“When we started, the aim of the program for mathematics was that Sweden would regain an internationally leading position in the field. I think we have progressed well. Swedish mathematics has had a very positive development, with several world-leading research environments, and it has become attractive for leading international researchers to come here,” says Peter Wallenberg Jr, chair of Knut and Alice Wallenberg Foundation.

## The randomness of fractions

Michael Björklund, professor in the Division for Analysis and Probability Theory, Department of Mathematical Sciences, receives funding to hire a postdoctoral fellow from abroad.The entire numerical line consists of real numbers, of which some comprise rational numbers, ones which can be written as a fraction of two integers. However, most real numbers cannot be expressed in this manner – they are irrational. The best-known irrational numbers include π and the square root of two, √2. How these irrational numbers should be best approximated using rational numbers has, in its simplest form, been well understood for more than a century. The field of diophantine approximation, which this project is about, examines how well an approximation can be performed using rational numbers for a given real number.

One way of approaching the question is to calculate the number of rational numbers that provide good approximations to a given real number and which have denominators below a specified high limit. It has been known since the 1960s that, for almost all real numbers, their quantity grows towards infinity at an almost exactly logarithmic rate with the given limit for the denominator. Deviations from this rate fulfil a specific form of the central limit theorem, i.e. a normal distribution, as recently proven by Michael Björklund and Alexander Gorodnik.

The purpose of the project is, in the next step, to move beyond the normal distribution. The basis for understanding deviations from the normal distribution is the already well-known analysis of the sums of independent random variables and the central limit theorem. In the case being studied, the random variables are only partially independent, but the analogy may bear fruit and lead to better estimations that those now known.

One way of approaching the question is to calculate the number of rational numbers that provide good approximations to a given real number and which have denominators below a specified high limit. It has been known since the 1960s that, for almost all real numbers, their quantity grows towards infinity at an almost exactly logarithmic rate with the given limit for the denominator. Deviations from this rate fulfil a specific form of the central limit theorem, i.e. a normal distribution, as recently proven by Michael Björklund and Alexander Gorodnik.

The purpose of the project is, in the next step, to move beyond the normal distribution. The basis for understanding deviations from the normal distribution is the already well-known analysis of the sums of independent random variables and the central limit theorem. In the case being studied, the random variables are only partially independent, but the analogy may bear fruit and lead to better estimations that those now known.

## The shape of the universe

Mingchen Xia, PhD student in the Division for Algebra and Geometry, Department of Mathematical Sciences, receives funding for a postdoctoral position with Professor Sébastien Boucksom at the École Polytechnique, Palaiseau, France.

Time and space shape our universe. The theory of how they are connected was developed in the eighteenth century by Isaac Newton. However, in his Principia Matematica, time and three-dimensional space are independent. They were first linked in a shared spacetime almost 200 years later, when James Clerk Maxwell realised that the speed of light in a vacuum was constant, regardless of who measured it. This was only possible if space and time were regarded as a unity – a spacetime.

Spacetime received its precise mathematical description in Albert Einstein’s masterpiece – the general theory of relativity, which was published just over a century ago. The theory describes the shape of the universe as a four-dimensional surface. Its shape is determined by the matter content of space – the more matter, the more curved the surface. But what is the shape of the universe if all matter is removed, how can a vacuum be described? If there is nothing at all, is the universe then entirely flat? The answer was astounding – it turns out that the theory of relativity actually permits many non-flat vacua.

In the theory of relativity, the exact relationship between matter and the shape of the cosmos is given a set of differential equations, Einstein’s field equations. These are notoriously difficult to solve and several different approaches have been developed over the past century to obtain direct solutions for spacetime. One of the most recent is pluripotential theory, an area within complex mathematical analysis. The plan is to use these methods to help explore the intricate geometry of the vacuum.

## About the Knut and Alice Wallenberg Foundation program for mathematics

Over the years 2014–2029, the program will provide SEK 650 million to allow Swedish researchers to receive international postdoctoral positions, as well as the international recruitment of visiting professors and of foreign researchers to postdoctoral positions at Swedish universities. The program also includes funding worth SEK 73 million for the Academy of Sciences’ Institut Mittag-Leffler, one of the world’s ten leading mathematics institutions.