The other day I went out for coffee with one of my friends who does theoretical math research. I actually ended up having my first cup of coffee ever.
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As was usual with this friend, we got onto the topic of his research and started debating the merits of our two majors. As an engineer I fundamentally believe that an idea's worth is based on its success of implementation in the real world. That is, unless it's solving a real world problem right now, I have difficulty valuing the idea.
However, all my mathematics-focused friends fundamentally disagree with this belief. One of the best examples I have heard that has countered my claim was the following. My friend believes that the work she does in theoretical mathematics is pushing the knowledge boundary of humanity and that it will only take time for the rest of the world to be able to understand her ideas and implement them. Her favorite example is Knot theory. Knot theory is the idea that all knots can be represented as a series of three different operations, known as the Reidmeister moves.
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There are essentially three basic operations, a twist, a poke, and a slide. Those three operations define every knot in existence. Take that Boy Scouts. As knot theory was being developed, especially in the 1970s, when studies shifted towards hyperbolic knots, many dismissed the field as a useless mathematical endeavor which could only contribute to the abstract concept of quantum field theory. Fast-forward twenty years and biologists are applying the theorems behind Knot theory towards protein folding and determining the chirality of molecules (chirality is essentially the difference between your left-hand and your right-hand). What was once an abstract concept and a mathematician's game was now helping further the study of DNA, since you can imagine DNA as a string and its folds and creases as knot operations.
Next time you are tying your shoes, imagine that you're playing with a piece of DNA and trying to unravel it. I don't think DNA gets tied into a bow though.