Last February he passed away Isadore Manuel Singer, Professor Emeritus at the Massachusetts Institute of Technology (MIT) and winner of the Abel Award. His most relevant contribution, the Atiyah-Singer index theorem, proposes a deep relationship between topology, geometry and analysis, the ramifications of which even reach theoretical physics. Beyond mathematics, Singer was a member of the Scientific Council of the White House and directed the Committee on Science and Public Policy of the American Academy of Sciences, where he dealt with topics such as the management of nuclear waste or privacy in the era of the computing, almost 20 years ahead of its time.
Singer was born in 1924 into a very humble family of Polish immigrants; his father was a painter and his mother, “the most intelligent person he ever met” – according to Singer himself – a seamstress. During the Great Recession and the years after, the family suffered severe financial straits and Singer had to combine several jobs to pay for his training. Although he got a scholarship to study Physics at the University of Michigan (USA), to survive he resold the football tickets that he got for free for being a student.
In 1944, after completing his degree in just two and a half years, he moved to the Philippines as a radar operator for the US Army. In the evenings, while his classmates plucked at poker, he tried to understand general relativity and quantum mechanics by following correspondence courses in Geometry and Group Theory offered by the University of Chicago.
Singer came from a very humble family. In college, he subsisted by reselling the football tickets that were given to him for being a student
Frustrated, on his return from the Philippines in 1947, he decided to enroll in a mathematics course at the same university. After a year of master’s degree, the discipline captivated him; felt that “it was his field” and decided to do the doctorate with the analyst Irving Segal, who would become one of his references – the other would be Shiing-Shen Chern whom he would meet in his senior year in Chicago. In 1950 he finished his thesis and moved to MIT as a postdoc; there he would develop a large part of his career.
The same day he arrived at MIT, he felt at home, something that had not happened in many years. In the department’s secretary, Singer met the mathematician, too Warren Ambrose and immediately began their friendship and collaboration, which would last a lifetime. Most nights they met in the Hayes-Bickford cafeteria to discuss math with infectious enthusiasm. Other times they would take refuge in jazz clubs where, if they were lucky, they could listen to Ella Fitzgerald or have coffee with Billie Holliday backstage. In spite of everything, it was not an easy time for Singer, as his son had been born with serious health problems. Singer never tired of admitting that he would not have survived as a mathematician without Ambrose’s constant support.
The collaboration paid off: Ambrose and Singer revolutionized the teaching of differential geometry at MIT and in 1953 they published a theorem that related curvature to holonomy and that would determine the development of differential geometry in the following decades.
The collaboration between Ambrose and Singer resulted in a theorem that would mark the development of differential geometry in subsequent decades.
However, his most fruitful collaboration was with Michael Atiyah, whom he had met in 1955 at the Institute for Advanced Study in Princeton (USA). Like everyone else, Singer had not been able to resist the portentous mix of brilliance, infectious enthusiasm and energy that characterized Atiyah and that would make him one of the most celebrated mathematicians of the 20th century. In 1962 Singer took a sabbatical from MIT, and decided to settle in Oxford. to work with Atiyah. As soon as he received it, the Englishman asked him a question: “Why is the Â-genus an integer for spin varieties?” Although Atiyah knew the answer, he sensed that there was a deeper reason. Singer was so intrigued by the question that he immediately went to work on the problem.
The Â-genus of a geometric space is a polynomial whose coefficients are rational numbers and its indeterminate variables are given by certain topological invariants. Since we operate with rational numbers, it would be expected that, after all the calculations, the Â-gender would be a rational number. However, in the early 1960s, mathematicians had used very sophisticated techniques from algebraic topology to prove that for a very specific class of geometric spaces – the spin manifolds -, surprisingly, the Â-genus is an integer.
In February 1962, Singer conjectured that this whole number should be the subtraction of two natural numbers that count mathematical objects. Through his training in analysis, Singer realized that they were actually accounting for solutions of differential equations, expressed as geometric operators. In March, Singer announced his discovery to Atiyah: the Â-genus arises by counting the number of solutions of the generalized Dirac operator to spin varieties.
They managed to give an explicit formula to calculate a topological invariant and generalized the Â-gender that had served as a guide.
From then on, they spent the next 18 months formalizing one of the fundamental theorems of the 20th century: the index theorem. Atiyah contributed his deep knowledge of topology and algebraic geometry, and Singer his in analysis and differential geometry. The final statement ensures that, given a differential equation of a certain type, its index – the difference between the number of parameters necessary to describe its solutions and the number of relations imposed by the differential equation – is a topological invariant. Perhaps most importantly, they managed to give an explicit formula to calculate it in terms of well-known topological objects, generalizing the Â-genus that had served as a guide.
More than a theorem, the result is a theory. For twenty-five years, Atiyah and Singer found new demonstrations that provided new approaches and explored their ramifications, especially in relation to the gauge theories in physics. Therefore, it is not surprising that Atiyah and Singer won the Abel Award in 2004, perhaps the most prestigious of the impressive list of awards that Singer received throughout his career.
Singer compared his collaboration with Michael Atiyah to the Cole Porter song “Anything Goes” (anything goes). However, those who knew him agree that this description fit Singer’s own approach to mathematics. In the end, Singer was unique in his straightforward, improvised style, and easy to get carried away by problem and intuition. Pure jazz.
David Fernandez Alvarez is a postdoctoral researcher at the University of Bielefeld (Germany)
Editing and coordination: Agate A. Timón G-Longoria (ICMAT)
Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Eddie is an Australian news reporter with over 9 years in the industry and has published on Forbes and tech crunch.