Jim Simons’ Return to Mathematics

This is an account of Jim Simons’ return to mathematics in the early 2000’s.

There is a “first indication” story:
After Jim left research per se in the late 1970’s, there was a math discussion at his 60th birthday fest in 1998 at Stony Brook, where Jim revealed to Maxim Kontsevich and me that he had been wondering about the meaning of two connections having (pointwise) the same invariant polynomials built on the curvature. He mentioned especially the case where one connection is flat.

I recall being shocked that this question made good sense and that, to my knowledge, no one had thought to ask it before.

A second stimulus:
This was Blaine Lawson’s preprint for the published paper The de Rham-Federer Theory of Differential Characters and Character Duality, giving a completely different geometric and analytic construction of the receptacle “differential characters” for the first Jim Simons’ invariant of 3-manifolds (this receptacle having been constructed by Jim with Jeff Cheeger in the early 1970s. This was contemporary with Pierre Deligne’s construction of Deligne Cohomology in algebraic geometry which transposed into differential geometry seemed equivalent to the Cheeger-Simons receptacle or theory called “differential characters”).

Jim had the idea to axiomatize such theories that resulted in the paper Axiomatic Characterization of Ordinary Differential Cohomology. These equivalent versions of differential characters being renamed there “Ordinary Differential Cohomology” because a seminal paper, Quadratic Functions in Geometry, Topology, and M-Theory, by Mike Hopkins and Isadore Singer defined a general construction for every “extraordinary cohomology theory”, for example complex K-theory, now referred to as differential K-theory.

This was the beginning of a dozen year mathematical collaboration of Jim with me that tied the above two stories together with a still unfinished story about a question raised by Isadore Singer which was motivated by theoretical physics. This being connected with the fact that the higher dimensional Chern-Simons versions of Jim’s first invariant of 3-manifolds (called “Chern-Simons terms”) are used frequently up to the present day in theoretical physics papers.

There were several papers [3-7] in this collaboration with Jim which were likely related to his being elected to the National Academy of Sciences (NAS) in 2014, because they showed Jim was still a true scientist besides being a great supporter of science.

I recall a gala dinner for the National Museum of Mathematics where Jim was introduced mentioning an article in the New York Times about his election to the NAS entitled “The world’s smartest billionaire”. As Jim walked to the podium, I wondered how he would react.

He was perfect. He said “How many billionaires are there in the world? I would rather be the world’s smartest millionaire!”. We all laughed and relaxed. It was Jim Simons at his best!

Regarding his support of science as compared to being a scientist, it has not been mentioned enough that Jim usually did much more than providing financial support for projects related to science and to other worthy endeavors in which he was involved.

Besides contributing capital, Jim worked hard day to day to ensure the health and success of these projects. In the case of the Simons Center for Geometry and Physics, I noted that he attended every meeting related to its design, its evolution and its maintenance until the very end of his capabilities.

He worked very hard all the time. I once guessed it might be about three times the normal busy schedule of a hard-working person.

Dennis Sullivan
New York, October 2024

References

[0] S.-S. Chern, J. Simons: Characteristic Forms and Geometric Invariants. Annals of Mathematics, Vol. 99, No. 1 (Jan., 1974), pp. 48-69

[1] B. Lawson, R. Harvey and J. Zweck: The de Rham-Federer Theory of Differential Characters and Character Duality. American Journal of Mathematics 125, No. 4, 791—847 (2003)

[2] M. J. Hopkins and I. M. Singer: Quadratic Functions in Geometry, Topology, and M-theory. J. Differ. Geom. 70, No. 3, 329—452 (2005)

[3] J. Simons and D. Sullivan: Axiomatic Characterization of Ordinary Differential Cohomology. J. Topol. 1, No. 1, 45—56 (2008)

[4] J. Simons and D. Sullivan: The Atiyah Singer Index Theorem and Chern Weil Forms. Pure Appl. Math. Q. 6, No. 2, 643—645 (2010)

[5] J. Simons and D. Sullivan: Structured Vector Bundles Define Differential K-theory. Clay Math. Proc. 11, 579—599 (2010)

[7] J. Simons and D. Sullivan: Differential Characters for K-theory. Prog. Math. 297, 353—361 (2012)

[7] J. Simons and D. Sullivan: Characters for Complex Bundles and their Connections. arXiv:1803.07129 (2018)
(containing an Appendix with a remarkable calculation/determination by Jim related to the adiabatic limit of certain distinct connections becoming “Chern-Simons” equivalent.)