# David Rydh has been awarded the Göran Gustafsson Prize

The Göran Gustafsson Prize in Mathematics 2021 for young researchers up to the age of 45, has been awarded to David Rydh, Professor of mathematics at the Department of Mathematics, KTH.

The winning motivation is "for his groundbreaking results on algebraic stacks" and the prize money of 5.1 million SEK will be distributed over the years 2022–2025.

– This largely means that I can devote myself to a greater extent to research, probably two doctoral students or postdoctoral fellows will be appointed. At the moment, I have two doctoral students who will graduate next year, so the research grant came at the right time, says
David Rydh
.

His research area belongs to the part of mathematics that is usually called moduli theory and a large part of his research pertains to moduli problems and the theory of stacks. By linking the theory of stacks to a seemingly completely different part of mathematics, monoidal categories, he has developed tools for solving moduli problems that previously could not be solved. Based on these, he has formulated a precise description of the local geometric structure of stacks that previously was missing.

Another important part of his research concerns the birational geometry of Deligne–Mumford stacks. It entails modifying stacks in a controlled way to be able to compare similar stacks. For this purpose, he has developed stacky blow-ups that can be used to both compactify stacks, which he compares to putting the lid on a jar, as well as removing and adding "stackiness": that which makes the difference between classic geometric objects and stacks.

David Rydh explains that a very attractive part of his work is trying to understand, explain and endorse "beautiful mathematics" as he describes it. So it's not just about pure problem solving.

– In research, there are vast numbers of unknown things to explore that attract me. Famous conjectures that no one has been able to prove and hitherto unknown structures that explain and unite different areas are undoubtedly tempting to work on, something I have done several times. To understand is probably the biggest driving force behind the research and the unexplained phenomena are endless, he says.

From the beginning, mathematics was not an obvious choice for him. His parents worked as music teachers, and as a child he spent a lot of time making music along with his six siblings. His first instrument was the cello, an instrument that appeals to him a lot as its sound is similar to the human voice. In school, he found almost all subjects equally interesting, but later became more interested in physics, mathematics and programming as he started to participate in various competitions in high school and secondary school. After that, programming began to take up more and more of his free time and when he started studying at KTH he participated in competitions such as ACM (ICPC) and TopCoder.

– Programming meant a lot to me and was probably an important reason why I later went further on the mathematics track when it came to choosing a specialization during the __engineering physics__ education. By this time I accidentally came in contact with algebraic geometry, says David Rydh.

Today, lack of time is challenging as his journey in mathematics is pleasantly combined with family life and everything that goes with it.

– Nowadays I often play the piano, it has been something of a release valve during the pandemic when I work at home, he says and gives a hint about something that could possible be his future research focus.

– At the moment I am quite interested in so-called "non-reductive geometric invariant theory". Then algebraic stacks appear with symmetry groups that are not "reductive", which means that they behave quite differently. Another relatively new area that has piqued my interest lately is "derived algebraic geometry". It is also motivated by moduli theory but is a little harder to explain, says David Rydh.

**Text: Sofia Nyström**

## David Rydh's results on algebraic stacks

For the past 30 years, moduli theory has attracted a great deal of interest in algebraic geometry, not least through influences from physics and string theory. An example of a seemingly simple question that had long eluded algebraic geometers is "How many rational curves in the plane of a given degree pass through n given points?". This question was elegantly solved by Kontsevich in 1994 through the study of intersection theory on moduli spaces of stable maps.

David Rydh has been particularly interested in algebraic stacks and has shown a number of results about them. Below are some of the most significant:

- A very concrete description of what an algebraic stack looks like near a point.
- Algorithms for how to modify an algebraic stack so that the symmetry groups become smaller so that in many situations we can transfer results from ordinary geometric objects to stacks.
- That it is possible to compactify Deligne–Mumford stacks (= stacks with finite symmetry groups). It roughly means that we start with an algebraic stack that has "holes" and show that it is possible to close these so that it remains an algebraic stack. Or as it was described earlier in the text above "like putting the lid on a jar".
- That some important moduli problems can be solved, i.e., they have corresponding moduli spaces (which are algebraic stacks).