Sunday, August 17, 2014

Prime Synchrony

Sunday reflection today, courtesy of Freeman Dyson and Hugh Montgomery (this reflection actually ties nicely into last week's Sunday offering as well)...:

"[Freeman] Dyson helped bring together the continuous and the discrete understandings of subatomic behavior. Similarly, by fusing his love of number theory with his expertise in creating the mathematical tools of physics, he would make the initial observation that would reinforce the connections between the discrete world of the integers and the continuous world of analysis, and thus galvanize research on the Riemann hypothesis.
"As Dyson recalls it, he and [Hugh] Montgomery [number theorist] had crossed paths from time to time at the [Princeton] Institute [for Advanced Study] nursery when picking up and dropping off their children. Nevertheless, they had not been formally introduced. In spite of Dyson's fame, Montgomery hadn't seen any purpose in meeting him. 'What will we talk about?' is what Montgomery purportedly said when brought to tea. Nevertheless, Montgomery relented and upon being introduced, the amiable physicist asked the young number theorist about his work. Montgomery began to explain his recent results on the pair correlation, and Dyson stopped him short -- 'Did you get this?' he asked, writing down a particular mathematical formula. Montgomery almost fell over in surprise: Dyson had written down the sinc-infused pair correlation function.
"Dyson had the right answer, but until that moment he had associated this formula with understanding a phenomenon that seemed completely unrelated to the primes and the Riemann hypothesis. In a flash he had drawn the analogy between the sinc-described structured repulsion of the zeta zeros and a similar tension seemingly exhibited by the different levels of energy displayed by atomic nuclei. Whereas Montgomery had traveled a number theorist's road to a 'prime picture' of the pair correlation, Dyson had arrived at this formula through the study of these energy levels in the mathematics of matrices. This connection is the source of most of the current excitement surrounding the Riemann hypothesis..."

-- from "Stalking the Riemann Hypothesis" by Dan Rockmore

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