Klein Project Blog

Originating author is Michèle Artigue.

Infinitesimals played an essential role in the emergence and development of differential and integral calculus. The evident productivity of this calculus did not prevent recurrent and fierce debates about the nature of these objects and the legitimacy of their use. At the end of the 19th century, when the construction of real numbers from integers and the modern definition of the concept of limit provided a solid foundation for differential and integral calculus, infinitesimals and the associated metaphysics was rejected and their use perceived synonymous with bygone and poorly rigorous practices. However, the language of infinitesimals continued to be used, for example in physics and even in mathematics. It never completely disappeared from the informal discourse and heuristic thinking of a number of researchers.

Is this language thus really incompatible with mathematical rigour? What does it offer that is interesting and specific, which explains its permanence? Non-Standard Analysis developed in the 20th century and provided answers to these questions and enabled infinitesimals to take their revenge.

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Originating author is Christiane Rousseau.
Mathematics offers tools for classifying objects. But is that of any practical use? More than we can imagine at first sight… It could allow us to conclude that a knot cannot be unknotted without cutting the rope, regardless how you move it in space. It could also tell you that you wrongly assembled your Rubik’s cube after dismantling it into pieces, and that there is no use trying to solve it.

In practice, classifying objects means grouping objects in classes of objects sharing some common properties. One very efficient way to do this is through an equivalence relation. Then, each object belongs to an equivalence class. But, this does not mean that we have an efficient way of describing an equivalence class! The mathematical notion of invariant offers an efficient way to do this. An invariant is some mathematical object (it could be simply a number) that is the same for all members of an equivalence class. Among the invariants we distinguish the complete invariants that characterize an equivalence class.

In this vignette, we will mostly work with examples and see how the notion of invariant is widely spread among mathematics, especially algebra and geometry. You will then be able to add your own examples.

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Originating author is João Pimentel Nunes

The Hairy Ball Theorem is from topology, that part of mathematics that is concerned with the form of spaces. For the most part, this result came from work at the end of the 19th century by Henri Poincaré[1], considered to be one of the founders of topology.

There are few mathematical results that are so familiar to us from everyday situations: many readers are faced every morning with the hairy ball theorem when they try to comb their hair and find a persistent whorl at the top of their heads. Stated simply, the Hairy Ball Theorem says that it is impossible to comb a spherical ball covered in hair so that there are no whorls.

Check out this cool video explanation of the theorem
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Originating author is Graeme Cohen.

Of all the familiar functions, such as trigonometric, exponential and logarithmic functions, surely the simplest to evaluate are polynomial functions. The purposes of this article are, first, to introduce the concept of a power series, which can be thought of as a polynomial function of infinite degree, and, second, to show their application to evaluating functions on a calculator. When a calculator gives values of trigonometric or exponential or logarithmic functions, the most straightforward way is to evaluate polynomial functions obtained by truncating power series that represent those functions and are sufficiently good approximations. But there are often better ways. We will, in particular, deduce a power series for and will see how to improve on the straightforward approach to approximating its values. That will involve Chebyshev polynomials, which are used in many ways for a similar purpose and in many other applications, as well. (For trigonometric functions, the Cordic algorithm is in fact often the preferred method of evaluation—the subject of another article here, perhaps.)

In the spirit of Felix Klein, there will be some reliance on a graphical approach. Other than that, we need only some basic trigonometry and calculus.

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