Contact: hypatia-math-csuni-koeln.de

Here we share the self-written experience reports of female students who have completed their project as part of HYPATIA.SCIENCE. We not only want to draw attention to ourselves, but also motivate more students, especially female students, to become part of such a project or to consider a doctorate for themselves.

2025
Janina Tikko - Swarm-based methods for non-convex optimization

"As part of the HYPATIA.SCIENCE project, I pursued the topic from my bachelor's thesis scientifically under the supervision of Prof. Angela Kunoth. I dealt with a very current and new field of research: "Swarm-based gradient methods for determining global minima of non-convex functions". The global optimization of non-convex functions is usually difficult because such functions often have many local minima (or maxima). As a result, the usual gradient methods get "stuck" in such minima. The swarm-based gradient method uses a swarm of agents that communicate with each other to overcome these local minima. In order to better understand the method and the swarm aspect, I first familiarized myself with the Cucker-Smale model. I then concentrated on other aspects of the process. For example, under my supervision, several other Bachelor students tested the method in two spatial dimensions and tried to optimize certain parameters. The similar swarm-based method with random descent directions was also examined in more detail. Despite promising results, doubts arose regarding the complexity of the random descent directions, as numerous householder reflections have to be applied. In the further course of the project, I therefore became more involved with the swarm-based gradient method in its original form. In particular, I created a poster for the 25th Rhine-Ruhr Workshop in January 2025 in Bestwig in collaboration with Moritz Danzebrink. My work during the project was very free. I was able to decide for myself which direction to take. I really enjoyed this type of scientific work and the topic. In addition, the research and collaboration with the Bachelor's students has now led to five further questions. I will therefore continue to work on this topic as part of my Master's thesis."

2023
Hannah Windgasse - Starting IPS: Implementation of a prototype for course planning

"As part of my research position at HYPATIA.SCIENCE, I was given the opportunity to get to know scientific work better. This research, which lasted six months, was a lot of fun and gave me an insight into scientific procedures and processes that you don't get in everyday studies. In our project, which I took on together with my supervisor Dr. Vera Weil, we worked on the idea of developing a prototype of a program that helps students to plan the course of their studies. This is because every student has to think again and again during the course of their studies about which courses they want or need to take in the next and subsequent semesters. In doing so, students have to consider the requirements and specifications of the examination regulations, but also their personal goals. These include, for example, the questions: "Which modules should be taken or even passed before others?", "Which modules are offered in which rotation?" or "How should I choose after failing this exam so that I can still complete my studies in seven semesters?". The prototype is designed to answer precisely these questions. It is intended to replicate the user's previous course of study and suggest possible plans based on the examination regulations and personal entries and choices. The aim is to provide students with assistance in the general planning of their studies and to help them better assess their own course of study. We have initially concentrated on the study of business mathematics, although the long-term goal is to make the program applicable to other degree courses. For me, the insight into everyday research was a comprehensive and great experience. I was able to gain many instructive insights into the content and had great fun working on the subject. Whether independently or in collaboration with my supervisor, I was also able to overcome unexpected problems and difficulties. These experiences gave me even more insight into the real everyday life of research. During my time on the project, I was always happy to work on the implementation. Admittedly, I was always itching not to close my laptop, but to complete or supplement one or two methods. All in all, I can only recommend every student to take this opportunity to gain experience and get a taste of "research air" as part of the HYPATIA.SCIENCE project."

Anahita Pouralijanki - Representation theory of finite-dimensional algebras

"In 2023, I started my research project as part of HYPATIA.SCIENCE under the supervision of Dr. Severin Barmeier and Prof. Dr. Sybille Schroll. My research focused on the deformation of gentle algebras, a special class of algebras that play an important role in representation theory and mathematical physics. One of my tasks was the in-depth study of the Auslander-Reiten quivers (AR-quivers) of these algebras. These diagrams are essential tools in representation theory and provide deep insights into the structure and representations of the algebras. By constructing surface models, I was able to make the AR-quivers more visible and accessible. In the course of my work, I learned three different methods to construct these surface models. The first method came from Karin Baur and Raquel Coelho-Simoes and offered an innovative approach to visualizing the structures. The second method was developed by Sebastian Opper, Pierre-Guy Plamondon and Sybille Schroll and provided a dynamic perspective on the deformations. Finally, I investigated a third method, which is a combination of the first two approaches and is based on the work of Wen Chang Through the close collaboration with my supervisors and the intensive engagement with the subject matter, I gained valuable experience in scientific work. It was particularly enriching to have the opportunity to explore new theoretical approaches myself and to develop innovative models. The HYPATIA.SCIENCE project was an extremely instructive and inspiring experience for me. It not only helped me to expand my mathematical and methodological skills, but also encouraged me in my decision to pursue a doctorate. The opportunity to work independently and contribute to the further development of such a fascinating field of research has further ignited my enthusiasm for science."

2022
Kira Hoffmann - Optimal management of the insurance surplus process through reinsurance

"Under the supervision of Prof. Dr. Dr. Hanspeter Schmidli and Dr. Leonie Brinker, I had the opportunity to gain my first insights into everyday research as part of a HYPATIA.SCIENCE position from August 2022 to February 2023.
During the first few weeks, my task was to familiarize myself with Dr. Leonie Brinker's dissertation and to understand the methods she had developed. She solved a stochastic control problem, which is formulated as follows: the subject of the analysis is the surplus process of an insurance company that pays out dividends according to a barrier strategy, i.e. whenever the surplus exceeds a certain limit q>0, all capital above this limit is paid out as dividends. Furthermore, the so-called drawdown process is considered, which measures the distance between the dividend barrier and the current surplus. A drawdown that is greater than a specified level d, where 0

2021
Chong-Son Dröge - Numerical spectral analysis on non-equilateral metric graphs

"Thanks to the HYPATIA.SCIENCE program of the University of Cologne, I had the opportunity to get an insight into scientific work in the winter semester 2021/2022. As part of the program, I worked on the topic of numerical spectral analysis on non-equilateral metric graphs under the supervision of Anna Weller. My interest in this topic was motivated by the research project "Neurodegeneration Forecasting - A Computational Brainsphere Model for Simulation of Alzheimer's Disease", funded by the Excellence Initiative of the University of Cologne. This research project was initiated in 2017 by Prof. Dr. Angela Kunoth and Prof. Dr. Yaping Shao from the Department of Meteorology and Prof. Dr. Alexander Drzezga from the Department of Nuclear Medicine. As part of this research project, the spread of two proteins in the brain that play a key role in Alzheimer's disease is being investigated. One of these proteins is the so-called tau protein, which spreads along the brain network, accumulates in the neurons and thus influences their function. A metric graph is used to model the spread of the tau protein as a reaction-diffusion process. Neurons form the nodes of the graph, the connections between the brain regions represent the edges of the graph, whose lengths describe the distance between the brain regions. The task is now to numerically determine the eigenvalues and eigenvectors of a suitable second-order differential operator on such a metric graph. If all edges have the same length, i.e. a so-called equilateral graph exists, the spectrum of the differential operator can be determined completely. As part of the HYPATIA.SCIENCE program, we have dealt with the more general case and considered so-called non-equilateral graphs, in which the edges do not all have the same length. At the beginning of the project, I familiarized myself with the basics and the already elaborated approach for equilateral graphs. This work consisted of familiarizing myself with the relevant literature and understanding the routines used for implementation. In weekly meetings, we presented our results, discussed problems and assigned new tasks. Once we had internalized the basics, we developed and implemented an initial approach for calculating the eigenvalues for the non-equilateral case. However, we realized that this approach leads to a highly non-linear problem, which can no longer be solved completely or only very inefficiently by classical numerical methods. After a joint meeting with Prof. Dr. Angela Kunoth, in which we discussed our problem and reflected on our findings, we came to the conclusion that a new approach should be chosen. This consisted of utilizing our findings from the equilateral case. Our new idea proved to be promising, so we are now pursuing it further. Participating in the HYPATIA.SCIENCE program was a great enrichment for me. I am very pleased that I can continue to work as a research assistant in Prof. Dr. Angela Kunoth's working group after the end of the program and that I can continue to work on this research project with my mentor, Anna Weller. At the end of my Bachelor's degree, I investigated numerical methods for solving the Schrödinger equation in my Bachelor's thesis with Prof. Dr. Angela Kunoth. This was tailored to my studies with a minor in physics. After that, I hadn't initially considered doing a Master's degree. But thanks to the research project, the HYPATIA.SCIENCE support and the planned close collaboration with Anna Weller, I am now giving the "Master's" option a chance after all. I would therefore like to take this opportunity to thank my mentor Anna Weller in particular. Through her, I was able to gain a good insight into the world of research as part of the HYPATIA.SCIENCE program. Working with her has awakened the joy of research in me. Over the last few months, I have realized that this topic is not only very relevant, but that our approach is at the forefront of international research. In the coming months, we want to prepare a first joint publication for a renowned journal. During my work, I have learned that not every approach is effective, but you can still achieve a result if you stay on the ball long enough. The weekly meetings with my mentor always helped me a lot personally. In line with Werner Heisenberg's words "Science is born in conversation", I realized during this time that discussions and exchanges are a large and important part of scientific work."

2020
Svenja Griesbach - Pandoras Box problems on metric spaces

"In May 2020, I was accepted into the HYPATIA.SCIENCE funding program under the supervision of Dr. Kevin Schewior. We had originally chosen the Pandoras box problem on metric spaces as our research project. The general Pandoras Box problem is defined as follows: There are n alternatives (boxes), each with a known non-negative price and an unknown non-negative profit (contents of the box) that comes independently from a known probability distribution. The agent sequentially opens a certain subset of the boxes, i.e. she pays the corresponding price to see the profit of the box. The goal is to find a strategy that maximizes the maximum profit found minus the costs paid in expectation. Weitzman showed in 1979 that the optimal strategy has a simple structure and can be computed efficiently.
Since the model has become very popular in theoretical computer science in recent years, we wanted to look at a generalization of the problem. Here, the boxes are located at different known positions in a metric space. Starting and ending at a fixed position, the agent now moves in the metric space, replacing the opening cost by the distance traveled. Unfortunately, this problem turned out to be quite difficult on closer inspection, which is why we rewrote the problem a little after a few months. However, as this is still our current research project, I don't want to go into more detail about the problem and our current research status.
Even though my official funding period has been over since the end of October, I still meet with Dr. Kevin Schewior every week to continue researching the problem. Since September, Dr. Felix Hommelsheim has also joined our small research team and is actively supporting us. In our weekly meetings, which last about two hours, we report on our new ideas for solving the problem and try to work out possible approaches together. I therefore pursued the same tasks and goals as my supervisor throughout the entire time. This gave me a very good insight into the everyday life of a doctoral student, even if you don't make any progress for a few weeks. Nevertheless, I really enjoy the research and especially the intensive teamwork, and the Hypatia.Science program has certainly helped to make research more accessible to me."

Antonia Thiemeyer - Local optima of the Gaussian mass

"As part of the HYPATIA.SCIENCE project at the University of Cologne, I had the opportunity to gain an insight into scientific work at the university for six months. The aim of our project was to find local optima for the Gaussian mass function over lattices. In contrast to the already investigated local minima of this function, there are no concrete results about the (local) maxima yet. By identifying grids by their Gram matrix, we were able to define the function over a Euclidean space and reduce the optimization problem to the evaluation of the gradient and the Hessian matrix. Finally, using techniques for theta series and modular forms, we were able to establish an expression for the Hessian matrix and develop methods to study the definiteness of this matrix. The assumption made at the beginning of the project about a possible maximizer turned out to be wrong in this step, but we were able to make new assumptions based on the results. It can be said that the methods developed could be used to find actual maximizers.
My tasks during the project were very varied. At the beginning, I had to familiarize myself with all the basics on my own. To do this, I studied the literature on grids and module shapes and presented my results in two lectures. Everything I had learned so far could be applied to the Gaussian mass function. After the initial theoretical results, we first examined the Hessian matrix numerically. We used the Sage computer algebra system for this. I was able to familiarize myself with it quickly, even without any previous knowledge. Finally, I had the task of compiling all the results and giving a presentation in the advanced seminar.
For me personally, taking part in the Hypatia.Science program was a great experience and an enrichment. I learned to work independently and to stay on the ball even when I had difficulties in the meantime. If I had any questions or problems, I could always turn to my supervisor, who always supported me promptly. In addition, I now have a more precise idea of scientific work. I am also very pleased that the results of this project will be used for further work. I am curious to see what results will emerge from further research. Overall, I leave this project with a very good feeling and could imagine doing a doctorate."

In the course of this project, a research paper was written in collaboration with supervisor Arne Heimendahl, among others: "Critical even unimodular lattices in the Gaussian core model"

Anna Nechamkina - Recurrence and transience in Boolean random graphs

"From June to November 2020, I took part in a research project as part of HYPATIA.SCIENCE and worked as a research assistant at the Chair of Stochastics and Probability Theory under the supervision of Prof. Dr. Mörters. My project topic was about recurrence and transience in Boolean random graphs. The question was whether a random wandering on a Boolean graph model would return to its starting point infinitely often, i.e. would be recurrent, or not, i.e. would be transient in this case. A Boolean random graph is a graph whose nodes are Poisson distributed and are assigned a random radius. The edges are formed on the basis of these radii, whereby two nodes are connected if the spheres of the two points intersect with the respective radii. The aim of the project was to prove the conjecture that recurrence exists in dimensions 1 and 2.
A paper in which the assertion had already been shown for a similar graph model served as the basis. My task was to understand the proof techniques in the paper and transfer them to my model. The second part of the project was to investigate transience for higher dimensions. Unfortunately, the university was closed for the entire period due to the coronavirus. As a result, it was not possible to meet at the Institute of Mathematics and work on the project on site. Nevertheless, there was a regular - almost weekly - exchange with the members of the teaching group about progress and difficulties that arose during the project.
During this period, I was able to get a good idea of how scientific work is carried out and what a doctorate involves. It involves a lot of independent work when you read up on new problems and try to pursue your research question and find a solution. However, it is just as important to regularly exchange ideas with the group, as this often leads to new ideas and approaches to problems that you are currently stuck on. I was able to gain a lot of experience and am now more aware than before of how a promotion could work."

Pauline Scharf - Local-harmonic dimensional forms in hyperbolic space

"The aim of the project was to construct local-harmonic dimensional forms on the hyperbolic space of dimension n. These are functions which are defined on higher dimensional generalizations of the complex upper half-plane, have infinitely many symmetries under Möbius transformations and are harmonic except for certain discontinuities. For the construction we have used an approach that utilizes the theory of singular thetalifts. The idea is to use a suitable integral operator to define a mapping from classical modular forms on the upper half-plane to locally harmonic dimensional forms on the hyperbolic n-space.
At the beginning of the project, I familiarized myself with the necessary basics. We then held regular Zoom meetings to discuss our results so far and the next steps, and to divide up the resulting tasks. During this second phase of the project, I either transferred proofs that had already been carried out for similar Thetalifts to the Thetalift we needed or carried out my own proofs and calculations. I always had a contact person who helped me with questions or problems.
Working on the project gave me a good insight into scientific work under coronavirus conditions. Unfortunately, this semester I only got to see how scientific work on a joint research project works under "normal" conditions through stories. Nevertheless, after completing the project, I have a good idea of what a doctorate would involve for me and I could well imagine doing it, but not immediately after my Master's degree, but only later, when working on site and direct exchange are possible again without any problems."

Anouk Weber - Models of mathematical epidemiology and scientific machine learning

"In my research position as part of HYPATIA.SCIENCE, I worked on the use of a scientific machine learning method for parameter identification in mathematical epidemiology models. I extended an already existing code of the working group of Professor Klawonn. The original code used Physics Informed Neural Networks (PINNs) and compartmental models to calculate the contact rates in a society during a pandemic. The differential equations resulting from the compartmental models were integrated into a neural network. In this code, it was assumed for the sake of simplicity that everyone in society has the same contact rate. My task was then to incorporate further differential equations into the code so that two groups with different contact rates could be taken into account. I carried out the calculations on a graphics card cluster.
Not only through my own work during the project, but also through the exchange with my supervisor, I gained a better insight into scientific work and the tasks and activities that one has during a doctorate. I enjoyed working on such a topical subject. For me, the HYPATIA.SCIENCE project was a very instructive experience, as I rarely have the opportunity to research new approaches myself during my studies. Instead, it is usually a matter of following theories that have already been researched and applying methods that have already been researched. The experience of working scientifically myself helped me a lot in deciding whether a doctorate was an option for me."