Planet of Mathematics

 "The first day of the calendar winter, the birthday of the great Russian mathematician Nikolai Ivanovich Lobachevsky, has been declared Mathematician's Day this year. How does this science enter our everyday life? To explain, the easiest way is to refer to the fact that this science is the language of physics, on which most of the technologies around us are based.

 

For example, quantum physics is formulated in terms of "Hilbert spaces", which were invented more than 100 years ago by the great German mathematician David Hilbert. At that time, no one knew about quantum electronic devices like transistors in a processor and LEDs on the screen of a smartphone, on which you may now be reading this text - but ultimately Hilbert spaces made their invention, calculation and production possible (as they will probably make a quantum computer possible).

 

However, this is not all the mathematics that appears in your hands as soon as you take out your smartphone. When you call or use it to pay for a purchase at the checkout, information theory and number theory work in a smartphone. Thanks to them, the sound of your speech is reliably encrypted from interference even with a weak signal, and the bank receives a crypto-protected command to write off the cost of the purchase from the account.

 

Such examples are easy to multiply, but their common drawback is that they paint mathematics as a set of weakly connected spells from some magical art, and the overall picture does not come together. To see this picture at least in general terms, let's try to compare this science with a planet - such as Earth.

 

Mathematics is about how to calculate something, isn't it? 

 

Calculation methods and algorithms, such as the just-mentioned noise-resistant codes and cryptographic protection algorithms, form the "crust" of this planet. These are its mountains, rivers, lakes and seas, quarries, mines and oil wells - everything that is visible to the naked eye, with which we began and which constitutes the applications of mathematics in other sciences and technological industries.

 

Under the crust of the planet is hidden a mantle - with it we can compare specific mathematical theories, in which the rationale for these methods and algorithms is rooted. At this level, invisible to the external observer, researchers work, creating the bulk of mathematical results, due to which its development as a science occurs.

 

But some of these results, by their appearance, change the structure of the mantle (and along with it the landscape of the crust) so much that they are correctly considered as the "core" of mathematics. Here is a simple example. In the 16th century, Italian mathematicians who came up with methods for solving algebraic equations discovered that if you subordinate "imaginary numbers" - square roots of negative numbers - to the usual rules of arithmetic, then with their help it is possible to find solutions to such equations that are not solved otherwise. This looked like an interesting algebraic invention - that is, from the point of view of our analogy, it seemed that they completed the construction of part of the "mantle" of the mathematical planet.

 

However, over the next few centuries it gradually became clear that complex numbers, that is, those that can be composed of real and imaginary parts, taken all together, have the richest geometric and topological properties. Studying the functions that map complex numbers to complex numbers, Riemann came to the concept of a "Riemann surface", and generalizing it, he obtained a "Riemann space", with the help of which, several decades later, Einstein formulated his general theory of relativity. In parallel with this, it turned out that in the language of complex numbers it is convenient to describe all sorts of oscillatory processes and calculate electrical circuits of alternating current.

 

Thus, a construction that at first glance was purely algebraic (and which one of its creators, Girolamo Cardano, considered "so subtle that it is unlikely to be of much use") turned out to be a bridge connecting completely different sections of mathematics, some of which simply did not exist at the time of the invention of complex numbers. Moreover, this entire world was already “hidden” in the original constructions of del Ferro, Tartaglia and Cardano, but it turned out to be so rich that only several generations of specialists were able to fully reveal it.

 

Recalling the calendar date that became the reason for these notes, we will say that Lobachevsky's geometry is, of course, also part of the “core” of mathematics. At that historical moment when the scientist presented his work on parallel theory to his colleagues, this was not obvious: moreover, many first-class mathematicians treated these works skeptically, not to say hostilely.

 

But by the end of the second third of the 19th century, not only Lobachevsky's geometry was recognized, but after it, many varieties of other non-Euclidean geometries had accumulated, and Felix Klein was able to formulate the main principle: each of the geometries corresponds to its own set of transformations of space, and the objects of each geometry are the corresponding "invariants", that is, what does not change under transformations: angles and distances under Euclidean rotations or straight lines and their intersections under projective transformations.

 

This discovery by Klein also brought algebra and geometry into conflict. This time, it was not algebra that revealed a geometric structure, as was the case with complex numbers, but, on the contrary, geometry turned out to be correctly described through algebraic groups of transformations. The consistent implementation of this program became one of the main topics of 20th century mathematics.

 

It is interesting that around the same years when Klein formulated his program, many people, such as Alfred Nobel, considered mathematics a completed science in which nothing more would be discovered. But the fate of Lobachevsky's discovery, as well as many others after him, shows that the potential for development has not been exhausted and, apparently, will never be exhausted.

 

This means that the magic spells with which technologies work and ultimately our everyday life is created and enriched will multiply."