The authors of this article have never heard of serious special school projects in Brazil for talented children and young geniuses. Not only are schools important, but societies should strive to fund the development of brilliant minds. Opposed to this are the lousy schools, unprepared factories for anything, and the dubious schools that produce the illusion of easy thinking, the illusion of quick work, and the illusion of consumerism.

A single brilliant mind can benefit humanity more than the sum of all hereditary captaincies, all possible and imaginable colonels, kings, and politicians together in a country's history. In 1995, Englishman Andrew Wiles finally demonstrated Fermat's Last Theorem that had been open since the 16th century. Wiles has devoted his mind to the problem since he was twelve years old. Around the age of seventeen he interrupted his dedication to occupy the mind with other notable problems of Number Theory, Algebra, and Algebraic Geometry. A few years before reaching the age of forty, Wiles had the historic opportunity to rededicate his brilliant mind to the most famous problem in the history of mathematics and solve it.

Not even a terrible and cruel mental illness can prevent a brilliant mind from producing a work of invaluable value to mankind. John Nash is a contemporary symbol of this possibility. A little can be known about your brilliant mind in the movie "A Bright Mind."

Indeed, the comparison between the work of a brilliant mind and the work of politician minds can be ridiculous. Compare the work of Isaac Newton with the work of all the kingdoms of England. Compare Albert Einstein's work to the political work of anyone. Compare the work of the AT&T employee, mathematician Peter Shor, to the work of all section chiefs and managers or directors of this or any other company anywhere in the world. Peter Shor has already written the mathematical basis of quantum codes. It is cowardly to extend this line of comparison further.

The work of more than seven hundred Microsoft scientists (mathematicians led by Michael Freedman, Fields Medal) will influence humanity more than all the wealth ever generated, and yet to be generated, by the company worth $ 400 billion today. Bill Gates himself knows this, for he was the one who had these brilliant minds hired.

We also do not know whether in Russia there is such a concern to cultivate talented minds and geniuses well. The fact is that, rarely, even without any social support or protection, do these minds survive and do a work that deeply affects not only their country but humanity and, who could deny, perhaps the universe itself. It seems that in this country of fantastic mathematical history came another phenomenon of brilliant mind. It's about Grigory (“Grisha”) Perelman. In November 2002, Perelman published an article that was soon recognized by mathematicians as relevant to the solution of the famous Thurston Conjecture and, in particular, to the even more famous Poincaré Conjecture. In March 2003, Perelman published the second article in this line of thought. From April to May 2003, he visited major mathematical research centers in the United States, such as MIT in Boston and New York University in Stony Brook.

Other brilliant minds are trying to find errors in Perelman's work. The same happened in 1994 with Wiles's work, when his own doctoral advisor, John Coates, found an unexplained point in the logical sequence leading to the conclusion of Fermat's Theorem. It cost Wiles another year of dedication to his mind, along with former student Richard Taylor, to achieve the famous French result Pierre de Fermat.

Another brilliant mind, Richard S. Hamilton of Columbia University in New York, was awarded by the Boston Clay Institute in late 2003 for his dedication and advances in the much broader Thurston Conjecture problem. , and much more difficult, than the Poincaré Conjecture. Mathematician William P. Thurston of Cornell University in Ithaca, New York, was also awarded the Fields Medal in 1983 for his brilliantly accomplished work.

A circle is a mathematical object that is very easy to imagine. It has interesting properties: (1) it is formed by a single piece, (2) infinite subsets of points always accumulate around some point, (3) there is no endpoint or initial point and (4) a segment perpendicular to it, pointed outwards, you can walk it back to the starting point pointing outwards just as you left. Another interesting fact is that it has dimension one. That is, to describe any part of it just use one variable. In fact, if we imaginarily remove a part of it, we see that this part is exactly equal to a range of real numbers, and just one variable *x* to go through this range. There is a noticeable peculiarity about the circumference: If it is represented by a rubber band, you will not be able to deform it without using the space around it until it is crumpled to a point.

The surface of an orange or a ball is very similar to the circumference. Mathematicians say that this surface called a sphere is a circumference of dimension two. If we imaginatively cut out a piece of the sphere, we will see that it sits perfectly on the table plane. We can remove a piece of orange peel in the shape of a rectangle and arrange it on a table. Therefore, we say that the sphere is locally flat. To describe a rectangle we need two variables *x* and *y*because we have to account for width and length. This is why mathematicians say that the sphere has dimension two. The same goes for the sphere: (1) it is formed by a single piece, (2) infinite subsets of points always accumulate around some point, (3) there is no end point or initial point and (4) a segment perpendicular to it. , pointed outwards, you can walk it back to the starting point pointing outwards just as you left. There is therefore a great similarity between the circumference and the sphere: they are (1) connected, (2) compact, (3) borderless, and (4) orientable.

If we imagine a rubber band around the orange, then we can easily displace it, without escaping the orange peel, crumpling it until it is compressed into one point. This becomes possible because the sphere has dimension two and the elastic is a circumference of dimension one. Dimension one objects can be deformed within a dimension two space. The sphere is said to be “simply connected” because it allows circumferences, without escaping it, to deform within it until it becomes a point. Remember that this does not happen with the circumference, as it cannot itself deform at a point without escaping itself into the surrounding space.

Note that the circumference can only be viewed in one plane, just as the sphere can only be viewed in three-dimensional space. That is why we cannot visualize the circumference of dimension three. It only fits in four dimensions. Poincaré stated that it is the only (1) connected, (2) compact, (3) borderless, (4) orientable, and (5) simply connected space of dimension three.

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