Mathematicians to share millions in research grants

Equations for integers in a new light

Simon L. Rydin Myerson, Assistant Professor at the Division of Algebra and Geometry from May 1, receives funding for a project about solutions to systems of Diophantine equations.

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).

Quantitative methods in semiclassical spectral theory

Simon Larson, Associate Senior Lecturer at the Division of Analysis and Probability Theory, receives funding for a project at the intersection of spectral theory, geometry and mathematical physics.

“I’m thrilled to have the opportunity to recruit a postdoc for this project. The new insights and perspectives that an additional member of our group will bring are likely to open up exciting directions of research. I enjoy collaborating and discussing mathematics with others, so bringing a postdoc into the project is something I very much look forward to.”, says Simon Larson.

Spectral theory belongs to classical mathematical analysis and has many applications in physics and engineering, beyond a purely mathematical interest. The connection between geometry and spectral theory was popularized by the Polish American mathematician Mark Kac. In 1966, he asked: can one hear the shape of a drum? The answer came almost 30 years later: no, one cannot hear the shape of a drum, as the same sound can come from different vibrating membranes. However, it is possible to determine some properties of a membrane, such as area and perimeter.

Mathematically, the question is formulated in terms of eigenvalues of operators (the frequencies with which the membrane oscillates) whether they can be used to recreate the underlying geometry. In the project, the focus is on the reverse problem of understanding the eigenvalues based on knowledge of the operator, especially the Schrödinger operators of quantum mechanics, where eigenvalues correspond to the energy levels of particles or systems of particles.

Of particular interest here is how the eigenvalues of the Schrödinger operators are distributed asymptotically towards infinity. The interest is motivated by the fact that it is precisely in this so-called semiclassical limit to infinity that quantum mechanics becomes classical mechanics. The proposed project aims to develop new tools for estimating the accuracy of semiclassical approximations under minimal assumptions on the underlying operators.

A great challenge in complex geometry

David Witt Nyström, Professor at the Division of Algebra and Geometry, receives funding for a project which aims to prove that the non-Archimedean Monge-Ampère equation is solvable.

“With this grant, I hope to employ Pietro Mesquita Piccione, who is currently a PhD student at the Université Sorbonne and is expected to defend his doctoral thesis this summer. Pietro is very talented and has expertise in non-Archimedean geometry that is essential for the project. He would make a positive contribution to our department both socially and research-wise.”, says David Witt Nyström.

A vital question in geometry is understanding when a geometric object can be given an optimal shape. It was not until 1907 that Paul Koebe and Henri Poincaré proved almost simultaneously that two-dimensional surfaces can always be deformed into a shape with constant curvature. The development of the proof of this uniformisation theorem had occured throughout the nineteenth century in parallel with the emergence of modern algebraic geometry, the birth of complex analysis and topology. Even more new tools were necessary to prove in 2003 a three-dimensional equivalent of the uniformization theorem – Thurston’s geometrisation conjecture.

Also in complex geometry, the subject of this project, the question of when objects can be given an optimal shape is raised. Here, the geometric objects are defined using complex numbers and the issue of uniformisation, known as the Yau-Tian-Donaldson conjecture, is one of the major unsolved problems in complex geometry.

A promising way to approach the solution is to use methods from non-Archimedean geometry, where the familiar real and complex numbers are replaced by abstract objects called non-Archimedean fields. These methods have proved successful in one important special case, but much remains to be done, including showing that the non-Archimedean Monge-Ampère equation is solvable. The aim of the project is to prove that the equation is solvable and thus get one step closer to proving the Yau-Tian-Donaldson conjecture.