How To Do Science

Spring has been knocking on the door as the second term slowly comes to an end, which from the viewpoint of my daily preparation for Tour de France is a most welcome change.

After two years at university I could not but notice that the most popular method in mathematics is to first prove an unnecessarily general statement and then conclude the original claim as a special case. This, however, turns out to be very efficient in the long run and we have actually seen several examples where a more general theory connects and explains various seemingly unrelated phenomena.

Incidentally, this has been one of the main goals in physics since the beginning of last century, as depicted in the recent film Theory of Everything. The abstract approach is very distinctive in algebra, where we often study abstract structures such as groups. The word group is essentially a label for a frequently appearing object. Let me explain the concept in terms of geometry where a triangle is an example of a label. The triangle label is given by a set of criteria determining whether a given object is a triangle or not, however two quite different things can have the triangle label at the same time. The label can further be specified by adding more conditions - for example in case of isosceles or even equilateral triangle - and for each set of criteria many nontrivial facts can be deduced without specifying the object of study.

The point of developing the theory abstractly is that we can then apply the results to any object that satisfies the determining criteria of the label - for instance once we derive the formula for an area of a general equilateral triangle, we can use it for any specific one.

Despite the fact that algebra (xy = yx) uses completely different language from geometry, the above principle applies in both cases. As for groups, these are crucial for many areas in mathematics and appear in many modules such as Algebraic Topology or Groups and Representations.

Outside of academics, I have been a member of a team taking part in so called One Pound Challenge. This is a competition to raise money for charity in one month starting with one pound. Our idea to sell homemade muffins was very successful and we have even attempted a variation of the Apfelstrudel, with rather surprising results. I have never thought this would be one of the things I learn while doing a degree in maths.

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