When Isadore Singer arrived at MIT in 1950 to teach in the Department of Mathematics, he found that his department had little contact with the physicists also housed in Building 2. More than 50 years later, mathematicians and physicists have much more to bring them together, thanks in large part to work done by Singer during his long career, which earned him this year's James R. Killian Jr. Faculty Achievement Award.
Singer, who was named an Institute Professor in 1987, gave the annual Killian Lecture in Kirsch Auditorium in the Stata Center on Thursday, March 23.
His talk, titled "Some Geometry of the Past Half Century and Its Historical Background," reflected on the mathematical ideas that have developed over the past 50 years, bringing together diverse areas of mathematics, including geometry, analysis and topology. That time period has also seen the development of a closer connection between math and physics.
"Mathematicians and physicists -- sometimes they have some (connection) and sometimes they don't. Certainly back in '66 there was not much," Singer said.
When the faculty awarded Singer the Killian Prize last May, Professor Marcus Thompson, chair of the selection committee, said, "Isadore Singer is one of the rare mathematicians who is able to communicate with theoretical physicists in their own language and engage them in genuine collaborations. Most important is the attitude he brings to these collaborations: not the usual mathematical disdain for the physicists' lack of rigor but a conviction that mathematicians must try to understand why the physicists' methods work and to abet them in their efforts."
The Atiyah-Singer Index Theorem, developed in the early 1960s, did much to bring mathematicians and physicists together. The theorem has had applications in many areas of theoretical physics, including gauge theory, monopoles, string theory and the theory of anomalies, among others. In 2004, Singer and Sir Michael Francis Atiyah of the University of Edinburgh shared the Abel Prize, considered the equivalent of the Nobel Prize for mathematics, for the development of their theorem.
The Abel Prize citation described the theorem as follows: "Scientists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas, called differential equations, involving their rates of change. Such formulas may have an 'index,' the number of solutions of the formulas minus the number of restrictions that they impose on the values of the quantities being computed. The Atiyah-Singer index theorem calculated this number in terms of the geometry of the surrounding space."
The index theorem "energized mathematicians and physicists and started a new liaison between the two subjects," Singer said.
Singer began his Killian talk with some reminiscences about his early days at MIT, which he said has "always been a very exciting place, and it seems to me it gets more exciting every year."
After earning his doctorate from the University of Chicago, Singer arrived at MIT in July 1950, ready to start teaching summer classes. His first day included a late-night trip with his new colleagues to a Boston coffeehouse "which at midnight was a meeting place for derelicts, drunks and, apparently, mathematicians." That was followed by a very early morning tour of Boston and Cambridge, and Singer started thinking that "maybe I had found a new home, and that turned out to be true."
Singer then described an example of the index theorem, known as the Gauss-Bonnet formula. The formula allows the large-scale shape of an object to be calculated from the local curvature of the object. As an example, he showed a slide of Earth, which is a solid sphere. A person on Earth's surface would not be able to see that shape, but could calculate the shape by measuring the curvature of the Earth at all its points.
In other words, "the average of local information gives global information," Singer said.
The Killian Award was established by the faculty in 1971 as a tribute to James R. Killian, former MIT president and chairman of the MIT Corporation. It is meant to "recognize extraordinary professional accomplishments by full-time members of the MIT faculty; provide a means for the communication of these accomplishments to the faculty, students, other members of the MIT community, and to the general public; and by so doing, honor the contributions made by Dr. Killian to the intellectual life of the Institute."
A version of this article appeared in MIT Tech Talk on April 5, 2006 (download PDF).
