Sir Michael Atiyah Interview [Stony Brook 2011]

[Music] hello my name is John Morgan I am director of the Simon Center for geometry and physics here at Stony Brook University and we are at the Simon Center today we have Rob Kirby Nigel Hitchin who will be interviewing Sir Michael attea about his life in mathematics and more generally his life gentlemen okay well Michael we're here in the Simon Center for geometry and physics do you think it's a golden age for the two subjects or was that long ago in Newton's time he they didn't recognize a golden age when you're inside it x-ray you when you get a bit perspective it helps but I think he's been going sufficiently active some time that we are living through something there we go they how big an age it will appear essentially later on remains to be seen right at the moment it looks an active subject area and that's why this Center has been built and I think correctly it says it's a focal point for a very exciting activity this moment but posterity may have different views so mathematicians are used to viewing all sorts of objects as spaces which means like importing the geometrical viewpoint to different areas of mathematics so you think the main import from physics is a language or a set of metaphors for us to look at our own subject or is it more than that I think it's more than that you know the notion of space is now front and sometimes fundamentally the where we live and move and has been for thousands of years it has different gone through different eras with the Greeks modern ideas are physicists and evolves it doesn't always mean the same thing increase its dimension by given one or two more here there you can meet it changes shape by curving so I think space is it but it's part of our visual imagination and we use it to great still because our brains are very well attuned to seeing things I'm thinking of three dimensions so I think it's it permeates mathematics and physics and it embodies the central parts of a lot of physics so I think it's more than just the but notation or symbolism something rather deep about it and then do you think physicists have a different vision that we can benefit from as pure mathematicians oh yes no question now I I mix a lot with physicists in the last decades if you didn't have a lot of intuition coming from the results of their training experimental data what happens the physical world I've had since they explained to me you know how some complicated phenomenon happens you put a magnet here it floats around in space and so know that they they they have a lot of wealth of intuition based on concrete experience based on real physical world are we safe honed and children developed love these mathematical but a lot of it is intuitive and that mathematicians don't generally have so part of this interaction is to let's learn from the physicists I don't mean learn in a formal way you learn by osmosis by talking to people and solving their instinct and I think that's sort of been going on now for some some decades and a lot of my students who've become a bit more NEPA say by the way physicists thinking aren't quite as critical physicists tend to thinking much more intuitive non-rigorous terms which doesn't mean it's not precise in any way methods to learn to accommodate that so they communicate do you think it's possible for or is it desirable for us mathematicians to go through the whole process of learning quantum theory the way that a physicist would as a as a graduate student or is it is that counter productive well for all of my decisions like me and even like you it's a bit late if you take what you're training in one sticks with you for life and makes helps to make your background later on you can learn a few bits here there you can widen your knowledge you can't really really hate yourself from the ground up with the younger generation it's very different they have a choice of what stage do you want to become learn the physics the mathematics and you make it a new generation your hybrids who are really half mansions how physicists in some deep sense but in general I think it's useful to have different points of view math position has a point of view based on his intuition his experience which uses the wide range of things Madison which are outside physics and so that's a compliment well the physicists bring another thing can be good idea for all my fishes to clothe themselves on physics go to lab put on their lab coats now get their hands dirty with what that's where you just saw that second-class physicists we don't even think it's important they think e their own well thank you would you say it's easier for physicists to morph into mathematicians to move in that direction than vice-versa something yes it depends of course we'd physics you're talking but you'd be talking about the kind of physics you meet and the mathematics corridor he has he who's a lot we already if you talk of physicists now there are turning in the world practical physicist who deals with wires and what do you call vows no because they've got a long way to go but the younger generation of physicists who become very theoretical and much fundamental busy is not extremely mathematically automatic they have to know mathematics to master their own trade and so it's a small step many of them to go over and learn a bit more about new mathematics and they learning much better if they know its relevance to their feet learning in abstract say I can learn a book on abstract algebra without motivation if a difficult learning about news only way to tell somebody tells you is highly relevant to what you're doing is much simpler you you can relate you connect up so but I think monks this or people on the frontier it's probably easier for the physicist to pick up mathematics because mathematics feel that it's a much more vague a notion how do you pick up intuition without having gone through it or how do you learn to get a feel for the things of it that is more difficult I think so I think you're right it's easier one way than the other what about geometry and physics in your own mathematical life when did that begin well you know when I was a student I was a student in Cambridge and in those days you know half the course was about pure mathematics Harvard was applied mathematics including Maxwell's equations so I got a basic training but and I went to lecture by it famous people like drak but I was going to be a pure math addition some of my friends did off the case of your physicists I became a petition we met again later many years later on and let's say I wasn't really raised in physics her III a little bit I have and later on I was in some aspect which I realized and they can hear affiliates but I was not really much in volatility until both slide turned 40 I don't know exactly what I mean it was then thing decided to happen which were brought the physics much closer into the mathematics and then I got more seriously involved so that probably going for that said here but the first part of my life I was definitely felt like who am i fishing with a mild innocent physics I would talk to me a little bit but not seriously did this did this changeover occur about the same time that we discovered the physicists were we're actually dealing with vector bundles and connections and in their own language with Christoffel symbols and say around nineteen seventy two or three period and I remember it was sometime in the early seventies I think I don't know going over to MIT to visit the physicists there and when we discovered what we metals were doing was very similar in technical terms to what totally different motivation with totally different description and so and then we suddenly realized that we were talking about the same thing from different angles and there are people around and say well this you should know their fingers and so very very short period of time Dictionary started to evolve people explained all the terminology and but it made even this earlier stage when we hope realized they were doing the same thing we had no idea where it came from in physics well the foundations were we made at the top top of the building said of the bottom of the building later on we had to go down and learn modes in the lower floors but it was it was meeting regularly on the very frontiers of the subject and that's one of these things that happened in the past - it's very interactive but usually the math lessons we interact with physicists Mattern comes along after physicists cause all the interesting results I said I'll clean it all up I'll give you some rigorous proofs this is I no longer in stay there so on to the next challenge in this case it happened differently they met on the common frontier what they were feeling and what they're doing but both the frontier of their subject meanwhile Vegas so there was really excitement numbers since didn't does go away we know all this you know they were they were same level as us and that actually a different experience from earlier times when the physicists were really ignored math editions they say you guys all you do is come and lay the foundations after we built the skyscraper so yes it was different different phenomenon then what about the geometry going back further were you always interested in geometry or did that develop I was always innocent geometry I think you know bends on your definition of geometry people like Plato says God was a geometry I mean to me geometry encompasses most things in mathematics as you get broader you simply bring it all in in your empire so but I would always a geometer I think when I was at school I lost a school are very enthusiastic about projective geometry good old-fashioned pretty much huge beautiful subjects and we became out of fashion very soon totally after I went to the school people say there's a month she wants geometries taught in schools so out Euclid now do them give them an algebra or computer that was so glad was old-fashioned German so I love geometry and then I had University we had to the other things but the next stage when I got a chance to learn the geometry the next level I I took it up drama - enjoy so I will always I think it dramas are in that sense I enjoyed geometrical things I liked divinity and there are very different headings they could think every part of an hour cynically portable but I would enjoy an illegally there's a matter of partly personality and temperament I liked it of seizing not literally of course what you see is a mathematical symbol or imagined imagination mathematical object putting it in pictorial plays out pictorial terms so it's waved but I said before I the visual skills of the brain are enormous we can see and if I look round I can absorb faster on information and fraction of a second and I know what it means now because the brain has been tuned as a evolutionary history to do that be very good at it and and whereas things evolving in computation on a piece of paper or algebraic a manipulation well we didn't evolve to do that no final evolutionary advantage if you could be the beat of monkey riding out of formula so therefore our brain still think in fundamental sense visually and if you can use that know about my body it's a very great advantage and it's a mistake to say that geometry only deals with certain things on the rest of it is symbolism you got it the other way around you want to try to take anything you got put it into pictorial form so you can get some imagination we're thinking of it that sense I'm a broad-minded geometers topology was a rapidly developing subject in their 50s and the 60s but you started more of as being an algebraic geometers but it's coming from a nice equal sides i I thought that is ok yeah I started as a classical algebraic Darwin simply but III I worked with my teachers Hodge and Todd both whom were on some modern fringe of Germany they both be involved with things which subsequently you recognize the topological topological aspects of algebraic geometry Hodge theory so I was also a good position that point of view I came from a classical background but moving towards the new frontier and there was a lot of topology around at the time when I was there Oxford and elsewhere new topological idea and when I went to born it's a lot of new new developments in the frontier zone between algebraic I don't think I'm gonna play a big role in the early development of some aspects of topology all the stuff that happens under in born and how about starving all my characteristic classes and numbers I know that always most large parts of it were motivated by algebraic geometry so yeah I had the right background for that and I thought no I didn't I wasn't trained as a topologist and since that lot of other topologies at I'm wearing formulas the Hamasaki theory but I came with a very good but Q of how topology relates to algebraic geometry and that's still very important part of his role was what you're first working out in real algebraic topology was at K theory and and why that particular subject well if that first he difficulty remember I mean basically kcat came out about to break geometry good antiques ideas this without a break JAMA tree through chief koval G and so on and then Crittenden came along with ki he disturbs K theory I picked it up put it back into topology because I'm interested in the interface I thought he could be useful combining that with topological ideas so it may be natural for me to latch on to kcat who came out obviously had me I may even have written some minor papers on topological aspects a bit before but in a - other way but Casey Lee was the natural frontier zone between algebraic geometry and topology in that period sintering bran was one of the first people to be you know fortunately invited to take part in these early meetings and all the idea of being flashed out so that that launch means that I direct I was fortunate to be the right Chapel the right time and I was these Sikhs go now they one of your major achievements was the index theorem which also taught you from geometry and topology into analysis so how did that begin yeah well again is a precise or continuous boom because I'm quite right estate the outer break geometry I had always had the aspect of complex analysis today going back to remap and in the great expansion after the war chief koval G by Aaron Carter that was formalized in a big way complex variable theory so that brought analysis complex variables he was part of the tradition of algebraic geometry and I loved it and I loved the new stuff so I knew all about the G bar operator all that what the new bit was having to sound move from that to real differential geometry we didn't know any complex structure and you had to write down differential equation but viola Laplace operator that's not that far away you had to make it though two or three key steps let him reach one of which you had to get away from the idea that you had chief koval groups every you mentioned you had to focus on the difference in the odds of the even ones I have a single operators and does that which was the famous direct copy writer and now once we realize that then we got involved with real analysis but it was kind of real announced it's so close to complex now it wasn't a subtly different field and yet a little bit new terminology but it's really quite old analysis potential theory elliptic equations so the easy enough to beat the big barriers to progress or to make we're not technical things or not how do you dig into an analysis from algebra they were the conceptual thing you know why did you think of going into different what made you think that this is a product even thing to do those other and that was actually much takes much longer for that to happen learning the technique is the is comparatively easy part that it takes books or experts but knowing what question to ask why you should be thinking this way that that's actually here there's no rules for that you just have to pick it up and make good guesses and say be the right place the right time have a bit inspiration very unpredictable patent so things don't work is the way subsequently people look back saying oh this was the obvious equals events yes but the obvious things for the look obvious now with a bit of a heart of the time and vice versa how did the the big question arise between you and singer say I mean did you have an idea about what the format of the theorem should be and when did singer come in and become involved in it did he start it well caveat one memory is always selective you remember what you did and you tend to forget what the other and I've no accepted of that so Freud a very good on these all that you suppress things that other people didn't you highlight your own contributions but let's see the point wasn't we we started off without a brake job machine we had the same as historic Rieman rocks the element is the culmination of century mathematics beautiful theorem fantastic result crazy Lee general and then Casey Lee came along good and he can push even further this was tremendous stuff and we all knew it and what motivated me was a small but appeared like a small sideline amongst all these formulas that hit smoke generated there was the famous formulas we calculated the dimensions of spaces of solutions chief ecology groups in terms of topological invariants but then he manipulated the topological invariants a great you a formal skill and so sometime tell me you work out a formula where the answer which appears to on face of it be a rational number because as denominators he's a key integer because you need to press it as the dimension of a certain vector space sometimes they come out to be without a vector space you don't know where it's come from so that intrigued me so we were trying to identify what so that he's missing integers were that was predicted by the formula and then we knew a lot of a lot of clues we knew it had something to do with spinners we knew the formulas the representation theory and we knew the shape of the arm sands eventually we could see what the answer was we didn't know the problem and we were looking for the problem which had this answer and I was talking about this thing he was long time as in singer was visiting Oxford at the time and I said we started discussing and I said to him you know we there should be something which explains this and then he he came out one day said well I think I remembered it it's the track opera Dixon he'd done more physics with me I mean interacts lectures but couldn't remember but he done bit more and differential geometry in the two together can use the Dirac operator we saw the operator hold up because we have everything waiting for it we were waiting with emitting things drop it we get all the ground prepares we knew how it did slot in how it related to complex variable theory we knew the answer we need lots of examples we had everything and well after that we had to have our proof but I said the proof is the comparatively trivial part of this operation it took a few years to produce and then we proved lots of different proofs so the proof is the last stage in the operation of the missing bit was to get the right foot view and to ours the right question surprisingly long time well so what's the Drakh operator not in the air before singer brought it up with you direct operating it was introduced by direct physics yes claim the quantum mechanics of the electron and particularly that's in a cop's key space was relativistic framework it had been generalized fizz's you knew about this through two curved space-time but never be knew by math addition to my knowledge there was no mass efficient Hodge to damage differential forms harmonic forms directed the Drakh operator they were in the same department they were colleagues for 30 years but didn't actually talk to each other about mathematics even they had done I had have been out of a job this is my job was to Philemon Wheatley what are they could have done together but didn't so nobody actually had done introduced adrak opera in geometry at all well sing appointed it out then you write it down but it may be helical sort of writing in Devon didn't do much with it now what can you do with it learn another motivation maybe maybe some people like this letter which you about the differential geometry they would certainly know it but if some point of you curves are in space but in romanian space nobody ever thought now what good's arm one experiment anybody harmonic for most apology but well i tell you these his first cc's was really to investigate from one experience and see what honkers why were they given that they didn't do in any natural relationship with topology whatever what good were they so it was it was that realization that same thing is going but mathematicians had so much fun to import in of course message to get up once I seen what you can do with it then it became popular became fashionable after a while with people look back now but obviously why these guys take so long to find it is to yoga and it is trivial you give it a graduate course it's almost the first thing you could do in exercise but when you're involving this theory doesn't look the obvious at all you would never occurred you know what and the reason is fundamentally spinners are I think are so mysterious things I don't like nobody any understand what spinners are I claim not the physics user for the electron of the algebra to use it for representation Theory not even the geometry is now sought these even they really have a good feeling for they don't know what they mean they use them always very fundamental but there's passage in the mind of her miles browse books but he says only in the level of spinners do he really reach that dim still understanding where he's in geometry which goes back to the Greek a beautiful passage but very hard to decipher but anyway ice claim we don't really understand and some strong sense the word what what what spinners are I had say wasn't surprising the way they took welcome to me introduce what were they for and now but they're only two people learning more about them and there's a ways which you can get better understanding but we still it still remains a bit of mystery there one of the deepest mysteries in geometry I think in a way I like to it this way it took years centuries for mass vision to understand the square root of minus one hundreds of years people have this cool yeah this number squared is minus one it didn't exist but it's very useful to use it you got wobble of series and you did the engine events yeah after 200 years of use it became respectable and Gauss gave a definition and modern complex variable theory was accepted that's fine now I claim spinners is like the square root of geometry geometry involves things like lengths areas volumes spinners involve the square root that's that's a much more difficult ocean than the square root of -1 and I don't think it's several hundred years I claim to understand what the school of the geometry really means I mean put it philosophically you know you see why it I like that's pretty that way to point out kind of issue it is square root of -1 was a serious problem it was called an imaginary number it did not exist it was imaginary Yeti was that means for and look what you could do it and then not only was used from algebra is useful analysis home physics eventually and the rack operator is to do with the square of the rack over the square root of a plus operator but a plus operator has a long tradition chilling history the square root that our prayer it took they started off with Hamilton with his quaternions name with Dirac to write down the square root the Laplace operator that's a good start but it's good if an opera is not the same as the geometry the operator is an aspect of geometry but scoot of the geometries we now beginning gradually with one thing another just a theory in ourselves to our theater we are getting beginning to piece together some sort of partial understanding of spinners in different contexts so we make being progress but give us not 100 years have this round table in a hundred years time is some of you young arrived I still be here let me can you explain why the physicists talk in terms of spin up and spin down I found this terminology well neither my birth is better than it's really under said it has to do with the relationship of spin he takes place in the spin speed with including the only direction in space and because of the minkovski space the light curtain relationship with the spinners you couldn't identify and that's why spin electron is ready to angular momentum anything like that so there is a connection between things in space we see up which are up and down and things in the spinning world which are ready but there are there is a overlap and so the spin up and spin down is part of that maybe Nigel can explain it better because if it's a lottery and they talk that's it I begin to wonder do I really understand I'll attend my own way when they talking their way it's not quite so clear to me what they really mean like gradually getting more confident on there but sometimes better not to ask one way we have it our own way might never be confused getting back to the index theorem you're your first proof of it was cocoa board isn't yes and yet the definitive series of papers use Cather did you always have in mind using K theories that was the first proof kind of rush job in some sense well he the key theory was very good digs work and we developed a theory but topological purposes to then the next era for the rat came up to give the formula for the Drac operating test case for that was for a single manacled wasn't general could go dick theory at all but kzd came in two places and it's important to realize is on the one hand if you wanted to study elliptic differential operators in general obligee order or whatever you like anyone a lord important to understand their look their symbol their local description by highest order terms and that was those are given by at any given point locally polynomial functions on the to cotangent space which is their values in the matrices and for elliptic operas they took their values in the non singular matrices so then you build a z from there there's a close relationship with the hub atop a theory of the classical groups and kaseylee comes in to understand the classification of the operators on a given manifold in some topological way and my qdk theory systematically you can show every operator on a manifold any order whatever in kaseylee terms can be reduced to the ones we are familiar with in differential geometry like the Dirac operator that's entirely separate contained to how you prove the theorem about the record just reduces the general case so the particular case by using these local symbols that bit we knew quite early on because we UK theory we knew once we learned about the general theory elliptic equations girlfriend the people we knew that one should formally at least study them and when we saw once the Katyn is kc8 to change charge of the symbol and to give you a formula and you could formulate result in kaseylee terms then didn't even prove then you had to go back and how you give the proof now you had a choice come what is it was most direct classical proof and having us to what hit his theorem and gave a proof but we knew that was not the idea of proof lots of reasons first of all what that didn't let him directly k theory secondly it didn't further on incorporate the Grodin big different generalization when you one man you have map between manuals like how much later so we knew that but so kdeeny coming in Christ he coming at once at the local level to do with the simply the way the symbol depends on cotangent vectors and it comes with a global level in terms of the analysis okay that's way and algebraic geometry the confusion is of these two think fused in algebraic geometry just fact the Sheep Syria resolutions of sheaves gives you a vector bundles Godin big Stickney you think I'm a fusion of the two it takes on the one hand you get the local serially to do with the resolution of a point and vector space by a sheath on the other hand it has the global theory or she's ecology so they're so closely intertwined you can't see them separate when you go to the real series they may come quite clearly different this is a global city but the operators I mean you focus on the ash want on the Dirac operator alone or you can look at the local theory and then you definitely with KC other symbols they really a mysterious way they're different complementary and yet in some I don't figured they get unified and eventually they when you go into further reaches in theory later on they can I get further unified but it's slightly confusing there are these two different way to reach Katie re-enters and they were both there what a lot of one of them was there from the beginning different formula to get to prove the theorem we focus on the fastest route but we knew that that wasn't really the best route others to generalize it we wanted in picture include the fixed point formula which had proved a bot okay I said it six points we want to generalize that so that Larry in case it says the KCl he began to take play a more important role as he went long and we formally abstractly but it was yeah the interplay between the two was an interesting and took a long time there's probably 10 years come with the spam of the time hi another people spent on variations on the same theme of the same formulas different proofs yeah so collaboration has been a feature of a lot of your papers well yeah what makes a good collaborator for you well good collaboration me is somebody who might collaborate with and that's not a trivial they have to have the same fundamental point of view in math what they must find interesting what you find interesting about - the same acidic values and judgment they must have the same kind of perspective looking to the future what they want in coming that's important without sharing and common for those off philosophy you might put it you're not gonna get very far they may help in the technical level so that's that we I share it I think all the people I collaborated with secondly it's important each person person parties as a collaboration brings a different perspective you have that same philosophy but different background you must have different expertise now one only one of you is good as bad as he has a good screwdrivers yeah you want to get the best of all worlds so my collaboration with Boston singer watch was much better bet on topologies and I was he knew more about lis groups hits a lot more about lis groups singer you need a lot more about analysis in general so all and then my younger collaborators like you and grab other expertise so it's important the fabric that has has something to contribute and so when you write a paper some time would you do one business partner does the rest usually the collaboration is saying to myth it's bad by that say you do it either way yourself but really the technique and the sort of understanding and the background is what you share you watch what you're exploiting why you can't do it on your own you also I mean advantages having two people two or three people collaborating is it you know you see around the corner you'll come up walking you hit a black wall you if you're coming on that side you see the other way around but you sick black he's an obstacle collaborator may be able to see the way around it that you don't see and say you get off faster that's a common practice you you have multiple points of view but there's a combination of a shared philosophy and also think is important that you should be they should be congenial companions now mathematics is a very very lonely game if you play entirely by yourself you could do like Andrew Wiles I'd lock yourself in your room sit there for seven years and come out with the proof of the theorem or fail as happens my friend Precure couplers who spent his life trying to prove the prank ringing injection and solidly thinker and deep thinker but he didn't succeed you know so you're putting a big gamble when you're diverting your entire life or by yourself to solving something so lowly business so being able to collaborate takes you out of your Hermits shell you talk to people keeps you it keeps you sane actually seriously a lot of mathematicians want the river to the edges of some kind of insanity literally and so it helps to keep you in touch with the world and then so good and that therefore the social aspect it'd be a good term to be able to share as a beer a lot of wine beach holiday together what it is is actually important part of the friendship which when you become friends of course with your collaborators that's part of the nature of good collaboration it's a human collaboration it's not just a soft technical of you maybe somebody you consult to how do you solve this problem but that's not a career collaboration that's a contract work how about a degree of skepticism is that important no collaborator well does happen I mean for example Rob but he loved formally until he saw formal a he didn't believe it and I was brought up with slightly more apps I'd say well look look Rao if we can prove that something is you know from toriel if it if he's buried under the depends only on the Romanian metric but then it has to be like this hello until I see the formal he do long calculations check it works out right you're right till he got the formula he was skeptical about abstract arguments I used to get slightly around it monthly cell but oh man later on I learned that sometimes it's married to the formal I mean you learn something on the formula he's outside the box human part of what you were doing and it goes off and so but that's how that helped to push me towards including formulae anything enough see it I was older than me by five years when I was younger people like Mike's student Graham Siegel he is the other way he was much more abstract than me see several I carry with him I would try to push him towards concrete and he would be no I don't need this big categories here I know they've got interesting point of view so yes some skepticism or that's quite skepticism in the same sense I mean I skepticism which means that you you can't prove something because it's not true wrong I haven't had much to most of my can I pretend to be optimists they believe they you know if it looks right and getting back to my teacher Todd no he would do lots of calculational things and he would say if there's any justice in this world this must be true he will evil injustice in the world like a Madonna you could have a butyl formula he wasn't true and that was he sort of inherited a bit of that if I find a beautiful formula it had to be true and most my collaborators tended to go along with it although I I do remember one occasion when Graham where he was the other way around either something copy I assisted something you know probably were put off I came back to live later on did work so I told you that all wrong [Laughter] from being skeptical to be enthusiastic overnight sometimes but no it's I don't think I did it but those skeptics outside for people I didn't work with who say oh you've caught this is Miller both work is so much to my collaborators we tended to be optimist yeah I read a quotation every day from James Joyce that errors are the portal of discovery did that ever feature in your work well I wish you tell my students me well you know you learn by your mistakes he was named to series you must make mistakes deliberately in order to make program but you've learnt because if you make a mistake and you can't find out where the mistake and you sweat like anything trying to understand why went wrong that understanding is then something you learn to gain you see something right before you died married ever and so yes you do learn from your errors I've made it I'm trying to think I mean these are so for the southern that error is misconception you it sort of failing to understand something that you should have done and then you've learned that I'm sure there are I'm something you're doing it all the time no technical level a little more difficult level you you you do things and then don't work and you go back look at it carefully and the simplest level you go back and find there's something very subtle question about signs you know it just always be Devils with you and then you get into it you find that actually this I could already quite delicate and require careful attention in a few places like that so yet you don't feel you're and if you didn't make any errors I think you what do you mean human first of all you make carrot or arias to Marnie I think he's I didn't and so and you love the mistakes because if you if you learned something too easily you also you open a book and read it they don't find in difficulty and you figure every night if you had to struggle couldn't follow it then it sticks so if you don't see the difficulties then you you we won't remember I heard you had to struggle through that then you really know every step of the way in every yard you he's built in so I totally believed and I mean he was the math this is a general statement about it maybe you can move on to more general issues I mean that's president of the Royal Society you had an overview of all scientific subjects and and the people that practice them do you think mathematicians are different from the rest yeah well I mean aim every every my station is different maybe I'm asking to be sad Sardi's different there's all sorts of ways so that the memories didn't cause different from other sizes in fact in some places even country and university with my wife with the mathematics you could do mathematics either spark jobs degree was politically you chose so and it's true this man's man he's both an art and a science and sometimes people doubt with the science and my view was always science essentially used mathematics mathematics out science justify or identify so I think it's but no less it's different you don't have to do experiments every physicist chemist biologist spends 90% of his time of the fields Oh doing experiments and temperature their time thinking about what a theorizing all the way around using ninety one time thinking and perhaps tips and every time do a calculation or pre on the computer or something like or talking to if somebody does experiments so that there's a balance between experiment and theory is very very different mathematics inherently the ultimate theoretical subject and so with Felix the difference is not enormous you be doing high-energy physics memory experiment cost one hundred billion dollars so you know there are the experiments you have to wait for ten years and they build the nexus cylinders I mean while ya got no choice but to do theory so highly physicists are vain almost indistinguishable from math positions in that sense but the good out of a solid-state physicist the electrician's different world chemists and the biologists even more they have not much very little theories go on what now they're giving a bit more so the mathematics is at the one end of the spectrum the only people were further out on their spectrum of the logicians and they of course by the back door have become community scientists there's interesting version I mean the wall the time if you trade PhD logic he was the one guy who couldn't get a job he was so abstract now they're they're the Kaiser brought up party computer companies so the logic became part of computer science and came became therefore integrated back into time but you all the time when you wait mathematics logic logic to philosophy I like person Russell like that all then philosophy when in theology those that was the hierarchy the old days theology was a key the pinnacle that was the ultimate thought thoughts love to me and of course science was called philosophy also called philosophy and called natural philosophy is good one philosophy so at that time I still philosophize and still but yes there was Prison Law Society I was I was aware that I came for a rather small niche event when they offer you present I said well I'm not really scientist even a pure is there are some pride masters and I said well but I'd be hesitant I didn't say yes immediately I said I need to be to think about it because I'm so far than one wing that you difficult for you to speak on behalf of the whole community take a central point of view but if in my best and of course you go back in time doodle President Roh sorry those days you could be my fisherman everything was there anybody between you and knew yes they were wanted to happen together before me there was a there was somebody who was sorry not very famous but a name escapes me now and then then the word of course people who are applied magicians much worse Kelvin was then Kelvin was a physicist but he would feel as president all sigh Stokes so people that ilk were in between but the last one before suddenly hundred years before and they haven't had any mathematicians even applied mathematicians well there's here what's the goodness is with you what climax wasn't suicide you only but now brotherhood jato top Thompson and Calvin Raley they were all present sorry do you think pure mathematics has a healthy future when funding bodies want more direct applications well I have you I only follow these things nowadays somewhat at a distance I think it's easy the phases they I'm sure they go in an app and some administrator Mixon is hearing on somebody then it reacts badly I mean I think we've been through good phases bad phases my past and in the old days that's writing mathematics wasn't regard that something could you put a lot of money into commercial benefit it wasn't it wasn't justified of this basis then the course numbers have grown much more so now people say what do you get for your buck and so they also Christian so they'll begin to be more expensive you are the more people participation so it's public confidence of success you might say because madman is now more widely studied movement which is around more help is given by investor then people ask questions and you go through a phase where there's it'll turn against you but I think visually ups and downs my main point when I was present all side he was try to persuade the mathematical community that their best help came from keeping closely to their scientific colleagues you know Jason feels engineers this astronomers Kimmie's these guys there's about value of mathematics they they they sufficiently mathematics you see they know without mathematics I saw you couldn't develop and so they in the broad sense are there across a large community and they carry a lot of clout so if you have if you're on the same they have is when you're outside you can argue a case for mathematics too if you go out on your own and say well you guys are doing furious stuff we do pure mathematics and you know we're superior to you you lose thank you because you carry no weight the weight is carried by the people who are to the world outside or some physicists are engineers are so I think my best bet always say in good terms and working relationships with colleagues in neighboring in both onyx and then they send the chart and the scientific community's health back them sign is outside yes you say you must support mathematically need mass money chain people we'd be advanced mathematics and there some of you may get varying degrees of support basically there are no aside and without them your said you really losing getting back to the mathematics but what do you think of the main challenges in geometry today and I mean they they differ from those in the 1950s when when you were beginning Pat I mean perhaps or perhaps not well you know when I was beginning I was a cusp point as this past it was after the war of all intervened before the war there was this great pain Raja said he was a big thing after the war there was this fantastic development in algebraic topology and chief kamalji you guys there and cars are on people who prints the kalila there was a tremendous great big he was tremendous anxiety period which launched of new directions I was fortunate to be around to catch the song drift to that we didn't think in other challenges I mean the word challenges I mean half the people would say well you have to and provide foundations for all the Italian algebraic geometers did but that's a little dull wave looking at I mean the Italians did good stuff they didn't have the right tools and like to come along but modern Theory did much more than provide the tools by the whole framework and structure which was part way beyond technical tools so it enlarge the scope and it combined putting lead groups as well and tell you unified large parts of mathematics and very healthy and then that's it the turn that's it though that the problem came out no people nobody brought a thought about studying differentiable structures on manifold there was a topology until came along with these exotic spheres and then that one's good even bigger so we need pollens weren't predicted nobody could predict nobody thought well there will realize it you had to prove something but they thought it would be odd you know you prove the obvious quite connecting you all eventually was proven so I think the challenge is usually the most exciting things I claim I'm always the unexpected if you know that there's a problem there to be solved you know what it is well tackle it may take a long time like 300 years is the firmest here you build a big machinery but most of the time he's like that you know unexpected things discovered on route and what something certainly the most surprising Donaldson teeny was totally unexpected open up totally new door and the relationship with physics which was interrelated with that well already totally unexpected so all the new things coming out of the interplay between geometry and physics are new but put them as challenges I think is under estimating them now we have to understand what it all means there's more than a challenge that's a philosophical conversion and then that'll keep us busy for hundred years I think understanding what it means and do what you do with a tiny developer theory how much you see how much Mattox we're in the middle of it now and probably one quarter long centuries worth and so you can see there are of course technical challenges proving this theorem which are good local objectives and sometimes focused attention and those can drive a subjects Fermat's Last Theorem led number theory is to develop algebraic number theory for hundreds of years and led to all sorts of things including finally the proof but even ahead little poofy still beat successful exercising stimulating ideas and well I suppose the Poincare conjecture some senses had a somewhat certainly 3-manifolds theory but and for manifold is still no one want to know what the biggest challenges understand for manifolds we don't so there was various times along the way we people thought ah yes fellas our theory shows it wasn't what you thought it was he's really algebraic geometry these guys then they said oh that's not that I said pregnant Goku's keep moving and not even bubbles given up their nose nobody is a sensible conjecture about what fun well unknown sorry how many new kinds of melons will be fine how many new invariantly produce I think that's it the people were there most about it are the least dogmatic about what you expect only guys who were don't know much say well here there's a nice conjecture we should be stopped couid though there's no good conjecture really four-dimensional priori conjecture of smooth manifolds is a loose small sub they won't get you very far so I think there are particular problems within it I don't think they highlight this eliminates conjecture so there are technical issues in the beginning but they just show our ignorance and I think understanding full manifold some deep sense and there's an experience and deep sense which make you along with that understanding the relationship of geometry and physics in deep sense what has all this got to do for your feelings he's all the string theory really physics mathematics you know there are people out the physical things we all say he's not physics it's mathematics if you want a job go to the mathematics park that's a big problem for young string theories and I sympathize as a physicist because string theory stuff is so far removed any present day an experimental test is impossible because the orders of magnitude says small or large nevertheless I think Theory soup has a lot of physical basis to it and low income in connection with cosmology any but where he's gonna go what in other words whether string theory will be a important part of series involves physics the next century or not remains to be seen and is it gonna be dominant part or a minor player with a billion new ideas coming along with string theory sidelined all these are big questions we don't I think again the people who know most about this I wouldn't be released there are those guys Damien Krause is one but I don't think they really solve the problem we have a theory it's called string theory and we just almost there around the corner you know just wait a year or two I think we didn't realize that's not true they close to solving some problems but I think they realize they've got a long way to go so I think it's long way to go so that's what what is fix their physical game is a big problem and what I am the mathematics that comes from physics a string which is sold well that is definitely mathematics and a big mathematical future even if fiza's decide they need something else that's happened in the past physics physics produces Theory managers you did physicist and forgot about it designing better math went on an OP series a good example so I think that's that'll keep mathematic busy for hundred years trying to understand the mathematics that goes behind that and I say I'd like to describe that is understanding physics are doing something like symmetry nonlinear Fourier transform theory special geometries these are all generalizations of previous geometry and in different dimensional moduli spaces function spaces and the old days called calculus of variations you can see all threads in that I suspect that will go into the big area and some series will follow out number theory here a bit inaudible there sorry about it so I see a very exciting future I think man about here it's a good place to be the bad ones not something we'll think any given stage we nearly finished close the door close the books back up take something else no you you've been a geometers and that's been closely tied to physics mechanics the real the real world now biology is in the upswing that's not clear to me what sort of mathematics is going to come out of that do you have some views well I mean I think one has to express there's any fence or caution the fact the mathematics been so successful to fix and my corollary chemistry and religion things in astronomy doesn't guarantee that he's going to equally successful apology that people who say the ball he's totally different field God made it written down the fundamental equations of physics but with biology just lay down the rules and threw the dice and let that evolution a is course and you know we we are here by a series of mistakes well natural selection which is not a mathematical struggle the form possess special selection is mathematically some sense but the outcome isn't gay doesn't lead itself to many laws well that that's to to some accept question - what is it there are they are doesn t certain fundamental facts from volunteer jar laws its gently code a very mathematical structure DNA how the jeddak's how things in contained and then so that's very differently as Matamata the big question i think is well mathematic play a significant role it's a really high higher level the one page a mind clearly grown is in understanding how the brain works now the big question for the 21st or 22nd century apologies how does the brain really work and we're scratching the surface people know a lot more than they did 25 years ago but it's just a tiny bit they did a very little idea and they went in week make analogy with computers totally naive so the brain is actually much more sophisticated thing that most people working in the field would like you to believe or pretend you know scientists think if jessica computer real good people working in a size know that it's they know an ear really think they'll made a lot of progress but in light of what it looked like hundred years later looking back you'll you will be done babies physics permanent so I think and that might require a very much more sophisticated set of ideas which might be mathematical and the mathematics might not intervene evolves probably but um when Mike I work with a colleague as a neurophysiologist and his view about the brain is that it it's it has a hierarchical structure it builds all levels of abstraction on itself just like mathematics mathematics the ultimate hierarchical subject which is why you can't drop out coming again every level it builds on track shops I don't like it I don't like Kenny greasy anybody and multiple categories but it didn't build on levels of abstraction and my friend said the brain really works in a similar way therefore there could well be interplay between the levels the abstractions hierarchies in mathematics and in the brain by some way which they can be useful templates for thinking the government in the brain that we we well it's love is all we'd like to say you know what is a chair a chair well millions of different kinds of chairs fact that the brain recognizes a chair as an object he's a miracle you know because well there's it the dogs with your bride's dog yeah we see a dog so we mate when the brain is abstracted out the notions that correspond to the objects we see even though the rod you see is incredibly detailed and different and they had to do that I couldn't we couldn't survive the evolutionary terms if we didn't know alive was a lot even if it didn't look quite like the other line so you had to be able to so the brain had to evolve levels of abstraction that's the kind of first level getting to the chair level I had to progress and our success the evolution has depended on our developing a brain then is better and better an abstraction and so that very general sense you might think mathematics might have a contribution to make it's a mistake to think that a person going say well I want a drawing board for the brain looks like and here it is no this is the theory of the brain forget you've got a start with the experimental data you know you'll know and you've got to work with all the you know physiologists you gotta breathe it into group of people who can talk to each other like this and eventually this is happening a bit but it may be a long way to go we may be talking about what will happen in a hundred years time and if that happens then people say oh yes there is a the mathematical theory which helps to understand how we think and then there's a lesson big massive contribution all the other stuff will be fiddling you know not some bolts moment we're still in the nuts and bolts here although molecular biology is moved a bit beyond that but I mean you are used by biology as being an outsider and a thorough biologist perhaps with those 100 year perspectives it might be good a good idea to wrap up thank you very much gentlemen okay I'll think thank you we look forward to meeting on hundred years time [Laughter] all right thank you very much thank you

Channel: Graduate Mathematics

Views: 3,289

Rating: 5 out of 5

Keywords: Sir Michael Atiyah Interview

Id: Vbn4ziRjiDQ

Length: 59min 55sec (3595 seconds)

Published: Fri Jun 08 2018