- What is Mathematics? - An Approximate Definition
- Why should I study Mathematics?
- Why should I study Mathematics in Cologne?
What is Mathematics? - An Approximate Definition
Many people have a distorted conception of mathematics due to unpleasant schoolday
memories of rote learning and stolid calculating. However, both in practice and in theory
quite different thing are expected of the mathematician. The job of the mathematician
can be best described as the "analysis of structures" and the "development of adequate models"
which in turn make it possible to solve complex problems "for example, through the use of computers).
For this purpose mathematicians have developed a special language (which continues to develop) that aids in
the unambiguous description of factual situation. On the one hand, it is more flexible and complex
than a computer language, on the other hand, in contrast to human language it avoids the discussion of
indistinct things. In this way the mathematician tries to reduce problems to their very essence.
The strengths of mathematics and thus of mathematicians are universality and flexibility. It is generally
quite difficult for new students, especially with today's often quite superficial preparation at school,
to become accustomed to this precision. They are often quite surprised that they are required to deal
actively with mathematics from the very beginning. Some new students may say to themselves, "I'll take a look at it"
and take their seats in the lecture halls as "observers". It has been our experience that this "wait-and-see"
attitude bears the seeds of failure. Just as one can succeed in sport only through active training, the study
of mathematics to a large part consists of independent and individual mathematical effort. You may say,
"That really sounds like a lot of work". Yes, we heartily agree, but we have experienced again and again
how fulfilling it is to have really understood a basic mathematical concept and thus to be able to solve problems.
Often it is the suprising interconnections that are really compelling aspects of mathematics. The following example illuminates just such a surprising interconnection.
The Radon Transformation:
At the beginning of the previous century the Czech mathematician Johann Radon (1887-1956)
studied the following question which on the surface seems to be rather theoretical: Given a function f, suppose that we know the integral f along every straight line G. Can we then reconstruct the function f.
Radon's mathematical answer to this question is known today as the Radon Transformation. It is the theoretical basis for modern computer tomography.
Roughly spreaking, the computer tomography device sends X-rays from all directions through the body and measures the rays leaving the body. The result is essential the integral of the mass density over all straight lines. Using a numerical variant of the Radon transformation it is possible to determine the mass distribution and thus the internal structure of the body and make it visible on the computer monitor.
(J. Radon: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. Nat. kl. 69 (1917), 262-277 nachgedruckt in P. Gruber et al. (Ed.) Radon, Johann: Gesammelte Abhandlungen. Verlag der Österreichischen Akademie der Wissenschaften, Wien; Birkhäuser, 1987.)
This example is informative in several ways. Radon had not the least inkling of computer tomography, but this motivation for contemplating
the problem was basically mathematical. The mathematical question that initially seemed completely theoretical found a very useful application
more than half a century later. This is a wonderful counter-example to the widespread shot-term const-benefit approach to science.
Similarly this example shows that the differentiation between "pure" and "applied" mathematics ist pointless; one could say said that there is only "good" mathematics and "bad" mathematics. Conversely, one could also say that many famous mathematicians have repeatedly drawn their motivation from very tangible problems.
It was a long journey from the theoretical Radon transformation to its implementation in the computer. An entire branch of mathematics, numerical
mathematics, is involved with the concrete implementation of theoretical results for which the computer is an indespensible tool.
A. M. Cormack and G. N. Houndsfield received the 1979 Nobel Prize for the introduction of computer tomography in medicine. As we know, there is no Nobel Prize for mathematics but there are the Fields-Medaille and the Abel Prize.
On the occasion of the millenium change, a panel of experts at the Clay Mathematics Institute (USA) selected seven millenium problems that involve basic questions in mathematics and which have resisted solution over a longer period of time. These problems show that, contrary to popular opinion, many basic mathematical problems cannot be solved with computers, however powerful they may be, but rather with human creativity alone. Three of these problems will be illuminated here. Look here for later developments, including the proof of the Poincaré conjecture.
There are more possibilities for browsing in mathematics here.
Why should I study mathematics in Cologne?
At our institute you will find a broad range of course offerings.
You will be supervised here by faculty with international reputation. Many colleagues already had professorial position before coming to Cologne. In addition, noted research prizes (Leibniz Prize, Hess Prize), prizes for young academics (Klaus Liebrecht Prize, University Prize), the Albertus Magnus Prize as well as the Alred Krupp Prize have been awarded to members of our institute. Moreover our institute traditionally has maintained good contacts with local industry. This is emphasised by teaching contracts that have been awarded to experienced practitioners.
The University of Cologne was founded in 1388 and, Heidelberg, is the second oldest university in Germany