A chapter from the book "Poincaré's Conjecture"

Petah Davar and the first chapter from the book The Poincaré Conjecture by: Donal O'Shea, Aryeh Nir Publishing. From English: Emmanuel Lotem

Courtesy of the websiteספרים TEXT

Henri Poincaré, the mathematical giant of the late nineteenth and early twentieth centuries, left us a persistent puzzle, which has implications for the possible shape of the universe. His hypothesis, formulated in 1904, was a stumbling block for the great mathematicians, who for generations were unable to prove it or disprove it.

In 2000, the Clay Institute of Mathematics defined the Poincaré theorem as one of the seven great and vital puzzles of the new millennium, and offered a million dollar reward for its solution.

Russian mathematician Grigory Perlman, who shies away from the limelight, shocked the world of mathematics in 2002, when he published his solution on the Internet and not in a peer-reviewed journal. After years of vigorous testing by several teams of mathematicians, it was found that his solution stands firm against any attempt to disprove it, and now it opens new horizons in the mathematical universe.

About the book

The Poincaré hypothesis is the fascinating story of the personalities, traditions and institutions that led for hundreds of years to Perlman's dramatic proof, and that in the process expanded our knowledge of how the universe works. The book opens a porthole for an instructive glimpse into the heart of our constant search for wisdom.

The end of the book is a suspense story in the full sense of the word, the story of the International Congress of Mathematicians in Madrid in 2006, during which Perlman was awarded the prestigious Fields award - and he immediately rejected it, on the grounds that he had no interest in public exposure.

Donnell O'Shea, a learned mathematician and master of writing, outlines the path of the theory of geometry from the ancient Babylonians to the new path-breakers in today's mathematics, a path that led to the celebrated Poincaré conjecture, and describes the fascinating and competitive attempts to settle it.

O'Shea is gifted with a special talent for clarifying complicated mathematical concepts to the general public, and thanks to him he breathes life into the achievements of Poincaré, Bernhard Riemann, William Thurston, Richard Hamilton and others - personalities whose spirit changed the face of mathematics in the last hundred years.

O'Shea is Dean of the Faculty and Vice President for Academic Affairs at Mount Holyoke College, where he also holds the Elizabeth T. Keenan Chair in Mathematics. He authored books and monographs in his professional field, and his research articles appeared in countless journals and anthologies. O'Shea is a member of the American Mathematical Society, the American Mathematical Union and the Canadian, London and French Mathematical Societies. He lives in South Hadley, Massachusetts.

"The history of the Poincaré theorem is the story of one of the most important fields in modern mathematics. Donal O'Shea tells the story in a style that is both entertaining and instructive - the concepts, issues and people thanks to whom everything happened. Highly recommended." Keith Devlin, Stanford University

The Poincaré hypothesis, the search for the shape of the universe by Donal O'Shea published by Aryeh Nir, from English: Emanuel Lotem, scientific editor for translation: Or Moshe Shalit.

introduction
This book deals with one and only one problem. Ever since the brilliant French mathematician Henri Poincaré formulated it, more than a century ago, this problem has fascinated mathematicians and tormented them, until it was solved - just recently. The Poincaré hypothesis concerns things that lie at the heart of our understanding of ourselves and the universe in which we live.

I am writing this book for the curious reader who remembers a little geometry from high school, but not much more than that—though I hope those with a richer mathematical background will also enjoy it. If you encounter passages that are difficult to understand, you can skip them without fear and continue from where the reading becomes easy again.

Go to any coffee shop. sit next to someone on the plane Listen to what your neighbors have to say about math. Some love her, but many more do not think fondly of her, and what they have to say about her sounds bad.

Some are convinced that they are unable, by nature of their creation, to control it. Some despise her, and many hate her from the bottom of their hearts, with the eagerness reserved for unrequited love.

How can a single subject, which has such an abundance of beauty, cause such different reactions from each other? Part of the aversion that some people feel towards mathematics, I think, comes from fear, and I don't delude myself that this one book has the power to change that.

But if you do not have a strong opinion against mathematics, I hope this book will motivate you to read more; And if you are students or students, and are thinking about continuing your path, you may find that you should take a few more units or courses in mathematics.

I sincerely hope that you will enjoy reading the following as much as I enjoyed writing it.

Donal O'Shea

Poincaré hypothesis | Cambridge, April 2003

In mathematics, revolutions proceed quietly. No military clashes, no cannons. Short articles in the press, under no circumstances on the main page. Far from catching the eye. And so it was in the dark afternoon of Monday, April 7, 2003, in Cambridge, Massachusetts, in the United States.

Young and old crowded the lecture hall at the Massachusetts Institute of Technology (MIT). They sat on the floor and in the aisles; They stood behind. The lecturer, a Russian mathematician named Grigory Perlman, dressed in a tattered dark suit and sneakers, paced back and forth as he was introduced. He was bearded and balding, with bushy eyebrows and dark, piercing eyes. He checked the microphone and began hesitantly: "I'm not strong in linear speech, so I'm going to sacrifice clarity for the spirit of things." Ripples of amused smiles passed through the audience, and the lecture began. The lecturer held in his hand a huge piece of white chalk and wrote on the blackboard a short mathematical equation, twenty years old: Its name is the Ritchie flow equation, and it treats the curvature of space as if it were some kind of exotic form of heat, a sort of molten lava flowing from areas of higher curvature and striving to spread over areas of lower curvature.

Perlman asked his listeners to imagine the universe as one member of the huge abstract mathematical set of all possible universes. He gave the equation a new interpretation, as if it described these possible universes in motion, like drops of water trickling down the sides of great hills in a vast landscape. During the movement of each member, the curvature changes within the universe it represents, approaching constant values ​​in several areas. In most cases, the universes develop proper geometries: some are identical to the Euclidean geometry we learned in school, some are very different. But some paths leading down the slope give rise to problems: the organs moving along them develop mathematically malignant zones, which are interrupted above the rest of the zones, if not worse. Not bad, the lecturer reassured his listeners, we can divert such paths; He drew a diagram that showed how to do it.

The audience was drawn to the lecture by an article Perlman published on a website in November of the previous year. In the last chapter of the article, Perlman presented an argument that, if indeed found to be valid, would be able to prove the most famous, most elusive and most beautiful conjecture in all of mathematics. It was presented in 1904 by Henri Poincaré, the greatest mathematician of his time and one of the most gifted of all time: the Poincaré conjecture is a bold guess dealing with the possible shape of our universe, nothing less and nothing more.

Still, it is nothing but a guess. The quest to prove it or disprove it held mathematicians by the gravitational forces of the siren song, and made it the most famous problem not only in geometry and topology, but also, one might say, in all mathematics. Only one problem besides her was similar to her in the reputation she got - the "Riemann hypothesis". In May 2000, the Clay Institute, an institution whose mission is to promote and disseminate mathematical knowledge, announced the "Seven Thousand Problems", and offered a prize of one million dollars to anyone who could solve one of them. The institute consulted many mathematicians when it compiled the list, and surprisingly, the Poincaré hypothesis and the Riemann hypothesis were the only two mentioned by each and every one of these consultants.

It must be assumed that more than half of the people in the hall tried their hand at the Poincaré conjecture at one time or another. All the listeners of the lecture, up to one - starting with the licked thirty-something year old with the curly hair, who wrote down his notes in Chinese, or the blonde in the tight shirt and too short skirt, and ending with the sporty type in the sloppy jogging pants and damp t-shirt, or the eye-trotting eighty-year-old whose conservative jacket was stained with chalk dust from For decades - everyone knew without a shadow of a doubt that they were present at one of the great moments of a three-thousand-year-old heritage. The mathematics spoken of was diligently passed down from generation to generation, from age to age, through days of great wealth and days of abject poverty, beginning with that unknown Babylonian who gave us the area of ​​the circle, on to the strict perfection of Euclid, and on to the flourishing of geometry and topology in the last two hundred years.

Two weeks and a few lectures later, on the most important campus of the State University of New York, in Stony Brook, a similar vision took place. The lecture hall was even more crowded, and this time several journalists were in the room. The reporters heard that Perlman had made an amazing discovery about the shape of our universe, and that he might win a million dollar prize as a result. And they also heard about his career shrouded in mists - how he disappeared from sight ten years before, a brilliant man by all accounts, but also a man whose promise was never fulfilled. Flash bulbs went off. "Don't do it," Perlman spat with obvious irritation.

The mathematician patiently answered all the questions asked by the audience at the end of his lecture, questions that fell on him like a shower of stones. "But that solution will explode at the end of time," said a voice from the middle of the hall. "It doesn't matter," replied Perlman; "We can trim it and start the flow again." Silence, followed by a few nods. The listeners were careful, and carefully considered what they heard. They would turn over what they heard from him for months to come, but for now, things sounded promising.

Much of the mathematics on which Perlman relied had not even occurred to him thirty years before. The technical tools he used were right at the limit of what was possible, and depended vitally on the work of some of those present in the hall. The atmosphere was electric. Everyone knew how sensitive the lecturer's arguments were, how convoluted they were, and how easily they could deviate from the right path. Everyone longed for his arguments to hold. A web site has already appeared, under the management of two professors from the excellent mathematics department of the University of Michigan, Bruce Kleiner and John Lott. The site contained links to Perlman's articles. Mathematicians around the world added notes and arguments to clarify obscure points, and expanded parts that seemed too concise to them.

Almost every mathematician, and not just the geometricians among them, knew someone who was present at the lecture, and expected to hear things from him. Most of the people sitting in the hall wrote lists, for themselves and their friends. Two of them - Christina Sormani, a young professor from Leyman College, and Yair Minsky, who later became a full professor at Yale - then uploaded their lists to the network's website, so that everyone could review them.

As at MIT, here too it was clear to all present, young and old - except for the journalists - that what they were hearing was the fulfillment of more than a hundred years of mathematical flourishing - the greatest flourishing that mathematical thinking has known since the existence of the human race. The lecture demanded the full attention of its listeners, leaving no room for stray thoughts. Nevertheless, many must have reflected on a particularly memorable event or article, either recently or many days ago, that was related to Poincaré's work; And we may have remembered someone who may have passed away years ago, and thought how happy he would have been to hear this lecture. All alike raved about this abundance of good ideas and promising ways worthy of an inquisitor.

The reporters, however, wanted to hear about the million dollars. How does Perlman feel, given the possibility that he will win such a large sum of money? When it gradually became clear to them that he was not at all interested, they changed direction and wrote about the withdrawn Russian who made an important mathematical discovery, and added their guesses that he might reject the prize. Perlman added more details in the following days, in hastily convened workshops. But he rejected all the journalists' requests to interview him, and returned to St. Petersburg a few weeks later, without responding at all to the job offers showered on him by American universities.

Poincaré's conjecture and Perlman's proof together constitute one of the greatest achievements of our time; They have a lot to teach us about the shape of our universe. The Ritchie flow equation that Perlman wrote on the board, a kind of heat equation, is very close to the Black-Scholes equation, which is used by stock market traders all over the world to price stocks and bond options. But curvature is a more complicated matter than temperature, or money. As will be explained in the following chapters, curvature is a geometric entity that requires more than one number to describe it, and the Ritchie flow equation that Perlman used is a shortened form of six interrelated equations—an impressive feat of elegance, a seemingly simple equation, but one that holds richness dizzying. It can be compared, in this respect, to the equation of Einstein's theory of general relativity, which expresses the curvature of space-time.

The Poincaré Conjecture is the story of the mathematics behind the conjecture and its proof. The story of every mathematical development, if it deserves its name, is not only the story of results, since we must also talk about the people who brought these results into the world. To the extent that achievements in mathematics come to the public's attention at all, they conjure up a romantic and heartfelt image - the genius locked in rooms, where he struggles with the indifferent cosmos and extracts a little understanding from his hands. Indeed, there are individuals whose insights appear out of nowhere, seemingly, and advance the field by decades at once. But despite all the colorfulness of genius and despite all its mysteries, progress in mathematics also depends on thousands of other people, and on the institutions and societies within which they live and work. It's time to tell this story, wide open. Its plot continues from Babylon three thousand years ago to St. Petersburg, to upstate New York and to present-day Madrid. It is a story of exploration, of wars, of scientific societies and of the growth of research universities, first in Germany and later in the United States. It traces the history of geometry over five thousand years, documenting the discovery of non-Euclidean geometry and the birth of topology and differential geometry. It involves dozens of human companies and institutions, and hundreds of people.

In between the discussions on mathematics, biographical, cultural and historical material is interspersed. The math will be too much for some and too little for others, but most people with a high school education will be able to handle the basic concepts included in the book, even if they struggle a bit with the more complicated points.

Can a person understand and appreciate the mathematics, and this famous hypothesis, without doing the calculations himself. For the convenience of the readers, a list of mathematical terms along with their definitions, biographical notes of personalities and a schedule describing the important events throughout history, a bibliography and a list of suggestions for further reading were included at the end of the book. Finally, place mirrors for the sources appear.

Part of mathematics was born in the distant past, thousands of years ago. Mathematical inquiry is one of man's oldest activities, since it dates back as far as carpentry, cooking and metal forging. But in practice, the amount of mathematics discovered since 1900 is greater than anything discovered since the dawn of history until then. Because of this, the pace picks up and the reliance on more detailed mathematical theories and scenes of places grows, inevitably, as the plot moves closer to the present day. You can skip the more mathematical sections; There will be no exam at the end. And you can always go back, if you want, and wonder about the things that seemed unclear to you at first. After all, the Poincaré conjecture has failed the most learned mathematicians for the past hundred years.