#### by Rob Kirby

Isadore Manuel Singer (“Is” or “I.M.”) was born in Detroit on 3 May 1924. His parents, Freda Rose and Simon Singer, came to Canada from Poland in 1917, and married there. Freda was an orphan, but was taught by three uncles, a writer, a lawyer, and a printer. Later she was a seamstress. Simon was a printer in Warsaw, and according to family lore he set the first published story of his cousin, the celebrated author Israel Joshua Singer (brother of the Nobel laureate Isaac Bashevis Singer).

Once in Canada, Simon and Freda intended to slip over the border into the US, crossing from Windsor to Detroit. Upon arriving in Detroit, Simon discovered that the printers were on strike, and they offered him the royal sum of \$5 a day to join the strike. Simon wrote back home to Poland to say that it was true that the streets of America were paved in gold!

As a young child, Is learned to read while watching and helping his father typeset, even before attending kindergarten, and he declares that his mother was the smartest person he ever met.

Is’ younger brother, Sidney Singer, was born in 1929. He attended Wayne State University and then earned his Ph.D. in physics and mathematics at the University of Illinois. He went on to a long career at Los Alamos National Laboratory.

#### School years

Is remembers the Depression years well; times were tough and his father was out of a job for a year. Those years in comparative poverty left Is with compassion for those in need.

Although a straight A student, Is found school boring, and preferred to
play sandlot baseball and other sports (he was an avid Detroit Tigers
fan). But in his junior year in high school he had excellent teachers
in English literature (they read *War and Peace*, and Is’ oldest
daughter is named
Natasha after the character in the book) and
chemistry. He found the periodic table of elements
especially fascinating. He remembers delivering *Forverts* (*Yiddish Daily
Forward*) before school and then reading novels at night.

Fascinated by Einstein, he recalled in his lecture for the 1979 Einstein centennial conference ([e3], p. 39):

I gave my first series of talks on relativity forty years ago to a high school science club. I can’t imagine what my peers thought of me at the time; it took me twenty years to understand what I was talking about.

Nearing graduation from high school, Is expected to go to Wayne State in Detroit and live at home. But he also applied to the University of Michigan, and in a competition won a fellowship paying \$50 a semester which exactly covered tuition. His mother told him that he couldn’t pass up \$50, so he matriculated at UM in 1941.

#### University of Michigan

His first year at Michigan was not academically challenging but Is’ interest was piqued when his calculus instructor explained “how Newton derived the elliptic orbits of planets from his inverse square law” ([e11], p. 25). “Awesome!” he remembers thinking. In high school, he had been equally interested in English literature and physics, and recalls that when he took a poetry class at UM, the other students instinctively understood what he did not, whereas in physics it was the other way around. So at UM he soon settled on physics, and though he was always obliged to work part-time, he took a heavy load of courses, including math and physics, and finished in just two and a half years.

#### War-time studies

When World War II began, Is joined the Army Signal Corps for they allowed him to stay in college until he completed his degree, though they required him also to take courses in electrical engineering. One reason that he finished his undergraduate degree so quickly was that he was skeptical the Army would let him study a full four years without calling him up to active duty. He subsequently came to believe that those fast-paced years at Michigan were good training for his time at MIT.

On his way from the east coast to the Philippines in 1944, he stopped in Chicago to inquire about extension courses at the University of Chicago, where he learned that he could take extension classes in geometry and group theory remotely, both useful for understanding physics better. Thus, while he ran a communications school for the Philippine Army during his war years, he devoted his off-hours not to poker with his fellow soldiers, but to his University of Chicago courses.1

In ([e3], p. 38), Is reminisced about the consequences of his extension studies for his future career:

My inclination toward intellectual pursuits was encouraged by my English and chemistry teachers in Central High School, Detroit. I majored in physics at the University of Michigan (1941–44) but regret not availing myself of their marvelous mathematics department. Puzzled by both relativity and quantum mechanics, I tried to master both these subjects while serving in the U.S. Army. Having failed, I concluded I needed a year of mathematics before I returned to physics, and I became a graduate student at the University of Chicago in 1947. That year stretched to 30.

#### From physics to a Ph.D. in math

With the war over, Is applied to the math department at Chicago, telling them he wanted more mathematical training before switching to physics. They admitted him in January, 1947, and he received a Masters Degree in 1948. Struck with the beauty of mathematics, he abandoned physics and decided to work toward his Ph.D. in math instead.

Although S.-S. Chern and Irving Segal were both mentors to Is at Chicago, Chern did not arrive until Is’ last year, 1949–50, so Segal became his Ph.D. adviser. Is describes Segal as a “wonderful and lousy teacher at the same time”. Wonderful because he was teaching great material in functional analysis, but lousy because he became stuck about halfway through most lectures. However “lousy” meant that Is would go home and work hard on the sticky points, often working things out for himself. This procedure meant that he was immersed in something akin to the Moore method in teaching, where students must work out concepts and principles for themselves as much as possible. Segal typically worked late into the evening in his office, and was thus often available to Is for math talk, and not just about thesis work. So the “Moore method” worked quite well in Is’ case.

Upon finishing his Ph.D., Is received two offers, one from the Institute for Advanced Study (IAS) and one from MIT, the latter a slightly better offer at \$50 more per year.

#### Employment at MIT

Is went to MIT as a Moore Instructor, a program with outside support begun under Ted Martin in 1948 which allowed for a lower teaching load.2

The day Is arrived at MIT and met Warren Ambrose is an oft-told story, nowhere better than in this tribute to Ambrose written by Is himself in 1966 [15].

I have not forgotten my first day at MIT. In 1950 Moore instructors had to teach summer school. On a sunny afternoon early in July, I crossed the bridge in search of Building 2. Math headquarters was on the second floor; it still is. I introduced myself to Ruth Goodwin, who handled all secretarial services, and I asked to see the chairman. When I gave Ruth my name, a chap sitting across from her, head buried in the Boston Globe lowered his paper and said: “Singer, I’m Ambrose. There is a seminar in Lie groups in five minutes. You can see Martin later. Come.”

I did, and met John Moore, Barrett O’Neill, and George Whitehead, who became lifelong friends. After getting my teaching assignment from Ted Martin, Ambrose told me the seminar met at midnight in the Hayes-Bickford coffee shop. He would pick me up at 11:45. Kay Whitehead joined us for these evening sessions; the coffee was deadly, the conversation lively. Ambrose gave me a tour of Boston that first night and by the time he dropped me off, we were close friends.

One day Ambrose said, “Singer, you listened to Chern’s lectures. What did he say?” At Chicago, I had passively taken notes of S. S. Chern’s course, while writing a dissertation in another subject. What with interpreting my notes, reading Chern’s papers, pouring over Élie Cartan, and Ambrose insisting on absolute clarity in every detail, we learned differential geometry together.

Ambrose designed the Geometry of Manifolds course, and we taught it in alternate years. It is pretty much the same today as it was then: standard manifold theory the first term and instructor’s choice of topics the second. Our students wrote some well-known graduate texts based on this course: Bishop–Crittenden, Hicks, and Warner.

With customary zeal, Ambrose changed the undergraduate program in pure mathematics. Whereas in 1948 André Weil explained differential forms to the faculty at the University of Chicago, less than a decade later we were using them in undergraduate differential geometry.

Ambrose taught the Lebesgue integral in the analysis course for juniors and seniors “because it’s simpler than the Riemann integral.” For almost twenty years Ambrose was the guiding spirit of pure mathematics at MIT. His efforts were key in making it a great department. Ambrose and I regularly drove around Boston late at night talking about mathematics and life. We knew every street; and to this day, with the inevitable traffic jam, I’ll remember an alternate route and, often enough, a special moment: yes, here is where we finally understood holonomy.

Ambrose taught me jazz. I had been an enthusiast of Dixieland and swing. But by 1950 bop was dominant. Charlie Parker was king, and everyone else his student. We heard all the great jazz musicians at all the jazz joints. There was a casualness and directness then that makes it difficult for me to hear jazz live today. It’s too rigid and formal now. Imagine watching Bird take on a long line of young sax players, listening to each intently and then playing their variation as it should be played. Teaching at its best, I felt. Imagine having coffee between sets with Billie and thanking her for her early records that meant so much to many of us. Dick Kadison and I used to hear Ella night after night at Birdland when we took a break from work. Ambrose explained jazz to me and talked about it in the fifties the way Wynton Marsalis does now.

Though mathematics was wonderful, I was heavily burdened. My oldest son was blinded at birth and, as I learned later, brain damaged. I could not have survived as a mathematician without Ambrose’s steady support and the steady support of my very good friend, Dick Kadison. Those who knew Ambrose know that expressing gratitude was forbidden. He walked rapidly away the one time I tried. Occasionally, I can provide special help to a young mathematician. I think of Ambrose and feel that by the time I have helped a hundred, I’ll have begun to pay my debt to him.

I loved Ambrose for his absolute honesty, his generosity, his wit, his energy, and above all for his tenderness, which he tried so hard to hide. I am sad that so few mathematicians knew what a great man he was. But I was happy for him when he found his wife, Jeannette, with whom he could be himself for twenty years.

Reflecting on the biggest difference between the University of Chicago and MIT, Is has said that at Chicago teaching of undergraduates was done by a designated faculty hired for the purpose. So Is saw very little of the undergrads. At MIT, by contrast, the teaching of undergraduates was shared by all faculty.

Ambrose and a few others were very eager to revitalize the curriculum, and they called upon Is to teach the faculty, and then the students, the latest mathematics from Chicago, especially the perspectives on differential geometry that Is learned from Chern [15].

Out of this intense work in geometry came the famous Ambrose–Singer
theorem relating holonomy to curvature. For a nice description of this
theorem, see
Hung-Hsi Wu’s
tribute in the same *Notices*
[15].

After two years at MIT, Is went to UCLA for 1952–54, where he was wont to tease Ted Martin, then chair at UCLA, by saying that MIT had sent him out to the minor leagues for further training. Is then spent a year at Columbia University, followed by a momentous year at IAS with Atiyah, Raoul Bott, Fritz Hirzebruch, J.-P. Serre and Arnold Shapiro, before he eventually returned to MIT in 1956.

The Kadison–Singer problem, posed in 1959 by Dick Kadison and Is
[1],
was motivated by Dirac’s work in quantum
mechanics. It was a problem in functional analysis about whether
certain extensions of certain linear functionals on certain
__\( \operatorname{C}^* \)__-algebras were unique. The uniqueness was proven in 2013
[e12],
although the proof was thought by Dick and Is and others to be
false.

Is got to know Michael Atiyah at IAS in 1955, and when he wished to go to England on a sabbatical, he asked Michael, who had just arrived in Oxford, if there was a spot for him. Michael immediately said yes.

#### The Atiyah–Singer index theorem

After arriving in Oxford in January 1962, and while warming himself at a little heater, Is looked up to see Michael in the doorway. An immortal three-line conversation followed:

Michael: Why is the\( A \)-roof genus an integer for spin manifolds?

Is: Michael, why are you asking me that, you know the answer?

Michael: Yes, I do know the answer, but I feel there is a deeper reason.

Is had other work on his mind (a joint project with Shlomo Sternberg on the infinite groups of Élie Cartan [3], but Atiyah’s question intrigued him. He spent the early spring, sitting on garden benches, thinking about the problem. By March he understood the issues and went to Michael’s office and said that integers arise as the count of the number of solutions of the Dirac operator generalized to spin manifolds. And in fact in all the other cases where integers occur (e.g., the signature of a manifold or the arithmetic genus), it is because of counting solutions of geometric operators.

Michael ran with that, proposed some formulas, and their collaboration began in earnest. Nine months later they had a complete proof of the Atiyah–Singer index theorem!

Atiyah gives his perspective on the index theorem in his commentary beginning in the third volume of his collected works [e9], excerpted here:

In the Spring of 1962, my first year at Oxford, Singer decided to spend part of his sabbatical there. This turned out to be particularly fortunate for both of us and led to our long collaboration on the index theory of elliptic operators. This had its origins in my work on

\( K \)-theory with Hirzebruch and the attempt to extend the Hirzebruch–Riemann–Roch theorem into differential geometry. We had already shown that the integrality of the Todd genus of an almost complex manifold and the\( \hat{A} \)-genus of a spin-manifold could be elegantly explained in terms of\( K \)-theory. For an algebraic variety the Hirzebruch–Riemann–Roch theorem went one step further and identified the Todd genus with the arithmetic genus or Euler characteristic of the sheaf cohommology. Also the\( L \)-genus of a differential manifold, as proved by Hirzebruch, gave the signature of the quadratic form of middle dimensional cohomology and, by Hodge theory, this was the difference between the dimensions of the relevant spaces of harmonic forms. As Hirzebruch himself had realized it was natural therefore to look for a similar analytical interpretation of the\( \hat{A} \)-genus. The cohomological formula and the associated character formula clearly indicated that one should use the spin representations. I was struggling with this problem when Singer arrived. Fortunately Singer’s strengths were precisely in differential geometry and analysis, the areas where I was weakest. With his help we soon rediscovered the Dirac operator! […]Once we had grasped the significance of spinors and the Dirac equation it became evident that the

\( \hat{A} \)-genus had to be the difference of the dimensions of positive and negative harmonic spinors. Proving this then became our main objective. By good fortune Smale passed through Oxford at this time and, when we explained our ideas to him, he drew our attention to a paper of Gel’fand on the general problem of computing the index of elliptic operators. […]Singer and I had some great advantages over the analysts investigating the problem. We were investigating a particular case, the Dirac operator, and we already ‘knew’ the answer. Also this case…encompassed all the global topological complications. Moreover, we had arrived at the problem starting from

\( K \)-theory and this turned out to be just the right tool to study the index problem. Finally the Dirac operator was in a sense the most general case, all others being essentially deformable to it. […][…] Singer and I eventually produced a proof of the general index theorem during my stay at Harvard in the Fall in 1962. […] I realized at the time the significance of the index theorem and that it represented the high-point of my work, but it would have been hard to predict that the subject would continue to occupy me in various forms for the next twenty years. I would also have been extremely surprised if I had been told that this work would in due course become important in theoretical physics.

The proof of the index theorem was presented in detail in an
Atiyah–Bott–Singer seminar at Harvard in Fall 1962, was announced in
the *Bulletin* in 1963
[2],
and was written up
in detail in the Borel–Palais seminar at the Institute for Advanced
Study in 1963
[e1].
This version was a proof using
cobordisms, where one showed that both the analytic and topological
index are invariant under cobordisms, and then checking that both
indices are equal on the generators of the relevant cobordism groups,
as had been done by Hirzebruch in his proof of the
Hirzebruch–Riemann–Roch theorem.

Bott explains this in his math review:

The authors here describe their solution to the index problem for elliptic operators on closed manifolds. Their result may also be thought of as a beautiful and far-reaching generalization of Hirzebruch’s Riemann–Roch theorem — both in statement and in the spirit of the proof.

The authors formulate the index problem in the following general setting. Let

\( E \)and\( F \)be vector-bundles over the compact manifold\( X \), where everything is\( C^{\infty} \)throughout. It then makes sense to speak of a differential operator\( D \)from\( E \)to\( F \), and such an operator induces a linear map\[ D:\Gamma(E)\to \Gamma(F) \]from the sections of\( E \)to those of\( F \). When\( D \)is elliptic, both the kernel and the cokernel of\( D \)are finite-dimensional, and the difference of these dimensions is by definition the index of\( D \). Alternately, one has the equality index\[ \sigma(D)=\sum (-1)^i\operatorname{dim}H^i(X;K), \]where\( K \)is the kernel sheaf of\( D \), as follows from rudimentary sheaf theory, and the local “onto” property of elliptic operators:\begin{align*} H^0 (X;K) &\simeq \operatorname{ker}D,\\ H^1(X;K) &\simeq \operatorname{coker}D,\\ H^i(X;K) &=0,\quad i\geq2. \end{align*}The “index problem” is to give a description of this integer in terms of the topological data implicit in the elliptic operator. To describe these one needs the following interpretation of the highest-order terms in

\( D \). Let\( T(X) \)be the cotangent bundle of\( X \), and let\( T_0(X) \)be the subset of nonzero vectors in\( T(X) \). The projection\( T(X)\to X \)will be denoted by\( \pi \). With this understood, the highest-order terms of\( D \)are seen to define a definite homomorphism\[ \sigma(D):\pi^{\ast} E\to \pi^{\ast}F \]of the pulled-back bundles on\( T(X) \). Further,\( D \)is elliptic if and only if this homomorphism, called the symbol of\( D \), is an isomorphism on\( T_0(X) \). This isomorphism — or rather its stable homotopy class\( [\sigma (D)] \)— is to be thought of as the topological “twist” of the elliptic operator\( D \).One may relate

\( [\sigma (D)] \)in various ways to more standard topological objects. Maybe the simplest construction is the following one. Let\( B(X) \)and\( S(X) \), respectively, stand for the unit ball and unit sphere bundle of\( T(X) \)endowed with some fixed Riemannian structure. Let\[ W(X)=B(X)\cup^{}_{S(X)} B(X) \]be the manifold obtained by glueing two copies of\( B(X) \)together along their boundary, i.e.,\( W(X) \)is the doubled manifold constructed from\( B(X) \). Now one uses\( \sigma(D) \)to construct a bundle\( E \cup_{\sigma(D)} F \)on\( W(X) \)by taking\( \pi^*E \)on one copy of\( B(X) \),\( \pi^*F \)on the other, and glueing them together over\( S(X) \)by means of the isomorphism\( \sigma(D) \). (Alternatively and really equivalently, one may use\( \sigma(D) \)to construct a difference element in\( K(B(X),S(X)) \)and that is the point of view taken in the paper under review.)A formula quite equivalent to the index formula of the paper now is of the form

\[\operatorname{index}(D)=\int_{W(X)} \operatorname{ch}(E \cup^{}_{\sigma(D)} F)\wedge \pi^*A(X). \]Here\( \operatorname{ch} \)denotes the Chern character and\( A(X) \)is a differential form on\( X \)which we will not specify here, but which is an explicit polynomial in the characteristic classes of\( X \).The index formula easily yields the results of the pioneers in this field such as Agranovič, Dynin, Gel’fand, Seeley, Vol’pert, etc. The formula is also seen to generalize the Riemann–Roch theorem of Hirzebruch to (not necessarily) algebraic complex manifolds and bundles. (To this one need only follow up our earlier expression for index

\( (D) \)as an Euler characteristic.)The proof of the index formula, as well as its various consequences in special cases, is clearly summarized in the note. For the sake of analysts we remark only that the general setting of operators on vector-bundles — as opposed to systems — is essential for the proof.

The proof using cobordism theory suffered because it did not work in the equivariant case, as the relevant equivariant cobordism groups were not all known. Atiyah writes [e8]:

For some years Singer and I searched for a better proof modelled more on Grothendieck’s proof of the generalized Hirzebruch–Riemann–Roch theorem. Eventually we found such a proof, based on embedding a manifold in Euclidean space and then transferring the problem to one on the Euclidean space by a suitable ‘direct image’ construction. This proof [4] worked purely in a

\( K \)-theory context and avoided rational cohomology. It therefore lent itself to various significant generalizations developed in the remaining papers of the series [e2], [5], [7], [6]. One generalization, the ‘equivariant’ index theorem, dealt with compact group actions preserving an elliptic operator. This extended the Lefschetz formula of my papers with Bott to the case of nonisolated fixed points (of isometries), and rested on the prior development of equivariant theory. This had been carried out by Graeme Segal in his thesis and the relevant applications to index theory were developed in [e2].

An Indian mathematician,
V. K. Patodi,
obtained remarkable results
applying the heat equation to the index theorem. Patodi joined Is and
Atiyah in a significant series of papers
[8],
[10],
[9],
[11],
establishing a signature theorem for manifolds with boundary, leading
to the __\( \eta \)__-invariant. In
[e9]
Atiyah remarked:

The problem of generalizing the Hirzebruch signature theorem to manifolds with boundary had long been an intriguing question. There had been many clues, notably the work of Hirzebruch on signature defects of cusps of Hilbert modular surfaces…. Singer and I therefore tried to analyse the problem in its more general form. Eventually we saw that the natural formulation was that of an index problem with a ‘global’ boundary condition. This was conceptually a major breakthrough.

In many ways the papers on spectral asymmetry were perhaps the most satisfying ones I was involved with. The way they stretched over differential geometry, topology, and analysis with a nod in the direction of number theory appealed greatly to me. At the time these papers had only a modest impact but, a few years later when contact was made with theoretical physics, they became extremely popular. In particular Witten’s work on global anomalies brought our

\( \eta \)-invariant into prominence, in a way which we could never have foreseen.

Victor Guillemin’s review of [8] gives a clear statement:

This paper announces some very exciting results concerning the so-called

\( \eta \)-invariant of a self-adjoint (not necessarily positive) elliptic operator. For such an operator one can form:\[ \eta_A(s)=\sum_{\lambda \neq 0} \operatorname{sign}\lambda \, |\lambda|^{-s}, \]with the sum taken over all eigenvalues in the spectrum of\( A \). This turns out to be a well defined meromorphic function of\( s \). One of the main results announced here is that\( \eta_A(s) \)is finite at the origin; hence the\( \eta \)-invariant of\( A \),\[ \eta (A) =\eta_A(0), \]is defined. In particular take\( X \)to be a compact oriented Riemannian manifold of dimension\( 4k-1 \)and let\( A \)be the operator acting on exterior differential forms of even degree given by\[ A(\phi)=(-1)^{p+1}d* \phi +(-1)^p *d \phi, \]“\( * \)” being the usual Hodge star operator. Suppose that\( X \)is the boundary of an oriented Riemannian manifold\( Y \)that is isometric near the boundary to the cylinder\( X \times [0,1] \). Let\( p_i(Y) \)denote the\( i \)-th Pontrjagin form of\( Y \). Then the authors link the\( \eta \)-invariant of\( A \)to the signature of\( Y \)by the formula\[ \operatorname{sign}(Y)-\int_Y L(p) = (-1)^{k+1} \eta (A), \]where\( L \)is the Hirzebruch\( L \)-polynomial. For\( Y \)boundaryless, the right-hand side is zero, and one retrieves the usual Hirzebruch signature formula. The authors also announce an index formula with boundary term for the Dirac operator involving an\( \eta \)-invariant. They conclude by giving a brief outline of the proof.

The __\( \eta \)__-invariant was partly motivated by Hirzebruch’s result
expressing the signature defect in terms of values of __\( L \)__-functions of
real quadratic fields. Is and Atiyah with
Harold Donnelly
extended
Hirzebruch’s formula to the case of totally real fields of any degree
[14].

In 1977, with Atiyah and
Nigel Hitchin,
Is showed that the moduli
space of __\( \mathrm{SU}(2) \)__ instantons (with Pontryagin index equal to one) over
__\( X^4 \)__ is hyperbolic 5-space
[12],
[13].
Then
Clifford Taubes
[e4]
extended this result to the case of 4-manifolds
__\( X^4 \)__ with positive definite intersection forms; he showed that the
5-dimensional moduli space has as its compactification __\( X^4 \)__
consisting of singular instantons at each point of __\( X \)__. This, together
with work of
Karen Uhlenbeck
paved the way for
Simon Donaldson’s
spectacular invariants
[e7]
distinguishing many smooth
4-manifolds which are homotopy equivalent and thus homeomorphic by
Freedman’s
work
[e5].
A remarkable corollary (first
observed by Freedman and the author
([e10], Chapter 10)
is that
__\( \mathbb{R}^4 \)__ has uncountably many smooth structures, some which embed
smoothly in standard __\( \mathbb{R}^4 \)__ and some which do not. The 1977 paper was
the forerunner of these startling results.

For personal reasons Is went to Berkeley in 1977. MIT was not happy about this decision, so they arranged for Is to retire but to be brought back easily if he was willing. Every year MIT would call and in 1984 he finally said yes, for they caught him at a time when he was particularly annoyed at departmental politics at Berkeley (interview with author, December 1, 2017).

Is’ wife Rosemarie and
daughters Emily and Annabelle did not
want to leave Berkeley, but things worked out well for both
daughters: Emily is now
biotech editor of *MIT Technology Review*
and Annabelle got her Ph.D. at UCSF in brain and cognitive sciences
and is now a postdoc at MIT.

#### The founding of MSRI

While at Berkeley, Is started his now famous geometry-math-gauge theory-physics seminar, partly due to a wish from some of the younger people at Berkeley, including three of his best Ph.D. students — Dan Freed, Dan Friedan, and John Lott — who wrote theses in this area.3

He also joined forces during this period with S.-S. Chern and Calvin Moore to put together a plan for a new institute, the Mathematical Sciences Research Institute, to be located on the West Coast on the grounds of the University of California (though retaining its legal independence) and to serve as a counterpart to the Institute for Advanced Study. With the combined weight of their reputations, they easily won an endorsement from the UC Chancellor, Mark Heyman, and then persuaded the National Science Foundation to fund MSRI, despite the likelihood that a new institute would reduce the number/size of the traditional individual grants. Chern and Moore became the first Director and Deputy Director of MSRI, respectively, but Is returned to MIT.

Meanwhile, the impact of the index theorem continued to grow. Is described its consequences for other fields ([e3], p. 38):

In 1962, while a Sloan Fellow…Michael Atiyah and I discovered an index theorem that combined topology, geometry, and elliptic partial differential operators in a new way. Our results extended, unified, and gave insights to some older theories. Exploring the consequences of this startling combination of different fields has kept us busy since then. Deep applications continue to be discovered. Some of the most interesting…are in high-energy theoretical physics. Mathematicians and physicists now realize that underlying gauge theories is the mathematical structure of fiber bundles. As a result, global geometry has application to elementary particle physics, and at the same time physicists are asking many new geometric questions. It is exciting to search for the answers.

#### Science and public policy

Is was elected to the National Academy of Sciences in 1968, and served as Chair of their Committee on Science, Engineering and Public Policy (COSEPUP). One of the first issues he tackled was what to do with nuclear waste. Another issue he brought up to the committee was the threat to privacy from the coming computer/internet revolution, but Is was way ahead of his time in this respect, and the rest of the committee was less interested.

While serving on the committee,
Is went to see Jerry Weisner, then president of MIT, who
congratulated him on being Chair of COSEPUP. Is expressed doubt about
his committee role, averring
that it would take away from his research. Jerry replied
that the experience would be valuable and would give Is
a different and much broader view of what he (Is) would want to do.
Is realized that Jerry was right, and instead of focusing merely on the
particular technical aspects of his work, he learned to step back and ask, *Do I want to
do this? Why do I want to do this? Where does it fit into my view of
mathematics?* — and this reflection often changed things.

Is also served on the David Committee, organized by Ken Hoffman (the then-recent chair of the MIT math department) who persuaded Ed David, Nixon’s Science Adviser (who had resigned in 1973 out of frustration), to be chair.

Is was a member of the White House Science Council from 1982–88. His experience in Washington left him with a favorable impression of the scientific talent summoned to duty in the capitol. For example, he stated that the choice of experts for science policy reflected greater soundness of judgment, in many cases, than did tenure decisions in academia. He also had great respect for the military, arising from his service in World War II.

Though Is later served on the Governing Board of the National Research Council (1995–99), it must be said that mathematicians are not frequently called to serve in policy-making roles in DC. Is remembers the White House Science Adviser, Jay Keyworth, coming in one morning after a long meeting the night before regarding some particularly intractable problem and saying (with a smile, one assumes): “Is, your suggestion solves the problem. How can you be so smart, you’re just a mathematician!”.

#### Teaching at MIT

For many years at MIT Is only taught graduate classes and a few high-level undergraduate classes. Then he decided to be a TA for the basic calculus class simply because he had never taught lower-division math courses before. He looks back on this experience fondly, and feels it made him feel part of MIT’s encompassing mission to educate undergraduates.

Is is an MIT “patriot”: He sums up MIT with one word, “generosity”, for they give faculty freedom to do their research and resources with which to do it. This was particularly true when Is was designated a “University Professor”, reporting directly to the Provost.

#### Mathematician and athlete

He still plays tennis twice a week — three times in summer — and at a very high level. “He really runs, at age 85!” marvels his colleague from Harvard, Michael Hopkins. His former student, Daniel Burns, notes, “Is has always had a very specific, jaunty walk. He bounces around on the balls of his feet. It’s part of his panache, like wearing cravats rather than ties. I have an idea that he likes movement like this, and it’s related to why he likes tennis.” Singer himself says simply, “I play tennis to learn. I think that’s true of everything I am interested in — I do it to learn. In mathematics and physics, when I’ve learned enough, I can explain it to others.”