Lattice points in circles
A classic way to count lattice points in circles is to associate the lattice points to an angular figure whose area and other properties can be compared with a larger and a smaller circle.
Photo: Kristian Holm

Coincidence or pattern?


Kristian Holm’s doctoral thesis deals with distributional questions for different types of mathematical objects. The two main types are lattices and zeros of L-functions, both of which connect to number theory, among other things.

In number theory, it is usually not the distributions of objects that are studied, but the objects themselves. The point of taking a step back is to get a larger overview and discover patterns that cannot be seen at a detailed level. These patterns, in turn, can reveal hidden secrets behind the objects that are studied. An early example of this was the non-trivial zeros of Riemann zeta function, where it was discovered that the distribution appeared to come from random matrices, a completely different mathematical area.

Kristian Holm
Kristian Holm
Photo: Setta Aspström

– The thesis contains interesting examples of a rather concrete probability theory as the objects that are studied are not abstract variables, but concrete objects. It succeeds in proving that there are patterns in the distributions, which gives you a better idea of how to think about, for example, zeros.

”Mathematics – extremely beautiful and powerful”

Two of the articles in the thesis deal with distributional questions for lattices. In the first, random variables associated with lattices are studied, namely the two concrete geometric attributes given by the lengths and angles of the lattice vectors. An interesting question is how lattices behave in relation to the space they belong to. There are many classical results here, like for example Gauss’ circle problem where you count the number of points with integer coordinates in a circle. When the circle grows, the number of lattice points is almost equal to the area of the circle, and the same is true for more general sets. In the second article a similar counting function is studied for a concrete increasing family of sets and random lattices. The main result is that the counting function satisfies a central limit theorem.

The fact that Kristian became a PhD student in mathematics is in itself something of a coincidence. In upper-secondary school he majored in languages, but it dawned on him that mathematics is extremely beautiful and powerful. Mathematics is indeed also a language and from that aspect a useful tool for other sciences, but with its built-in truths, it is much more than that. Kristian worked in a shop in Ringkøbing, and one day when there was little to do at the cash-point, he was playing with a mathematical problem. A customer saw this, it turned out that he was an Australian mathematician, and after a few more visits to the shop he said: I think you should study mathematics.

The dream of doing both research and teaching

Said and done, Kristian took his master’s degree in Aarhus and realised that he wanted to continue as a PhD student. He tried applying in Sweden as well, and got an open PhD position in Gothenburg. Number theory was something that he would like to work with since he sees it as an accessible mathematical discipline where it is easy to appreciate the results.

–  I have really enjoyed doing my PhD here. Corona was a sad interruption, but I had had time to get to know people, and was pleased to be here in Sweden where you could still meet under certain forms. In Denmark, there is often talk about how much is forbidden in Sweden, but now it was the other way around! It is good that the PhD is spread out over five years. Even if you are not actively doing research all the time, you have time to mature. Teaching has been real fun and I would like to continue doing that, I actually worked as an upper-secondary teacher for a year before I started my PhD. Now, I was responsible for a mathematics course for future primary school teachers.

Next, however, research awaits in the form of a postdoctoral position in Kiel. And this, too, comes from a lucky coincidence. Kristian was at a conference a couple of months ago when a speaker mentioned a mathematical concept that he had heard but forgotten about. He felt like searching for the term, and chose a hit a bit further down the list. On the page was a small announcement for this position in Kiel that he had otherwise not seen. There was no deadline and only an email address, and when Kristian reached out after a couple of days it turned out that they had already started going through the applicants. But if he could submit an application within 24 hours, his application would be considered. It turned out that the position had strong connections to what Kristian had worked on – and after another 24 hours he was informed that the position was his!

Read Kristian Holm’s thesis

Kristian Holm will defend his PhD thesis Limit Theorems for Lattices and L-functions on January 20 at 9.00 in lecture hall Pascal, Hörsalsvägen 1. Supervisor is Ander Södergren, assistant supervisor Michael Björklund.