The most important mathematician you’ve (probably) never heard of

Ask someone to call the most important physicist out of 20^tour Century, and they will almost certainly tell Albert Einstein. However, ask the same question about the field of mathematics, and you will probably meet empty glances – that is why I am going to introduce you to Alexander Grootentik.

Einstein, both the inventor of the theory of relativity and the key figure in the development of quantum mechanics, had a huge impact on physics, and he surpassed science to become a genuine global celebrity. Grothendik may have played a similar role in the transformation of mathematics, but he disappeared from academic circles, and then society as a whole, before he died, leaving his heritage is written only in his revolutionary work.

Prior to this, the work of the Grozitelik and his personality made it a difficult sale as a public figure compared to Einstein's exhibition approach. Of course, Einstein’s ideas about the nature of space, time and the universe were incredibly complex, not least when they expressed mathematically, but he had dexterity to tell stories that made his work available. Examples, how Double paradoxIn which the astronaut traveling at high speed discovers that their twin has withstood more than upon returning, is a great way to understand relativity.

On the contrary, even a description of what Groothendik spoke about, an immersion in the mess of abstract and unfamiliar concepts is required. I will do everything possible to explain some of this, but in fact I will only be able to create a superficial impression.

Let's start at the top level. Grothendieck is the most famous among mathematicians for rethinking the foundations of algebraic geometry. Very wide, this is a field associated with the relationship between algebraic equations and forms. For example, the values ​​of the equation x² + y² = 1, when constructed on the graph, form a circle of radius 1.

One of the first who really began the formalization of this connection between algebra and geometry was the philosopher of the 17th century Rene Descartes, whose decent coordinates we still use for graphs of equations on graphs today. But these relationships can go much deeper. Mathematicians love to generalize, in the sense of perceiving the idea and stretching of it in order to be as wide as possible, establishing connections that were not previously obvious. Grothendik was a master in this – indeed, Book about his life And the work described the “search for the maximum community” as part of his personal mathematical signature.

Referring to the above equation of the circle, a set of points that solve the equation and make up a circle is what mathematicians call “algebraic diversity”. The algebraic variety should not be a set of points on the Cartesian aircraft. It can also be points in the 3D space (for example, those that make up the sphere) or even higher dimensions.

This was not enough for Grothendieck. For example, take equations X² = 0 and X = 0. Both have one solution, setting up X to 0, which means their set of points – their algebraic varieties are the same. And yet, it is clear that the equations are different, so something is lost here. In 1960, in the framework of his search for the maximum community, Grotentik presented the concept of “scheme”, which intended to receive this additional information.

How does this work? Here we need another concept: a ring. In confusion, this has nothing to do with the shape of the circle we talked about. Instead, what mathematicians call the “ring” is a collection of objects that, when adding or multiplying, remain in this collection – in a sense they are attached, or circle back on themselves, like a ring, although the name is only a free metaphor.

The simplest example of the ring is integers: all negative integers, all positive integers and zero. Regardless of how you add or multiply an integer, you will always get an integer. Another important property of the ring is “multiplier identity”, which means that this is an object that, when multiplying on another object, always creates a second object again. In integers, this is simple – multiplier identity is 1, because any whole number multiplied by 1 has not changed. It also gives us a convenient example of something that is not a ring – a set of only integers does not have 1, so there is no multiplier identity.

Introduction Schemes, Grotheendieck Combined the Idea of ​​Algebraic Varieties with that of Rings (Note: I'M Slichtly Hand Waving Complexites Here!) Encode the Missing Information for Equations Like X² = 0 and X = 0. This Turned Out to BE Incredibly PowerFul, Secause It Allowed Mathematicians to Turn Problems From A Variety of Subdisciplines Into Geometrical Problems Without Losing Any Crucial Details or Structure, and the ATTACK Them with Geometric tools.

I will exhibit two important problems that fell on mathematicians with a new sword of schemes. The first is the Weil assumptions, four statements proposed in 1949 by mathematician Andre Weil, which concerned the counting of the number of decisions for specific types of algebraic varieties. Returning to our example of a circle, there are an infinite number of values ​​that correspond to the equation X² + Y² = 1 (you can roughly think about it as to say that the circle has an infinite number of “sides”). But Wail was interested in varieties where only the final number of decisions possible. He suggested, but could not prove, the equation known as the dzet function could calculate these solutions.

Using the schemes, Grothendieck and his colleagues were able to prove the three assumptions of Weil in 1965, and the fourth proof came After some time in 1974 from Pierre DelogyHis former student, who also used schemes. The evidence of divide was considered one of the greatest results 20^tour Century Mathematics, responding to a challenge that puzzled mathematicians for 25 years. It also fixed how powerful Grandeck schemes can be in connecting the areas of mathematics, in this case the theory of number and geometry.

The schemes also played a vital role in hacking the notorious last theorem of the farm, the problem in the theory of numbers that has lived mathematics for more than 350 years, while it is. was decided by Andrew Wiles in 1995The Pole theorem says that there are no positive integers of the numbers A, B and C, which satisfy the equation^Not + b^Not = C.^Not If N is a whole number more than 2 (case 2 is just a Pythagoras theorem, you can notice). It was written 17^tour Century Matematian Pierre de farmat on the edge of a book in mathematics, which, according to him, was too small to restrain his evidence. But the farm almost certainly had no evidence, given that Wiles's decision relied on mathematics, developed much later, including Grosandik. For example, Wiles used algebraic geometry tools to translate the problem into one of the elliptical curves, a particularly useful type of algebraic diversity that can be studied with the language of the schemes, but in fact its whole approach was inspired by a new way of thinking introduced by a grocerian.

There is much more Grothendieck, which I did not illuminate, which makes up fundamental tools that many mathematicians use today. For example, he generated the idea of ​​“space” (in a mathematical sense) to “topos”, allowing you to consider not only points in space, but also many additional levels of information while trying to solve the problem. He and his colleagues also wrote two huge volumes about algebraic geometry, which still serve as a Bible for an object today.

So, with all this influence, why didn’t you hear about him? As I may have demonstrated, his work requires some effort to understand. But he also avoided attention for various reasons. Being a devoted pacifist, he refused to attend the 1966 ceremony for the award of the Fields medal, one of the highest awards in mathematics because it was held in Moscow and he objected to the hostilities of the Soviet Union (he had similar views on the US military operations, it should be mentioned). “Fertility is measured by offspring, not honors,” he said, preferring to allow his mathematical work to speak for himself.

In 1970, he left academic circles, leaving his position at the Institute of Scientific Sciences in France in protest against his military funding. Initially, he continued his mathematical work outside the official scientific circles, but he became more and more isolated. In 1986, he wrote an autobiography, Harvest and cropsAbout his experience in fulfilling mathematics and his disappointment in the mathematical community. Next year he made a philosophical manuscript called Dream keyIn which he described how God sent him prophetic dreams. Both texts spread among mathematicians, but were officially published only in recent years.

The next decade Grothendik retreated even further from society. Living in a remote French village, breaking all ties with a mathematical community, at some moment he tried to exist only in a dandelion soup, until the locals intervened. It is believed that he continued to write a lot about mathematics and philosophy, but not one of the works was published. Indeed, in 2010 he began to send mathematicians a letter demanding that nothing of this beDespite the field, despite all the connections that he established – and made it possible – in the world of mathematics, he ultimately rejected them in his personal life. He died in 2014, leaving behind a huge mathematical heritage.

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