The concept “spacetime” received its precise mathematical description in Albert Eninstein’s general theory of relativity, published just over a century ago. In this, the shape of the universe is described as a four-dimensional surface where the shape is determined by the matter content of space. The laws that govern the movement of spacetime can in mathematical terms be reduced to equations, called Einstein field equations.

Mingchen Xia studies the geometry of the vacuum that would appear if all matter content of space would be removed. The problem is that the equations are nonlinear, which makes it difficult to find a direct solution. Several different approaches have been developed to obtain solutions, and one of the most recent is to use techniques from pluripotential theory where notions and special metrics from complex geometry and complex analysis are used.

## Singular instead of regular functions

In the three articles of the thesis, the first deals with some open problems of the area. In the second, Mingchen has made use of the results from the first to gather conditions that guarantee the existence of the metrics, and in the third he explores what happens if there are no such metrics. The real innovation in the thesis is the applications of singular quasi-plurisubharmonic functions to the vacuum problem, whereas previously the emphasis has been on the regular functions.

Mingchen’s interest for mathematics began when he was about ten years old, although he then did not know very much about it. He started to study more advanced mathematics and was most fascinated by the rigorous proofs and the structure behind it all. When he did his master’s studies in Paris all students were supposed to have an internship. His advisor suggested an available colleague as his future thesis advisor, who turned out to be Robert Berman in Gothenburg. After taking his master’s degree, Mingchen decided to come to Sweden for doctoral studies.

## Great possibilities to do the research you want

– I did some research before, so I knew what to expect. Gothenburg has been a good place to live in and to do a PhD, although I do not like the winter! Compared to a big place like Paris there is a limited number of mathematicians if you are interested in a very specific topic, which can cause difficulties. But it is very cozy here and you have great possibilities to do the research you want, the advisors do not press too hard to make you go in a certain direction. In my opinion, to find the direction you want to go in yourself is one important part of being a PhD student.

After the thesis defence, Mingchen is returning to Paris where he will have a postdoctoral position, financed by the Wallenberg Foundation.*M**i**ngchen Xia will defend his PhD thesis “Pluripotential-theoretic methods in K-stability and the space of Kähler metrics” on October 5 at 14.30 in lecture hall Pascal, Hörsalsvägen 1. Supervisor is Robert Berman, assistant supervisor David Witt Nyström.***Text and photo**: Setta Aspström