Аннотация:
How would you continue the sequence 1,2,4,8,...? The obvious answer "16, 32,.." is given by 99 per cent of our colleagues, pure mathematicians. Surprisingly, the rate is much lower among applied mathematicians, physicists and engineers! If you answer "9", you can skip the first lecture...
The sequence "1,2,4,8,9,10,12,16,17,18,20,24,25,26,28,32, ..." is identified by the Sloane encyclopedia of integral sequences as the Hurwitz-Radon function evaluated on powers of $2$. This function was discovered in about 1920 independently by A. Hurwitz and J. Radon in the context of so-called "square identities", a subject that historically belongs to number theory. The Hurwitz-Radon function then appeared, sometimes unexpectedly, in many different areas, such as algebra and representation theory, geometry and topology, combinatorics, relating them in a beautiful manner. Most surprisingly, the Hurwitz-Radon function was recently used in the context of multi-antennas wireless communication.
The main goal of these lectures is to collect various topics related to the Hurwitz-Radon function. We defend the following general idea: whenever the numbers 1,2,4,8 (or perhaps 0,1,3,7) show up as exceptional values in some mathematical problem, one should systematically look for the Hurwitz-Radon function . We will also learn why the quaternions are commutative and the octonions associative.
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