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Head of his department of mathematics, member of the scientific council (). Got degree bachelor's degree in a year at Merton College, Oxford University. He began his scientific career in the summer under the guidance of Professor John Coates at Clare College, Cambridge University, where he received his doctorate. During the period from to Wiles held positions as an Associate Fellow at Clare College and Associate Professor at Harvard University. Together with John Coates, he worked on elliptic curve arithmetic with complex multiplication using the methods of Iwasawa theory. In the year Wiles moved from the UK to the US.

One of the highlights of his career was the announcement of the proof of Fermat's Last Theorem in 1993 and the discovery of an elegant method to complete the proof in 1994. He began his professional work on Fermat's Last Theorem in the summer of 1986 after Ken Ribet proved the conjecture about the connection between semistable elliptic curves (a special case of the Taniyama-Shimura theorem) and Fermat's theorem.

History of proof

Andrew Wiles was introduced to Fermat's Last Theorem at the age of ten. Then he made an attempt to prove it using methods from a school textbook. Later, he began to study the work of mathematicians who tried to prove this theorem. After entering college, Andrew gave up trying to prove Fermat's Last Theorem and took up the study of elliptic curves under John Coates.

In the 50s and 60s, a connection between elliptic curves and modular forms was suggested by the Japanese mathematician Shimura, who built on ideas expressed by another Japanese mathematician, Taniyama. In Western scientific circles, this hypothesis was known thanks to the work of André Weil, who, as a result of a thorough analysis of it, found a lot of fundamental evidence in its favor. Because of this, the theorem is often referred to as the Shimura-Taniyama-Weil theorem. The theorem says that every elliptic curve over the field of rational numbers is a modular. The theorem was fully proven in 1998 by Christoph Broglie, Brian Conrad, Fred Diamond and Richard Taylor, who used methods published by Andrew Wiles in 1995.

Relationship between Fermat's and Taniyama-Shimura's theorems

Let p be a prime number and a, b and c be natural numbers such that a p +b p =c p . Then the corresponding equation y 2 = x(x - a p)(x + b p) defines a hypothetical elliptic curve, called the Frey curve, which exists if there is a counterexample to Fermat's Last Theorem. Gerhard Frey, based on the work of Helleguark, pointed out that if such a curve existed, it would have extremely unusual properties, and suggested that it might not be modular.

The connection between the Taniyama-Shimura and Fermat theorems was established by Ken Ribet, who built on the work of Barry Mazur and Jean-Pierre Serra. Ribet proved that the Frey curve is not modular. This meant that the proof of the semistable case of the Taniyama-Shimura theorem confirms the truth of Fermat's Last Theorem. After learning about Ken Ribet's 1986 proof, Wiles decided to devote his full attention to proving the Taniyama-Shimura conjecture. While many mathematicians were extremely skeptical about the possibility of finding this proof, Andrew Wiles believed that the conjecture could be proved using twentieth-century methods.

At the very beginning of his work on the Taniyama-Shimura conjecture, Wiles casually mentioned Fermat's Last Theorem in a conversation with his colleagues, which aroused increased interest on their part. But Wiles wanted to focus as much as possible on the problem, and too much attention could only get in the way. To prevent this from happening, Wiles decided to keep the true essence of his research a secret, entrusting his secret only to Nicholas Katz. At that time, Wiles, although he continued teaching at Princeton University, did not engage in any research unrelated to the Taniyama-Shimura hypothesis.

Reflection in culture

Wiles' work on Fermat's Last Theorem was featured in the musical Fermat's Great Tango by Lessner and Rosenbloom.

Wiles and his work are mentioned in the Star Trek: Deep Space Nine episode "Facets".

Awards

Andrew Wiles is the recipient of many international awards in mathematics, including:

Shock Award (1995)
Cole Award (1996)
National Academy of Sciences Mathematics Award from the American Mathematical Society (1996)
Ostrovsky Prize (1996)
Royal Medal (1996)
Wolfskel Prize (1997)
Silver plate from the International Mathematical Union (1998)
King Faisal Prize (1998)
Clay Mathematical Institute Award (1999)
Knighting of the British Empire (2000)
Show Prize (2005)

In 1993, Wiles published a 130-page solution to the equation and, oh! sorry, found a bug. The dramatic history of the proof of this "insidious" theorem is mentioned in many books, films, scripts. It also served as the subject for the musical "Farm's Last Tango", staged at the Clay Institute of Mathematics in 2000 as an afterword to a 350-year-old epic. The main character is the brilliant Andrew, but under a different name.

Andrew John Wiles stubbornly believed in the provability of Fermat's Theorem. It was necessary to study and verify the provability of the entire mathematical arsenal accumulated by mankind. When the theorem Taniyama-Shimura , Japanese mathematicians were able to connect with Fermat's Theorem (this was done by mathematician Ken Ribet), Andrew realized that the techniques of the 20th century would prove Fermat's Last Theorem, although many of Wiles's colleagues were skeptical about this.

The very origin of Fermat's theorem can be called mystical.

"Pierre de Fermat was a man brought up in the traditions of the Renaissance, who lived at a time of rethinking ancient Greek knowledge. But Fermat managed to pose a question that the ancient Greeks did not think to ask. As a result, he became the author of the most difficult task on Earth, which had to be solved by others. As if teasing descendants with false hopes, Fermat left them a brief message in which he informed them that he knew the solution, but kept silent about what exactly it consisted of. Thus began a “race” that lasted three centuries,” writes John Lynch, author of the film “Farm’s Last Theorem”, who is also the editor of the TV Horizon series BBC.

Mathematics is not without reason called the language of God or Nature. Andrew John Wiles, penetrating the secrets of the divine language, made his way to the cherished goal. The Great Theorem did not give rest, the "race", which began three hundred years ago, obliged not to stop. The ingenious mathematician needed solitude, complete concentration.

Andrew John Wiles by that time was a professor of mathematics at Princeton University, a member of the scientific council of the Clay Institute of Mathematics. I had to leave my career, give all my strength to the completion of the last stage of the proof of the Great Theorem.

Wiles' genius, manifested in the solution of one but great theorem, was able to connect many seemingly incompatible ideas, helped to develop new bold concepts. The solution of one problem helped bring mathematics back to its original unity. Mathematicians again spoke in one divine language.

Andrew John Wiles(b. April 11, 1953, Cambridge, UK, Knight Commander of the Order of the British Empire since 2000) - an outstanding English and American mathematician, professor and head of the Department of Mathematics at Princeton University, member of the Scientific Council of the Clay Institute of Mathematics.
He received his bachelor's degree in 1974 from Merton College, Oxford University. He began his scientific career in the summer of 1975 under the guidance of Professor John Coates at Clare College, Cambridge University, where he received his doctorate. From 1977 to 1980, Wiles served as an Associate Fellow at Clare College and Associate Professor at Harvard University. Together with John Coates, he worked on elliptic curve arithmetic with complex multiplication using the methods of Iwasawa theory. In 1982, Wiles moved from the UK to the US.
One of the highlights of his career was the proof Fermat's Last Theorem in 1993 and the discovery of a technical method that allowed him to complete the proof with the help of his former graduate student, R. Taylor, in 1994. He began working on Fermat's theorem in the summer of 1986 after Ken Ribet proved the conjecture about the connection between semistable elliptic curves (a special case of the Taniyama-Shimura theorem) and Fermat's theorem. The basic idea of such a connection belongs to the German mathematician Gerhard Frei. Fermat's Last Theorem states that there are no natural solutions to the Diophantine equation x n + y n = z n for natural n > 2.
Andrew Wiles learned about Fermat's Last Theorem at the age of ten. Then he made an attempt to prove it using methods from a school textbook; Naturally, he didn't succeed. Later, he began to study the work of mathematicians who tried to prove this theorem. After entering college, Andrew gave up trying to prove Fermat's Last Theorem and took up the study of elliptic curves under John Coates.
In the 1950s and 1960s, a connection between elliptic curves and modular forms was suggested by the Japanese mathematician Shimura, who built on ideas expressed by another Japanese mathematician, Taniyama. In Western scientific circles, this hypothesis was known thanks to the work of André Weil, who, as a result of a thorough analysis of it, found partial evidence in favor of it. Because of this, the conjecture is often referred to as the Shimura–Taniyama–Weil theorem. The theorem says that every elliptic curve over an algebraic number field is automorphic. In particular, every elliptic curve over rational numbers must be modular (certain analytic functions of a complex variable are modular). The last property was fully proved in 1998 by Christoph Broglie, Brian Conrad, Fred Diamond and Richard Taylor, who tested some degenerate cases, supplementing the most general case considered by Wiles in 1995. Certainly, Wiles's work is fundamental. However, his method is very special and only works for elliptic curves over rational numbers, while the Taniyama-Shimura conjecture covers elliptic curves over any algebraic number field. Based on this, it is reasonable to assume that there is a more general and more elegant proof of the modularity of elliptic curves.
Andrew Wiles is the recipient of many international prizes in mathematics, including:
Shock Award (1995).
Cole Award (1996).
National Academy of Sciences Mathematics Award from the American Mathematical Society (1996).
Ostrovsky Prize (1996).
Royal Medal (1996).
Wolf Prize in Mathematics (1996).
Wolfskel Prize (1997).
MacArthur Fellowship (1997).
Silver plate from the International Mathematical Union (1998).
King Faisal Prize (1998).
Clay Mathematical Institute Award (1999).
Knight Commander of the Order of the British Empire (2000).
Shaw Award (2005).

In the last twentieth century, an event occurred on a scale that has never been equal in mathematics in its entire history. On September 19, 1994, a theorem formulated by Pierre de Fermat (1601-1665) over 350 years ago in 1637 was proved. It is also known as "Fermat's Last Theorem" or as "Fermat's Great Theorem" because there is also the so-called "Fermat's Little Theorem". It was proved by 41-year-old, up to this point in the mathematical community, nothing particularly unremarkable, and by mathematical standards already middle-aged, Princeton University professor Andrew Wiles.

It is surprising that not only our ordinary Russian inhabitants, but also many people who are interested in science, including even a considerable number of scientists in Russia who use mathematics in one way or another, do not really know about this event. This is shown by the incessant "sensational" reports about "elementary proofs" of Fermat's theorem in Russian popular newspapers and on television. The latest evidence was covered with such information power, as if Wiles's proof, which had passed the most authoritative examination and received the widest fame all over the world, did not exist. The reaction of the Russian mathematical community to this front-page news in the situation of a rigorous proof obtained long ago turned out to be amazingly sluggish. Our aim is to sketch out the fascinating and dramatic story of Wiles' proof in the context of the fairy tale of Fermat's greatest theorem, and to talk a little about the proof itself. Here, we are primarily interested in the question of the possibility of an accessible presentation of Wiles' proof, which, of course, most mathematicians in the world know about, but only very, very few of them can talk about understanding this proof.

So, let's remember Fermat's famous theorem. Most of us have heard of her in one way or another since we were in school. This theorem is related to a very significant equation. This is perhaps the simplest meaningful equation that can be written using three unknowns and one more strictly positive integer parameter. Here it is:

Fermat's Last Theorem states that for values of the parameter (the degree of the equation) greater than two, there are no integer solutions to this equation (except, of course, the solution when all these variables are equal to zero at the same time).

The attractive power of this Fermat's theorem for the general public is obvious: there is no other mathematical statement that has such simplicity of formulation, the apparent accessibility of the proof, as well as the attractiveness of its "status" in the eyes of society.

Before Wiles, an additional incentive for fermatists (as people who maniacally attacked Fermat's problem were called) was the German Wolfskell's prize for proof, established almost a hundred years ago, though small compared to the Nobel Prize - it managed to depreciate during the First World War.

In addition, the probable elementality of the proof was always attracted, since Fermat himself "proved it" by writing on the margins of the translation of Diophantus' Arithmetic: "I have found a truly wonderful proof for this, but the margins here are too narrow to accommodate it."

That is why it is appropriate here to give an assessment of the relevance of popularizing Wiles' proof of Fermat's problem, which belongs to the famous American mathematician R. Murty (we quote from the soon-to-be-published translation of the book "Introduction to Modern Number Theory" by Yu. Manin and A. Panchishkin):

Fermat's Last Theorem holds a special place in the history of civilization. With its external simplicity, it has always attracted both amateurs and professionals ... Everything looks as if it was conceived by some higher mind, which over the centuries has developed various directions of thought only to then reunite them into one exciting fusion to solve the Big Fermat's theorems. No person can claim to be an expert on all the ideas used in this "wonderful" proof. In an era of general specialization, when each of us knows "more and more about less and less", it is absolutely necessary to have an overview of this masterpiece ... "

Let's start with a brief historical digression, largely inspired by Simon Singh's fascinating book Fermat's Last Theorem. Around the insidious theorem, alluring with its apparent simplicity, serious passions have always boiled. The history of her proof is full of drama, mysticism and even direct victims. Perhaps the most iconic victim is Yutaka Taniyama (1927-1958). It was this young talented Japanese mathematician, who in life was distinguished by great extravagance, created the basis for Wiles's attack in 1955. Based on his ideas, Goro Shimura and André Weil a few years later (60-67 years) finally formulated the famous conjecture, proving a significant part of which, Wiles obtained Fermat's theorem as a corollary. The mysticism of the story of the death of the non-trivial Yutaka is connected with his stormy temperament: he hanged himself at the age of thirty-one on the basis of unhappy love.

The whole long history of the enigmatic theorem was accompanied by constant announcements of its proof, starting with Fermat himself. Constant errors in an endless stream of proofs comprehended not only amateur mathematicians, but also professional mathematicians. This has led to the fact that the term "fermatist", applied to Fermat's theorem provers, has become a household word. The constant intrigue with its proof sometimes led to amusing incidents. So, when a gap was discovered in the first version of Wiles' already widely publicized proof, a snide inscription appeared at one of the New York subway stations: "I found a truly wonderful proof of Fermat's Last Theorem, but my train came and I do not have time to write it down."

Andrew Wiles, born in England in 1953, studied mathematics at Cambridge; in graduate school was with Professor John Coates. Under his guidance, Andrew comprehended the theory of the Japanese mathematician Iwasawa, which is on the border of classical number theory and modern algebraic geometry. Such a fusion of seemingly distant mathematical disciplines was called arithmetic algebraic geometry. Andrew challenged Fermat's problem, relying precisely on this synthetic theory, which is difficult even for many professional mathematicians.

After graduating from graduate school, Wiles received a position at Princeton University, where he still works. He is married and has three daughters, two of whom were born "in the seven-year process of the first version of the proof." During these years, only Nada, Andrew's wife, knew that he alone stormed the most impregnable and most famous peak of mathematics. It is to them, Nadia, Claire, Kate and Olivia, that Wiles's famous final article "Modular Elliptic Curves and Fermat's Last Theorem" is dedicated in the central mathematical journal Annals of Mathematics, which publishes the most important mathematical works.

The events around the proof unfolded quite dramatically. This exciting scenario could be called "fermatist-professional mathematician."

Indeed, Andrew dreamed of proving Fermat's theorem since youthful years. But unlike the vast majority of fermatists, it was clear to him that for this he needed to master entire layers of the most complex mathematics. Moving towards his goal, Andrew graduated from the Faculty of Mathematics of the famous University of Cambridge and began to specialize in modern number theory, which is at the junction with algebraic geometry.

The idea of assaulting the shining peak is quite simple and fundamental - the best possible ammunition and careful development of the route.

As a powerful tool for achieving the goal, Wiles himself develops the already familiar Iwasawa theory, which has deep historical roots. This theory generalized Kummer's theory - historically the first serious mathematical theory to storm Fermat's problem, which appeared back in the 19th century. In turn, the roots of Kummer's theory lie in the famous theory of the legendary and brilliant romantic revolutionary Evariste Galois, who died at the age of twenty-one in a duel in defense of the honor of a girl (pay attention, remembering the story with Taniyama, to the fatal role of beautiful ladies in the history of mathematics) .

Wiles is completely immersed in the proof, even stopping participation in scientific conferences. And as a result of a seven-year seclusion from the mathematical community in Princeton, in May 1993, Andrew puts an end to his text - it's done.

It was at this time that a great occasion turned up to notify the scientific world of his discovery - already in June a conference was to be held in his native Cambridge on exactly the right topic. Three lectures at the Cambridge Institute of Isaac Newton excite not only the mathematical world, but also the general public. At the end of the third lecture, on June 23, 1993, Wiles announces the proof of Fermat's Last Theorem. The proof is saturated with a whole bunch of new ideas, such as a new approach to the Taniyama-Shimura-Weyl conjecture, a far advanced Iwasawa theory, a new "deformation control theory" of Galois representations. The mathematical community is looking forward to the verification of the text of the proof by experts in arithmetic algebraic geometry.

This is where the dramatic twist comes in. Wiles himself, in the process of communicating with reviewers, discovers a gap in his proof. The crack was given by the mechanism of "deformation control" invented by him - the supporting structure of the proof.

The gap is discovered a couple of months later by Wiles' line-by-line explanation of his proof to a colleague in his Princeton department, Nick Katz. Nick Katz, being on friendly terms with Andrew for a long time, recommends him cooperation with a promising young English mathematician Richard Taylor.

Another year of hard work passes, connected with the study of an additional tool for attacking an intractable problem - the so-called Euler systems, independently discovered in the 80s by our compatriot Viktor Kolyvagin (already working at New York University for a long time) and Thain.

And here is a new challenge. The unfinished, but still very impressive result of Wiles's work, he reports to the International Congress of Mathematicians in Zurich at the end of August 1994. Wiles fights with all his might. Literally before the report, according to eyewitnesses, he is still feverishly writing something, trying to improve the situation with the “sagging” evidence as much as possible.

After this intriguing audience of the largest mathematicians of the world, Wiles's report, the mathematical community “exhales joyfully” and applauds sympathetically: nothing, the guy, with whomever he happens to, but he advanced science, showing that it is possible to successfully advance in solving such an impregnable hypothesis, which no one has ever done before. didn't even think about doing it. Another fermatist, Andrew Wiles, could not take away the innermost dream of many mathematicians about proving Fermat's theorem.

It is natural to imagine the state of Wiles at that time. Even the support and benevolent attitude of colleagues in the shop could not compensate for his state of psychological devastation.

And so, just a month later, when, as Wiles writes in the introduction to his final proof in the Annals, “I decided to take a last look at the Euler systems in an attempt to revive this argument for proof,” it happened. Wiles had a flash of insight on September 19, 1994. It was on this day that the gap in the proof was closed.

Then things moved at a rapid pace. Already established cooperation with Richard Taylor in the study of the Euler systems of Kolyvagin and Thain made it possible to finalize the proof in the form of two large papers already in October.

Their publication, which occupied the entire issue of the Annals of Mathematics, followed already in November 1994. All this caused a new powerful information surge. The story of Wiles's proof received an enthusiastic press in the United States, a film was made and books were published about the author of a fantastic breakthrough in mathematics. In one evaluation of his own work, Wiles noted that he had invented the mathematics of the future.

(I wonder if this is true? We only note that with all this information flurry, there was a sharp contrast to the almost zero information resonance in Russia, which continues to this day).

Let's ask ourselves a question - what is the "inner kitchen" of obtaining outstanding results? After all, it is interesting to know how a scientist organizes his work, what he focuses on in it, how he determines the priorities of his activity. What can be said in this sense about Andrew Wiles? And surprisingly, in today's era of active scientific communication and collaborative style of work, Wiles had his own way of working on superproblems.

Wiles went to his fantastic result on the basis of intensive, continuous, many years of individual work. The organization of its activities, speaking in official language, was extremely unscheduled. This was categorically not an activity within the framework of a specific grant, on which it is necessary to regularly report and again plan to receive certain results by a certain date each time.

Such activities outside of society, not using direct scientific communication with colleagues, even at conferences, seemed contrary to all the canons of the work of a modern scientist.

But it was individual work that made it possible to go beyond the already established standard concepts and methods. This style of work, closed in form and at the same time free in essence, made it possible to invent new powerful methods and obtain results of a new level.

The problem facing Wiles (the Taniyama-Shimura-Weil conjecture) was not even among the nearest peaks that modern mathematics could conquer in those years. At the same time, none of the experts denied its great importance, and nominally it was in the "mainstream" of modern mathematics.

Thus, Wiles' activities were of a pronounced non-systemic nature and the result was achieved thanks to the strongest motivation, talent, creative freedom, will, more than favorable material conditions for working at Princeton and, most importantly, mutual understanding in the family.

Wiles's proof, which appeared like a bolt from the blue, became a kind of test for the international mathematical community. The reaction of even the most progressive part of this community as a whole turned out to be, oddly enough, rather neutral. After the emotions and enthusiasm of the first time after the appearance of the landmark evidence subsided, everyone calmly continued their business. Experts in arithmetic algebraic geometry slowly studied the "powerful proof" in their narrow circle, while the rest plowed their mathematical paths, diverging, as before, farther and farther from each other.

Let's try to understand this situation, which has both objective and subjective reasons. The objective factors of non-perception, oddly enough, have roots in the organizational structure of modern scientific activity. This activity is like a skating rink going down a slope with tremendous momentum: its own school, its established priorities, its own sources of funding, and so on. All this is good from the point of view of an established system of reporting to the grantor, but it makes it difficult to raise your head and look around: what is really important and relevant for science and society, and not for the next portion of the grant?

Then - again - I do not want to get out of my cozy mink, where everything is so familiar, and climb into another, completely unfamiliar hole. It is not known what to expect there. Moreover, it is obviously clear that they don’t give money for the invasion.

It is quite natural that none of the bureaucratic structures that organize science in different countries, including Russia, did not draw conclusions not only from the phenomenon of the proof of Andrew Wiles, but also from the similar phenomenon of the sensational proof of Grigory Perelman of another, also famous mathematical problem.

The subjective factors of the neutrality of the reaction of the mathematical world to the "millennium event" lie in quite prosaic reasons. The proof is indeed extraordinarily complicated and lengthy. To the layman in arithmetic algebraic geometry, it seems to consist of a layering of the terminology and constructions of the most abstract mathematical disciplines. It seems that the author did not at all aim at being understood by as many interested mathematicians as possible.

This methodological complexity, unfortunately, is present as an inevitable cost of the great proofs of recent times (for example, the analysis of Grigory Perelman's recent proof of the Poincaré conjecture continues to this day).

The complexity of perception is further enhanced by the fact that arithmetic algebraic geometry is a very exotic subfield of mathematics, causing difficulties even for professional mathematicians. The matter was also aggravated by the extraordinary syntheticity of Wiles's proof, which used a variety of modern tools created by a large number of mathematicians in the most recent years.

But it must be taken into account that Wiles did not face the methodical task of explaining - he constructed new method. It was the synthesis of Wiles' own brilliant ideas and a conglomeration of the latest results from various mathematical fields that worked in the method. And it was such a powerful design that rammed an impregnable problem. The proof was not accidental. The fact of its crystallization fully corresponded to both the logic of the development of science and the logic of cognition. The task of explaining such a super-proof seems to be absolutely independent, a very difficult, although very promising problem.

You can test public opinion yourself. Try asking mathematicians you know about Wiles' proof: Who got it? Who understood at least the basic ideas? Who wants to understand? Who felt that this is the new mathematics? The answers to these questions seem to be rhetorical. And it is unlikely that you will meet many who want to break through the palisade of technical terms and master new concepts and methods in order to solve just one very exotic equation. And why for the sake of this task it is necessary to study all this?!

Let me give you a funny example. A couple of years ago, the famous French mathematician, Fields laureate, Pierre Deligne, a prominent specialist in algebraic geometry and number theory, when asked by the author about the meaning of one of the key objects of Wiles' proof - the so-called "ring of deformations" - after half an hour of thought, he said that he was not completely understands the meaning of this object. Ten years have passed since the proof.

Now you can reproduce the reaction of Russian mathematicians. The main reaction is its almost complete absence. This is mainly due to Wiles' "heavy" and "unaccustomed" mathematics.

For example, in classical number theory you won't find such long proofs as Wiles's. As number theorists put it, "the proof must be a page" (Wyles's proof, in collaboration with Taylor, is 120 pages long in the journal version).

It is also impossible to exclude the factor of fear for the unprofessionalism of your assessment: in reacting, you take responsibility for assessing the evidence. And how to do it when you do not know this mathematics?

Characteristic is the position taken by direct specialists in number theory: "... and awe, and burning interest, and caution in the face of one of the greatest mysteries in the history of mathematics" (from the preface to Paulo Ribenboim's book "Fermat's Last Theorem for Amateurs" - the only one available today day to source directly on Wiles' proof for the general reader.

The reaction of one of the most famous contemporary Russian mathematicians, Academician V.I. Arnold on the proof is “actively skeptical”: this is not real mathematics - real mathematics is geometric and has strong connections with physics. Moreover, Fermat's problem itself, by its very nature, cannot generate the development of mathematics, since it is "binary", that is, the formulation of the problem requires an answer only to the question "yes or no". However, mathematical work recent years V.I. Arnold's works turned out to be largely devoted to variations on very close number-theoretic topics. It is possible that Wiles, paradoxically, became an indirect cause of this activity.

At the Mekhmat of Moscow State University, nevertheless, proof enthusiasts appear. The remarkable mathematician and popularizer Yu.P. Solovyov (who died untimely) initiates the translation of E. Knapp's book on elliptic curves with the necessary material on the Taniyama–Shimura–Weil conjecture. Alexey Panchishkin, who is now working in France, in 2001 reads lectures at the Mekhmat, which formed the basis of the corresponding part of his work with Yu.I. Manin of the excellent book mentioned above on modern number theory (published in Russian translation by Sergei Gorchinsky with editing by Alexei Parshin in 2007).

It is somewhat surprising that at the Moscow Steklov Institute of Mathematics, the center of the Russian mathematical world, Wiles' proof was not studied at seminars, but was studied only by individual specialized experts. Moreover, the proof of the already complete Taniyama-Shimura-Weil conjecture was not understood (Wyles proved only a part of it, sufficient for proving Fermat's theorem). This proof was given in 2000 by a whole team of foreign mathematicians, including Richard Taylor, Wiles's co-author on the final stage of the proof of Fermat's theorem.

Also, there were no public statements and, moreover, no discussions on the part of well-known Russian mathematicians about Wiles' proof. A rather sharp discussion is known between the Russian V. Arnold (“a skeptic of the method of proof”) and the American S. Leng (“an enthusiast of the method of proof”), however, its traces are lost in Western publications. In the Russian central mathematical press, since the publication of Wiles' proof, there have been no publications on the subject of the proof. Perhaps the only publication on this topic was the translation of an article by the Canadian mathematician Henry Darmon, even a still incomplete version of the proof in the Mathematical Advances in 1995 (it's funny that the full proof has already been published).

Against this "sleepy" mathematical background, despite the highly abstract nature of Wiles's proof, some intrepid theoretical physicists have included it in the area of their potential interest and began to study it, hoping sooner or later to find applications of Wiles's mathematics. This cannot but rejoice, if only because this mathematics has been practically in self-isolation all these years.

Nevertheless, the problem of adapting the proof, which greatly aggravates its applied potential, has remained and remains very relevant. To date, the original, highly specialized text of Wiles' article and the joint article by Wiles and Taylor has already been adapted, though only for a fairly narrow circle of professional mathematicians. This was done in the mentioned book by Yu. Manin and A. Panchishkin. They succeeded in smoothing over a certain artificiality of the original proof. In addition, the American mathematician Serge Leng, a fierce promoter of Wiles' proof (unfortunately passed away in September 2005), included some of the most important constructions of the proof in the third edition of his now classic university textbook Algebra.

As an example of the artificiality of the original proof, we note that one of the bright features that give this impression is the special role of individual prime numbers such as 2, 3, 5, 11, 17, as well as individual natural numbers such as 15, 30 and 60. Among other things, it is quite obvious that the proof is not geometric in itself in the usual sense. It does not contain natural geometric images that could be attached to for a better understanding of the text. The super-powerful "terminologized" abstract algebra and "advanced" number theory purely psychologically hit the perception of the proof of even a qualified reader-mathematician.

One can only wonder why, in such a situation, the experts of the proof, including Wiles himself, do not “polish” him, do not promote and popularize an obvious “mathematical hit” even in the native mathematical community.

So, in short, today the fact of Wiles's proof is simply the fact of the proof of Fermat's theorem with the status of the first correct proof and the "some super-powerful mathematics" used in it.

The well-known Russian mathematician of the middle of the last century, the former dean of the Mekhmat, V.V. Golubev:

“... according to the witty remark of F. Klein, many departments of mathematics are similar to those exhibitions of the latest models of weapons that exist at firms manufacturing weapons; with all the wit invested by inventors, it often happens that when a real war begins, these innovations turn out to be unsuitable for one reason or another ... The modern teaching of mathematics presents exactly the same picture; students are given very perfect and powerful means of mathematical research ... but further students cannot bear any idea of where and how these powerful and ingenious methods can be applied in solving the main task of all science: in understanding the world around us and in influencing him the creative will of man. At one time, A.P. Chekhov said that if in the first act of the play a gun is hanging on the stage, then it is necessary that at least in the third act it should be fired. This observation is fully applicable to the teaching of mathematics: if any theory is presented to students, then it is necessary to show sooner or later what applications can be made from this theory, primarily in the field of mechanics, physics or technology and in other areas.

Continuing this analogy, we can say that Wiles's proof is extremely favorable material for studying a huge layer of modern fundamental mathematics. Here students can be shown how the problem of classical number theory is closely related to such areas of pure mathematics as modern algebraic number theory, modern Galois theory, p-adic mathematics, arithmetic algebraic geometry, commutative and non-commutative algebra.

It would be fair if Wiles's confidence that the mathematics he invented - mathematics of a new level was confirmed. And I really don’t want this really very beautiful and synthetic mathematics to suffer the fate of an “unfired gun”.

And yet, let us now ask ourselves the question: is it possible to describe Wiles's proof in sufficiently accessible terms for a wide interested audience?

From the point of view of specialists, this is an absolute utopia. But let's still try, guided by the simple consideration that Fermat's theorem is a statement about just integer points of our usual three-dimensional Euclidean space.

We will sequentially substitute points with integer coordinates into Fermat's equation.

Wiles finds an optimal mechanism for recalculating integer points and testing them for satisfaction of the equation of Fermat's theorem (after introducing the necessary definitions, such a recalculation will just correspond to the so-called "modularity property of elliptic curves over the field of rational numbers", described by the Taniyama-Shimura-Weyl conjecture").

The recalculation mechanism is optimized with the help of a remarkable discovery by the German mathematician Gerhard Frey, who connected the potential solution of Fermat's equation with an arbitrary exponent to another, completely different equation. This new equation is given by a special curve (called the Frey elliptic curve). This Frey curve is given by a very simple equation:

The surprise of Frey's idea was the transition from the number-theoretic nature of the problem to its "hidden" geometric aspect. Namely: Frey compared to any solution of Fermat's equation, that is, to numbers satisfying the relation

the above curve. Now it remains to show that such curves do not exist for . In this case, Fermat's Last Theorem would follow from here. It was this strategy that was chosen by Wiles in 1986, when he began his enchanting assault.

Frey's invention at the time of Wiles's "start" was quite fresh (85th year) and also echoed the relatively recent approach of the French mathematician Hellegouarch (70s), who proposed using elliptic curves to find solutions to Diophantine equations, i.e. equations similar to Fermat's equation.

Let's now try to look at the Frey curve from a different point of view, namely, as a tool for recalculating integer points in Euclidean space. In other words, our Frey curve will play the role of a formula that determines the algorithm for such a recalculation.

In this context, it can be said that Wiles invents tools (special algebraic constructions) to control this recalculation. As a matter of fact, this subtle instrumentation of Wiles constitutes the central core and the main complexity of the proof. It is in the manufacture of these tools that Wiles's main sophisticated algebraic discoveries arise, which are so difficult to perceive.

But still, the most unexpected effect of the proof, perhaps, is the sufficiency of using only one "Freev" curve, which is represented by a completely simple, almost "school" dependence. Surprisingly, the use of only one such curve is sufficient to test all points of the three-dimensional Euclidean space with integer coordinates for satisfaction of their relation of Fermat's Last Theorem with an arbitrary exponent.

In other words, the use of only one curve (albeit one that has a specific form), which is understandable even to an ordinary high school student, turns out to be tantamount to building an algorithm (program) for the sequential recalculation of integer points in ordinary three-dimensional space. And not just a recalculation, but a recalculation with simultaneous testing of the whole point for “its satisfaction” with the Fermat equation.

It is here that the phantom of Pierre de Fermat himself arises, since in such a recalculation what is usually called "Ferma't descent", or Fermat's reduction (or "method of infinite descent") comes to life.

In this context, it immediately becomes clear why Fermat himself could not prove his theorem for objective reasons, although at the same time he could well “see” the geometric idea of its proof.

The fact is that the recalculation takes place under the control of mathematical tools that have no analogues not only in the distant past, but also unknown before Wiles even in modern mathematics.

The most important thing here is that these tools are "minimal", ie. they cannot be simplified. Although in itself this "minimalism" is very difficult. And it was Wiles's realization of this non-trivial "minimalness" that became the decisive final step of the proof. This was exactly the same “flash” on September 19, 1994.

Some problem that causes dissatisfaction still remains here - in Wiles this minimal construction is not explicitly described. Therefore, those who are interested in Fermat's problem still have interesting work to do - a clear interpretation of this "minimality" is needed.

It is possible that this is where the geometry of the “algebraized” proof should be hidden. It is possible that Fermat himself felt exactly this geometry when he made the famous entry in the narrow margins of his treatise: "I found a truly remarkable proof ...".

Now let's go directly to the virtual experiment and try to "dig into" the thoughts of the mathematician-lawyer Pierre de Fermat.

The geometric image of the so-called Fermat's little theorem can be represented as a circle rolling "without slipping" along a straight line and "winding" integer points around itself. The equation of Fermat's little theorem in this interpretation also acquires a physical meaning - the meaning of the law of conservation of such motion in one-dimensional discrete time.

We can try to transfer these geometric and physical images to the situation when the dimension of the problem (the number of variables in the equation) increases and the equation of Fermat's little theorem turns into the equation of Fermat's big theorem. Namely: let us assume that the geometry of Fermat's Last Theorem is represented by a sphere rolling on a plane and "winding" on itself whole points on this plane. It is important that this rolling should not be arbitrary, but "periodic" (mathematicians also say "cyclotomic"). Rolling periodicity means that the linear and angular velocity vectors of a sphere rolling in the most general way after a certain fixed time (period) are repeated in magnitude and direction. Such a periodicity is similar to the periodicity of the linear velocity of a circle rolling along a straight line, modeling the “small” Fermat equation.

Accordingly, Fermat's "large" equation acquires the meaning of the law of conservation of the above motion of the sphere already in two-dimensional discrete time. Let us now take the diagonal of this two-dimensional time (it is in this step that the whole complexity lies!). This extremely tricky and turns out to be the only diagonal is the equation of Fermat's Last Theorem when the exponent of the equation is exactly two.

It is important to note that in a one-dimensional situation - the situation of Fermat's Little Theorem - such a diagonal does not need to be found, since time is one-dimensional and there is no reason to take a diagonal. Therefore, the degree of the variable in the equation of Fermat's little theorem can be arbitrary.

So, rather unexpectedly, we get a bridge to the "physicalization" of Fermat's last theorem, that is, to the appearance of its physical meaning. How can one not remember that Fermat was not alien to physics.

By the way, the experience of physics also shows that the conservation laws of mechanical systems of the above form are quadratic in the physical variables of the problem. And finally, all this is quite consistent with the quadratic structure of the laws of conservation of energy in Newtonian mechanics, known from the school.

From the point of view of the above "physical" interpretation of Fermat's Last Theorem, the "minimal" property corresponds to the minimal degree of the conservation law (this is two). And the reduction of Fermat and Wiles corresponds to the reduction of the laws of conservation of recalculation of points to the law of the simplest form. This simplest (minimum complexity) recalculation, both geometrically and algebraically, is represented by the rolling of the sphere on the plane, since the sphere and the plane are “minimal”, as we understand it, two-dimensional geometric objects.

The whole complexity, which at first glance is absent, here lies in the fact that an accurate description of such a seemingly “simple” movement of the sphere is not at all easy. The point is that the "periodic" rolling of the sphere "absorbs" a bunch of so-called "hidden" symmetries of our three-dimensional space. These hidden symmetries are due to non-trivial combinations (compositions) of the linear and angular motion of the sphere - see Fig.1.

It is precisely for the exact description of these hidden symmetries, geometrically encoded by such a tricky rolling of the sphere (points with integer coordinates "sit" at the nodes of the drawn lattice), that Wiles's algebraic constructions are required.

In the geometric interpretation shown in Fig. 1, the linear movement of the center of the sphere “counts” integer points on the plane, and its angular (or rotational) movement provides the spatial (or vertical) component of the recalculation. The rotational motion of the sphere is not immediately possible to "see" in the arbitrary rolling of the sphere on the plane. It is the rotational motion that corresponds to the hidden symmetries of the Euclidean space mentioned above.

The Frey curve introduced above just “encodes” the most aesthetically beautiful recalculation of integer points in space, reminiscent of moving along a spiral staircase. Indeed, if we follow the curve swept by some point of the sphere in one period, we will find that our marked point will sweep the curve shown in Fig. 2, resembling a "double spatial sinusoid" - a spatial analogue of the graph. This beautiful curve can be interpreted as a graph of the "minimum" Frey curve. This is the graph of our testing recalculation.

Having connected some associative perception of this picture, to our surprise we will find that the surface bounded by our curve is strikingly similar to the surface of the DNA molecule - the "corner brick" of biology! It is perhaps no coincidence that the terminology of DNA-encoding constructs from Wiles' proof is used in Singh's book Fermat's Last Theorem.

We emphasize once again that the decisive moment of our interpretation is the fact that the analogue of the conservation law for Fermat's Little Theorem (its degree can be arbitrarily large) is the equation of Fermat's Last Theorem precisely in the case of . It is this effect of "minimality of the degree of the law of conservation of rolling of a sphere on a plane" that corresponds to the statement of Fermat's Great Theorem.

It is possible that Fermat himself saw or felt these geometric and physical images, but at the same time he could not assume that they are so difficult to describe from a mathematical point of view. Moreover, he could not assume that to describe such a non-trivial, but still sufficiently transparent geometry, it would take another three hundred and fifty years of work by the mathematical community.

Now let's build a bridge to modern physics. The geometric image of Wiles' proof proposed here is very close to the geometry of modern physics, trying to get close to the mystery of the nature of gravity - quantum general theory relativity. To confirm this, at first glance unexpected, interaction of Fermat's Last Theorem and "Big Physics", let's imagine that the rolling sphere is massive and "presses through" the plane under it. The interpretation of this "punching" in Fig. 3 strikingly resembles the well-known geometric interpretation of Einstein's general theory of relativity, which describes precisely the "geometry of gravity."

And if we also take into account the present discretization of our picture, embodied by a discrete integer lattice on a plane, then we are completely observing “quantum gravity” with our own eyes!

It is on this major "unifying" physical and mathematical note that we will finish our "cavalry" attempt to give a visual interpretation of Wiles' "super-abstract" proof.

Now, perhaps, it should be emphasized that in any case, whatever the correct proof of Fermat's theorem, it must necessarily use the constructions and logic of Wiles' proof in one way or another. It is simply not possible to get around all this because of the mentioned "minimality property" of Wiles' mathematical tools used for the proof. In our "geometro-dynamical" interpretation of this proof, this "minimality property" ensures the "minimum the necessary conditions” for the correct (i.e. “converging”) construction of the testing algorithm.

On the one hand, this is a huge disappointment for amateur fermatists (unless, of course, they find out about it; as they say, “the less you know, the better you sleep”). On the other hand, the natural "irreducibility" of Wiles' proof formally makes life easier for professional mathematicians - they may not read periodically appearing "elementary" proofs from amateur mathematicians, referring to the lack of correspondence with Wiles's proof.

The general conclusion is that both of them need to “strain themselves” and understand this “savage” proof, comprehending, in fact, “all mathematics”.

What else is important not to miss when summing up this unique story that we have witnessed? The strength of Wiles' proof is that it is not just formal logical reasoning, but is a broad and powerful method. This creation is not a separate tool for proving one single result, but an excellent set of well-chosen tools that allows you to "split" a wide variety of problems. It is also of fundamental importance that when we look down from the height of the skyscraper of Wiles' proof, we see all the previous mathematics. The pathos lies in the fact that it will not be a "patchwork", but a panoramic vision. All this speaks not only of the scientific, but also of the methodological continuity of this truly magical proof. It remains “just nothing” - only to understand it and learn how to apply it.

I wonder what our contemporary hero Wiles is doing today? There is no special news about Andrew. He naturally received various awards and premiums, including the very famous depreciated during the first civil war German Wolfskel Prize. For all the time that has passed since the triumph of the proof of Fermat's problem until today, I managed to notice only one, albeit as always large, article in the same Annals (co-authored with Skinner). Maybe Andrew is hiding again in anticipation of a new mathematical breakthrough, for example, the so-called "abc" hypothesis - recently formulated (by Masser and Osterle in 1986) and considered the most important problem in number theory today (this is the "problem of the century" in the words of Serge Leng ).

Much more information about Wiles' co-author on the final part of the proof, Richard Taylor. He was one of four authors of the proof of the complete Taniyama-Shmura-Weil conjecture and was a serious contender for the Fields Medal at the 2002 Mathematical Congress in China. However, he did not receive it (at that time only two mathematicians received it - the Russian mathematician from Princeton Vladimir Voevodsky "for the theory of motives" and the Frenchman Laurent Laforgue "for an important part of the Langlands program"). Taylor published during this time a considerable number of remarkable works. And just recently, Richard achieved a new great success - he proved a very famous conjecture - the Tate-Saito conjecture, also related to arithmetic algebraic geometry and generalizing the results of German. 19th century mathematician G. Frobenius and 20th century Russian mathematician N. Chebotarev.

Let's finally fantasize a little. Perhaps the time will come when mathematics courses in universities, and even in schools, will be adjusted to the methods of Wiles's proof. This means that Fermat's Last Theorem will become not only a model mathematical problem, but also a methodological model for teaching mathematics. On its example, it will be possible to study, in fact, all the main branches of mathematics. Moreover, future physics, and perhaps even biology and economics, will be based on this mathematical apparatus. But what if?

It seems that the first steps in this direction have already been taken. This is evidenced, for example, by the fact that the American mathematician Serge Leng included in the third edition of his classic manual on algebra the main constructions of Wiles' proof. The Russian Yuri Manin and Aleksey Panchishkin go even further in the mentioned new edition of their "Modern Number Theory", setting out in detail the proof itself in the context of modern mathematics.

And now how not to exclaim: Fermat's great theorem is "dead" - long live the Wiles method!

Andrew Wiles is a professor of mathematics at Princeton University, he proved Fermat's Last Theorem, over which more than one generation of scientists struggled for hundreds of years.

30 years on one task

Wiles first learned about Fermat's last theorem when he was ten years old. He stopped by on his way home from school to the library and became interested in reading the book "The Last Task" by Eric Temple Bell. Perhaps without knowing it yet, from that moment on he devoted his life to finding proof, despite the fact that it was something that eluded the best minds on the planet for three centuries.

Wiles learned about Fermat's Last Theorem when he was ten years old.

He found it 30 years after another scientist, Ken Ribet, proved the connection between the theorem of Japanese mathematicians Taniyama and Shimura and Fermat's Last Theorem. Unlike skeptical colleagues, Wiles immediately understood - this is it, and seven years later he put an end to the proof.

The process of proof itself turned out to be very dramatic: Wiles completed his work in 1993, but right during a public speech he found a significant "gap" in his reasoning. It took two months to find an error in the calculations (the error was hidden among 130 printed pages of solving the equation). Then, for a year and a half, hard work was carried out to correct the error. The entire scientific community of the Earth was at a loss. Wiles completed his work on September 19, 1994, and immediately presented it to the public.

frightening fame

Most of all, Andrew was afraid of fame and publicity. He is very long time refused to appear on television. It is believed that John Lynch was able to convince him. He assured Wiles that he could inspire a new generation of mathematicians and show the power of mathematics to the public.

Andrew Wiles turned down TV appearances for a long time

A little later, a grateful society began to reward Andrew with awards. So on June 27, 1997, Wiles received the Wolfskel Prize, which was approximately $50,000, much less than Wolfskel had intended to keep a century earlier, but hyperinflation has reduced the amount.

Unfortunately, the mathematical equivalent Nobel Prize- The Fields Prize, Wiles simply did not get due to the fact that it is awarded to mathematicians under the age of forty. Instead, he received a special silver plate at the Fields Medal ceremony in honor of his important achievement. Wiles has also won the prestigious Wolf Prize, the King Faisal Prize and many other international awards.

Opinions of colleagues

The reaction of one of the most famous modern Russian mathematicians Academician V. I. Arnold to the proof is "actively skeptical":

This is not real mathematics - real mathematics is geometric and has strong connections with physics. Moreover, Fermat's problem itself, by its very nature, cannot generate the development of mathematics, since it is "binary", that is, the formulation of the problem requires an answer only to the question "yes or no".

At the same time, the mathematical works of V. I. Arnold himself in recent years turned out to be largely devoted to variations on very close number-theoretic topics. It is possible that Wiles, paradoxically, became an indirect cause of this activity.

real dream

When Andrew is asked how he managed to sit in four walls for more than 7 years, doing one task, Wiles tells how he dreamed during his work thatthe time will come when mathematics courses in universities, and even in schools, will be adjusted to his method of proving the theorem. He wanted the very proof of Fermat's Last Theorem to become not only a model mathematical problem, but also a methodological model for teaching mathematics. Wiles imagined that on her example it would be possible to study all the main branches of mathematics and physics.

4 ladies without whom there would be no proof

Andrew is married and has three daughters, two of whom were born "in the seven-year process of the first version of the proof."

Wiles himself believes that without his family he would not have succeeded.

During these years, only Nada, Andrew's wife, knew that he alone stormed the most impregnable and most famous peak of mathematics. It is to them, Nadia, Claire, Kate and Olivia, that Wiles's famous final article "Modular Elliptic Curves and Fermat's Last Theorem" is dedicated in the central mathematical journal Annals of Mathematics, which publishes the most important mathematical works. However, Wiles himself does not at all deny that without his family he would not have succeeded.

All elementary math - average math internet school - great mathematicians - wiles. Simon Singh

History of proof

Reflection in culture

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