I was sorry to hear this morning of the death yesterday at the age of 96 of Is Singer, a mathematician who led much of the interaction between mathematics and physics during the 1970s and 1980s. In the early stages of my career, among mathematicians investigating the amazing relations between mathematics and the quantum field theories describing fundamental physics there were three towering figures: Atiyah, Bott and Singer. That the last of them has now left us marks the end of an era.

Each of the three had a huge influence on me, both intellectually and personally. Reading their papers and listening to their lectures were great intellectual experiences, shaping early on my understanding of what is central to mathematics and how it fits together with physics. Especially inspirational was the way that they brought together very different fields of mathematics, with Atiyah having his roots in algebraic geometry, Bott in topology and Singer in analysis. Their work together makes a strong case for the unity of mathematics and the relation to physics makes an equally strong case for the unity of mathematics and physics.

On a personal level, at a time when I was tentatively moving from a career in physics to one in mathematics, getting to meet and talk to each of them had a big impact. Much as I respected the great theoretical physicists I had met, rarely had I found them to be particularly friendly or encouraging, and their attitudes influenced the general atmosphere of the field. Atiyah, Bott and Singer struck me each in their own way as wonderfully warm and enthusiastic personalities, and I believe this influenced the atmosphere among mathematicians working in their fields. They were among the most respected figures in the math community, so their enthusiasm for ideas coming out of physics generated a lot of interest in these ideas among a wide variety of mathematicians.

Singer had always had an interest in physics, majoring in physics as an undergraduate at the University of Michigan, then after the war going to graduate school in mathematics at the University of Chicago. I highly recommend reading or watching this long interview with him from 2010, where you can learn the story of his career.

A mathematical high point of this career was his work during the early 1960s with Atiyah that led to the Atiyah-Singer index theorem. A crucial part of this story was Atiyah in 1962 asking Singer why the A-roof genus was integral. Singer realized that this was because it counts the number of solutions of an equation, and that the equation was the Dirac equation. This example in some sense generates a huge amount of mathematics which is described by the index theorem, and which links together very different mathematical fields. On this and other topics, well-worth reading is the 2004 interview with Atiyah and Singer after they were awarded one of the first Abel Prizes.

One can trace much of the history of the modern interaction of mathematics and quantum field theory to an origin back in the summer of 1976, when Singer visited Stony Brook and talked to physicists there about gauge theories, geometry and the BPST instanton (Simons and Yang a year earlier had started to realize how gauge theory, geometry and topology were linked). The next year he was in Oxford working with Atiyah and Hitchin on instantons, which really set off an explosive development of new ideas, inspiring and fascinating both mathematicians and physicists.

Singer spent the years from 1977 to 1983 at Berkeley, which he turned into a major center for this new mathematical physics. During this time he was also one of the founders of MSRI, which to this day plays a major role in worldwide mathematical research. After 1983 he returned to MIT, from which he retired in 2010. I believe the last time I saw him was at his 85th birthday conference, which I wrote about here.

**Update**: The New York Times has an obituary here.

**Update**: Dan Freed (who was a graduate student of Singer’s) has a piece about Singer at Quanta magazine here.

I am sorry for your loss, and the world’s.

Now we (those of us in our age group, each in our respective disciplines) are the elders.

We can only hope to play the role so well.

Is there a good, reader-friendly source for understanding what the index theorem means and what its implications are? My student self — who desperately tried to read Lawson and Michelson back in graduate school but found it inscrutable — would love to know!

Davide,

When I was learning this stuff I found Atiyah’s own expository articles clear and inspiring. See for instance

1966 ICM lecture

1967 Algebraic topology and elliptic operators

1973 The index of elliptic operators (colloquium lectures)

Classical Groups and Classical Differential Operators on Manifolds

For the heat equation method and later developments, one place to look is Dan Freed’s unfinished notes on Dirac operators (he was Singer’s student)

https://web.ma.utexas.edu/users/dafr/DiracNotes.pdf

For relation to quantum mechanics:

Orlando Alvarez, Lectures on Quantum Mechanics and the Index Theorem

Thanks for that – I met Feigenbaum once, and had the same experience that mathematicians at the high level were more engaging than physicists. Was just thinking about Irving Segal and his conformal c0smology and then find out here he was Singer’s advisor. Small universe.

-drl

drl,

Feigenbaum was a rather unusual character, and many would consider him a physicist, not a mathematician.

I was interested to see a nice tweet by Erik Verlinde about Singer:

https://twitter.com/erikverlinde/status/1360328234743836676

where he also notes that Singer was much friendlier than the people he was used to dealing with

“[Singer was] one of the friendliest people I have met.”

Verlinde though also shows the difference between theoretical physics and math culture when he writes:

“In my view, he was the only mathematician with a deep understanding of theoretical physics.”

Among mathematicians I know, even the few who might think there is only one physicist with a deep understanding of mathematics would never say that out loud…

Well written, authentic obituary. Many thanks, Peter. I am looking forward to read the 2010 interview with Prof. Singer tonight. Isadore Singer has contributed a lot to the renewed fertile interaction between physics and mathematics initiated by physicists’ discoveries of selfdual field configurations in gauge theories in the 1970ies, and it makes me feel sad that another grand master of the field is no longer among us. The index theorem and how it counts the zero modes of the Dirac operator on an instanton is wonderful stand-alone math stuff which, moreover, has deep physical implications in pointing to the essentials of dynamical chiral symmetry breaking and quantum anomalies in Yang-Mills. In addition, its significance for the physics of gauge theory is expressed by the fact that Nahm’s duality transformation between instantons on T4, which vastly generalises the important ADHM construction, significantly relies on the index theorem. Nahm’s transformation, in turn, is the venue to calorons of non-trivial holonomy which mediate the emergence of mass and charge from pure energy in Yang-Mills.

Davide wrote:

I don’t know the perfect one for you. John Rognes wrote a popular account for the Abel prize which might be good for people who know less math than you. You might find it unsatisfyingly vague.

Briefly put, the Atiyah-Singer index theorem gives a topological formula for the dimension of the space of solutions of a differential equation on a manifold. As such, it’s a great way to use topology to learn something about the solutions of differential equations without actually solving them. And conversely, it’s a great way to use differential equations to learn things about the topology of manifolds!

It was extremely important sociologically. It built a bridge between two fields — partial differential equations and topology — which had not existed before. And because the differential equations it applies to are important in physics, it connected physics more firmly to both these fields.

More precisely, the Atiyah-Singer index theorem is a formula for the “index” of a differential equation, where the index is the dimension of the space of solutions (the “kernel”) minus something else (the dimension of the “cokernel”). So, you have to be clever to use it to get precise information about the dimension of the space of solutions. But it easily gives a lower bound.

Furthermore, it only works when the manifold is compact and the differential equation is “elliptic” (like the Laplace equation, not like the wave equation or heat equation).

On the bright side, it works for differential equations where the solutions are not just functions, but sections of vector bundles — like the elliptic version of the Dirac equation, which turns out to be the key to the general case. Another interesting thing is that because Dirac equation involves spinors and Clifford algebras, the Atiyah-Singer index theorem highlights the role of spinors and Clifford algebras in topology.

I spent a lot of time in college reading

Seminar on the Atiyah-Singer Theorem, by Richard Palais. It had great explanations of the analysis prerequisites, like pseudodifferential operators on vector bundles, Sobolev spaces, and chain complexes of Hilbert spaces. Somehow all that seems second nature to me by now. But I still don’t feel I have a great intuitive feel for the topology: that is, why the formula for the index in terms of characteristic classes is what it is.When I went to grad school at MIT, Singer was one of the stars there: he ran a seminar on mathematics connected to quantum field theory, and a huge crowd attended, including Raoul Bott and many other bigshots. When visitors like Witten or Atiyah came to town, we’d all go to their talks.

At that time — say, 1982-1986 — it seemed that everyone wanted to learn the Atiyah-Singer index theorem and generalize the heck out of it. I attended Dan Quillen’s seminar where he was trying to find a really simple proof of the Atiyah-Singer theorem. It was very exciting, because he was working in real time. Each class he would start flawlessly, and then continue until he got stuck on something. In the end he was scooped by Ezra Getzler, who however used some fancy analysis that Quillen would have wanted to avoid.

Just want to add, for those interested in the topic, that learning about the index theorem is the best way to prepare if you want to study noncommutative geometry in the sense of Connes, which pretty much involves (among other things) trying to generalize the index theorem to the noncommutative setting. So, that’s one modern spinoff of the Atiyah-Singer index theorem.

@Davide

I also attempted to go through Lawson & Michelsohn “Spin Geometry”, and also found it heavy going back in the day.

Years later, I found Rosenberg’s “Laplacian on a Riemannian Manifold” slightly less heavy going.