mathematicians-from-japan

"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."

"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."

"as quoted by Allyn Jackson: (quote from p. 1015)"

"... 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."

"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."

"Nature is doing everything in perfect harmony... [W]herever you see and whichever [way] you look, there is something fundamental happening, and there are thousands and tens of thousands of laws of nature that are being satisfied at the same time... [M]any of those laws of Nature are... yet unknown to humans, but it's amazing how coordinated Nature is. It's working all the time! So even when you are fed up, and you close your books, and your professor leaves the room, and go into vacation time, and your internet is down, and so on, you think science stops existing and it stops existing for humans, but Nature keeps going."

"The other side... of this observation is that.... whichever [way] you look, and whatever you listen to, and wherever you cast your mind... in that part of the universe that you are observing Nature is doing something. So nothing is easier... to discover than science because science is happening all around you. It's just a matter of opening up your mind a little bit, and making a little bit of effort, and... you have to have an eye for surprises, but humans are born to be surprised, and programmed by Mother Nature to be curious... So you just pay attention, and pause, and relax... [E]specially, you shouldn't worry about what other people think, and what your social standing is. If you are interested in something, it's interesting, and if you're not interested in something, it's not interesting. But you should just keep looking, and everywhere you look, you reach out with your right arm, you reach out your left arm. You stick out your left foot and right foot. Everywhere you reach, there is a bit of science in there... [S]o you meet science all over the place. It's very easy."

"The study of languages, some... call it linguistics, but the nuance is quite different. Linguistics, since especially Chomsky and that school, became very... analytical and almost mathematical, and so I'm absolutely not interested in or... an analytical study of languages. ...I'm a mathematician, and if I wanted to that... I'll just do... straight mathematics... [I]nstead, philology in the glory days of the 19th century meant primarily the reconstruction of the Indo-European family. So people knew lots and lots of languages, and their peculiarities, and their accidentals and evolution in Greek, Latin, Sanskrit... [I]t was practiced outside the European family, for notably the Semitic family, especially languages that have a lot of written records that go way back, and you can do science. So that's what philology means and that's what they used to do, but I do emphasize that I'm... absolutely not interested in mathematical aspects of linguistics. I'm interested in the languages themselves."

"[E]very human life... is unique, especially seen from the inside. ...You might look like the boring doldrums and the standard... path to somebody else, but for each individual... that person is living only once, and unique experiences... are not... replaceable by anything else... I'm not sure that my experience is qualitatively different from other people's... [P]eople struggle through various difficulties and have moments of joy and... discovery and sometimes... get fed up and... want to leave... [T]hen sometimes they come back and so on... I don't think that it's that different. ...People should realize that their experience is unique and it's interesting, if you make it interesting. ...[I]f you decide that, "Oh, I'm a boring person..." of course you become, ', a boring person... [O]ther people will not help you out. They'll say you're boring, but... you live only once, and... I'm sure there's lot's going on in your brain that the rest of the world cannot see, naturally... [Y]ou should cherish it..."

"[T]he one lesson that I drew from coming in from... lots and lots of detours. ...[T]he other side of the coin. ...I started doing mathematics seriously quite late ...That had interesting consequences. ...Most people in mathematics came into mathematics early ...typically in your teens ...[T]he phenomenon of exists ...only in music and mathematics. ...[C]hild prodegies exist primarily as performers, and the mathematical equivalent is problem solvers, rather than theory builders, and the music equivalent would be s. It's true that Mozart was a child prodigy in composition, but on the whole... performers and problem solvers are the dominant types of child prodigies... [T]his phenomenon only exists in music and mathematics, and correspondingly, they come in quite early... Innate or not... it involves a lot of ... Well, maybe there is such a thing as talent... but one necessary condition for a child prodigy... is... the... that could bear with long long long hours of enormous amounts of training, and sometimes it becomes an obsession. ...I'm aware that child prodigies exist in chess and in ... and in go and so on, but that's... a small variation on mathematics... I don't know about innateness and... I'm not sure about... talent. ...[T]he human brain is a very complicated machine and it would be very surprising if there is no... innate difference between one brain and another... after all, there are innate differences between one body and another... I have seen lots and lots of mathematics students... who are very talented... by the standard judgement... But ultimately... on the whole, I am simplifying... it's really the effort, and how much you really like the subject that made a difference as to ultimate success."

"So I'm not such a great believer in talent. Maybe I'd rather use the word ... because... I find that the people talk too much about the genius, and... correspondingly to the... in music and... mathematics... That's a dangerous word, and I'm not sure that it's socially beneficial to talk about... whether it exists or not. ...Maybe scientifically you can talk about the concept of genius, but it... does more damage sociologically, than it does good."

"It's possible to argue... [T]here are wonderful musicians, let's talk about... western classical music... nowadays... but... compared with the time of Bach, Mozart and Beethoven, composition in classical music has gone down. ...It's possible to argue that the concept and the belief in genius killed classical music composition. It's extremely... discouraging to be told, "You shouldn't compose unless you can be Beethoven." ...[T]he concept of the genius is an invention of the... German idealistic philosophers. People didn't talk about genius before, and people... simply didn't have the concept and it wasn't part of people's thinking. ...[L]ook at ...Italian Baroque composers... They didn't care about talent or genius but they're creating wonderful music... just on the spot. ...[T]hey didn't have to be tragic. They don't have to have a dramatic life. ...You just do it because you like it, and the same with mathematics."

"I'm fortunate enough to be friends and personally acquainted with some of the leading mathematicians of our age, but... I have never met a genius. They are all understandable. They are wonderful, wonderful people, and they really love what they are doing, and their creations... open up a whole world for you... but I don't think... I ever met a genius. ...In practical terms, it's much more useful to ...focus on other things. So that's why I'm skirting around your question on innate ability."

"[J]ust as many people... correctly worry about biodiversity, I get... emotional and upset... whenever linguistic diversity in particular, and cultural diversity in general, decreases... is threatened... [T]he history of evolution tells us that... you get interesting diversity and... life forms because of . Whenever diversity decreases and one single species or... idea or... way of doing things starts taking over, usually the world is headed for destruction. ...Monkeys that call themselves humans might do some optimization calculations... in their foolishness, and they say, "Oh, ...we have times this ...and that means we have to do it this way. Everyone should be behaving this way..." and so on... [T]hen they end up doing this and in some sense the invention of money doomed us to go in that direction. But I do believe that that way lies madness. ...[F]or me, madness means you abandoned diversity and ...everyone started running in the same direction, and that's really dangerous. So I am... a great partisan of people doing things their own cultural ways, and I don't want, for example, English to take over the entire world."

"[L]ife is very short and the universe is a wonderful place, and there is so much to see, and so much to experience, and so much to become more intelligent about... [T]he only way you can do it is to have your personal (however modest)... experience of various phenomena... happenings and... events. ...[T]elling about something (this is a meta-comment about something) instead of doing the actual thing, is the worst way to approach science... I'm not going to tell anyone about this and deprive them of the pleasure of seeing the phenomenon themselves."

"You can Google my name, and ... There are lots of videos that they put out online... and you can watch these. However, even that is only a substitute. There's nothing that replaces your own touch. Trying things yourself, and indeed noticing curiosities in nature itself."

"Watching videos is much better than a stupid person like me telling you about this, so that's really the worst possible way to proceed, but even watching videos is no good. You should really try those things yourself and... discover things yourself. There's so much, so much out there."

"People say... discovering things is difficult and... extracting science from everyday life, and the mundane facts... requires talent and special aptitude and so on. I believe that's wrong for the following reason. The reason presupposes a certain belief and outlook on the universe. My outlook is... people say... "I don't like science." That's fair enough... and "Oh, I like science, but... I get tired after a while and I can't continue for so long"... [T]hat's very very reasonable. Or, "I try very very hard but I can't get through some difficulties." Well, what's more human than that? Sure, but... however fragile and... weak humans are, there's... one... creature (anthropomorphically speaking)... who keeps practicing science very very successfully, in fact with 100% success, 24 hours a day, 7 days a week with no stop... and has been doing it for ages and ages... everywhere you go, and that's Nature herself."

"[T]here are lots of things that one does which are essential, indespensable for survival and which is foundational for everything else, about which people never ask... "What's exciting about it?" What's exciting about breathing for example. ...[I]f you stop breathing, you are no longer. ...You're aware of breathing sometimes. It's not that you're completely unconcsiously invisible, but you don't ask that question. What's exciting about... living itself? Of course there are ups and downs. There are dramas in life, but people don't live because it's exciting. People live because it's natural for them and because that's what they want to do, despite everything sometimes, or in some lucky cases, because of some things. ...But people live because it's a basic and natural way of existing as humans, as indeed, biological creatures... [S]cientists, when they are unhampered and unencumbered by those dictates of sociology... where you have to publish in certain ways because you want to enhance your career, because you want to achieve some status, because you want to... ensure you have a certain standard of living and so on. If they are doing science where they do science because, almost, they have to, because that's their existence... If I lost my job... I have to be able to live somehow, but let's assume that I have some kind of income, and I have to move to and live in isolation. I think after... the initial period of being really depressed... "Why am I stuck here?" and so on, I think I'd end up doing science, because that's... who I am."

"[T]he question I wanted to be asked and which I have never been asked is, "What is the question you have been wanting to be asked, but you have not been asked?""

"When you learn mathematics, you learn a lot of definitions. And let's say that you have a certain number of definitions - maybe you learn 100 definitions. Also, there are a number of theorems, and a number of examples to which the theorems apply. Now, a good piece of mathematics should have many more theorems than definitions. And you should have many more examples than theorems - that's a good situation. Unfortunately, everywhere in the world, it happens that in textbooks, in classrooms - You learn 100 definitions (you have memorize definitions!) And then you learn 10 theorems; And then there is only one example. It should be that you have one definition, ten theorems, and 100 or even 10,000 examples to which the theorems apply."

"Draw pictures, draw pictures! As a guideline, when I do research in any area (fluid mechanics, geometry, dynamical systems, topology, combinatorics, representation theory), I always draw pictures when I'm doing research. I'm drawing one picture every few minutes, so by the end of the day, after maybe six or seven hours, I have maybe 30, 40 pictures, if not more. So in a week that's hundreds. You should be drawing lots and lots of pictures, trying lots of pictures. Some of these pictures can be in your head, but you should start by drawing lots of real pictures, on paper. But not a few pictures. Not tens of pictures. Hundreds of pictures, please, hundreds. Because that's how we can, eventually, listen to Mozart's music - in mathematics!"

"In science we... tend to be interested in things that have been already labeled "interesting." ...[W]e ...think science happens in institutionalized contexts, and that the latest fashions and the "cutting edge" ...is where science occurs... [P]eople who get interested in science often read... books that tell you about the cutting edge... and they get excited, but it's not that they have had an intimate contact with science and got excited. It's rather that sociologically they have been told to be excited about something that's supposed to be exciting. ...They haven't had any exposure to "theory X" but they... are told hero stories... romantic stories... But those... "sciences in flower"... are already blossoming. ...[T]hey have lots and lots of intricate structures up there and [are] connected to lots and lots of things. But at the same time, as with plants, we should look at "sciences in sprout." ...We have the impression that when we stop doing science and go on holiday, or close up our offices and shut down the laboratories for the weekend, science stops happening. And when you close the textbooks and the professor says, "OK, end of class!" you can forget about science... and it stops happening. But that's not true. There's something that keeps practicing science 24 hours a day, 7 days a week, nonstop everywhere in the world, everywhere in the universe, and that's called Nature! If you take... even this blob of air in front of me... there are so many beautiful and intricate and unbelievably complicated, complex s that are dancing together and trying to satisfy one another, and succeeding and satisfying this huge... network of patterns. That is science! It's amazing, and conversely, there is nothing easier to discover than science. ...Everywhere I look, there must be science, because we live in this universe. We cannot even escape living in this universe. We need... imagination and a little bit of patience because you often fail, but especially... we need to look, imagine and maybe a willingness to be trained in acquiring better vision, which is called scientific education. ...It's wonderful that people get interested in science because they're supposed to be interested in science and read... and so on, but that they also try in daily life, phenomena around them, however modest. It would be nice if they started noticing sciences in sprout."

"[W]hen I did my PhD, it was in very very pure mathematics and I still love that field, ...algebraic topology ...but then ...I started moving into more physical subjects and... started doing experiments... I thought this was an opportunity. Until then I had lots of... friends and family who were not scientists, and who didn't have any mathematical background, but... with pure mathematics it's very difficult to convey the excitement of computing s... [W]ith physics I decided that every time I finish a project, write a paper or even figure out something, I should design a toy that... captures... joy that I have had, and can share it with people... [T]hen it became quite successful, and it became the other way around. Now I look around and... realize that there's science all around and so I start from the toys... I try to... discover one every month... I was given most of them because my good friends send me toys... but I have... stumbled on... 1/3 of them myself, and... depending on the public lecture... I got... maybe a dozen and... try to tell a story. ...[S]ome of the collections are more cohesive stories than others, but ...whatever I take, I start seeing connections ...And as with the larger nature, so with my toy collection, there are lots and lots of inner connections that I'm waiting to discover; and I usually can."

"Different cultures... until recently, used... science in different styles... [M]athematics, which is supposed to be the most universal of these... If you do zoology or geology or things like that which are geographically constrained, different countries might so things differently, but mathematics is... as universal as any human endeavor can get... [N]onetheless... the Russians... write and think about mathematics in a way very different from how Americans thought and wrote, and the French wrote... and thought mathematics in a way very different from how the Japanese did, and so on... [Y]ou can tell instantly which school, which culture, it came from."

"Many people say mathematics is very difficult to learn, and so it is, and it's probably one of the most difficult things that you can learn, and besides, human brains are not really well adapted to mathematics. It's designed for doing other things, but a lot of mathematical difficulties that people encounter... are actually linguistic. ...[T]here is a definition, a very very precise way of thinking about the limits, and continuity and so on, which... goes under the name of epsilon and delta. So for every epsilon there exists a delta such that... and blah, blah, blah... [T]his is a stumbling block for just about everyone, but when I came into mathematics as an adult... I felt no difficulty whatsoever. In fact I didn't even notice that it was supposed to be difficult. That's because I had been very rigorously trained in the use of languages, as a linguist. ...[S]o the idea that if you change the order quantifiers, of course the meaning changes completely. It was trivial, of course... Compared with the task of taking apart the syntax of somebody like Thucydides... whose sentence continued for a page, with subordinate clause upon subordinate clause... By the way, he writes really clearly, but in a complicated . ...[C]ompared to that kind of thing, the language of mathematics was very very easy. ...[T]here is nothing to it."

"[M]ost people don't have sufficient mastery of their native language. They never had the experience. They don't have... enough practice of careful use of their own native language. ...[Y]ou speak really carefully, making sure that you understand absolutely everything that you are saying and every word and every phrase counts... [N]o... people just blah, blah, blah... talk away. ...[I]f you have a really careful habit of careful use of language... most of the difficulties of mathematics will go away."

"[I]t's just that mathematics is an unforgiving subject where any misunderstanding, any lack of understanding shows immediately, whereas in the rest of human endeavors you can keep going by faking for quite a long time. So in that way, yes, the language frames how you understand mathematics, but in that very very practical way. ...[T]he best way to improve your chance of future advance in mathematics is to practice, and improve your native language."

"I don't believe that I'm a good communicator. I believe that lots of other people are simply very bad communicators... I don't think other people are thinking. It's completely common sense. I have no intention of claiming any credit for what I do, and if you think this passionately, and if your agenda is not some of the other things I described earlier... [I]f your agenda is to share surprises and to share, if possible, some of the joy to make people understand, there are obvious things that you can do... I'm very surprised that people aren't doing it... [I]t's absolutely obvious to anybody."

"If you... came back from a very nice trip... Lots of adventures and lots of wonderful experience... and... relaxing one evening... with your family, and you tell your stories, your family are drawn in... I can already...imagine hearing... laughter and... clapping hands and gasps of breath... [Y]ou're communicating very well... [Y]ou do the same thing with science. It's not difficult at all. Absolutely not! In fact the onus is on the other side. Why are people so incompetent? ...[B]ecause their agenda is somewhere else and... who can blame them? ...As humans you want to have a comfortable life. You want to have some... socially recognized position and... security... and the society requires that you communicate in a certain way, which is not at all the way science should be communicated, if your agenda is not one of those."

"The maxim "A university professor is the next easiest profession after a beggar" really rings true."

"The Fourier expansion of an elliptic modular form has been fruitfully utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms."

"Andrew Ogg:"

"The subject that might be called “explicit class field theory” begins with Kronecker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x) = exp(2πix), i.e., with x ∈ Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. ... Aside from some results of Hecke in 1912, the only progress on the twelfth problem was made by Shimura and Taniyama in the 1950s. They achieved complete results concerning the abelian extensions of number fields arising from abelian varieties, with complex multiplication, of arbitrary dimension n ..."

"Though he was by no means a sloppy type was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes."

"One fundamental and frequently discussed theme in the further development of is the issue of increasing the number of professional mathematicians who have a solid, technically accurate understanding of the details of inter-universal Teichmüller theory. ... in order to achieve such a solid, technically accurate understanding of the theory, it is necessary to devote a substantial amount of time and effort over a period of roughly half a year to two or three years, depending on various factors. It also requires the participation of professional mathematicians or graduate students who are • sufficiently familiar with numerous more classical theories in ... • sufficiently well motivated and enthusiastic about studying inter-universal Teichmüller theory, and • sufficiently mathematically talented, and who have a • sufficient amount of time to devote to studying the theory."