The Knut and Alice Wallenberg Foundation has since 2014, together with the Royal Swedish Academy of Sciences, supported the mathematical research in Sweden through an extensive mathematics programme. The aim is for Sweden to recover its position at the international cutting edge by giving the best young researchers international experience and by recruiting young as well as more experienced mathematicians to Sweden.

Press release from the Knut and Alice Wallenberg Foundation >>

#### New solutions to ancient problems

Julia Brandes plans in her project to make use of the improved precision of the circle method to study the number of integer solutions to certain equation systems under certain extra conditions. Two cases are particularly interesting. On the one hand, she studies such equation systems where the variables lie in widely differing ranges – some variables are large while others are small. The other part of the project is about solutions which, if they are written in a number system with a certain prime base p, avoid certain digits. Such numbers are of interest because they have a strange but regular fractal structure. By making use of this special structure it is expected that it will be possible to predict how many such solutions there are.

– I think that research in mathematics should be supported for two reasons. On the one hand, there is a certain romance in fundamental mathematical research, as in fundamental research in terms of space travel or particle accelerators. That romance is not to be underestimated as the source of inspiration for future researchers. How many young people interested in mathematics do not read popular science books on, for example, Ramanujan or Fermat’s last theorem? The second reason is less romantic: A large part of the science and engineering research had not been imagined without mathematics.

The whole interview with Julia Brandes at the Faculty of Science web site, in Swedish >>

#### Cosmic plasma in a mathematical suit

Stephen Pankavich is an associate professor at the Colorado School of Mines, Golden, USA. He will be a visiting professor at the Department of Mathematical Sciences and together with researchers here develop new methods for solving different mathematical problems in the kinetic theory of plasma dynamics. Plasma is a special kind of gas in which electrons are stripped from the atoms, making the gas electrically charged. Plasma is therefore of practical interest; for example, plasma engines have been developed to drive probes that are sent far out in space. Plasma is regarded as the fourth form of matter, after gases, liquids and solids, and is the most common state of matter in the universe. Galactic clouds, tails of comets and the solar wind, among many other things, consist of plasma’s electrically charged particles.

The motions of plasma are described by a number of complicated partial differential equations. The purpose of this project is to show that the equations have realistic solutions, and to determine the properties of these solutions, such as development over time, and calculate their sensitivity with respect to the plasma’s state, such as its mass, charge, or temperature. Since the mathematical models always have physical counterparts, the challenge of analysing a problem mathematically also becomes a challenge in understanding the physical phenomena it describes at a deeper level. Therefore, a discovery of a specific behaviour in solutions to partial differential equations can be translated into real knowledge of problems in plasma physics or astrophysics.

#### New tools for capturing the unruly

Jakob Hultgren receives a postdoctoral position at a foreign university and funding for two years after his return to Sweden. The project’s title is “New notions of canonical metrics and stability in complex geometry” and contains two separate parts. The first is based on the new types of canonical metrics and stability conditions that Jakob introduced together with David Witt Nyström when he was a doctoral student (interview before the thesis defence at Chalmers University of Technology last year, in Swedish). There are many different things that needs investigation, but the overall goal is to establish new connections between geometric analysis and other fields such as algebraic geometry and probability theory. The second part of the project is about a question related to mirror symmetry in string theory that was posed in the early 2000s (the Gross-Wilson conjecture). This is a very algebraic problem and the new part of the approach is an analytical tool that Jakob developed together with Magnus Önnheim at Chalmers.

– At the University of Maryland, I will mainly work with Yanir Rubinstein. He has made many important contributions to the field but is also very broad and flexible in his interests, so there will probably appear many unexpected and interesting angles during the time of the project. Many of our ideas and tools come from geometric analysis where Karen Uhlenbeck, who received the Abel Prize last week, is active.

For example, one of the inspirations for the first part of the project is one of her results: the Donaldon-Uhlenbeck-Yau theorem.**Texts**: Carina Eliasson, Simone Calogero, Jakob Hultgren**Photos**: Johan Bodell (Julia Brandes), private (Stephen Pankavich, Jakob Hultgren)