Mathematical demonstrations Is chaos an exceptional state? Can order be brought to the infinite variety of abstract shapes in mathematics? Can seemingly unsolvable equations be solved? Using novel concepts, mathematics provides clues to these questions. Chaos is everywhere Some systems follow a perfectly deterministic law of evolution. And yet it is impossible in practice to predict how they will develop from a given initial condition: such systems are chaotic. Now, two French mathematicians have shown that the probability of the appearance of chaos in certain particularly simple systems governed by what is known as holomorphic dynamics is not always nil. This unexpected result shows that chaos is by no means exceptional. Annals of Mathematics September 2012 First 3D image of a flat torus Just as a terrestrial globe cannot be flattened without distorting the distances, it was not thought possible to visualize abstract mathematical objects called flat tori in ordinary three-dimensional space. However, a team of mathematicians and computer scientists has succeeded in constructing a three-dimensional visual representation of a flat torus. This is a smooth fractal, half way between fractals and ordinary surfaces. The construction could provide new tools to help solve very complex equations, as well as offering stunningly beautiful images. Proceedings of the National Academy of Sciences April 2012 online 32 A tool to classify higher dimensional shapes A cup and a donut don’t have the same shape, but they do share the same topology. Although mathematicians know how to classify different types of two- and three-dimensional surfaces, doing this in higher dimensions is extremely difficult. Yet a team has now shown that an essential tool known as the “fundamental group”, which is used to undertake such classification in lower dimensions, is to a certain extent also effective for higher dimensions. This should help bring some order to the infinite variety of abstract shapes in mathematics. Inventiones Mathematicae September 2012 online A year at CNRS 2012
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