Institute for Advanced Study researcher mines polynomial patterns in dense sets

IAS - November 2021
Sarah Peluse has spent the past few years looking for patterns in numbers. Specifically, she’s been looking for polynomial progressions, which involve sequences of numbers with a fixed common ratio plus a fixed value (x, x + y, x + y2), such as 3, 3 + 3, 3 + 32. Prior work had proved that once a set of numbers gets big enough, it reliably contains these polynomial progressions. But nobody knew just how big exactly.
Then, in 2019, Peluse, now a Veblen research instructor at Princeton University and the Institute for Advanced Study (IAS), and Sean Prendiville, a lecturer in mathematics at Lancaster University in the United Kingdom, provided an answer. Building on Peluse’s earlier work on three-term polynomial progressions in subsets of finite fields and the polynomial Szemerédi theorem in finite fields, they devised a formula that defines the explicit bounds for how small one configuration of polynomials could be and still have this polynomial pattern.
Their proof represented a major step forward, but it had limitations. “We had solved it for one polynomial pattern, but we couldn't see how it could possibly be made more general,” Peluse says.  
Then, in work that she says took her by surprise, she did just that: she figured out how to generalize the proof. “I was kind of shocked that I was able to do it. It was the most complicated thing I had ever found,” she says of the solution, which is now published in Forum of Mathematics, Pi 8.  
Kannan Soundararajan, professor of mathematics at Stanford University and past member of the IAS who was Peluse’s doctoral advisor, says that her work on polynomial patterns in dense sets—5 proofs in all now—surmounts a variety of difficulties. “In general, it turns out to be very difficult to show that patterns that we expect to find can in fact be found in suitably large sets,” he says, noting that this difficulty is sometimes referred to as ‘finding hay in a haystack!’” In the case of polynomial progressions, where the polynomials have different degrees, he adds, “we are trying to find patterns where the parameters are of vastly differing sizes, which is not the case for the already difficult problem of finding arithmetic progressions. This is one of the difficulties that is surmounted in Peluse's work.”
Receiving recognition for her contributions
Recently, the work was formally recognized: Peluse was awarded the 2022 Maryam Mirzakhani New Frontiers Prize for her contributions to arithmetic combinatorics and analytic number theory, particularly her work with polynomial patterns in dense sets. The $50,000 prize for early-career women mathematicians is one of the three annual mathematics prizes from the Breakthrough Prize Foundation. 
Although Peluse has already published numerous proofs, she completed her doctoral degree only in 2019, and so she certainly is still in the early stages of her mathematics career. She took a somewhat unusual journey to get here, being raised by a single mom and attending a liberal arts college starting at age 15 after taking all of the math courses at her high school and local community college. Lake Forest College was close to her home, so her mom could drop her off on the way to work and bring her home at the days’ end, and financial aid made it an affordable option to continuing her education in mathematics. 
Within two years, Peluse had completed all the available mathematics classes at Lake Forest while also competing on the cross-country team. She continued with both pursuits at the University of Chicago—where she thrived both as a mathematician and runner, earning recognition as a Division III All-Academic athlete and the Alice T. Schafer Mathematics Prize from the Association for Women in Mathematics. While there, she developed her interest in analytic numbers theory—which uses analytical techniques to investigate properties of integers and prime numbers—and then went deep into it during her doctoral program at Stanford University, where she found mentors who were also working in the area. 
Her commitment to mathematics persisted even when she saw others in her classes drop out, discouraged after a poor grade on an exam. “There’s a tendency for women to drop out at that point, even though there may be lots of men who did worse than them on the test who don’t,” she says. As things stand, upper-level mathematics continues to be largely male, with just a small proportion of the very top women who continue with it. Peluse wishes colleges found more ways to support women to stay in mathematics even when the going gets extra tough.
A willingness to cope with setbacks and frustrations is critical to a career in mathematics, she notes. She also notes the importance of being willing to obsess over a conjecture and to spend long hours writing out formulas and testing the proof again and again. 
Even then, she finds, sometimes ideas and solutions wait to arrive until she’s doing something else, such as running errands or taking a walk. Other times, they take collaboration. She cites the representation theory work that she published last yearexploring the statistical properties of integers divided into character tables of symmetric groups, or squares. Working alone, she was able to show that almost all entries in the squares are eventually multiples of prime numbers up to 13. But she couldn’t decipher a way to make the proof work for all primes. Then Soundararajan read the proof and noticed a way it might be possible. Together they developed a formula proving that almost any number in these squares, even very large ones, is a multiple of any fixed prime. 
Finding a new fascination in multi-dimensions
Peluse has approximately 2 years remaining on her Veblen research instructor position at IAS and Princeton University. During this time, she will teach calculus to undergraduates and co-direct a weekly number theory seminar, in addition to working on her math. 
She is currently focused on completing a paper on multidimensional patterns in subsets of integer grids. She’s also thinking about new directions for her work, confessing that she feels wholly finished with polynomial patterns for the time being and is now focusing on multi-dimensional patterns. 
“I feel like I don't really have that much choice,” she says of the direction of her mathematical investigations. “I can only work on things that I'm really interested in—and I can't really control what I'm interested in.”
—Jennifer Fisher Wilson