The Power of Involutions

zagier2.pngTime and again, there are moments when mathematics just feels magical.
For me, one example for this is given by generating functions (and that is why they can be found on this blog).
Today, I want to talk about another such example: involutions. We will look at how they are used to prove in one sentence that primes of the form p=4n+1 can be written as a sum of squares, in the proof of the wonderful Lindström-Gessel-Viennot lemma, and in the proof of Euler’s pentagonal number theorem.

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The Beats of Bitterns

bitternIn my PhD thesis and in the paper Detecting bird sounds in a complex acoustic environment and application to bioacoustic monitoring, I have proposed a simple yet robust algorithm for automatically detecting calls of the Eurasian bittern in the wild.

The Eurasian bittern is a threatened bird species found in reed beds and other wetland habitats. Its sound is very simple, very deep, and very characteristic. It is pretty similar to what you can produce by blowing over the opening of an empty bottle (use a large one) in regular intervals.

Recently, I have been asked whether I still have the Matlab code for the algorithm. A bit of digging on an external hard drive has shown that I indeed still have it.

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How Algebra Sheds Light on Things

causticEgbert Brieskorn‘s mathematics books contain a lot more flesh than those structured strictly by definitions, lemmas, and theorems. What might be a hindrance to those looking for a quick grasp of a theory is a treasure trove for others who relish an extra of motivation, history, and examples. Currently, I am reading his book Plane Algebraic Curves (authored together with Horst Knörrer) which, already in the introductory chapter, contains many wonderful examples such as linkages, envelopes, and the Hopf fibration. I will come back to some of these topics in later posts. Today’s post contains a quick glimpse of caustics.

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Beautiful Binomials

binomialsWhile leafing through the book Geometric Trilogy I – An Axiomatic Approach to Geometry, I came across two nice geometric depictions that are probably widely known but which I would love to have seen back in school. The first one is the geometric illustration of the algebraic fact that (a+b)^2 = a^2+2ab+b^2, the quadratic case of the binomial theorem. This can already be found in Book II of Euclid’s Elements. From this, it is easy to come up with a three dimensional version of the construction giving (a+b)^3=a^3+3a^2b+3ab^2+b^3. Creating a good picture of this is somewhat tedious so I am very glad that I can use this opportunity to link to a wonderful blog where this and other wonderful mathematical illustrations and animations can be found: Hyrodium’s Graphical MathLand.

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Every now and then, mathematics and the sciences allow us glimpses of a fascinating world. Here, I wish to share some of those glimpses which are particularly fascinating to me.

Stay tuned for my views on things I deem worth a look such as

Virus capsides,
Soap bubbles,
Protein structure,
Snow flakes,
Butterfly wings,

This blog gives my personal opinions and view of the world.
In my professional life, I am a computer scientist and you can learn more about this facet at LinkedIn, Google Scholar, and ResearchGate.