Exotic geometry will shed light on a century-old conjecture

Associate Professor Christian Johansson will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematical Sciences at Chalmers University of Technology and University of Gothenburg.

In January 1967, Robert Langlands, then in his early thirties, composed a letter to a fellow mathematician, André Weil, where he outlined an ambitious project in which he linked two completely different mathematical fields – number theory and geometry. He formulated a great many conjectures, which are hypotheses that have not yet been proven, although most experts believe they could be true. The planned project is part of the now very comprehensive Langlands Program. 

The plan is to study L-functions, a vital component of the Langlands Program. L-functions provide information about Diophantine equations, for which only integer solutions are allowed for polynomials with integer coefficients. Despite being known since Antiquity, Diophantine equations are notoriously difficult, but not always impossible, to solve. Nor is it possible to determine in advance whether a given equation has a solution at all, which the Russian mathematician Yuri Matiyasevich proved in 1970.

One way of using L-functions to approach Diophantine equations is to explore the 100-year-old but still unsolved Artin’s conjecture. This deals with basic properties of L-functions and was an important inspiration for Langlands’ ideas. The project aims to use exotic geometry to make progress in the exploration of Artin’s conjecture in particular, and the Langlands Program in general.