# International Postdoctoral Grant to three young mathematicians

The Independent Research Fund Denmark makes it possible for three researchers, educated at the Department of Mathematical Sciences, to do research on arithmetic statistics in Paris, graphs and operator algebra in Glasgow, as well as control systems at a particle accelerator in Lund.

Asbjørn Christian Nordentoft, Kevin Aguyar Brix and Søren Wengel Mogensen are among the 18 recipients of this year's international postdoc grants from The Independent Research Fund Denmark (DFF). DFF has granted DKK 26 million to young researchers with original ideas. The aim is to enable the grant recipient to consolidate his or her individual research profile by managing a specific research project in an independent manner at a research institution outside of Denmark.

## Arithmetic statistics and automorphic representations

Asbjørn Christian Nordentoft defended his PhD dissertation in December 2020, supervised by Morten Risager. He was then employed as a postdoc at Bonn University. He has received DKK 1.4 million from DFF for his research at the Université Sorbonne, Paris Nord, where he will start after the summer holidays of 2021. His supervisor will be Farrell Brumley.

Asbjørn calls his project for *Arithmetic statistics and automorphic representations*. He describes the project as follows:

“Arithmetic statistics is the mathematical theory concerned with the statistical and distributional properties of the integers. As an example, one could ask how the last digits of primes distribute among the four possible digits 1,3,7 and 9 (forgetting about the primes 2 and 5). It turns out that the last digits of primes equidistribute among the four possible outcomes (i.e. there will be approx. 25% of primes with each of the four last digits). In my project, I will investigate the distribution of so-called “automorphic periods”, which are certain numerical quantities intimately tied to the primes and their generalizations (via the so-called “L-functions”). In one of my projects, I have a conjecture that certain automorphic periods will distribute according to the normal distribution. To prove this I will use tools from mathematical analysis and representation theory.“

## C*algebras and topological dynamics

Kevin Aguyar Brix defended his PhD thesis in September 2019, supervised by Søren Eilers. He then spent a year at the University of Wollongong, Australia, funded by the Carlsberg Foundation Internationalization Scholarships. They also paid for next year's research at the University of Glasgow, Scotland. From September 2021, The Independent Research Fund Denmark has secured his continued studies in Glasgow with a grant of DKK 1.3 million. Xin Li is his host in Glasgow.

Kevin's research project is called *Fine structure of C*algebras associated with topological dynamics*. He describes his research as follows:

"A graph is just a collection of vertices and arrows but this simple abstraction is capable of modelling a surprising number of real systems (think neural networks). By running around the arrows of the graph, we create a dynamical system as time passes, and different graphs will usually produce different systems. Symbolic dynamics is a branch of mathematics that has studied such systems for over a century with great success. On the other hand, operator algebra is a mathematical discipline that grew out of quantum physics but which stands independently today with great influence from Danish mathematicians. In 1980, these two disciplines were connected to the benefit of both fields. We have learnt to encode and study the dynamical systems in the setting of operator algebras.

"The fundamental problem 'when do two graphs generate the same dynamical system' is still an open problem. Today we know how to translate many properties back and forth between symbolic dynamics and operator algebras, and in this project, I aim to use operator algebraic methods to learn more about the underlying dynamical systems in order to approach this notoriously difficult problem. If successful, this might also shed light on the various dynamical systems, which surround our everyday lives."

## Causality and control in dynamic systems

Søren Wengel Mogensen was affiliated with the Copenhagen Causality Lab when he was a PhD student at MATH. He had Niels Richard Hansen as supervisor and defended his PhD thesis in July 2020. Has since worked as a postdoc at DTU (Technical University of Denmark). Søren has been granted DKK 1.3 million for studies at Lund University in Sweden. His research project is called *Causality and control in dynamical systems*. He describes it as follows:

”Humans' effect on global temperature is an important topic. What would the temperature have been, had humans not emitted greenhouse gasses? This is an example of a causal question, a question about cause and effect. Many scientific questions are causal. We wish to not only describe the world but to understand it. Causal questions are important when controlling various processes and systems. In a modern power grid, it's necessary to intervene to secure stable access to power. Control theory studies how to intervene in such systems to ensure their functioning. Understanding how the system works, that is, a causal understanding, is crucial to be able to intervene and create the desired outcome.

"Causal methods are important within many scientific disciplines. In this project, we study systems that evolve over time and develop new methods to learn about cause and effect. It's complicated to answer causal questions and often passive observation will not be sufficient. In some applications, e.g., in many control systems, it's possible to experiment. This can allow us to give answers to causal questions even in complex systems and this project will advance our understanding of this. We will apply the new methods to control systems in particle accelerators that are used in experimental physics."

## Project details

**Project period:**

2021 - 2023

**Support:**

Independent Research Fund Denmark - International Postdoc Grants

**Contact:**

Asbjørn Christian Nordentoft

Kevin Aguyar Brix

Søren Wengel Mogensen