Though Robert Langlands did not emphasize the connection between periods and L<\/em>-functions,\u00a0 Sakellaridis and Venkatesh viewed them as central to broadening and deepening the connections between seemingly distant areas of mathematics that Langlands had proposed.<\/p>\n

In their 2012 book, they developed a machine of sorts, which took a period as an input, performed a long computation, and output an L<\/em>-function. Not all periods produce corresponding L<\/em>-functions, though, and the main theoretical advance of their book was to understand which ones do. (This built on earlier work by Atsushi Ichino and Tamotsu Ikeda at Kyoto University.)<\/p>\n

But their approach had two limitations. First, it did not explain why a given period yields a given L<\/em>-function. The machine that turned one into the other was a black box. It was as if they\u2019d constructed a vending machine that often reliably yielded something to eat each time you put in money, only there was no telling what it would be in advance, or if the machine would eat the money without dispensing a snack.<\/p>\n

For any given case, you\u2019d put in your money \u2014 your period \u2014 then \u201cgo and do a long computation and see which among a zoo of L<\/em>-functions you got,\u201d Venkatesh said.<\/p>\n

The second thing they didn\u2019t manage to accomplish in their book was to come to an understanding of which L<\/em>-functions have associated periods. Some do. Others don\u2019t. They couldn\u2019t figure out why.<\/p>\n

They kept working after the book came out, trying to figure out why the connection worked and how to run the machine in both directions \u2014 not only getting an L<\/em>-function from a period, but also the other way around.<\/p>\n

In other words, they wanted to know that if they put $1.50 into the vending machine, it meant they were going to get a bag of Cheetos. Moreover, they wanted to be able to tell that if they were holding a bag of Cheetos, it meant they\u2019d put $1.50 into the vending machine.<\/p>\n

Because they link objects that, on their face, have nothing in common, dualities are powerful. You could stare at a lineup of mathematical objects forever and not perceive how L<\/em>-functions and periods match up.<\/p>\n