Young Mathematicians Get To Know the Professional World

Ecosystems are composed of animate and inanimate components whose populations form a community. But what laws determine the dynamics of these fascinating systems? And how do predator and prey species interact with each other?
 

Simualtion of Populations

In this project, students consider mathematical models to describe the evolution of populations. Starting from intuitive growth models of a species to the interactions of a predator-prey relationship, they try to describe, understand and simulate the relationships on the computer.

»The big difference to the approach of problems in school is that you have a topic and a question that you deal with the whole time for a week,« Lisa Fesser tells us enthusiastically. »We try to find a more and more precise solution together. At school, the context is not always clear; sometimes it's just numbers and calculations. I like that too, but I find it super interesting when the connection to reality is recognisable.« The 18-year-old is in the twelfth grade in Trier.

The group is supervised by our colleagues Tim Nicolai and Davide Manfredo from the division »Mathematics for Vehicle Engineering« at Fraunhofer ITWM.

At some stage in life, many people reach the point where they start thinking about buying and selling real estate. But what is currently a realistic selling price or purchase price of a similar house in a similar location?

Meanwhile, there are various websites to answer this question. First, one gives details of location and amenities, and then the website gives an estimated appropriate purchase price. But how is the displayed price calculated? Do the queried features have any significant impact on the price at all, or can we possibly find a model that is just as good, but simpler?
 

Models for Buying and Selling a House

In this project the students ask themselves the following questions:

• What are so-called linear regression models anyway?
• Is it possible to calculate the best choice of model parameters by hand without using a computer?
• How can I check if my model is a »good« model?

»I always wanted to work on a bigger project and have computer science involved right away - that has already been pretty fulfilled,« Benedikt Graeve tells us in the interview. »We're also a really good group, where everyone is committed, talks to each other and is really motivated. It's just a great experience.«

The group is supervised by Chiara Fendt from the working group »Statistics«, department »Mathematics« of the RPTU in Kaiserslautern.

Who doesn't know the situation? – You drive your car too fast and get a speeding ticket. Or you take a photo of a bill and the banking app reads out the data for the transfer. Or you're writing on a tablet and it translates what you've written into text.

All of these situations (and there are certainly many more) have one thing in common: letters and numbers must be recognized from a given image.
 

At the End of the Learning Process We Get an AI

In this project, the students will learn how to teach the computer to recognize and correctly assign letters. The goal is to program a small AI for text recognition that can read the letters from simple pictures. 

»We are working on the recognition of text and images - specifically images of flashers,« reports Noah Gomolka. »We are developing a small AI of our own. I had thought about how to approach something like this before, but I didn't think it was possible. But when you work together as a team, you can achieve so much in a short time.« The 17-year-old comes from near Mainz and already programmes smaller projects himself in his free time.

The group is supervised by Eva Schmidt from the working group »Optimization« of the department »Mathematics« of the RPTU in Kaiserslautern.

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We live in an abundant society. But what happens when a resource or a self-image, such as access to WLAN, is no longer available to everyone without restriction? In such a situation, a division must be found that not only takes into account wishes and needs for usage times, but is also as fair as possible.

A real-world issue in which the same questions play a major role is the work scheduling of nurses and other shift workers. Here, preferences regarding time off on weekends or holidays or work during certain shifts must be taken into account.
 

Defining Fairness Mathematically

In this project, students will explore how to define fairness mathematically, how to calculate a fair schedule using an optimization problem, and how to use game theory methods to facilitate a fair exchange. Additionally, we will write a small program to help us with this division process.

»The teamwork was all about what can be a fair distribution in the first place. One of the biggest challenges here was to define what fair means and to come up with a sensible solution,« says Jakob Kaiser. »We tested and simulated different definitions of fairness in different systems to then see what is the fairest system.« The 21-year-old lives in Koblenz and is particularly interested in studying at RPTU Kaiserslautern.

The group is supervised by Mareike Witzig from the division »Optimization« at Fraunhofer ITWM. She is also currently a Felix Klein scholarship holder herself and studies mathematics at the RPTU Kaiserslautern, so she can also answer questions from the teenagers about this.