tony Phillips had joined Stony Brook the same year as Jim, so they had a nucleus of young mathematicians, and right off Jim was able to hire James Ax, Jeff Cheeger, Ron Douglas, Detlef Gromoll, and Roger Howe (who later slipped away to Yale).

While at Stony Brook, Jim decided he wanted to embark
on something new, having finished his work on the Bernstein
conjecture. So began his study of characteristic classes, and his
attempt to
find a combinatorial formula for the first Pontryagin class __\( p_1 \)__ of a
4-manifold. Only later did he discover that many mathematicians
had so tried — and failed.

As Chern and Simons stated ([3], page 48), “The hope was that by integrating the characteristic curvature form […] simplex by simplex, and replacing the integral over each interior by another on the boundary, one could evaluate these boundary integrals, add up over the triangulation, and have the geometry wash out, leaving the sought after combinatorial formula. This process got stuck by the emergence of a boundary term which did not yield to a simple combinatorial analysis”, and which “seemed interesting in its own right”.

This boundary term, a 3-form on the frame bundle over the link of a
simplex, the 3-sphere, could also be considered for any orientable
3-manifold. Jim saw that the 3-form was closed, thus representing
a real cohomology class on the frame bundle. A cross-section of the
frame bundle would pull the cohomology class down to the base
3-manifold, and different cross-sections would change the answer by
an integer, so Jim now had an interesting cohomology class over __\( \mathbb{R}/\mathbb{Z} \)__
on the 3-manifold, which, evaluating on the 3-manifold, gives an
invariant in __\( \mathbb{R}/\mathbb{Z} \)__.

But all this depends on a metric, so Jim wondered how the invariant would
change under a conformal change in the metric — though, in fact, it
turned out *not*
to change. Then came an epiphany during a flight to Minnesota with
Jeff Cheeger.
Jim wanted to calculate what happens if the 3-manifold
is conformally embedded in __\( \mathbb{R}^4 \)__, and he discovered to his
disappointment that the invariant was zero. But Jeff thought it was a
“*fantastic*” result, for then the invariant gives an obstruction to
a conformal embedding of a conformal 3-manifold
into __\( \mathbb{R}^4 \)__.

That’s when Jim realized he was onto something with his bundle of results. Chern got very excited when Jim showed it to him, and that was the beginning of their collaboration. As Jim explained, “I never published the 3-manifold results because it was subsumed in this bigger bundle.”

The formula for the 3-form on the frame bundle is the now famous
__\[
\operatorname{Tr}\mkern2mu[A \wedge dA + \tfrac{2}{3} A \wedge A \wedge A],
\]__
whose integral
gives the *action* in Chern–Simons theory, a 3-dimensional
topological quantum field theory discovered by
Ed Witten.
Jim recalls, “The physicists love that term. I remember telling
Yang
at a certain
point that I’ve got a bunch of numbers, but he didn’t want to pursue
it, and I didn’t know any physics. It wasn’t until six or seven years
later that Witten picked it up. So it was a fluke; I didn’t set out
to change physics but you never know what you’re going to get when
you set out with a good problem.”

Jim was honored in 1975 by the eighth Veblen Prize in Geometry for the abovementioned mathematics. The other Veblen Prize that year went to Bill Thurston.

After his collaboration with Chern, Jim worked with Jeff Cheeger on what turned into differential characters [4], and this led to further work (mentioned in [e3]) which was quite frustrating to Jim and partly explains his decision to leave math for the third time.

The differential characters were secondary characteristic classes of
__\( G \)__-bundles which could be described more geometrically in the special
case of __\( \mathit{SO}(n) \)__-bundles over a simplicial complex, where __\( \mathit{SO}(n) \)__ is
considered as a discrete group. Dividing out by __\( \mathit{SO}(n-1) \)__ to get an
__\( (n-1) \)__-sphere fiber, and taking a section over the boundary of an
__\( (n-1) \)__-simplex, one gets (generically) an __\( (n-2) \)__-sphere determined by
the __\( n \)__ vertices and the geodesic faces they describe. Normalizing
the metric so that the volume of the sphere is one, the volume of the
__\( (n-1) \)__-simplex is an interesting real number mod __\( \mathbb{Z} \)__.

Assuming the dihedral angles are rational, Jim and Jeff expected the volumes to be irrational, despite some atypical cases where the volume is rational. They were unable to find any such cases, however, and this problem remains open today.

Jim spent 1975–76 at the University of Geneva doing math research, but also some trading on the stock market. When he returned in 1976–77, he realized he was good at investing, and that he liked it, so he decided to go into it full time. This decision led to the birth of Renaissance Technologies.

Meanwhile, Jim’s other entrepreneurial ventures flourished. The jazz club Mt. Auburn 74 was a resounding success. His friend Steve Kuhn who was at Harvard with him played there often and has since become a famous jazz pianist. Joan Baez got her start there. And the South American project went on just as planned: the equipment arrived, the manufacturing plant opened and the partners now have a network of businesses operating there. There are other ventures one could mention, but the most famous is Renaissance Technologies, especially the Medallion Fund, arguably the most successful hedge fund ever.

Through the Simons Foundation, Jim has also been active as a philanthropist in a wide variety of fields. The Foundation’s contributions to the mathematical sciences alone are considerable: 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, the construction of MSRI’s Chern Hall, Brookhaven National Laboratory, and the endowing of several university chairs.

With his wife, Marilyn, Simons created the Avalon Park and Preserve in Stony Brook under the auspices of the Paul Simons Foundation (established in memory of their son, Paul), and the couple has made significant contributions to autism research (SFARI).

The Simons’ philanthropy has an international scope as well: the Nick Simons Institute in Nepal, founded in memory of their son Nick who had traveled widely in the region, sustains a rural health care initiative that was launched in 2006.

Jim retired from Renaissance Technologies in 2010 and is yet again working on math, having embarked on a series of joint papers with Dennis Sullivan.