Hours of contemplation, approaching a solution in small steps, and a healthy scepticism towards computer-generated results. In these guest articles, two mathematicians at the University of Münster, Dr. Franziska Jahnke und Dr. Mira Schedensack , explain what mathematical research means to them and what the aim of this research is. Both undertake research and teaching at the University of Münster within the framework of the MATHRIX junior professorships, which were established in 2017 to promote gender equality in mathematics.
"Moments of true insight make all the apparently fruitless hours worthwhile"
For me, mathematical research means staring for hours into space, lost in thought, or repeatedly scribbling on a sheet of paper. Of course, I also read mathematical papers, discuss with my coauthors or get my ideas down on paper - but the centrepiece of my work is nonetheless these hours of contemplation. These are often very frustrating, and my thoughts go around in circles ... However, they serve to make those rare moments even more valuable in which an insight starts to grow. Then the trembling uncertainty as to whether the thought will withstand all the necessary checking which now follows! These moments of true insight make all the previous hours worthwhile, even if they seemed fruitless before, and they always call for more.
Since I was a child, I have loved numbers, puzzles and riddles. The good weather conditions in southern Germany led me to study mathematics in Freiburg. Here, I came into contact with the abstract world of pure mathematics which has not let go of me since. Most of all I was intrigued by algebra, the study of solving equations in a variety of number systems, and by mathematical logic, a foundational mathematical discipline. After graduation, I did my doctorate in model theory, an area of mathematical logic, at the University of Oxford. Since I had a scholarship from an EU network, I was able to travel to conferences all over the world, meet model theorists and learn about their research. But all the travelling can only complement -and is no substitute for - the hours spent in deep thought. I moved to Münster after my PhD, where I now teach and learn, initially as an assistant and now as a junior professor. I also have a two-year-old daughter - and this has taught me it is also possible to do mathematics while sitting on a bench in a playground!
In my work I’m studying a certain kind of number systems stemming from algebra (valued fields) with model-theoretic methods. In particular, I’m working on questions like “When does the arithmetic of the field describe the valuation, i.e. which valuations are definable?” and “Is there a connection between the combinatorial patterns which are encoded in the arithmetic of a field and properties of the definable valuations on it?”.
Regarding the second question, there is a big unsolved conjecture by Shelah (a famous Israeli logician) which, it seems at the moment, can be neither proven nor disproven, but which inspires and motivates my research.
Special cases of the conjecture and further evidence for it have been proven in the last few years, some thanks to my work. This nourishes the hope that many more hours spent staring into space, and many more small steps, might give rise to a solution after all.
"There are a lot of applications for which our research is relevant"
How does a pocket calculator calculate the square root of 2? Even a simple task such as this requires the use of numerical algorithms. And practically most applications are unthinkable without these algorithms. But the extent to which the result can really be trusted is one of the fundamental questions of numerical mathematics - and this means of course that it is all the more important for more complex problems. My field of work is the numerical treatment of partial differential equations such as occur for example in describing phenomena in natural sciences or engineering. One example of such phenomena is the distribution of heat in a certain space. The temperature in a certain place depends on the temperature in the whole neighbourhood. Because of this connection, it is generally not possible to calculate an exact solution. As a result, approximate solutions are arrived at by using numerical algorithms for differential equations.
I have been working on the development and analysis of such methods since the time I was doing my PhD at Humboldt University in Berlin. Of course, such algorithms have to meet certain criteria. For example, the amount of time and effort needed when implementing in a computer programme should not be excessive. As computing capacity is limited and an economical approach should be taken as regards available resources - for example, the memory or power consumption of high-performance computers - the focus of my research is also on the development of efficient numerical methods.
Since October 2017 I have been working at the University of Münster’s Institute for Analysis and Numerics, where, together with my colleagues, I am undertaking research into these methods - in order, for example, to find answers to the following questions: If an approximation is given, is it then possible to estimate the error without knowing the exact solution? And is it possible, without knowing the exact solution, to determine how to select a resolution in order to find the optimum approximation to the solution? To a lot of people, this initially sounds very abstract, but there are in fact a lot of applications for which our research has practical relevance - for example, if you want to simulate the deflection of a bridge with a certain load in order to calculate the limits for critical loads.
Many algorithms which are used specially in complicated problems in industry and in other applications - for example, in weather forecasting - can’t as yet be analysed well with today’s level of knowledge. For this reason, it is always important to retain a healthy scepticism as far as results provided by a computer are concerned. In such complicated problems, and for want of any mathematically provable error estimates, heuristic arguments are often used. This can lead to numerics being difficult to categorize at the beginning of any mathematics degree course.
In my degree course, I too initially did numerics as a compulsory topic. In the process I discovered that the focus was on mathematical proofs. The provable results in numerics are often restricted to prototypical equations - for which, however, error estimates can be mathematically deduced instead. It is these particularly theoretical aspects that I am working on in my current research. The mathematical proof that I am focusing on in my work leads not only to rigorous statements, but also to a growing understanding of the methods under consideration which still have to be developed.
One example of my research is a current collaboration with engineers in which we are attempting to deal with certain models from elasticity theory using mathematically substantiated algorithms, in the process of which we are developing the mathematical foundations for the simulation of these problems.