San Francisco Chronicle - (Sunday)
Isadore Singer — brought math, physics together
Isadore Singer, who unified large areas of mathematics and physics in becoming one of the most important mathematicians of his era, died on Thursday at his home in Boxborough, Mass. He was 96.
His death was confirmed by his daughter Natasha Singer.
Singer created a bridge between two seemingly unrelated areas of mathematics and then used it to build a further bridge, into theoretical physics. The achievement created the foundation for a blossoming of mathematical physics unseen since the time of Isaac Newton and Gottfried Wilhelm Leibniz, when calculus first provided tools to understand how objects moved and changed.
Singer’s work with the British mathematician Michael Atiyah allowed for the development of critical areas of physics, like gauge theory and string theory, that have the potential to revolutionize our understanding of the most basic structure of the universe.
Singer was an influential voice in scientific matters outside the theoretical realm as well. From 1975 to 1980 he was chairman of the Committee on Science and Public Policy at the National Academy of Sciences, the most important scientific advisory committee for the president and other government officials.
In that post, he organized a report on the disposal of nuclear waste, defended President Ronald Reagan’s “Star Wars” missile defense initiative despite being politically opposed to many Reagan policies, and, decades before it became a pressing public issue, warned about the loss of privacy with the growth of the internet. Under Reagan, he was a member of the White House Science Council from 1982 to 1988, and from 1995 to 1999 he was on the governing board of the National Research Council.
Singer was awarded the National Medal of Science in 1983 and the Abel Prize in 2004, often considered the Nobel of mathematics.
He studied physics at the University of Michigan, graduating in 2½ years to join the Army as a radar officer during World War II. Stationed in the Philippines, he ran a communications school for the Filipino army during the day. At night, he filled in the gaps of his abbreviated education, studying mathematics in correspondence courses to learn the prerequisites for relativity and quantum mechanics.
After leaving the Army, he spent a year studying math at the University of Chicago. Though he had planned to return to physics, he fell in love with mathematics and stayed to earn his doctorate. He did a postdoctoral fellowship at the Massachusetts Institute of
Technology, where he ended up teaching for almost his entire career.
During an interlude at UC Berkeley, he helped found the Mathematical Sciences Research Institute. He also began proving a number of important theorems, leaving mathematics literature peppered with his name: the KadisonSinger conjecture (formulated in 1959 and proved only in 2013), the AmbroseSinger theorem, the McKeanSinger formula and RaySinger torsion.
But all those were dwarfed by his singular contribution, the AtiyahSinger Index theorem. Together with Atiyah, he created an unimagined link between the mathematical subfields of analysis and topology — and then united those fields with theoretical physics.
Singer was the expert in analysis, which is the study of differential equations, used to describe physical phenomena in the language of calculus. Such equations are extremely useful for describing realworld situations, but they have a problem: No one knows how to solve them precisely. Scientists are stuck with approximation.
Atiyah, meanwhile, specialized in topology, which studies the shapes of abstract mathematical objects, often in many more dimensions than our ordinary three. Topology considers shapes to be elastic, so that objects can be pulled or squished without changing their fundamental nature.
The two fields seemed to be nearly irremediably divided, because topology twists objects around, and analysis needs them to be rigid. Nevertheless, in the early 1960s, Singer and Atiyah sought to figure out if Atiyah’s topological tools could solve Singer’s analytical problem and find the solutions to differential equations.
Finding the exact solutions was too hard. But they found a way to figure out the number of solutions to the equations, even without their exact values. This was the AtiyahSinger Index theorem.
The result created a bridge between topology and analysis that Atiyah, Singer and others widened and built upon over the next decade, creating an entirely new field called index theory.
That was just the beginning. In 1975, James H. Simons, a mathematician and a close collaborator of Singer’s (and later a prominent hedge fund manager and philanthropist), and Chen Ning Yang, a Nobelwinning physicist, were discussing their work. They realized that in their own scientific languages they were each talking about a common underlying structure. What the physicists called a “gauge theory” was what the mathematicians called a “fiber bundle.”
Through this connection, the AtiyahSinger Index theorem applied to physics just as it did to mathematics. The revolution it had brought to mathematics now carried over to physics, too.
“This was the Big Bang of late 20th century unification between mathematics and physics,” said the mathematician and economist Eric Weinstein. “It was Is Singer who lit the spark that caused the fire.”