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Interview with Klaus Kröncke

Published May 30, 2022
Klaus Kröncke

Tell us a bit about yourself and your educational background; what inspired your interest in your field?

I am a Mathematician specialized in Differential Geometry and Geometric Partial Differential Equations. I was born and raised in Vienna, where I did my Diploma studies in Mathematics. I moved to Potsdam for my PhD, to Regensburg as a postdoc and then to Hamburg as an assistant professor. After spending 11 years in Germany, I started working as an associate professor at KTH in January 2022.

Already as a child, I read many books about astronomy and cosmology. As I got older, I became more and more fascinated with general relativity and the fact that the universe is actually a curved space-time. Eventually, I realized that I was even more intrigued by the purely geometric aspects of this theory. I think this is what drove me to my field of research.

Receiving the Göran Gustafsson Prize must be exciting. What did you receive the prize for, and what is your plan been in utilizing the grant?

It is indeed very exciting, and it feels almost unreal to get such a big prize not even half a year after I have moved to Sweden. I could not have wished for a better start.

I got the price for getting new insights into stability aspects of the so-called Einstein equation. My plan is to deepen these insights and to establish new connections between various notions of stability and geometric properties of solutions of the Einstein equation. The prize will be used to hire postdocs I will be working with on these problems.

What is your research about?

My research focuses to a large extent on curved spaces which satisfy the aforementioned Einstein equations. These are stationary points of a geometric evolution equation which is called the Ricci flow. As such, there is a natural notion of stability under this flow. I am looking for criterions which imply stability or instability of such a space. Moreover, these spaces can be used to construct interesting spacetime models in general relativity. I am trying to link stability aspects of the underlying spaces to properties of the constructed spacetimes.

What are the biggest challenges your work/research combats?

When working on a difficult mathematical problem you often go through mixed emotions. One moment, you think you might have an idea to solve the problem, but the next moment, you realize that this solution either does not work at all or leads to another problem that you must solve again. During a research project you go through this process many times and you learn something new each time. The more experience you gain, the more aware you become of these processes and the more relaxed you deal with them.

Are you planning to develop your research in new directions?

I want to better understand singularity formation under the Ricci flow. This is connected to the stability problem and in the long term, it can be very useful for classifying curved spaces.

What do you view as your most important research accomplishment?

I think my most important result so far concerns the stability of spaces of so called locally asymptotically Euclidean geometry under the Ricci flow. These spaces are of infinite size, but the spaces become more and more flat as one approaches infinity. This result not only proves stability with respect to perturbations with slow decay at infinity, but it also establishes a precise convergence behaviour of the Ricci flow. Various technical obstacles needed to be solved and it took two papers with about 120 pages in total to write down the complete solution.

What advice would you give to youngsters interested in becoming scientists?

To choose a field of interest, is the most important thing of all; you should be enthusiastic about your work. Take some time to develop your interests; they may and will change in time. Be aware that you will have to move from one place to another several times, but don't be afraid too much of that.

Is it important for you to communicate your research to the world? If yes, why and how do you do that?

For making colleagues aware of your results, it is very important to regularly give scientific talks in research seminars or conferences. On average, I give six to eight talks per year in such a framework. Such talks are usually followed by discussions which can give a lot of inspiration for your future work. Moreover, many collaborations I had were initiated through such discussions.

Apart from giving scientific talks, I find it also important to give talks for a non-scientific audience, especially to raise the awareness and acceptance of science in public. Such general talks, given to a youth audience, can also help to attract more young minds into the field of sciences.

As a Mathematician, it is particularly tough to communicate your results, but it can work up to a certain extent. I gave a talk for a general audience (about the geometry of space and time), and I gave a few courses in Mathematics in summer camps for students in high school and university.