Mathematics

Fields of Mathematics

What are the tasks of mathematics? That is a difficult question! Mathematics cannot that easily be divided exactly into different branches (especially if it should still be understandable for some outsider), and there ist no "typical mathematician". Nevertheless, I will try to present a rough classification of the different fields of mathematics and to discuss the job outline of a mathematician (further below):

• Algebra originally dealt with solving simple equations. Starting from the 19th century, the general study of algebraic structures (groups, rings, fields, etc.) became increasingly important.
• Geometry, which is closely related to algebra, is concerned (amongst other things) with the "classical" Euklidian geometry, but also with non-Euklidian geometries, metric properties (such as length and angle), etc.
• Fundamental for Analysis is the theory of convergence (concerning sequences and series) as well as differentiating and integrating real functions. Therefore, analysis deals with functional and differential equations, the measure of sets (length, area, volume, etc.), and other topics.
• The task of Numerical Mathematics is to solve problems approximately, which cannot be solved accurately (at least without any extravagant expenses), and to estimate the speed of convergence. Error analysis (concerning errors occuring when using computers) is also a part of this field.
• In Stochastics one would like to describe random processes with the aid of mathematical modells and then find methods in order to draw some conclusions which might be relevant in practice.

Moreover, there are a lot of application areas for which mathematics is fundamental:

• In Natural Sciences, mathematics (especially analysis) does play an important role – not only in physics, but e.g. also in biology, medical science or chemistry.
• In Computer Science, mathematics is used in order to design and analyze algorithms as well as in the field of data encryption.
• In Economics, mathematical modelling does play a major role. Furthermore, mathematical optimisation is used extensively.

After reading the short discriptions above, some may think that mathematics cannot provide results which are useful in everyday life (apart from some minor exceptions). The general impression of mathematicians which most people might have, is shown quite well by the following joke:

Two men are sitting in the basket of a balloon. For hours, they have been drifting through a thick layer of clouds, and they have lost orientation completely. Suddenly, the clouds part, and the two men see the top of a mountain with a man standing on it. They shout down to him: "Hey! Can you tell us where we are?!"
The man doesn't reply. The minutes pass as the balloon drifts past the mountain. When the balloon is about to be swallowed again by the clouds, the man on the mountain shouts: "You're in a balloon!"
Why can you be sure that this man is a mathematician?

1. He thought long and thoroughly about what to say.
2. What he eventually said was irrefutably correct.
3. His statement was of no use whatsoever.

In short: Mathematicians are considered to be queer fish whose results of research are useless for everyone but themselves. Perhaps the "prejudice" of the "absent-minded professor" is not a prejudice at all, but is rather in accordance with the truth?

Most people can imagine, what tasks an architect has to deal with, what job a carpenter has to do, and what a physician has to accomplish every day (at least roughly) – but what exactly is the task of a mathematician? Maybe a mathematician has indeed nothing "reasonable" to do?

No, quite the contrary! Mathematics plays a very important role in a lot of different parts of everday life, e.g. insurance companies and consulting agencies, but also in the IT-sector, in medical service and, finally, in astronautics. Due to the fact that mathematicians are working in all these different fields, there is no definite occupational image of a mathematician.

In some cases, mathematicians have to develop useful modells for certain issues (e.g. for stock prices, for climate or for bloodstream) in order to make some progonosis and to be able to do without (often quite tedious and expensive) experiments as far as possible. In computer science, mathematics plays a decisive role for instance at data encryption. In some line of business (e.g. in consulting agencies) a mathematician does not only benefit from his technical knowledge, but also from his skills concerning abstraction and recognition of coherences which were boosted during his studies.

You may now laugh at jokes about and for mathematicians – both jokes which make fun of mathematicians (just like the one above) as well as "insider jokes" which may only be comprehensible to someone familiar with mathematics. Furthermore, I have collected some sayings and quotations of and about mathematicians.

Personal Experiences

And why did I choose mathematics as my field of study? Actually, I have been interested in this subject ever since. My parents have always tried to support me – without putting pressure on me.

At high school, I fortunately had the opportunity to take part in the Mathematical Olympiad, a yearly competition for high school students. Over the years, I was able to improve my mathematical skills, so that I eventually managed to qualify for some international competitions:

• In eleventh grade, I was qualified for the Austrian-Polish Mathematical Competition which took part in Toruń (Poland) in June/July 1998.
• In twelveth grade, I was finally allowed to take part in the International Mathematical Olympiad (as one of six pupils from all over Austria). This competition took place in Bucharest (Romania) in July 1999.

After my experiences at the Mathematical Olympiad, the decision for the study of mathematics was quite a natural one: My achievement at the competitions and the fun we all had (especially at the preparation courses which were held every year in Raach) merely made me all the more determined to do it. As the study of mathematics at the Vienna University of Technology has the reputation of being more application-oriented than the one at the University of Vienna, and I did not want to work as a "theoretician" for the rest of my life (even though I enjoyed the theoretical mathematics needed for the Mathematical Olympiad), I decided in favor of the University of Technology.

After finishing my studies, I can say that the experience I was able to gain at the Mathematical Olympiad during my time at high school turned out to be very advantageous as the difference between mathematics at school and the study of mathematics is quite large: At school, one usally deals with problems which can be solved simple by applying certain rules, whereas at university the main focus is rather on accurate reasoning. But thanks to my good "preparations", I did rarely encounter any problems and did never regret my decision to study mathematics.

In the first three years of my work life, I did not need a lot of theoretical mathematics, but rather the logical, structural way of thinking that is also aquired when studying mathematics. At the research company PROFACTOR in Steyr (Austria), I was working on discrete event simulation of production processes. As part of my work, I designed and implemented a Java based framework for discrete event simulation. Therefore, I also needed some skills in programming. Hence, I could greatly improve these skills during the past three years (more or less as a side effect to my job).

After my RTW trip, I left my mathematical roots even further behind. I am now working as a software developer in healthcare IT. Even though true mathematical problems only seldomly arise, I have again found an interesting job.

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