Paul Nelson is a world leader in the analytic theory of automorphic forms. This is an important area of modern number theory. He recently made groundbreaking progress on the so-called subconvexity problem, in great generality, and this work was featured in Quanta Magazine, which is an online publication by the Simons Foundation whose goal is to enhance public understanding of science. Now Paul Nelson’s paper on groundbreaking progress on the subconvexity problem has also been published in Inventiones Mathematicae.

Paul Nelson explains:

"The paper concerns L-functions, mathematical objects central to the study of prime numbers and their distribution. Despite being studied for centuries, L-functions remain the subject of fundamental open problems, such as the famous Riemann Hypothesis. One of the key challenges is to understand how quickly L-functions grow. Surprisingly, this relates to many apparently unrelated questions in number theory and quantum chaos, whose solutions boil down to establishing 'subconvex bounds' for L-functions that constrain the growth beyond what can be inferred from a general argument known as the convexity principle.

L-functions come with a numerical invariant called the degree. Until recently, subconvex bounds had been established only in the case of low degree L-functions. The just-published paper establishes subconvex bounds for a wide class of L-functions of arbitrarily large degree. The approach combines techniques from microlocal analysis, representation theory and invariant theory, but is conceptually simple and direct."

The paper is published in Inventiones Mathematicae via the following link

• https://link.springer.com/article/10.1007/s00222-023-01180-x

Furthermore, Department of Mathematics is very happy to announce that from May 1st 2023, Paul has been promoted to professor at the Department of Mathematics at Aarhus University. Congratulations Paul.