My field of work within mathematics is algebra, or more precisely group theory. Already in a basic algebra lecture, one usually gets to know the concept of a simple group. These are the elementary building blocks (or "atoms") that make up all finite groups. As it has turned out in the course of development, such groups are anything but simple in the sense of "easy to deal with", but the classification of finite simple groups is one of the most complex theorems in all of mathematics ever---with a proof estimated at about 10000 pages (ten thousand, the number of zeros is correct!).In this section, I deal with a particular class of finite simple groups, the so-called Chevalley groups or Lie-type groups. These are something like algebraic analogues of the real or complex Lie groups. Instead of differential geometry, algebraic geometry is needed to study them, over algebraically closed bodies of positive characteristic.
What fascinates you about mathematics?
What fascinates me about it is that, on the one hand, different mathematical theories converge here and, on the other hand, computers can also be used profitably, whereby the goal is to calculate exactly or symbolically. With the help of modern computer algebra systems (such as GAP, Magma, Sage, ...), one can, for example, systematically search for counterexamples, collect material to support conjectures, or even come up with the right idea for a theoretical argument in the first place. Methodologically, this often has an experimental character, i.e. something that many do not actually associate with pure mathematics in the classical sense. For many years, I have been involved in the development of such a system ("CHEVIE") and actively use it in my research.
I also find it remarkable that such partly very abstract, supposedly application-resistant theories have concrete applications after all. Just think of the development of certain codes for data transmission, some of which are interwoven with finite simple groups in a very interesting way (keyword: Golay code).
And finally, a word about mathematics in general. I had the chance to live and work in France and Scotland for several years, which I found very enriching. It is often said that the language of mathematics is universal and does not depend on whether it is expressed in, say, French, English, Chinese, Swabian or any other language. In fact, or so I have learned, there is also something like a language- and culture-independent, i.e. literally borderless international community of mathematicians---which is actually a beautiful, remarkable thing.
Prof. Dr. Meinolf Geck
Vice-Dean of the Department of Mathematics
Chair of Algebra
Institute for Algebra and Number Theory