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Pondering a Miracle

By Graham Farmelo

Fellow, Churchill College, University of Cambridge, U.K.
Adjunct Professor of Physics, Northeastern University, Boston, MA

Miracle of miracles—that’s how the great physicist Frank Yang described the discovery that some of the beautiful structures of modern mathematics precisely describe our universe’s underlying order. This link between the concrete world of physics and the abstractions of mathematics is at the heart of some of the most exciting research now being done at the overlap between the two worlds.

Yang, a Nobel-prize winner, was on the Faculty at Stony Brook University from 1966 to 1999. During that time, he did path-breaking research on the relationship between mathematics and physics. In the early 1970s, after an especially enlightening discussion with Jim Simons, head of Stony Brook’s mathematics department, Yang made one of his impressive leaps of imagination. He identified several connections between advanced concepts long familiar to geometers and the well-established gauge theory of sub-atomic particles that he and physicist Robert Mills had proposed about fifteen years earlier. The connection between gauge theories and geometry became a vibrant research topic, especially after a series of brilliant papers by the mathematician Michael Atiyah and his colleagues. Atiyah later said that he had morphed into a “quasi-physicist.”

One of the key lessons of twentieth-century science—that geometry is central to our understanding of fundamental physics—was soon clearer than ever. It is fitting that the charitable foundation established in 1994 by Jim Simons and his wife Marilyn later made it possible to set up at Stony Brook the Simons Center, now one of the world’s leading centers of research into topics in geometry and physics, which are inextricably connected in ways that continue to puzzle experts.

I first visited the Simons Center for Geometry and Physics one summer morning in 2016, mainly to talk with the mathematician Simon Donaldson. My purpose was to seek his advice about topics covered in a book I was researching on the relationship between cutting-edge pure mathematics and the search for the most fundamental laws of nature. Although I was at the Center for only a day, I saw that it is an embodiment of the key message of my book, titled The Universe Speaks in Numbers: by working alongside each other and even collaborating, physicists and mathematicians can profoundly enrich their understanding of both subjects. The symbiotic relationship between the subject was expressed crisply by the British theoretical physicist Paul Dirac in his lecture On the Relation between Physics and Mathematics, which he delivered in Edinburgh eighty years ago. He noted: “as time goes on it becomes increasingly clear that the rules that the mathematician finds interesting are the same as the ones that Nature has chosen.” When I showed extracts of that lecture to Donaldson, he seemed—like many contemporary mathematicians and physicists—to be taken aback by the clarity and far-sightedness of Dirac’s vision. 

Dirac would have loved Donaldson’s work, especially the extraordinarily rich insights Donaldson gave into four-dimensional manifolds—loosely speaking, four-dimensional spaces—using Yang and Mills gauge theory. This advance wonderfully exemplifies how an idea generated in the course of physics research can help advance the boundaries of mathematics. Michael Atiyah told me a few weeks before he died last January that Donaldson’s work on four-dimensional manifolds was “the most important development in mathematics in the last quarter of the twentieth century.”

Donaldson later made many other contributions that turned out to be extremely useful to physicists, especially to gauge theorists. This is an excellent example of the truth expressed by a catchphrase now often used by mathematicians and physicists: not only is mathematics unreasonably effective in physics—as the theoretician Eugene Wigner memorably remarked in 1959—physics is also unreasonably effective in mathematics.

Last November, I returned to the Simons Center for a week to give a public lecture about Dirac and view that mathematical beauty is the lodestar for research into fundamental theoretical physics. It was a privilege for me to have the opportunity to meet some of the researchers based at the Center—mathematicians, physicists and the specialists sometimes known as physical mathematicians (who seek to shed light on both new mathematics and the laws of Nature). I was especially pleased to record interviews with Simon Donaldson, the physicist Zohar Komargodski and the Center’s director Luis Álvarez-Gaumé. These recordings will shortly be released online as part of the series that explores some of the themes of my new book, and include discussions with several other leading figures in the field, including Edward Witten, Nima Arkani-Hamed and Greg Moore. Research at the Simons Center is very much in the interdisciplinary tradition that dates back at least to the publication of the Principia in 1687, written by Isaac Newton. A precursor of today’s theoreticians, he was a consummate mathematician and a peerless thinker about the inner workings of the universe, not to mention a world-class experimenter.

At the end of my stay, Luis Álvarez-Gaumé invited me to return for a longer visit, and I eagerly accepted. I look forward to witnessing the progress made by the mathematicians and physicists at the Simons Center as they help to explore Yang’s “miracle of miracles”— and, perhaps one day, to understand it. Or is it, as Einstein suggested three years before he died, “an eternal mystery?” •