## Ayla Gafni will use grant to further her analytic number theory studies

OCTOBER 27, 2020 BY SHEA STEWART

Ayla Gafni views the solving of complex mathematical problems as deciphering a puzzle or searching for the solution to a centuries-old mystery.

The University of Mississippi mathematics professor fell in love with the abstract science of numbers, quantity and space at a young age. That enthusiasm for mathematics has followed Gafni into her Ole Miss classrooms and propels her research endeavors that, in turn, make her a more effective classroom professor.

“The concepts I teach in my classes are the same tools I use in my research,” said Gafni, an assistant professor of mathematics who joined the faculty in 2019. “I can give the students a deeper context for the abstract material that we’re learning.

“Sometimes students miss the importance of mathematics because they are looking for a real-world application. By showing them the way a mathematician would approach a subject, I try to convince them that real-world applications aren’t the goal. The goal is to start with a simple idea and see how much you can learn just by asking the right questions and taking the logic as far as it will go.”

To help further her mathematics research and enrich her classroom instruction, Gafni recently earned a Ralph E. Powe Junior Faculty Enhancement Award, a prestigious competitive research award that provides seed money for junior faculty members.

“This is my first research grant, so it’s a big milestone for me toward establishing my research career,” said Gafni, whose research interests include analytic number theory, which focuses on integers, and combinatorics, which is the study of counting arrangements of objects.

“This award is especially significant because the competition is across all science, technology, engineering and mathematics fields,” she said. “It’s encouraging to know that the abstract problems I like to think about are of interest beyond my little research group and even beyond mathematics.”

Gafni is among 35 award winners chosen from 167 applicants nationally for this year’s round of Ralph E. Powe Junior Faculty Enhancement Awards.

“The Department of Mathematics is proud of Dr. Gafni’s award of a prestigious Ralph E. Powe Junior Faculty Enhancement Award,” said James Reid, professor and chair of mathematics at UM. “Dr. Gafni is involved in cutting-edge research in number theory and has published in some of the best mathematical journals in the world, such as the Proceedings of the American Mathematical Society, Acta Arithmetica and the Journal of Number Theory.

“Her work is internationally acknowledged with presentations occurring at the University of Toronto, Cornell University, the University of Bristol, the University of Warwick and the Mathematical Sciences Research Institute in Berkeley, California.”

Granted by Oak Ridge Associated Universities, or ORAU, the award provides $5,000 for the one-year grant, with the Office of Research and Sponsored Programs at UM providing a $5,000 match. The award is open to full-time assistant professors at ORAU member institutions within two years of their tenure-track appointment at the time of application.

“One of ORAU’s historic purposes has been to assist our member universities and their faculty,” wrote Andy Page, ORAU’s president and chief executive officer, in a letter informing Gafni of the award. “This award is clearly in that tradition. It also represents public recognition by your academic peers of the quality and promise of your research.”

Gafni will use her Powe award to study properties of divisor functions in joint work with two colleagues: Steve Gonek, professor and chair of mathematics at the University of Rochester, and Trevor Wooley, Andris A. Zoltners Distinguished Professor of Mathematics at Purdue University.

Her research proposal is titled “Correlated Divisor Sums and Moments of the Riemann Zeta Function.”

“A divisor function counts how many ways a number can be split up into a product of integers,” Gafni said. “We study these objects by writing an integral that is equal to what we want to count. We then use advanced calculus to get an estimate for the number of solutions.”

Difficult mathematical problems don’t get solved all at once, Gafni said, but with important open problems, many mathematicians make incremental progress by studying related objects.

“We need a deeper understanding of the behavior of the divisor functions so that we can estimate certain integrals, called moments,” she said. “This will lead to a better understanding of the Riemann zeta function, which in turn would lead to a better understanding of the primes.

“This research project is one small cog in the big machine of analytic number theory.”

Gafni is the second UM mathematics faculty member in four years to receive a Ralph E. Powe Junior Faculty Enhancement Award.