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*By *Daniel S. Freed

*March 4, 2021*

A former graduate student reflects on how Isadore Singer, who died on February 11, brought together mathematicians, physicists and anyone else interested in the deeper connections between diverse fields.

Isadore Singer in his office at the Massachusetts Institute of Technology in 2006.

John Nikolai

The mathematics community lost a titan with the passing last month of Isadore “Is” Singer. Born in Detroit in 1924, Is was a visionary, transcending divisions between fields of mathematics as well as those between mathematics and quantum physics. He pursued deep questions and inspired others in his original research, wide-ranging lectures, mentoring of young researchers and advocacy in the public sphere.

Mathematical discussions with Is were freewheeling. He wanted to get to the essence of the matter at hand, to understand and present new ideas in his own way. He constantly asked questions and provoked others to look deeper and more broadly. He knew no boundaries, which led him to forge deep connections between fields. Above all, he valued his freedom — the freedom to explore, the freedom to err, the freedom to create. A social mathematician by nature, he sought out and nurtured friendships with like-spirited collaborators.

Is came to mathematics relatively late, having majored in physics as an undergraduate at the University of Michigan before going off to war in 1944. Upon his return, he went to graduate school in mathematics at the University of Chicago to better understand relativity and quantum mechanics, only to discover that mathematics was his true intellectual home.

One of his first loves in mathematics, differential geometry, employs calculus to study smooth shapes, called manifolds. (By contrast, algebraic geometry uses algebra — the number side of mathematics — to study shapes defined by polynomial equations.) Local quantities, such as curvature, are the province of the differential geometer, and questions in the subject quickly lead to differential equations, whose solutions carry geometric meaning. Global questions — for example: How many holes of a particular dimension does a given manifold have? — are the province of topology, which uses a whole different set of tools. The province of analysis, another early focal point for Is in mathematics, involves function spaces and the study of differential equations that special functions satisfy.

But Is’ world did not admit provinces. Local differential geometry, global topology and analysis are different aspects of geometry. Their interplay is where the real fun is, and it lies at the heart of his most celebrated achievement: the Atiyah-Singer index theorem. Very roughly, this result produces a count of solutions to special differential equations on curved spaces in topological terms. It led to immediate advances in topology, geometry and analysis; surprisingly, it has ramifications in quantum physics as well.

Starting in 1950, Is spent most of his professional life at the Massachusetts Institute of Technology, save for a few years in the late 1970s and early 1980s at the University of California, Berkeley. After arriving at MIT, Is and fellow Midwesterner Warren Ambrose reshaped how we understand, teach and carry out research in differential geometry. In the mid-1950s, Is spent a year at the Institute for Advanced Study in Princeton, New Jersey, then a locus of radical advances in topology and algebraic geometry. He encountered new vistas and met future collaborators, including Michael Atiyah, a kindred spirit whose mathematical homes in algebraic geometry and topology perfectly complemented Is’ expertise in differential geometry and analysis. Michael had a similar freewheeling, anything-goes working style, and he too constantly sought deeper meanings and connections. The Atiyah-Singer index theorem was born of this instinct, the direct result of a penetrating question Michael put to Is when the latter arrived for a sabbatical visit to the University of Oxford in 1962: Why is the A-roof genus of a spin manifold an integer?

The “A-roof genus” is a topological invariant of manifolds that is part of 1950s topology, guaranteed a priori only to be a rational number — a ratio of whole numbers. But topologists had proved that it is actually an integer for manifolds with a particular geometric feature: a spin structure. (Spinors, which are a kind of square root of vectors, had been introduced in algebra and also in physics as part of Paul Dirac’s theory of the electron. A spin structure on a manifold allows such square roots to exist.) Even though Michael and his collaborator Fritz Hirzebruch had proved the integrality as a consequence of their development of *K*-theory, an important innovation in topology, Michael still sought insight that the proofs do not provide. The desire for a more profound understanding resonated deeply with Is, who was immediately hooked by the question. A positive whole number should count something, and a whole number which can be negative may be the difference between two positive whole numbers, each of which counts something. In the Atiyah-Singer worldview, geometry is paramount, so whatever is being counted should be geometric.

Here, Is’ mastery of differential geometry came into play. Presumably inspired by the geometry of other integer invariants of the period, and based on his knowledge of Dirac’s theory, Is devised a version of the Dirac equation in differential geometry — it requires the spin structure which is at the heart of Michael’s question — and he conjectured that the A-roof genus measures the existence and uniqueness of solutions to that equation. This was Is’ response to the question. Michael immediately saw how to incorporate it into the *K*-theory that he and Fritz had developed, and he quickly had the statement of the general index theorem in hand. The first proof, which involves a large dose of analysis, was done within the year.

The statement, the proof and the immediate applications which flowed from them brought together mathematical subdisciplines which in the prevailing ethos of the day often existed in noninteracting orbits. The circle of mathematical ideas around the Atiyah-Singer theorem grew in the ensuing years. When the scope expanded even further in the mid-1970s, Is again played a central role.

The impetus this time was the self-duality equation, a nonlinear differential equation that arises in quantum field theory. Is was very familiar with the so-called Wu-Yang dictionary, which related gauge theory in physics to the structures in differential geometry he had learned from Shiing-Shen Chern, one of his teachers at the University of Chicago a quarter century earlier. The dictionary had grown out of dialogues at Stony Brook University with Jim Simons, whom Is had mentored years before at MIT. (Simons went on to found the Simons Foundation, which also funds this editorially independent publication.) Is saw a role for geometry in illuminating what the physicists had discovered, and on a trip to Oxford in 1977 he posed the question, laying out the problem in a series of lectures. Those lectures and similar ones elsewhere inspired a burst of activity. Significantly, the nonlinearity of the equation led to a fascinating new web of mathematics in which algebraic geometry, topology, differential geometry and analysis are beautifully entwined, now with physics in the mix as well.

Is went further. He grasped early on, at a time when this was in no way apparent, that the “quantum” in quantum field theory is something that we geometers need to engage with directly and incorporate into our mathematics. It is difficult to convey how visionary Is was at that time. He led the way by grappling with the physics and presenting it on his own terms in courses on quantum field theory, supersymmetry and string theory at both Berkeley and MIT. His Tuesday seminar at Berkeley often featured physicists explaining the latest results, followed by a Chinese dinner at which our communal education in physics continued. Is’ position in the mathematics community and the force of his ideas brought more and more mathematicians on board. As a result of his leadership, by the mid-1980s there was a vigorous interaction between quantum field theorists and geometers. As Is foresaw, the relationship grew and deepened. It continues to bear fruit for both fields.

Is’ leadership in mathematics and science extended to policy and community, where he engaged at a high level. To mention just one achievement, in the early 1980s Is teamed up with Chern and Cal Moore to found a new home for mathematics research, the Mathematical Sciences Research Institute. The fact that research institutes throughout the world now emulate this model is a tribute to the founders’ vision.

Of course, Is also enjoyed many aspects of life beyond mathematics and physics. His devotion to his family, and theirs to him, was paramount. Ambrose taught him to love jazz, another lifelong passion. And there was always tennis — Is played vigorously and enthusiastically into his 90s, constantly working on his game.

In fact, his longtime coach and close friend Jeff Bearup conjured an image of Is that perfectly captures the effect he had on people. He would arrive at his tennis club — or it could easily have been a math or physics department or a conference — and everyone would turn, smile, and shout out with admiration and respect: “Is!”