Equations for integers in a new light

Dr Simon L. Rydin Myerson will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.

Can you imagine that musical harmony and “y squared = x cubed” are connected? This begins with the famous work of Srinivasa Ramanujan on the circle method in the 1910s. It goes like this: imagine all the possible values that could appear on each side of an equation as the pitches of musical notes which all play at once. If we discover the shape of the resulting sound wave, we can deduce how common the solutions of the equation are.

Questions about integer solutions to polynomial equations were asked already in Antiquity, by the Greek mathematician Diophantus of Alexandria. Ever since, Diophantine equations have occupied the minds of many of the greatest mathematicians, such as Fermat, Euler and Gauss, and have been the source of entirely new fields of mathematics. In 1970, Yuri Manin made a profound conjecture about how common the solutions should be for algebraic equations with many variables. Many researchers are working to refine and test his theory.

The aim of the project is to obtain new results for the number of solutions to systems of Diophantine equations. For this, the circle method will be used. The method is based on estimates of specific integrals which result from the equations. For systems of Diophantine equations of degree larger than 2, a new idea from mathematics in the 1930’s will be tested. This will also provide new insights into Diophantine approximation, which states how well irrational numbers (such as the square root of 2) can be approximated by rational numbers (fraction numbers like 3/2 or 7/5).