86 quotes found
"I think I'll stop here."
"I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days."
"I loved doing problems in school. I'd take them home and make up new ones of my own."
"But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem -- Fermat's Last Theorem."
"Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it."
"I realized that anything to do with Fermat's Last Theorem generates too much interest."
"I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal."
"Young children simply aren't interested in Fermat. They just want to hear a story and they're not going to let you do anything else."
"Fermat couldn't possibly have had this proof."
"I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof."
"But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof."
"Fermat was my childhood passion."
"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future."
"But perhaps that's always the way with math problems, and we just have to find new ones to capture our attention."
"Certainly one thing that I've learned is that it is important to pick a problem based on how much you care about it."
"However impenetrable it seems, if you don't try it, then you can never do it."
"Always try the problem that matters most to you."
"I had this rare privilege of being able to pursue in my adult life, what had been my childhood dream."
"I know it's a rare privilege, but if one can really tackle something in adult life that means that much to you, then it's more rewarding than anything I can imagine."
"Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them."
"Originally the Kolyvagin-Flach method only worked under particularly constrained circumstances, but Wiles believed he had adapted and strengthened it sufficiently to work for all his needs. According to Katz this was not necessarily the case, and the effects were dramatic and devastating. The error did not necessarily mean that Wiles's work was beyond salvation, but it did mean that he would have to strengthen his proof. The absolutism of mathematics demanded that Wiles demonstrate beyond all doubt that his method worked for every element of every E-series and M-series."
"At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste . . . Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics."
"In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun)."
"It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: “There is a t_1<0 such that the image of t_1 under the natural mapping t_1 \mapsto {\rm Petya}(t_1) belongs to the set of dirty hands, and a t_2, t_1, such that the image of t_2 under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.”"
"A person, who had not mastered the art of the proofs in high school, is as a rule unable to distinguish correct reasoning from that which is misleading. Such people can be easily manipulated by the irresponsible politicians."
"In the last 30 years, the prestige of mathematics has declined in all countries. I think that mathematicians are partially to be blamed as well—foremost, Hilbert and Bourbaki—the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms."
"All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.)."
""In almost all textbooks, even the best, this principle is presented so that it is impossible to understand." (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition."
"When you are collecting mushrooms, you only see the mushroom itself. But if you are a mycologist, you know that the real mushroom is in the earth. There’s an enormous thing down there, and you just see the fruit, the body that you eat. In mathematics, the upper part of the mushroom corresponds to theorems that you see. But you don’t see the things which are below, namely problems, conjectures, mistakes, ideas, and so on. You might have several apparently unrelated mushrooms and are unable to see what their connection is unless you know what is behind."
"Such axioms, together with other unmotivated definitions, serve mathematicians mainly by making it difficult for the uninitiated to master their subject, thereby elevating its authority."
"Mathematics is the part of physics where experiments are cheap."
"The axiomization and algebraization of mathematics, after more than 50 years, has led to the illegibility so such a large number of mathematical texts that the threat of complete loss of contact with physics and the natural sciences has been realized."
"Let me just say that Arnold was a geometer in the widest possible sense of the word, and that he was very fast to make connections between different fields."
"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."
"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."
"Although there are at present many occupations that require a good deal of skill and training in advanced mathematics, mathematics itself is still often regarded as a curious profession demanding singular talents and a singular personality."
"Mathematical maturity is anyhow an uncertain concept, for the mind’s natural competence seems to change with age, its purview variable."
"What I have achieved has been largely a matter of chance. Many problems I thought about at length with no success. With other problems, there was the inspiration—indeed, some that astound me today. Certainly the best times were when I was alone with mathematics, free of ambition and of pretense, and indifferent to the world."
"Langlands' life has been by no means as extravagant as Grothendieck's, but his romanticism is evident to anyone who reads his prose; the audacity of his program, one of the most elaborate syntheses of conjectures and theorems ever undertaken, has few equivalents in any field of scholarship."
"He was a visionary. He pointed us into a direction where we can go and find the truth, find out what’s really going on. It’s about seeing the world in the right light."
"He’s like a modern-day Einstein. But everybody knows about Einstein and nobody knows about Langlands. Why is that?"
"He’s clearly one of the most important living mathematicians. His legend precedes him. But the question is, ‘Do mathematicians really know what he has done?’ It’s like having a famous writer but no one has read his books."
"He would become fluent in French, Russian, German and Turkish, and well-versed in their literature. Frenkel, who exchanges emails with Langlands in Russian, speculates that his versatility with languages may have had something to do with his ability to see connections in disparate fields of mathematics."
"Langlands spent every morning, seven days a week, for five years working on the paper he delivered in Oslo. It is written entirely in Russian and dedicated in large part to reformulating the geometric program championed by Frenkel. This new paper is an attempt to shift the field toward a more traditional approach: it proposes a new mathematical basis for the geometric theory that relates more closely to Langlands’s own conjectures by using similar tools to the ones he used in the ’60s—in the process, restoring his work back to its original arithmetic purity."
"Since I was a teenager I had an interest in number theory. Fortunately, I came across a good number theory book by L. E. Dickson, so I knew a little number theory. Also I had been reading Bell’s histories of people like Gauss. I liked number theory. It’s natural, in a way, because many wonderful problems and theorems in number theory can be explained to any interested high-school student. Number theory is easier to get into in that sense. But of course it depends on one’s intuition and taste also."
"It doesn't matter how long it takes, if the end result is a good theorem."
"A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect."
"Let θ be an algebraic integer and assume that all conjugates of θ, except θ itself, have an absolute value less than 1. Then –θ also has this property; on the other hand, θ is real. Without loss of generality, we may therefore suppose θ ≥ 0. Since the norm of θ is a rational integer, we have θ ≥ 1, except for the trivial case θ = 0. Recently, R. Salem ... discovered the interesting theorem that the set S of all θ is closed and that θ = 1 is an isolated point of S. Consequently there exists a smallest θ = θ1 > 1. We shall prove that θ1 is the positive zero of x3 – x – 1 and that also θ1 is isolated in S. Moreover we shall prove that the next number of S, namely the smallest θ = θ2 > θ1, is the positive zero of x4 – x3 – 1 and that θ2 is again an isolated point of S. Since θ1 = 1.324..., θ2 = 1.380..., both numbers are less than 2½; therefore our statements are contained in the following: . Let θ be an algebraic integer whose conjugates lie in the interior of the unit circle; if ±θ ≠ 0, 1, θ1, θ2, then θ2 > 2."
"I am afraid that mathematics will perish by the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up."
"Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations."
"The theory of functions of several variables turns out to be essentially more difficult than the theory of one variable because of the existence of points of indeterminacy. In the case n > 1, a mere glance at the poles already indicates a behavior which is completely different from that in the case n = 1. The reason is that, in case n > 1, the poles are not isolated and, in general, there does not exist a Laurent expansion. In a neighborhood of a nonregular point we are forced to view meromorphic functions as quotients of power series."
"One of the many importants ideas introduced by Minkowski into the study of convex bodies was that of gauge function. Roughly, the gauge function is the equation of a convex body. Minkowski showed that the gauge function could be defined in a purely geometric way and that it must have certain properties analogous to those possessed by the distance of a point from the origin. He also showed that conversely given any function possessing these properties, there exists a convex body with the given function as its gauge function."
"Ein Bourgeois, wer noch Algebra treibt! Es lebe die unbeschrankte Individualitat der transzendenten Zahlen! ["It's a bourgeois, who still does algebra! Long live the unrestricted individuality of transcendental numbers!"]"
"... Siegel kept working well into his eighties, after he had returned to Göttingen."
"... when Carl Ludwig Siegel announced that he would hold class on a University holiday, his students left the room empty on the appointed day, hiding nearby to see what he did. “Sure enough, Siegel got up front in the empty room, started in with the beautiful lecture as though he had a full room,” said Merrill Flood *35. After he had continued for a while, “we sheepishly trooped in, and listened to his lecture.”"
"In 1966 Siegel oversaw the publication of his collected works in three volumes. He spent the rest of his life editing (and writing) a fourth volume. As the story goes, he burned everything else, fearing that a historian—as he himself had done with Riemann—would get into his papers."
"It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind."
"In 1917 Levi-Civita discovered his celebrated parallelism which is an infinitesimal transportation of tangent vectors preserving the scalar product and is the first example of a connection. The salient fact about the Levi-Civita parallelism is the result that it is the parallelism, and not the Riemannian metric, which accounts for most of the properties concerning curvature."
"Integral geometry, started by the English geometer M. W. Crofton, has received recently important developments through the works of W. Blaschke, L. A. Santaló, and others. Generally speaking, its principal aim is to study the relations between the measures which can be attached to a given variety."
"Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan."
"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves."
"The treatises of Darboux (1842–1917) and Bianchi (1856–1928) on surface theory are among the great works in the mathematical literature. They are: G. Darboux, Théorie générale des surfaces, Tome 1 (1887), 2 (1888), 3 (1894), 4 (1896), and later editions and reprints. L. Bianchi. Lezioni di Geometria Differenziale, Pisa 1894; German translation by Lukat, Lehrbuch der Differentialgeometrie, 1899. The subject is basically local surface theory."
"I have no doubt that future historians of differential geometry will rank Chern as the worthly successor of Elie Cartan in that field."
"Recently, having refreshed my understanding of the mathematics of relativity theory, I called one of my old Berkeley professors to ask him some questions about the geometry of general relativity. S. S. Chern is arguably the greatest living geometer. We spoke on the phone for a long time, and he patiently answered all my questions. When I told him I was contemplating writing a book about relativity, cosmology, and geometry and how they interconnect to explain the universe, he said, "It's a wonderful idea for a book, but writing it will surely take too many years of your life ... I wouldn't do it." Then he hung up."
"S. S. Chern revolutionized differential geometry with the use of moving frames, the invention of characteristic classes, the modern concept of a connection and so much more, but he’ll probably always be most remembered for the yellowing University of Chicago mimeographed lecture notes from the 1950s. An entire generation of geometers learned the elements of differentiable manifolds from those notes."
"The theory of functions of several complex variables has gone from its infancy with the work of Hartogs, Levi and Poincaré shortly after the turn of the century to its current role as a central field of modern mathematics, much as its predecessor, function theory in one complex variable, did in the 19th century. A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc. The major developments in the theory from 1930 to 1950 came from Cartan and his school in France, Behnke's school in Münster, and Oka in Japan. The central ideas up to that time were synthesized in Cartan's Séminaires in the early 1950's, and these were very influential to the next several generations of mathematicians. Cartan's accomplishments were broad and he influenced mathematics through his writing, his teaching, his seminars, and his students in a remarkable manner."
"The theory of s in approximately one century old, although its origin may be traced back much further. As originally formulated by and subsequently used throughout his work, the theory was intended as a tool to be used in the study of geometric problems. After two periods of theoretical development, one in the 1930s and the other in the 1960s, there has recently been a renewed interest in exterior differential systems as providing a systematic framework for the study of geometric problems. It is my opinion that this development is just beginning, and that exterior differential systems should become a standard tool for geometers, especially for questions where the differential equations expressing the problem are overdetermined systems, and for global questions. When used properly, the theory has a marvelous ability to reveal the underlying geometry in a complicated problem."
"One of the points I have tried to make is that mathematics is extremely useful to our society. If this is true, one would think that we as a society would vigorouly support the research that leads to new uses and that students would be at an all time high. Today that is not the case. The mathematics community has yet to effectvely demonstrate to the public and their elected representatives that our subject is dfferent from the sciences. We do not design widgets or cure diseases, yet our impact on engineering and medicine is enabling and significant. But the community has dwelled so long in splendid isolation that the public poorly understands what we do."
"for a smooth algebraic curve includes both the Hodge structure (period matrix) on cohomology and the use of that Hodge structure to study the geometry of the curve, via the . extended the theory of the period matrix to smooth algebraic varieties of any dimension, defining in general a Hodge structure on the cohomology of the variety. He gave a few applications to the geometry of the variety, but these did not attain the richness of the Jacobian variety. In recent years, Hodge theory has been successfully extended to arbitrary varieties, and to families of varieties."
"... 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”."