# Douglas Stanford and Pedro Vieira: ‘The Universe Speaks in Numbers’ interviews

Douglas Stanford and Pedro Vieira are two of the most brilliant young theoretical physicists, both seeking to understand nature at its finest level. In these two interviews, Stanford and Vieira talk about their work and explain why they are content to spend their times developing fundamental theories of nature, with little or no surprising new inputs from new observations and experiments.

### Bonus for specialists

During the interview with Douglas Stanford, he mentioned a remarkably illuminating mathematical suggestion made by Edward Witten. After Graham asked for a little more detail about this, Douglas kindly supplied the following note, reproduced here with his permission:

‘The problem [we were considering] was something called the * Schwarzian theory*, which describes the low-energy physics of near-extremal black holes, and also the

*, which is a simple quantum mechanics model people have studied a lot in recent years. In previous work with other people, we had found a complicated way of analyzing this theory, and had a good reason to believe that there was a very simple answer for the thermal partition function, but we couldn’t find a direct proof of the formula.*

**Sachdev-Ye-Kitaev model**I talked with Edward about this for a little while, and as a final thought as he was about to walk out the door, he suggested that although there was a risk that it might send us on a bit of a wild goose-chase, we could think about it backwards: just ask what types of integrals have simple answers, and then check to see if the Schwarzian theory was one of that type. What he had in mind was a math tool called the ** Duistermaat-Heckman formula**, which I had never heard of before, and which says that a special class of integrals have very simple answers. It turned out to work perfectly. This was, I think, my first encounter with fancy mathematics and its relevance to simple physics, and honestly, it was a significant shock to my physics worldview. We did write a paper about it, which is here

Fermionic Localization of the Schwarzian Theory (PDF)

The punchline is equation (2.37).’