123 quotes found
"It is very possible that a proper understanding of string theory will make the space-time continuum melt away.... String theory is a miracle through and through."
"Vibrating strings in 10 dimensions is just a weird fact... An explanation of that weird fact would tell you why there are 10 dimensions in the first place."
"I don't think that any physicist would have been clever enough to have invented string theory on purpose... Luckily, it was invented by accident."
"String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory."
"Most people who haven't been trained in physics probably think of what physicists do as a question of incredibly complicated calculations, but that's not really the essence of it. The essence of it is that physics is about concepts, wanting to understand the concepts, the principles by which the world works."
"Quantum mechanics... developed through some rather messy, complicated processes stimulated by experiment. While it's a very rich and wonderful theory, it doesn't quite have the conceptual foundation of general relativity. Our problem in physics is that everything is based on these two different theories and when we put them together we get nonsense."
"In Newton's day the problem was to write something which was correct - he never had the problem of writing nonsense, but by the twentieth century we have a rich conceptual framework with relativity and quantum mechanics and so on. In this framework it's difficult to do things which are even internally coherent, much less correct. Actually, that's fortunate in the sense that it's one of the main tools we have in trying to make progress in physics. Physics has progressed to a domain where experiment is a little difficult... Nevertheless, the fact that we have a rich logical structure which constrains us a lot in terms of what is consistent, is one of the main reasons we are still able to make advances."
"String theory at its finest is, or should be, a new branch of geometry. ...I, myself, believe rather strongly that the proper setting for string theory will prove to be a suitable elaboration of the geometrical ideas upon which Einstein based general relativity."
"I think one has to regard it as a long term process. One has to remember that String theory, if you choose to date it from the Veneziano model, is already eighteen years old... that quantum electrodynamic theory towards which Planck was heading [in 1900], took fifty years to emerge."
"I would expect that a proper elucidation of what string theory really is all about would involve a revolution in our concepts of the basic laws of physics - similar in scope to any that occurred in the past."
"It's been said that string theory is part of the physics of the twenty-first century that fell by chance into the twentieth century. That's a remark that was made by a leading physicist about fifteen years ago. ...String theory was invented essentially by accident in a long series of events, starting with the Veneziano model... No one invented it on purpose, it was invented in a lucky accident. ...By rights, string theory shouldn't have been invented until our knowledge of some of the areas that are prerequisite... had developed to the point that it was possible for us to have the right concept of what it is all about."
"It was clear that if I didn't spend the rest of my life concentrating on string theory, I would simply be missing my life's calling."
"Even though it is, properly speaking, a postprediction, in the sense that the experiment was made before the theory, the fact that gravity is a consequence of string theory, to me, is one of the greatest theoretical insights ever."
"Good wrong ideas are extremely scarce... and good wrong ideas that even remotely rival the majesty of string theory have never been seen."
"Generally speaking, all the really great ideas of physics are really spin-offs of string theory... Some of them were discovered first, but I consider that a mere accident of the development on planet earth. On planet earth, they were discovered in this order [general relativity, quantum field theory, superstrings, and supersymmetry]... But I don't believe, if there are many civilizations in the universe, that those four ideas were discovered in that order in each civilization."
"If supersymmetry plays the role in physics that we suspect it does, then it is very likely to be discovered by the next generation of particle accelerators, either at Fermilab... or at CERN... Discovery of supersymmetry would be one of the real milestones in physics, made even more exciting by its close links to still more ambitious theoretical ideas. Indeed, supersymmetry is one of the basic requirements of "string theory," which is the framework in which theoretical physicists have had some success in unifying gravity with the rest of the elementary particle forces. Discovery of supersymmetry would would certainly give string theory an enormous boost."
"Just around the same time that the string picture was formed, asymptotic freedom was discovered and made possible, in QCD, a more precise and successful theory of the strong interactions. Yet there has always been a striking analogy between QCD and string theory. If the hypothesis of quark confinement in QCD is true in its usual form, than a widely separated quark and antiquark are joined by a “color flux tube.” This has an obvious analogy to the notion of a meson as a string with charges at its ends, as assumed in string theory. Explaining this analogy would mean understanding quark confinement. This would be quite a nice achievement, since it is a longstanding sore point in theoretical physics that despite real experiments and computer simulations supporting the quark confinement hypothesis and despite a lot of ingenious work explaining qualitative criteria for quark confinement and why this notion is natural, there is no convincing, pencil and paper demonstration of quark confinement in QCD."
"Even before string theory, especially as physics developed in the 20th century, it turned out that the equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle."
"It was found [in the 1970s], unexpectedly and without anyone really having a concept for it, that the rules of perturbation theory can be changed in a way that makes relativistic quantum gravity inevitable rather than impossible. The change is made by replacing point particles by strings. Then Feynman graphs are replaced by Riemann surfaces, which are smooth - unlike the graphs, which have singularities at interaction vertices. The Riemann surfaces can degenerate to graphs in many different ways. In field theory, the interactions occur at the vertices of a Feynman graph. By contrast, in string theory, the interaction is encoded globally, in the topology of a Riemann surface, any small piece of which is like any other. This is reminiscent of how non-linearities are encoded globally in twistor theory."
"Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. ...2. It greatly reduces the number of possible theories. ...3. It gives the first hint that string theory will change our notions of spacetime. Just as in QCD, so also in gravity, many of the interesting questions cannot be answered in perturbation theory. In string theory, to understand the nature of the Big Bang, or the quantum fate of a black hole, or the nature of the vacuum state that determines the properties of the elementary particles, requires information beyond perturbation theory... Perturbation theory is not everything. It is just the way the [string] theory was discovered."
"We know a lot of things, but what we don't know is a lot more."
"... one thing that's worth mentioning, though, it that apart from the dream of understanding physics at a deeper level involving gravity, work in string theory has been useful in shedding lights on more conventional problems in quantum field theory and even in and as well with applications to mathematics. Apart from its intrinsic interest, those successes are one of the things that tend to give us confidence that we're on the right track. Because, speaking personally, I find it implausible that a completely wrong new physics theory would give rise to useful insights about so many different areas."
"The past decade has seen a remarkable renaissance in the interaction between mathematics and physics. This has been mainly due to the increasingly sophisticated mathematical models employed by elementary particle physicists, and the consequent need to use the appropriate mathematical machinery. In particular, because of the strongly non-linear nature of the theories involved, topological ideas and methods have played a prominent part. ... In all this large and exciting field, which involves many of the leading physicists and mathematicians in the world, Edward Witten stands out clearly as the most influential and dominating figure. Although he is clearly a physicist (as his list of publications clearly shows) his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has suprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical ideas."
"For example, Ed Witten recently derived a formula for Donaldson invariants on Kähler manifolds using a twisted version of supersymmetric Yang-Mills theory in four dimensions. His argument depends on the existence of a mass gap, cluster decomposition, spontaneous symmetry breaking, asymptotic freedom, and gluino condensation."
"In the spring of 1985 Ed Witten, one of the most brilliant of young physicists at Princeton University, announced that he would give a talk. ...it was clear that this talk would be an extraordinary occasion. ...our seminar room was packed with people, some old and famous, some young, all eager with expectations. Witten spoke very fast for an hour and a half without stopping. It was a dazzling display of virtuosity. It was also, as Witten remarked quietly at the end, a new theory of the universe. ...When Witten came to the end... The listeners sat silent. ...There were no questions. Not one of us was brave enough to stand up and reveal the depths of our ignorance. ...I describe this scene because it gives a picture of what it means to explore the universe at the highest level of abstraction. Ed Witten is taking a big chance. He has moved so far into abstraction that few even of his friends know what he is talking about. ...He did not invent superstrings. ...Ed Witten's role is to build superstrings into a mathematical structure which reflects to an impressive extent the observed structure of particles and fields in the universe. After they heard him speak, many members of his audience went back to their desks and did the homework they should have done before, reading his papers and learning his language. The next time he talks, we shall understand him better. Next time, we shall perhaps be brave enough to ask questions."
"Witten's excitement arose from the fact that the theory passed several crucial tests which other theories had failed. To have found a theory of the universe which is not mathematically self-contradictory is already a considerable achievement."
"Our Witten, which art in Princeton, Hallowed be thy name. Thy Nobel come, Thy will be done, In CERN as it is in the US. Give us this day our daily string, And forgive us our theory, As we forgive those who do phenomenology. Lead us not into experiment, And deliver us from tests. For thine is the arXiv, Hep-th and math-AG, For ever and ever, Amen"
"Enter superstring theory. The concept that particles are really tiny strings dates from the 1960s, but it took on wings in 1974, when John Schwarz... and Joel Scherk... came to terms with what had been an ugly blemish in their calculations. String theory kept predicting the existence of a particle with zero mass and a spin of two. Schwarz and Scherk realized that this unwelcome particle was nothing other than the graviton, the quantum carrier of gravitational force (Although there is no quantum theory of gravity yet, it is possible to specify some of the characteristics of the quantum particle thought to convey it.) This was liberating: The calculations were saying not only that string theory might be the way to a fully unified account of all particles and forces but that one could not write a string theory without incorporating gravity. Ed Witten... recalled that this new constituted "the greatest intellectual thrill of my life.""
"In the high carrels of theoretical physics, where intelligence is taken for granted, Witten is regarded as preternaturally, almost forbiddingly, smart. ...he wears the habitual small smile of the theoretician for whom sustained mathematical thinking has something like the emotional qualities that mystics associate with meditation. He speaks in a soft, high-pitched voice, floating short, precise sentences punctuated by witty little silences—the speech pattern of a man who has learned that he thinks too fast to otherwise be understood. Though he is the son of a theoretical physicist, he came to science in a roundabout fashion."
"The Theory of Everything, if you dare be bold, Might be something more than a string orbifold. While some of your leaders have got old and sclerotic, Not to be trusted alone with things heterotic, Please heed our advice that you are not smitten— The Book is not finished, the last word is not Witten."
"A crucial observation, central to the second superstring revolution initiated by Witten and others in 1995, is that string theory actually includes ingredients with a variety of different dimensions: two-dimensional Frisbee-like constituents, three-dimensional blob-like constituents, and even more exotic possibilities to boot."
"In the mid-1990s, Witten, based on his own insights and previous work by Michael Duff... and Chris Hull and Paul Townsend... gave convincing evidence that... String theory... to most string theorists' amazement, actually requires ten space dimensions and one time dimension, for a total of eleven dimensions."
"Edward Witten is fond of declaring that string theory had already made a dramatic and experimentally confirmed prediction: "String theory had the remarkable property of predicting gravity." What Witten means by this is that both Newton and Einstein developed theories of gravity because their observations of the world clearly showed them that gravity exists, and that, therefore, it required an accurate and consistent explanation. On the contrary, a physicist studying string theory—even if he or she was completely unaware of general relativity—would be inexorably led to it by the string framework."
"Work by Strominger and Witten showed that the masses of the particles in each family depend upon... the way in which the boundaries of the various multidimensional holes in the Calabi-Yau shape intersect and overlap with one another. ...as strings vibrate through the extra curled-up dimensions, the precise arrangement of the various holes and the way in which the Calabi-Yau shape folds around them has a direct impact on the possible resonant patterns of vibration. ...as with the number of families, string theory can provide us with a framework for answering questions—such as why the electron and other particles have the masses they do—on which previous theories are completely silent. ...carrying through with such calculations requires that we know which Calabi-Yau space to take for the extra dimensions."
"In the mid-1980s Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten... discovered that each hole—the term used in a precisely defined mathematical sense—contained within the Calabi-Yau shape gives rise to a family of lowest-energy string vibrational patterns. ...among these preferred Calabi-Yau shapes are ones that also yield just the right number of messenger particles as well as just the right electric charges and nuclear force properties to match the particles [listed in the book] ..."
"In the spring of 1995... Drawing on the work of a number of string theorists (including Chris Hull, Paul Townsend, Ashoke Sen, Michael Duff, John Schwarz and many others), Edward Witten—who for decades has been the world's most renowned string theorist—uncovered a hidden unity that tied all five string theories together. Witten showed that rather than being distinct, the five theories are actually just five different ways of mathematically analyzing a single theory. ...The unifying master theory has tentatively been called M-theory."
"Much as Kaluza found that a universe with five spacetime dimensions provided a framework for unifying electromagnetism and gravity, and much as string theorists found that a universe with ten spacetime dimensions provided a framework for unifying quantum mechanics and general relativity, Witten found that a universe with eleven spacetime dimensions provided a framework for unifying all string theories."
"Between sessions at a physics conference, I asked a number of attendees: Who is the smartest physicist of them all? ...the name mentioned most often was Witten's. He seemed to evoke a special kind of awe, as though he belonged to a category unto himself. He is often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind since Newton."
"Edward Witten... dominates the world of theoretical physics. Witten is currently the "leader of the pack," the most brilliant high-energy physicist, who sets trends in the physics community the way Picasso would set trends in the art world. Hundreds of physicists follow his work religiously to get a glimmer of his path-breaking ideas."
"The boundaries of physics have been changing. Now scientists ask not only how the world works (a question the Standard Model answers) but why it works that way (a question the Standard Model cannot answer). Einstein asked "why" earlier in the century, but only in the past decade or so have the "why" questions become normal scientific research in particle physics, rather than philosophical afterthoughts. One ambitious approach to "why" is known as string theory, which is formulated in an eleven-dimensional world. Work on string theory has proceeded so far by study of the theory itself, rather than via the historical fruitful interplay of experiment and theory. As Edward Witten remarks... string theory predicts that nature should be supersymmetric. Supersymmetry is a surprising and subtle idea—the idea that the equations representing basic laws of nature don't change if certain particles in the equations are interchanged with one another."
"He never does calculations except in his mind. I will fill pages with calculations before I understand what I'm doing. But Edward will sit down only to calculate a minus sign, or a factor of two."
"The positive energy theorem was for half a century or more an open challenge to relativists. Many attempts were made to prove flat spacetime was stable, but none completely succeeded completely until a majestic tour de force of geometric reasoning of Shoen and Yau. This was followed two years later by a proof of Witten, which was as elegant as it was short. It is this proof of Witten’s that we take as a template... for the quantum theory."
"We shouldn't toss comparisons to Einstein around too frequently, but when it comes to Witten... He's head and shoulder above the rest. He's started whole groups of people on new paths. He produces elegant, breathtaking proofs which people gasp at, which leave them in awe."
"My stay was nearly over when one day Ed Witten said to me, "I just learnt a new way to find exact S-matrices in two dimensions invented by Zamolodchikov and I want to extend the ideas to supersymmetric models. You are the S-matrix expert, aren't you? Why don't we work together?" I was delighted. All my years of training in Berkeley gave me a tremendous advantage over Ed—for an entire week."
"The MacArthur Foundation chose Witten in 1982 for one of its earliest “genius” grants, and he is probably the only person that virtually everyone in the theoretical physics community would agree deserves the genius label. He has received a wide array of honors, including the most prestigious award in mathematics, the Fields Medal, in 1990. The strange situation of the most talented person in theoretical physics having received the mathematics equivalent of a Nobel Prize, but no actual Nobel Prize in physics, indicates both how unusual a figure Witten is, and also how unusual the relationship between mathematics and physics has become in recent years. When I was a graduate student at Princeton, one day I was leaving the library perhaps thirty feet or so behind Witten. The library was underneath a large plaza separating the mathematics and physics buildings, and he went up the stairs to the plaza ahead of me, disappearing from view. When I reached the plaza he was nowhere to be seen, and it is quite a bit more than thirty feet to the nearest building entrance. While presumably he was just moving a lot faster than I was, it crossed my mind at the time that a consistent explanation for everything was that Witten was an extraterrestrial being from a superior race who, since he thought no one was watching, had teleported back to his office. More seriously, Witten’s accomplishments are very much a product of the combination of a huge talent and a lot of hard work. His papers are uniformly models of clarity and of deep thinking about a problem, of a sort that very few people can match. Anyone who has taken the time to try to understand even a fraction of his work finds it a humbling experience to see just how much he has been able to achieve. He is also a refreshing change from some of the earlier generations of famous particle theorists, who could be very entertaining, but at the same time were often rather insecure and not known always to treat others well."
"After Einstein’s dramatic success with general relativity in 1915, he devoted most of the rest of his career to a fruitless attempt to unify electromagnetism and gravity using the sorts of geometric techniques that had worked in the case of general relativity. We now can see that this research program was seriously misguided, because Einstein was ignoring the lessons of quantum mechanics. To understand electromagnetism fully one must deal with quantum field theory and QED in one way or another, and Einstein steadfastly refused to do this, continuing to believe that a theory of classical fields could somehow be made to do everything. Einstein chose to ignore quantum mechanics despite its great successes, hoping that it could somehow be made to go away. If Witten had been in Einstein’s place, I doubt that he would have made this mistake, since he is someone who has always remained very involved in whatever lines of research are popular in the rest of the theoretical community. On the other hand, this example does show that genius is no protection against making the mistake of devoting decades of one’s life to an idea that has no chance of success."
"It is less than four years since cohomological methods (i.e. methods of Homological Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper[11], and it seems certain that they are to overflow the part of mathematics in the coming years, from the foundations up to the most advanced parts. ... [11] Serre, J. P. Faisceaux algébriques cohérents. Ann. Math. (2), 6, 197–278 (1955)."
"The introduction of the digit 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps..."
"The question you raise “how can such a formulation lead to computations” doesn’t bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand – and it always turned out that understanding was all that mattered."
"I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance."
"We should ask our fellow physicists to invent a principle of anti-interference, which would bring light out of two obscurities (Leray and Grothendieck)."
"Alexandre Grothendieck was very different from Weil in the way he approached mathematics: Grothendieck was not just a mathematician who could understand the discipline and prove important results—he was a man who could create mathematics. And he did it alone."
"No one but Grothendieck could have taken on algebraic geometry in the full generality he adopted and seen it through to success. It required courage, even daring, total self confidence and immense powers of concentration and hard work. Grothendieck was a phenomenon."
"Many people who knew Grothendieck during his time at I.H.E.S. speak of his kindness, his openness to any kind of question, his gentle humor. He was often barefoot. He fasted once a week in opposition to the war in Vietnam. Mazur recalled that Grothendieck had met a family at the local train station with nowhere to stay, and he invited them to live in the basement apartment of his home. He had a machine installed that helped make taramasalata—a fish-roe spread—so that they could sell prepared food at the market."
"He really never worked on examples, I only understand things through examples and then gradually make them more abstract. I don’t think it helped Grothendieck in the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract possible way. It’s just very strange. That’s the way his mind worked."
"Weil's new mathematical language, algebraic geometry, had enabled him to articulate subtleties about solutions to equations that hitherto had been impossible. But if there was any hope of extending Weil's ideas to prove the Riemann Hypothesis, it was clear they would need to be developed beyond the foundations he had laid in his prison cell in Rouen. It would be another mathematician from Paris who would bring the bones of Weil's new language to life. The master architect who performed this task was one of the strangest and most revolutionary mathematicians of the twentieth century - Alexandre Grothendieck."
"Applications in arithmetic geometry (such as Weil conjectures, Ramanujan conjecture, Mordell conjecture, Shafarevich conjecture, Tate conjectures) are unthinkable in the classical style, these really need Grothendieck's foundations of algebraic geometry."
"Many mathematicians are rather childlike, unworldly in some sense, but Grothendieck more than most. He just seemed like an innocent—not very sophisticated, no pretense, no sham. He thought very clearly and explained things very patiently, without any sense of superiority. He wasn’t contaminated by civilization or power or one-up-manship."
"In Récoltes et Semailles, Grothendieck counts his twelve disciples. The central character is Pierre Deligne, who combines in this tale the features of John, "the disciple whom Jesus loved”", and Judas the betrayer. The weight of symbols!"
"Jean Dieudonné and Laurent Schwartz were able to discipline Grothendieck just enough to prevent him from running off in all directions, and to restrain his excessive attraction to extreme generality."
"Grothendieck’s undertaking throve thanks to unexpected synergies: the immense capacity for synthesis and for work of Dieudonné, promoted to the rank of scribe, the rigorous, rationalist and well-informed spirit of Serre, the practical know-how in geometry and algebra of Zariski’s students, the juvenile freshness of the great disciple Pierre Deligne, all acted as counterweights to the adventurous, visionary and wildly ambitious spirit of Grothendieck."
"When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?"
"Bombieri is one of the guardians of the Riemann Hypothesis and is based at the prestigious Institute for Advanced Study in Princeton, once home to Einstein and Gödel. He is very softly spoken, but mathematicians always listen carefully to anything he has to say."
"The nice thing about mathematics is doing mathematics."
"From Grothendieck] and his example, I have also learned not to take glory in the difficulty of a proof: difficulty means we have not understood. The ideal is to be able to paint a landscape in which the proof is obvious."
"In Récoltes et Semailles, Grothendieck counts his twelve disciples. The central character is Pierre Deligne, who combines in this tale the features of John, “the disciple whom Jesus loved”, and Judas the betrayer. The weight of symbols!"
"Deligne’s method was totally perpendicular to Grothendieck’s: he knew every trick of his master’s trade by heart, every concept, every variant. His proof, given in 1974, is a frontal attack and a marvel of precision, in which the steps follow each other in an absolutely natural order, without surprises. Those who heard his lectures had the impression, day after day, that nothing new was happening–whereas every lecture by Grothendieck introduced a whole new world of concepts, each more general than the one before–but on the last day, everything was in place and victory was assured. Deligne knocked down the obstacles one after the other, but each one of them was familiar in style. I think that this opposition of methods, or rather of temperament, is the true reason behind the personal conflict which developed between the two of them. I also think that the fact that “John, the disciple that Jesus loved” wrote the last Gospel by himself partly explains Grothendieck’s furious exile that Grothendieck has imposed upon himself."
"Usually mathematicians are either theory builders, who develop tools, or problem-solvers, who use those tools to find solutions... Deligne is unusual in being both. He’s got a very special mind."
"It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathematicians have a good ear; some not (and some prefer the slangy expressions such as 'iff'). That's life."
"You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one."
"If Serre was a Mozart, Grothendieck was a Wagner."
"Je pourrais dire, en exagérant à peine, qu’entre le début des années cinquante jusque vers l’année 1966, donc pendant une quinzaine d’année, tout ce que j’ai appris en "géométrie" (dans un sens très large, englobant la géométrie algébrique ou analytique, la topologie et l’arithmétique), je l’ai appris par Serre, quand je ne l’ai pas appris par moi-même dans mon travail mathématique. C’est en 1952 je crois, quand Serre est venu à Nancy (où je suis resté jusqu’en 1953), qu’il a commencé à devenir pour moi un interlocuteur privilégié - et pendant des années, il a été même mon seul interlocuteur pour les thèmes se plaçant en dehors de l’analyse fonctionnelle. - Grothendieck, Récoltes et Semailles."
"One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. [...] When I got to the room, fifteen or so researchers were there, along with a few students seated in the rear. A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things. [...] At the end of the seminar, Serre came up to me and said—and here I quote verbatim: “You’ll have to explain that to me again, because I didn’t understand anything.” [...] the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension. It takes a lot of nerve to listen closely to a presentation, then go up to the speaker, smile, and tell him that you “didn’t understand anything.” I never would have dared. Why did he do it? I first told myself it must be one of the things you have the right to do when you’re Jean-Pierre Serre. Then I realized that could also work the other way: what if this technique had actually helped him become Jean-Pierre Serre?"
"In the winter of 2008, Jenifer and I visited Chennai Mathematical Institute. This remarkable Institute is the creation of Seshadri. It is a unique blend of an American style liberal arts college with traditional Indian guru one-on-one teaching, adding physics, computer science, history and music to its maths curriculum. Only in India could an intellectual with no business or management experience, who spends all his spare time singing classical south Indian music, have been the catalyst for such a unique educational experiment."
"On a more personal note, I see many similarities between India's Dalit problems and the African-American problems that have rocked the US since its beginnings. For this reason, I personally take Dr. Ambedkar as one of my heroes."
"There is only one other survey, Datta and Singh’s 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics."
"John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective "genius". I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on "s". His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me."
"I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate."
"The beauty of mathematics only shows itself to more patient followers."
"I don’t think that everyone should become a mathematician, but I do believe that many students don’t give mathematics a real chance."
"It's not only the question, but the way you try to solve it."
"I think it's rarely about what you actually learn in class it's mostly about things that you stay motivated to go and continue to do on your own.""
"A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail"
"Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson's paper appeared, both Kapranov and I had strong reputations. Simpson's paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it."
"It soon became clear that the only real long-term solution to the problems that I encountered is to start using computers in the verification of mathematical reasoning."
"Today we face a problem that involves two difficult to satisfy conditions. On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics. On the other hand, we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics."
"Voevodsky's construction makes it possible to obtain an “incarnation” of the motivic cohomology but it does not, however, find a solution to the standard conjectures, which are still today – along with the Hodge conjecture – the fundamental open question in modern algebraic geometry."
"Within mathematics itself, Voevodsky's proposal, if adopted, will create a new paradigm. In his “fairy tale” and some of his other papers, Langlands made deft use of categories and even 2-categories, but number theory is only superficially categorical, and so is the Langlands program. In the event that Univalent Foundations could shed light on a guiding problem in number theory — the Riemann hypothesis or the Birch Swinnerton-Dyer conjecture, which is not so far removed from Voevodsky's motives — then we could easily see Grothendieck's program absorbing the Langlands program within Voevodsky's new paradigm."
"Infinity-groupoids encode all the paths in a space, including paths of paths, and paths of paths of paths. They crop up in other frontiers of mathematical research as ways of encoding similar higher-order relationships, but they are unwieldy objects from the point of view of set theory. Because of this, they were thought to be useless for Voevodsky’s goal of formalizing mathematics. Yet Voevodsky was able to create an interpretation of type theory in the language of infinity-groupoids, an advance that allows mathematicians to reason efficiently about infinity-groupoids without ever having to think of them in terms of sets. This advance ultimately led to the development of univalent foundations."
"His mathematical prose is simple, spare, and exceedingly beautiful. His prose style is to mathematics what Hemingway's is to English or Simenon's to French."
"Fields medalists are nothing out of the ordinary at Princeton—you sometimes find yourself seated next to three or four of them at lunch!"
"If paparazzi specialized in mathematical celebrities they'd camp outside the dining hall at the IAS and come away with a new batch of pictures every day."
"In some situations however, when you are deeply with your problem, you feel at home anywhere just thinking about your problem. Some of my best work was done in hotels, on the train, and there's no rule. More important when you think is what goes on inside rather outside. [...] The best thoughts can be nearly everywhere."
"When you are into mathematics, you have been so high on the scale of complexity of reasoning that you are living in some kind of altered reality. You think everybody on the street is able to understand complicated reasoning [...]. And you get very frustrated, when you discover that's not the case."
"If the proof is correct then no other recognition is needed."
"By the end of 2006 it was generally believed that Perelman’s proof was correct. That year, the journal Science named Perelman’s proof the “Breakthrough of the Year.” Like Smale and Freedman before him, the forty-year old Perelman was tapped to be a Fields Medals recipient for his contributions to the Poincaré conjecture (in fact, Thurston also received a Fields Medal for his work that indirectly led to the final proof). The countdown for the $1 million prize had begun (some wonder if Perelman and Hamilton will be offered the prize jointly)."
"Revolutions in mathematics are quiet affairs. No clashing armies and no guns. Brief news stories far from the front page. Unprepossessing. Just like the raw damp Monday afternoon of April 7, 2003, in Cambridge, Massachusetts. Young and old crowded the lecture theater at the Massachusetts Institute of Technology (MIT). They sat on the floor and in the aisles, and stood at the back. The speaker, Russian mathematician Grigory Perelman, wore a rumpled dark suit and sneakers, and paced while he was introduced."
"In Tehran university, at the math club, where I was studying the pictures of Fields medalists were lining the walls. I looked at them and said to myself, ‘Will I ever meet one of these people?’ At that time in Iran, I couldn’t even know that I’d be able to go to the West."
"To go from the point that I didn’t imagine meeting these people to the point where someday I hold a medal myself — I just couldn’t imagine that this would come true."
"I read all these books and I had the feeling that just reading things is not enough. I also wanted to create my own stuff, to create something new."
"Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improved clarity. I'm happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion.Over the years, this has helped me develop clarity in some things, but I remain muddled in many others."
"Mathematics is primarily a tool for human thought."
"The most important thing about mathematics is how it resides in the human brain."
"People can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality...computers are far better at formal computation and formal reasoning, but humans are far better mathematicians."
"When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze."
"The term `geometry'...refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics."
"I was really amazed by my first encounters with serious mathematics textbooks...I could appreciate that the mathematics was an impressive intellectual edifice, and I could follow the steps of proofs. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited. It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)n remark)m style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind."
"... the dialogue between mathematical physics, geometry, and algebraic topology. The interaction between these subjects has been such a dominant feature of research developments in the past few years that it seems scarcely necessary to recite a list of examples: in fact, on the mathematical side it is quite hard to think of active areas in geometry and topology which have not been noticeably influenced by insights from physics—where by "physics" we mean particularly quantum field theory—and on the other hand the geometrisation of fundamental physical concepts is a profound and pervasive development."
"Geometers have studied the topology of closed surfaces and their higher- dimensional analogues (manifolds) for a long time. But a remarkable breakthrough came in the early 1980s when Simon Donaldson found some totally new and unexpected invariants of four-dimensional manifolds. These were based on the Yang–Mills equations of physics but it was not until later that Edward Witten again showed how to interpret Donaldson's invariants in terms of quantum field theory. Later still, using duality ideas from string theory, Witten and Seiberg made a significant improvement of Donaldson theory which led to solutions of old problems."
"... It is true — certainly in mathematics; what I say now will not apply to the School of Historical Studies where people tend to make their main contributions at a much later age — in mathematics and physics, the prime period in one's life in probably over by, say, 45 or so. That's a bit conservative. Some might put it a bit earlier. Of course one can continue to work, and do very good work afterwards, but certainly the most productive period would be before that, between 25 and 45."
"I think in some sense much has to do with luck. If you are lucky many times, then you are a genius, of course. You may be lucky just a few times or some people might not have any great luck at all. I don't know really what is the reason for this. I think what lies behind having luck is first of all if you have a background that is a bit different from what everybody else has so that you are not encumbered with precisely the same knowledge and are not thinking exactly the same way. It also helps if you can benefit by accidents, facts that you come across quite accidentally and start thinking about and see there is something more. I would say that most of the better things I have done all came about not because I set out from the beginning to do them. Something shifted the focus of my attention completely and I ended up doing something rather different. One has to be able to see opportunities and learn to utilize them. Real, original work, I think, comes about in this way."
"Selberg’s work in automorphic forms and number theory led him naturally to the study of lattices (that is, discrete subgroups of finite covolume) in semi-simple Lie groups. His proof of local rigidity and, as a consequence, algebraicity of the matrix entries of cocompact lattices in groups such }}, n > 2, marked the beginnings of modern . His results were followed by proofs of local rigidity for cocompact lattices in all groups other than the familiar , where its failure reflects the well-known local deformation theory of Riemann surfaces. These results inspired to find and prove his celebrated “strong rigidity” results for such lattices in groups other than . From his work on local rigidity and algebraicity, Selberg was led to the bold conjecture that, in the higher rank situation, much more is true; namely, that all lattices are arithmetic (i.e., they can be constructed by some general arithmetic means). He was able to prove this conjecture in the simplest case of a non-cocompact irreducible lattice in the product of at least two ’s. The full Selberg arithmeticity conjecture in groups of rank at least two was established by , who introduced measure- and p-adic theoretic ideas into the problem, as well as what is now called “super-rigidity”."
"The classification of s shows that every finite simple group either fits into one of about 20 infinite families, or is one of 26 exceptions, called . The is the largest of the sporadic finite simple groups, and was discovered by and ... Its order is 8080,17424,79451,28758,86459,90496,17107,57005,75436,80000,00000 = 246 ⋅ 320 ⋅ 59 ⋅ 76 ⋅ 112 ⋅ 133 ⋅ 17 ⋅ 19 ⋅ 23 ⋅29 ⋅ 31 ⋅ 41 ⋅ 47 ⋅ 59 ⋅ 71 (which is roughly the number of elementary particles in the earth). The smallest irreducible representations have dimensions 1, 196883, 21296876, ... The has the power series expansion j(τ) = q−1 + 744 + 196884q + 21493760q2 +... where q = e2π iτ, and is in some sense the simplest nonconstant function satisfying the functional equations j(τ) = j(τ + 1) = j(−1/τ). noticed some rather weird relations between coefficients of the elliptic modular function and the representations of the monster as follows: 1 = 1 196884 = 196883 + 1 21493760 = 21296876 + 196883 + 1 where the numbers on the left are coefficients of j(τ) and the numbers on the right are dimensions of irreducible representations of the monster. At the time he discovered these relations, several people thought it so unlikely that there could be a relation between the monster and the elliptic modular function that they politely told McKay that he was talking nonsense. The term “monstrous moonshine” (coined by ) refers to various extensions of McKay’s observation, and in particular to relations between sporadic simple groups and modular functions."
"... if you take the s, we have a classification of them ... And then we've got a very simple explanation of why this list turns up, that they more or less correspond to finite reflection groups. And we know who to classify finite reflection groups. ... we can give single uniform construction of all the compact Lie groups. But there's nothing like that for the sporadic groups."
"(quote at 35:35 of 1:36:06 in video)"
"as quoted by Allyn Jackson: (quote from p. 1015)"
"The notion of infinitely near s is classical and well understood for s. We generalize the notion to higher dimensions and to develop a general theory, in terms of idealistic exponents and certain s associated with them. We then gain a refined generalization of the classical notion of first characteristic exponents. On the level of technical base in the higher dimensional theory, there are some powerful tools, referred to as Three Key Theorems, which are namely Differentiation Theorem, Numerical Exponent Theorem and Ambient Reduction Theorem."
"Les théorèmes démontrés à l'aide du théorème de Hironaka ne se comptent plus. Pour la plupart, on a l'impression que la résolution des singularités est vraiment au fond du problème, et ne pourra être évitée par recours à des méthodes différentes."
"trans.: The theorems demonstrated using Hironaka's theorem are countless. For the most part, one has the impression that the is really at the heart of the problem, and cannot be avoided by resorting to different methods."
"... I once was in Japan and eating alone. A Japanese couple came and wanted to practice their English. They asked me what I did. I said I was a mathematician but could not get the idea across until I said: “Like Hironaka”. Wow! It’s as though in America I’d said “Like ”, or , or . Perhaps Hironaka’s name is ... the only one known, but in America I don’t think any mathematician’s name would get any response."
"When a student picks a research topic, the decision is influenced by the research interests of the advisor, the state of the field in which the problem lives, the personality of the advisor, the chemistry between them ... who else in working on whichever problem. It's immensely complicated. And asking for a sort of simple prescription for how to assign research problems — it reminds a little of the question of how to decide who should marry whom."
"First of all, there are problems that no one knows how to solve. There are problems that have been studied but untouched, or problems on which there is partial progress. There are problems that sound compelling when formulated, but which no one has thought of yet. There are concepts which are very useful in solving problems .. or which perhaps ... sound very natural and compelling when formulated, but have not been formulated yet. And all of these things interact. So, by solving problems, one is led to concepts — and, by thinking about concepts, one is led to problems."
"Charles Fefferman (Charlie) is a mathematician of the first rank whose outstanding findings, both classical and revolutionary, have inspired further research by many others. He is one of the most accomplished and versatile mathematicians of all time, having so far contributed with fundamental results to harmonic analysis, s, , , quantum mechanics, fluid mechanics, and , together with more sporadic incursions into other subjects such as neural networks, financial mathematics, and crystallography."