What are your tasks in the department?
I'm at the Lehrexportzentrum Mathematik, short LExMath, which provides the required mathematics for engineering students and which coordinates various things for the math courses for other study paths. Moreover, I'm helping with the training of math teachers.
Every now and then, there is space for lectures on an algebraic topic.
Is it possible to ask you to supervise a bachelor or a master thesis?
Yes. As long as you are ready to try something new.
This also holds for me: during the projects I was involved in, I have learnt a lot.
Is Mathematics for Engineers interesting for you as a lecturer?
Such a course deals with mathematics developed in the period from around 1750 to 1900, of course in a version prepared for today's requirements.
In this period, tools were developed that one can enjoy again and again: the determinant, complex numbers, the Hessian matrix, Fourier series... In order to develop something comparable today, mathematicians have to make an effort.
What are mathematicians supposed to develop?
Mathematics means moving from a concrete level to an abstract level, reasoning and calculating on this level, and then applying the results back to the concrete level.
For example, the introduction of coordinates in the abstract model IR 2 enables the computational treatment of geometric problems.
What you see as concrete and what you see as abstract depends on your point of view. The procedure can therefore be iterated. For example, IR 2 is a concrete vector space.
So mathematicians create abstract models in which you can move around free of concrete baggage and from which you can apply results to the concrete level. Create, expand and apply.
That's pretty general. And what is your favourite occupation?
Dealing with things that give the impression they are waiting to be treated. Such as, for example, the n-complexes of Veronika Klein, which generalise the usual complexes of Homological Algebra, representing the case n = 2.
When moving around in such a context for a while, it appears increasingly concrete over time. The trust in interconnections increases, the need for constant reassurement by examples decreases.
How can a practical person be convinced of the existence of infinitely small quantities?
If you need a snapshot, you shouldn't use too long an exposure time. An infinitely short exposure time would be best, just longer than zero.
If you want to know the condition of an elastic workpiece at a point, it is not enough to look at this mere point, as it does not have a contact surface. Instead, you can cut out an infinitesimal cube and look at its state of tension.
Later, Cauchy, Bolzano and Weierstraß reinterpreted these infinitely small quantities as finite, but arbitrarily small quantities, in order to fit them into the existing framework of mathematics. They have thus provided a solid foundation for the intuitive understanding of infinitesimal calculus of Leibniz and Euler.
In practice, familiarity with this foundation helps to decide whether, in a given situation, a mathematical tool can be used or not. After all, the manual is written in that language.
Do we really have π = 3,2 ?
You're referring to the cartoon on my door that makes fun of an Indiana bill of 1897. In fact, the number π is one of the oldest and most enigmatic beasts in mathematics. When I recently had the opportunity to look at the current state of the argument for the transcendence of π, I was amazed.
Thank you for the interview.
Priv.-Doz. Dr. Matthias Künzer
Stellvertretender Leiter des Lehrexportzentrums Mathematik