New insights when perspectives shift

James Wright is a professor at the University of Edinburgh, UK. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will be a visiting professor at Centre for Mathematical Sciences, Lund University.

When the French mathematician Joseph Fourier demonstrated just over 200 years ago how the function for heat conduction could be described using an infinite sum of simple trigonometric functions – a Fourier series – he guessed that any function could be written in the same way. Despite the guess not being entirely correct, it is correct in principle. Nowadays, it is impossible to imagine developments in mathematics, physics, or engineering without Fourier series. 

In the theory of Fourier series, Fourier analysis meets real analysis, i.e. the study of real numbers and functions, and complex analysis, which investigates functions of complex numbers (two-dimensional numbers). For decades, both real and complex analyses have been used to advance Fourier analysis.

In the 1930s, many studies of Fourier series were conducted using complex analytic methods. However, these methods were limited to functions of a single variable.
Fundamental questions about functions of several variables remained unanswered until real methods were developed in the 1950s. Unlike earlier, when complex analysis was used to explore Fourier analysis, real variable methods now have a significant influence on complex analysis, both in one and in several variables. However, some problems are still limited to functions of a single variable. To move to multiple variables, the planned project will focus on the exchange of methods and ideas between Fourier analysis and complex analysis and address fundamental problems which have until now been out of reach. 

Photo University of Edinburgh