21/06/2019

Perelman and the solution to one of the millennium's problems.

Grigori Perelman, em foto antiga; matemático russo é considerado tão brilhante quanto recluso

BBC reproduction

A decade ago, Grigori Perelman, one of the great minds of the 21st century, bid farewell to his profession and public life.

At the time, he was already world-famous for solving one of the most difficult mathematical enigmas of the millennium, whose origin dates back to the 18th century and is rooted in the ancient Prussian city of Königsberg (now Kaliningrad, in Russia).

The city had seven bridges over the Pregel River, connecting not only the two sides of the city but also two small islands within the river's course. Legend has it that the people of that time formulated a question that became a famous problem:

Is it possible to leave home in one of the four regions of Königsberg, cross all the bridges only once, and return to the same starting point?

The solution is not only more difficult than it seems, but it has also led to the creation of new branches of mathematics, including topology.

Pontes de Königsberg / Creative Commons

In 1735, the great mathematician Leonhard Euler provided the answer: it was not possible. But what is most curious is that, in solving the problem, he made a conceptual leap.

Euler realized that the distances between the bridges were irrelevant. What really mattered was how the structures were connected to each other, which means that the theory is not limited solely to the city of Königsberg, but rather applies to all topologically similar configurations.

This is the beginning of the concepts of topology, which today underpin virtually all subway map designs in the world, to clearly communicate to users what they need to know: how to get where they want to go.

Do dilema das pontes nasceu a topologia, usada em trajetos de metrô / Getty Images

Although the origins of topology can be traced back to the bridges of Königsberg, it was only in the hands of the most famous and respected mathematician of the late 19th century, the Frenchman Henri Poincaré, that the subject was transformed into a new and powerful way of understanding form.

Topology

The main idea behind topology is that, when studying an object, its properties are most important, not the object itself. And, if two objects share the same properties, they should be studied, because the results can be scaled to all objects that share the same properties – that is, homeomorphic objects.

Some people refer to this important field of mathematics as "flexible geometry" because, according to this view, two shapes are the same if it is possible to transform one into the other without breaking it.

So, for example, topologically a soccer ball and a rugby ball are equivalent, because one can be molded to become the other.

Na topologia, uma bola de futebol e uma bola de rúgbi são equivalentes, porque uma pode ser moldada para se transformar na outra

That's why it's joked that a topologist can't tell the difference between a cup of coffee and a donut.

The thing is, although it sounds strange, topographically a cup and a donut are the same.

Um donut se converte em xícara (e vice-versa) sem ser quebrado / Science Photo Library

But while it's possible to deform a donut to turn it into a cup and vice versa, it's impossible to deform a ball enough to turn it into a donut, because we can't create the hole in its center without changing the sphere's properties.

The problem

Poincaré came to know all possible two-dimensional topological surfaces. Furthermore, he developed all possible ways in which this flat, two-dimensional universe could be enclosed.

But the fact is that we live in a three-dimensional universe. This led the mathematician to ask himself in 1904: what are the possible shapes that our universe could have?

He died in 1912 without finding the answers. The problem became known as the "Poincaré conjecture (or hypothesis)" and remained as a legacy for future generations of mathematicians, who for decades were unable to solve the problem for 3D surfaces.

Henri Poincaré (1854-1912) levou o problema adiante, mas não conseguiu resolvê-lo para superfícies em 3D / Science Photo Library

Thus, Poincaré's hypothesis was included in the list of the seven mathematical problems of the millennium, whose solution would be awarded US$1 million by the Clay Mathematics Institute of Massachusetts, in the USA.

Then, in 2002, the internet site arXiv published the first of three parts of an article with the intricate title "The entropy formula for Ricci flow and its geometric applications".

The text was 39 pages long and was authored by Grisha Perelman.

Unorthodox

Grigori "Grisha" Perelman had been studying the subject in his hometown of St. Petersburg, to which he had returned after living for some years in the USA. According to a colleague, Perelman returned because he realized that his work flowed better in Russia.

He was no stranger to the mathematical community: in 1994, he had already proven the "soul conjecture," according to which one can deduce the properties of a mathematical object from small regions of those objects, called souls.

Grigori Perelman resolveu o problema por conta própria, mas recusou qualquer tipo de reconhecimento por isso / Getty Images

After that, he received job offers from some of the world's leading universities, such as Stanford and Princeton, but he preferred to become a researcher at the Steklov Institute in St. Petersburg, a position that paid less than US$100 a month.

During his time in the US, he said he had earned enough money to live comfortably.

But he had also managed to make progress on a question raised by an American mathematician he admired: Richard Hamilton.

Flows that did not flow

In 1982, Hamilton had published an article on an equation called the "Ricci flow," which was suspected to be a way of proving the Poincaré conjecture.

But the task was extremely technical and its execution complicated.

Fluxo de Ricci, acima em 2D, foi usado por Perelman para encontrar suas respostas / CBM

In 1993, Perelman had accepted a research fellowship at the University of California, Berkeley, where he attended several of Hamilton's lectures.

At the end of one of them, Hamilton explained to Perelman the obstacles he had encountered in trying to prove the conjecture; the Russian replied that he had done a study that could help him overcome these obstacles. Hamilton, however, didn't pay much attention.

Two years later, Perelman wrote to Hamilton again explaining his ideas, but the American never replied.

Perelman ended up working alone, and in 2002 he published the results of his efforts online. This publication ended up sparking enormous interest among mathematicians.

The resolution

Although the article did not even mention Poincaré, four years later a consensus emerged that Perelman had, in fact, solved the conjecture.

And while four years may seem like a long time, it's good to remember that we're talking about mathematics.

Unlike other fields of knowledge, where theories can always be revised, the proof of a theorem is definitive. In Perelman's case, at least two teams of experts examined his article to confirm that there were no loopholes or errors, and from this they produced studies of hundreds of pages (while the original article was only 39 pages long).

Furthermore, Perelman's proposal was so complex that even experts had difficulty understanding it.

The silence of genius

After more than a century of failed attempts, the hypothesis of a brilliant mathematician had been proven by another equally brilliant, albeit more eccentric, mathematician.

Perelman received a new barrage of offers—awards, positions, honors, cash payments, invitations to conferences, and research grants—which he reportedly found deeply offensive.

"Monetizing success is the ultimate insult to mathematics," he stated.

Consequently, he even rejected the Fields Medal, the mathematical equivalent of a Nobel Prize, for "his contributions to geometry and his revolutionary ideas"; an award from the European Mathematical Society; and the million dollars that the Clay Institute wanted to give him for solving one of the Millennium Problems.

"If the theory is correct, it doesn't need any other kind of recognition," Perelman stated.

He soon stopped speaking to the press, announced his intention to abandon his profession, and retired to live with his mother as a semi-recluse in a modest apartment. Reports indicate that he only leaves the house to buy basic necessities or to attend opera and classical music concerts.

"I'm not interested in money or fame. I don't want to be on display like an animal in a zoo," he once said.

Some acquaintances claim that he is simply interested in proving theorems, not in winning prizes.

In the scientific world, many lamented his complete abandonment of mathematics. Unless, at some point, he surprises the community with another brilliant publication online.