# Three PhD students from the Department of Mathematics are attending the 3MT

In the 3 Minute Thesis Competition (3MT), PhD students present their research project in only three minutes with just one static slide for an audience of non-specialists.

The three PhD students are Ragnhild Ørbæk Laursen, Emil Dare and Nikolaj Nyvold Lundbye.

The three PhD theses presented:

## Ragnhild Ørbæk Laursen

### Title: Are you confident in your data analysis?

"It is very important to make the right analysis of your data to obtain the correct conclusions. In my PhD thesis, I have looked at one of the most popular methods to analyze big data called non-negative matrix factorization. This method is, for example, used to analyze big collections of DNA from different blood samples and determine whether patients have different cancers in their body. One mostly unknown fact is that this model can give different solutions for the same data set. This is not because the model is used in a wrong way, but simply because the model contains a lot of free parameters. It is important to know all of these solutions to be sure to make the right conclusions about your data. One solution could, for example, tell you that you have a certain cancer in your body and another could tell you that you do not. Only knowing one of these solutions would be critical, but knowing both will inform you about the possible instabilities in the solution. In my PhD thesis, I have developed an algorithm you can apply to your solution, and it will let you know of all the other solution that exist, if there is any."

## Emil Dare

### Title: Stereologically Adapted Crofton Formulae

"In my PhD thesis, I delve into stochastic geometry, where I derive Crofton formulae. In a beautiful interplay between geometry and integral theory, these formulas connect geometric properties of an object with intersections of the object, which have a lower dimension. In the construction of these formulae, multiple branches of mathematics are intertwined, making the subject highly intriguing. The derived formulas can also find practical applications. They give rise to methods for estimating the properties of 3-dimensional figures from, for example, a collection of 2-dimensional sections. This is particularly relevant in microscopy, where cells are observed in a microscope. The 3-dimensional cell is not directly visible; instead, 2-dimensional sections are observed. It is precisely the Crofton formulas that enable the estimation of properties such as volume, surface area, and general orientation of cells based on a collection of such 2-dimensional images.

## Nikolaj Nyvold Lundbye

### Title: How many holes should your data set have?

“In my PhD thesis, I am studying collections of points which are scattered completely at random. The points could represent anything - cells in the blood, trees in a forest or stars in the universe. Rather than being interested in the precise location of the points, I am more interested in their overall shape, for example, if they form a circle or not. In particular, in the field of”Topological Data Analysis" that I am working in, we are interested in the number of holes. I am currently working on proving that in this completely random setup, how many holes would we expect and where are the holes likely to be. This result can be used to determine confidently if points seen in a microscope or on a satellite image are random or if there is some underlying structure. Or in other words, if you’re looking at a random part of the universe or not."