2024-25 Frederick L. Hovde Distinguished Lecturer

 

Trevor Wooley

Andris A. Zoltners Distinguished Professor of Mathematics

Department of Mathematical Sciences

 

Tuesday, April 8, 2025

MATH 175
2:30-3:30PM - Lecture
3:30PM - Reception

The Music of Diophantine Equations — Harmony or Cacophony?

Abstract

Diophantine equations are polynomial equations to be solved in integers. They were considered by the ancients, including Diophantus of Alexandria in the 3rd century AD, after whom they are named. Although simple-to-state Diophantine equations are often fiendishly difficult to solve, the mathematics that has been developed in studying their solubility has found widespread application, proving useful in such real world applications as data encryption and security. In this talk we describe the role that Fourier analysis has played in recent developments surrounding Diophantine equations in many variables. It transpires that these equations satisfy constraints modulo powers of prime numbers, influencing associated Fourier series, and in harmonious situations the statistical properties of the solutions may then be interpreted elegantly as a product of factors associated with these prime numbers. The great challenge for researchers in this Diophantine frontier of analytic number theory is to (provably) tune in to this harmony amongst the cacophony of apparent randomness in the integer solutions of these equations. We describe very recent progress on such examples as Waring's problem by the speaker and his coauthors, allude to applications concerning equidistribution, and point to challenges and problems for the future. 

Bio

Trevor Wooley, Andris A. Zoltners Distinguished Professor of Mathematics has made fundamental contributions on a variety of topics related to number theory, most notably related to the Hardy-Littlewood circle method and Waring’s problem. Waring’s problem is a particular example of a Diophantine equation, which is a polynomial equation to be solved using only integers. Such equations influence the development of codes and cryptosystems for use in DVDs, in mobile phones, and in banking security.  Key to their practical utility is the illusion of randomness; although deterministic in nature, the solutions of these equations in integers should appear randomly distributed to an outsider. One therefore seeks to provide assurance that hidden patterns underlying these solution sets do not unravel their usefulness.  A significant portion of Wooley’s work has focused on the study of Diophantine equations using seemingly unrelated methods from Fourier analysis to make deep connections between harmonic analysis and number theory. 

In one major direction of research, Wooley greatly improved the bounds on the number of summands required in Waring’s problem, which is concerned with the ways in which a positive integer can be written as the sum of the fewest possible number of kth powers of positive integers.  Wooley developed a method called efficient differencing that reduced by a factor of two the relevant bound.  This was the first significant progress in 30 years on this well-studied problem;  this method led to further advances in related problems by many researchers.  Some of the recognition he received for this research includes a 45-minute invited address at the 2002 International Congress of Mathematicians, election to the Fellowship of the Royal Society (the UK version of the National Academy of Sciences), the Salem Prize, and the Junior Berwick Prize of the London Mathematical Society. 

In another direction, Wooley developed a method called efficient congruencing that provides bounds on certain sums of exponential functions.  Special cases of such sums occur in Waring’s problem and in the analysis of the zero-free region of the Riemann zeta function.  Wooley’s approach led eventually to a complete understanding of Vinogradov’s mean value theorem, settling conjectures originating in 1935.  This theorem relates sums of fixed powers of integers to mean values of an associated sum of exponential functions.  Wooley developed his method of proof over a period of years and received substantial recognition for his contributions in this area, including a second 45-minute invited lecture at the 2014 International Congress of Mathematicians, the Fröhlich Prize of the London Mathematical Society, and a 5-year European Research Council Advanced Grant of 1.9 million euros.

Wooley’s research accomplishments have been recognized in the US and abroad throughout his career.  In addition to the distinctions indicated above, he has been supported by a Sloan Fellowship and a Packard Fellowship (given at that time to 20 scientists and engineers across the US, with roughly one per year to a mathematician), a Royal Society Wolfson Merit Award, and several NSF grants in the US.  He has more than 140 publications in excellent journals and has a remarkably high profile in the mathematics community as indicated by his lectures at top universities and conferences throughout the world.