The Essence of Things

Emerging from graduate school in 2002 without having yet achieved this self-awareness, Venkatesh was eager to prove himself. A collaboration with Ellenberg soon gave him the opportunity. The pair took on the task of counting “number field extensions,” the simplest of which are built by adding to the rational numbers a handful of irrational numbers that satisfy some polynomial equation. These number systems “have a little bit of irrationality, but not too much,” said Ellenberg, who is now a professor at the University of Wisconsin, Madison.

There are infinitely many number field extensions, but there’s a natural way to measure how complex each one is, and the question then becomes to count how many there are below each complexity level. Venkatesh and Ellenberg figured out a new upper bound on these counts that greatly improved on the previous state of the art, which hadn’t budged for decades.

The project, Venkatesh said, was “psychologically very important” to him. “We got into this problem ourselves, and we really made progress on it.”

The pair originally tackled the problem using some new mathematical tools they’d heard about at a conference. But as they worked to improve the exposition of their proof, they eventually found that they had simplified away all the sophisticated new tools, Ellenberg said.

“It speaks to what I now recognize as being very characteristic of Akshay,” Ellenberg said. Their first, more technical version of the paper did prove their theorem. But for Venkatesh, Ellenberg said, it’s “not so much about whether you answered the question or whether you proved the theorem. It’s about: Did you understand what’s actually going on?”

Many mathematicians besides Ellenberg are struck by Venkatesh’s ability to distill ideas down to their essence. “When he’s finished, often his proof is the proof that would be in any textbook from here on,” Sarnak said.

“There have been lots of times in mathematics when someone has explained to me the right way to think about a certain piece of mathematics,” said Frank Calegari, of the University of Chicago. “But Akshay is one of the very few people who has done that with my own work.”

Soon after his work on number field extensions, Venkatesh carried out a new study that Sarnak calls his first “home run.” It concerned generalizations of the Riemann zeta function, which maps each number s to the infinite sum 1/1s + 1/2s + 1/3s + 1/4s + …. In 1859, Bernhard Riemann showed that knowing which values of s make this function spit out the number zero would tell mathematicians how many prime numbers there are that are smaller than any given number. But no one has succeeded in proving his hypothesis about where these “zeros” lie. The Riemann hypothesis, which has hundreds of consequences beyond explicating the distribution of prime numbers, is widely regarded as the most important unsolved problem in mathematics.

Since the mid-19th century, mathematicians have considered variants of the zeta function in which the ones in the numerators of the infinite sum are replaced by a more complicated sequence of numbers, generally a mix of positive and negative terms. Each of these “L-functions” has its own version of the Riemann hypothesis, which, if proved, would unlock other prime number mysteries, such as how prime numbers are distributed within various sequences of numbers.

These generalized Riemann hypotheses are also extremely hard to prove, so for decades, mathematicians have looked for ways to sidestep the hypotheses and prove some of their many consequences directly. One of the most important of these consequences is something called subconvexity, which says, roughly speaking, that the positive and negative numbers in the sequence of numerators of an L-function quickly start balancing each other out. Subconvexity estimates for L-functions, when they can be proved, yield statistical information about patterns in whole numbers — for example, one subconvexity estimate gives a description of the variety of ways any given large number can be written as the sum of three perfect squares.

Before Venkatesh turned his focus on L-functions, subconvexity estimates were commonly done on a case-by-case basis, often involving long papers full of technicalities, Kowalski said. But in 2004, Venkatesh sent the draft of a long paper to Philippe Michel, a mathematician now at the Swiss Federal Institute of Technology Lausanne who had studied subconvexity in depth. In the paper, Venkatesh used ideas from dynamical systems — the study of systems that change over time — to solve the subconvexity problem in much greater generality than had previously been accomplished. “The method was completely new,” Michel said. “It was a big shock to me.”

Venkatesh and Michel teamed up on a second paper that used this new approach to find subconvexity estimates for a huge family of L-functions. The pair of papers, along with several others Venkatesh wrote around that time, Sarnak said, “made him already one of the leading people in the world” in number theory and dynamics.

Collaborating with Venkatesh should perhaps feel intimidating, given the breadth of his knowledge and the depth of his insights — but somehow, it doesn’t. “He’s very strong technically, but talking to him doesn’t have a technical feel at all,” said Yiannis Sakellaridis, of Rutgers University, Newark. “You just talk about the essence of things.”

Searching Through the Fog

When I met with Venkatesh, he told me proudly that he had put up his daughter Tara’s hair in a “masterpiece” of a bun for her ballet dress rehearsal the day before. “I’ve written papers about the braid group,” he said. “Actual braids? Much harder.”

We strolled from his office to the institute’s preschool to pick up his younger daughter, Tuli, and then to their apartment in the visitor housing complex. Tara arrived soon after on the school bus, and Venkatesh settled in on the sofa, tickling Tuli while Tara eagerly described a science expo at her school.

Venkatesh was on solo parent duty, since his wife, Sarah Paden, who is finishing up her doctoral dissertation in musicology at Princeton, was away at a college reunion. It’s a role he slips into easily when she needs to travel, Paden said: He follows a comfortable groove that runs through the library, the coffee shop and home. Venkatesh is still an avid reader, although these days his book selection process usually involves simply snatching whatever looks interesting off the library shelf before his daughters drag him to the children’s section. At the moment, he’s rereading War and Peace. The writing is as wonderful as he remembers from his previous time through, but “it has these didactic passages that I find more annoying now than when I was younger,” he said.

Paden describes Venkatesh as a homebody. “Routine, consistency and home — these are comforts for him,” she said. They give him the freedom he needs, she said, to be adventurous in his mathematical life. “I feel like he takes leaps in his work that he doesn’t necessarily in life.”