Robert Langlands - The Abel Prize interview 2018

so professor Lang manse firstly we want to congratulate you for being awarded the habla prize for 2018 and you will receive the award tomorrow by His Majesty the King of Norway we would like to start asking you a question about aesthetics and beauty in mathematics you gave a talk in 2010 at the university of notre Dom in the US with the intriguing title is their beauty in mathematical theories the audience consisted mainly of philosophers so non mathematicians the question can be expanded upon does one have to be a mathematician to appreciate the beauty of the proof of a major theorem or to admire the edifice erected by mathematicians over thousands of years could we expand on this what are you thoughts on on this that's a difficult question part at the level of the level of Euclid why not I think I should say I think in the article the article was in a collection of essays on beauty you will notice that I avoided that in the very first lines I said basically I do not know what beauty is and went on to other topics and I discussed the difference between theories and theorems and I think that my response is this same today beauty is not so clear for me it's not so clear that one should speak of beauty and mathematics at the same time mathematics is an attraction want to call it beauty that's fine but I it even if you say you want to compare with the beauty of architecture architecture beauty I think is different than mathematical beauty Oh unfortunately as I say I just avoided the question in me in the article that if you forgive me I'll avoid it today yeah as you are well aware of Edward Frenkel who you had worked with and who is going to give a talk on the a couple of days from now on our lectures talking about the aspects of the diagonals program he wrote a best-seller of a book with the title art and mathematics yes and with the subtitle the heart of hidden symmetry yes and the Langlands program features prominently in that book he makes a valiant effort I would say to try to explain to the lame and what is the Langlands program and I'm very intrigued also by the preface that Franklin writes he says there's a secret world out there a hidden parallel universe of beauty and elegance intricately intertwined with ours it's the world of mathematics and it's invisible to most of us do you have you probably read the book do you have any comments five thumbs through the book I have never read it now I'm going to see something which is rather which is probably not relevant to your question but we secrete something that but which I've never thought but we are scientists we ask you what we think of me listen that leads to a scientist say in particular about the history of the earth the history of the creatures in it history of the universe and we even discussed sometimes the beginnings of the universe but a question that something that puzzles me I've never thought about it except sometimes when I was taking a walk but how did it start at all it doesn't make any sense it you have to either something came out of nothing or there always was something but I somehow or other you have to it seems to me that if you were our philosopher are you a philosopher we'd have to ask yourself how is it that something can can be there it's very complicated but it's not irrelevant that the world is very complicated it's simply the fact that it is there and how did how do something come of nothing well you might think maybe with numbers it can happen but I I have to beyond that I don't know but you have had your creative moments where where all the sudden you have had a revelation yeah and hasn't that been feeling of immense beauty before you or the when suddenly things fit together exactly not quite like looking at the clouds or looking at the CEO we had a child it's something else it just works it works and it didn't work before it's very pleasant but series have to be structural I mean there has to be something some structural some appealing structure in theory but you know the beauty the women are beautiful men are beautiful children are beautiful dogs are beautiful and force a beautiful sky is beautiful but what numbers on the page or diagrams on the page it's not quite the right word and it's satisfying it's intellectually satisfying that things fit together but beauty misplace when things fit together they said in the article I avoided the word beauty and I because I don't know what it means to say that a mathematical theorem is beautiful that it's elegant its gracious that is surprising but I understand but beauty but we can at least agree upon that the Frankel's endeavor this book was a valiant effort to really explain to the layman what beauty is in mathematics and especially what the Langlands program is a beautiful thing yes I wish Franco were here so that I could present my views and he could present his I I've read Franco and I studied Franklin but because he explained the geometric theory so I wasn't interested so much in the beauty I was Mathura his his description of the geometric theory and I got quite a bit from it but I also had the feeling that it wasn't quite right so I if I wanted to say no I would want to fail in front of him so he could contradict me okay we will see how that works out so perhaps it seems like you agree with Tolkien who said that beautiful was one of the ugliest words in English so you have an intriguing background from from British Columbia in Canada and and as I understand it at school you had an an almost total lack of academic ambition yeah at least you say so and so unlike very many other Albert laureates mathematics meant nothing to you as a child the fact that I could add subtract and multiply very quickly in Vancouver oh it was actually I was in New Jersey but the interviewer was in Vancouver and he asked me questions along the following line along those lines and I didn't answer I answered rather frivolously but no I stole the experience I had with mathematics of arithmetic I depart from elementary school and so on I liked the countenance one was that I worked in my father's lumberyard and those were the days when you piled everything on the truck by hand and you and you tallied it right you counted the number of two by fours I do pie for us is that a concept here two by fours 10 feet 12 feet 8 feet 16 feet and then you you multiplied you used to that and then your number you added up the number of 10 x 10 plus the number of 12-foot lanes fine you multiplied by 12 and so on and so forth you got a number and converted it the board feet or something and then you knew how much it was worth now the only experience that is that of course I would be load the truck with some elderly carpenter or some elderly farmer from the vicinity and he would have one of these small carpenters pencils and he would be very painfully well marking one two three four five one two three four five and so on and then he would have to add it all up and of course that was 12 or 13 or 14 and I could tell him the answer before even I started and then I wait patiently he did that so that was my only experience with mathematics except for one or two things 20th new tricks that he used in building window frames to guarantee that the angles are right angles I although that was just a trick right that the diagonals have to be of equal length of the rectangle is going to be right angled so why did you then move towards mathematics no why not language or other things I studied actually when I went to university I was so I thought in those days it was just that was almost an immediate post-war period it was still regarded as necessary for a mathematician to learn several several languages so French English Russian and maybe even I tell you now that fascinated me I had to hit almost no the the instruction in French in Canada in english-speaking Canada was rather formal it nobody paid too much attention to but that rather fascinated me and the fascination has remained with me all my life so that but that was an incidental to mathematics yeah but but why did you start at the university at all well there are two things I come to the second in just a minute but so went to high school it was there were children from from the neighborhood and from the surrounding countryside and so they tested me I was indifferent so I was it didn't pay too much attention but they also gave IQ tests and my conjecture has always been that I I had probably an unusually high IQ I mean quite unusually high IQ I don't know it didn't mean much to me but that's my conjecture in retrospect so we had at the time a very many of our of our teachers were just former members of the of the army in world war ii who were given a position and as matthias more or less in gratitude for their service in the army this fella was young he probably had a university degree and he took once an hour last time to say that he must absolutely go to the university so III noticed that but I noticed that for another reason namely I had acquired a mild interest in in science I because I had a book of my future father and I had a book about it was a sort of speak it was a rather leftist book about eminent scientists of course mark wasn't included Marx was included and Arwen was included Iceland was included in Solon various scientists from the 1617 18th century and he gave it to me now he had it he himself could had a childhood was basically no education so he learned to to read at age of about 37 during the Depression when when the labor parties were recruiting unemployed people and so he learned to to read but never very well and I think never really could write but he always had a good memory so he remembered a number of things and he also had a library and in particular he had this book which was a book that was very popular in pre-war so I began to read in that book and of course my wife had my shoot your wife had there a more idea of what one might do as an adult than I did and she did influence me so and so I had that so there I had this just book in which I read about standard people like Darwin and so on and that and that influenced me a little it gave me some idea what one might do and also there was the accident that I always wanted to leave school and hitchhike across the country but I was I was when I was turned 15 which was a legal age at which you could stop going to school I'd only one year left and my mother made a great effort and persuaded me stay another year and during that last year there was this so things were changing for various reasons during that last year Kandra this you know that lecture of the English teacher and the introduction to one or two books and so I decided go to university yes so you you go on to have a master's thesis in British Columbia yes you marry your wife and then you go to Yale and start on a PhD so that's quite a journey that you're having there and how did you how did you choose a thesis topic do you remember for you--for you for your PhD at Yale first of all there was I know he'll I had a book you may know I know and semigroups and I was an avid reader of that book and I had a course from Phoenix Browder on differential equations now Felix brought here may not know but Felix Browder was an abysmal lecturer and so you had to spend about two or three hours after every lie he was a he knew what he was talking about but he took him a long time to get to the point or to find to remember this or that detail of a proof well I made I went home and I wrote out everything he I talked about and so I had this background and partial different differential equations from his course I had all of I know he'll his book on semigroups and I put the 2d I just put together I think about these things in other words I really like to think about these things I thought about them so you formed your own topic you found your but from there on we have what I like to think of as a journey towards and discovery after your thesis we have your work on an Eisenstein series and and so forth working with the theory of Harishchandra and and so forth would you care to would you care to explain to us what this background was before your your Advent on to the Langlands program on our series yeah a Nigerian fellow would immigrated to the US after the difficulties in Hungary all right so that was in the mid 50's and Excel Burke's wife was Hungarian and I so I think he was invited to the Institute by Selberg he and his wife maybe their children too so he had come into the u.s. sponsored more or less by Selberg and then he was giving a graduate course at Yale and he talked about cell Berg's paper this was just basically at the time i Selberg second thing was a second Indian paper number didn't write very many papers about that time but I think there were two and he talked about that so I thought began to talk and and also I I think I mentioned this is we were talking earlier that there had been a promise seminar on convexity in the theory of functions of several complex variables well if you so if you hear about selbo you hear about Eisenstein series and then you want to prove things well this this theory of convexity and B we did you move instantly more or less to analytic continuation of Eisenstein series in several variables so I'd already thought about that but I thought about it in a rather restricted context no algebraic numbers for example and then so I went to Princeton but not happy not because of anything I thought about a Eisenstein series but because of work on one parameter semigroups but so I gave up I gave a lecture one seminar over which Berkeley powder didn't run it but he kept an eye on it and I think he was impressed simply because I I was talking about something that wasn't my thesis because I talked about this work with Eisenstein series so I think he was impressed by me and then he talked there was someone who had no worse of course he was he study he was a family was in Berlin not that when he was born but when he was a child he went to universities and he had connections with Emmy neutering and and Haase for example so he took an interest in anything that had to do with algebraic number theory so he encouraged me to think about Eisenstein series in a more general context that's not for say groups over over the rational number but for groups of algebraic number fields so Buckner was almost like a mentor for you for a while not not a mentor but he Buckner was like a foster father if you like so I encouraged me he was it was more an encouragement he so pushed me maybe I mentioned I had already mentioned this once to somebody else today I think in other words Bachner so he encouraged me to work over algebraic number theory so or number fields rather than just over the rational number field so algebraic number field you know the basically learned from hecka and because I I read papers by Siegel because their ways of analytically continue series that you could take from the Siegel's papers called little eggs eagle and so I started to read a bit in the literature of these two Haase and Siegel and write about Eisenstein theories basically using these very classical methods in any case one one year just about a week before classes were to start I had given no thought to class field theory and their art had been in Princeton had been a specialist on class field theory but he had gone away back to Germany and they were one or two disappointed students who had come with him to learn a little bit about class field theory but there was no no real information to be to be to be obtained from the courses in classical theory and I came back a week before I'm so I knew nothing about it I tended a seminar arranged by these disappointed students but it wasn't such a good seminar so I was quite innocent darkness it was week before could marketed you into give a course on class fields here how can I do it I don't know anything about it there's only a week left well he insisted so I gave a course on classical Theory from Chevrolet's paper which is a more modern view and well it was I got through it yeah and there were three or four students who said I learned something from it so that began I began to think about the fact that apart I was aware of the fact that there was no non abelian class field theory in it some people like our didn't expect it to be any and so I was just aware of it that's all they would say and so that was that I don't know where we were probably working in 1564 something like that - oh that late already okay yes I would could go back and decide were 6465 and you already had a position at Princeton at the time yes that was all that time except for I took a year I was the Institute was thanks to sell burgers at the Institute for a year and I was in California okay so I was away - two times so I had a comfortable position at the University I went up the ladder of reasonably rapidly I think by it's 67 hours and maybe an associate professor something like that and all this while while you were doing this you were contemplating on the trace formula is that correct or well let me go back yeah so yeah I forgotten something the so I was concerned with the Tres formula and I wanted to apply it I mean the obvious thing to anyone to apply the trace for me to calculate the dimensional space a lot of morphic forms that simplest things I wanted to do it and so you you plug in a matrix coefficient of understood but it doesn't look like a matrix coefficient of a infinite dimensional representation into the trace formula and you calculate I didn't know quite what to do with this and then I spoke to someone he died really young David loud slugger and he said well people are saying that there's something to be found in Harris Chandra so so I started to read Harris Chandra and what I discovered I because the reading has changed it was it this integral that was appearing in the trace formula was an orbital integral of a matrix coefficient and the orbital integral of a matrix coefficient we know from Syria representation of finite groups is a character so and basically you learn from Harris Chandler's paper that this is the key this is the case so that meant that I had to start to read her agenda as I did once you start to read her agenda it goes I don't know but but but but but that was the crucial stage this this observation of lounge lager that people are beginning to think that Hearst renders relevant so so there we are we have it all and so I began to think of all these things slowly and sometimes it worked out sometimes they didn't work I could calculate I can apply the trace formula successfully but that's ultimately neither here nor there so the dates are a little obscure right because when was an International Congress of early 60s 62 or 60 62 was in Stockholm all right so the year after so see Gelf one gave a talk the year after at that at that kong at that conference and year later his talk was circulating now he gave his views of the matter the point was that he introduced the notion of cusp one explicitly transform as a critical is a critical notion and if the notion that I think appears in rather obscure papers of Harishchandra and go to mom hmm but it's hard to you have to look for it but with Gelfand it was it was clear why that was so fundamental now the question it's answer than the question I don't think selber ever grasp the notion of supper of course didn't read other people's papers and I don't think he ever grasped the notion of custom and I think that was an obstacle that he never overcame but soon as ice I mean it won't hate me once you read galvan you can do it you could prove the general theory of Eisenstein series you had to know something in other words you had to be someone who's who knew what about the operators you know unbounded operators in Hilbert space so you had to have someone with this kind of background or it didn't mean anything to you but if you had that backup that background then you saw immediately what was to be done and so take what Selberg had done in in rank one it to the general case I only talk really mathematics with selber once in my life mmm that was before I came to the Institute was a university in vitam at Partners instigation I was I so I'm sure he invited me over and he explained to me the proof of the analytic continuation in ranked one now of course the proof analytic continuation in rank one it's just the theory I mean it's like herman balls theory of differential equations our second order and a half line if you know I'd read it Cottington 11 not too long before so I could just sit there and listen to Selberg this is the kind of thing he knew very well and he explained it to me whether he regretted that afterwards I can't say but he explained to me how it worked in rank 1 we're very impressed by his presentation or what I had never spoken mathematics with him with the mathematic with the mathematician at that level before in my life I never really spoken with mathematics with mark nur and and and he's the only one would and what impression did he make on you what impression as I think it was the back I never it was clearly the best mathematician with whom I never spoken in my life up until that day that's what but even so you didn't have further conversations with him afterwards no he wasn't a talkative man and I you see what I did have occasional conversation because then I would I would still continuing to try to prove the fun'll analytic intention right Eisenstein series comes a little early early enough and I would say to him well I've done it and this is that case what do you say well we don't care about one case of that case we want to do the general theory so he didn't listen to me so I never really although I was a colleague and officers were basically side by side we say hello may I interject here because you came to the Institute of Advanced Study in 72 I gather yes yeah and you had the office that Albert Einstein used to have well yeah we have to be careful about that first of all I did that was not in 72 okay occupied before me by billing all Bjoerling had it before you okay anyway I mean and cell Berg was virtually adjacent office for my mom used to have before yeah I don't know about but anyway it's sort of interesting you spent time together a lot of years and you never spoke mathematics at the time you know Bert you must know didn't speak with very many people about mathematics and he spoke with one or two I think but but not not many and I'm not sure how much he thought about mathematics in his later years I just don't know but even so there's this work of yours on Isis night series it really had some consequences in hindsight didn't it yes so mrit occurred in hindsight right because so that took me about a whole year and I think I was exhausted after that it was okay so you think you have it there was for example an induction proof and the you know induction proofs first you have to know what to assume and if you assume too much it's not true and if you assume too little then it doesn't work so it took me an in fact there was a problem in other words things were happening that I didn't recognize something could happen in other words there could be a second-order pull you naturally assumed there was only a first-order pole and so I so I took me a long time to reach that stage in other words you it's with g2 it's where the group g2 and you think this is this is going to work and then and then you try it and it doesn't work come on doesn't work in general and then you think about it what where could it really go wrong and it turns out you only go wrong for g2 so you make a calculation with g2 and buying it you know and what do you see you see there's a second-order pole oh no or a new kind of first-order pool and so that changes the game you have a different case you have a different kind of out of Norfolk forms so but it eventually worked it was a whole an exhausting year and it eventually worked but but then then you went to Berkley is that correct then I went to Berkley pretty much exhausted from the particular adventure and where you really so let's say exhausted that you thought really about quitting mathematics we're quitting math America is a strong statement right so but I I did this I spent a year in Berkeley and then I got some things done in retrospect I had more done more than our taught I thought I was too demanding so next year I was really trying to I think do something with class field theory and I I didn't see anything I had a whole year in which I don't feel Berkeley I did something in retrospect but the year afterwards doing I didn't at first do anything and I was growing discouraged so I decided a little bit of foreign adventure and so I pretty much decided that the time was I should just go away and maybe think of doing something else well this opportunity I had a friend a Turkish friend and he explained to me possibilities of going to Turkey so I decided to do that so I decided to do that there were various things to do I wanted to learn some Turkish and and I wanted to I went back to the studying Russian I had a very nice teacher but I still had a little bit of time to spare and I didn't know I didn't know quite what to do and I began to calculate the constant term of Eisenstein series just for the fun of this for something to do and that's why calculating them and you'll be doing it and I begin to so I looked at it and I just calculated it for various groups and then I noticed that it was always of the form F basically f of X / f of X plus 1 something like that so but if you can continue the Eisenstein series then you can consider consider the continued the constant term and if it's f of X or f of X plus 1 then you can continue f of X and so and these things are all the products right so you have new euler products and that's of course analytic number theory which there is just loved euler products so you had it you had something brand new you know they had a analytic continuation functional equation so there you were you had something brand new and and you could do basically you could do one of these for every group of every group of you could just do it for a lot of groups every every Cartland class had even the reductive groups you could you could even do it over reductive groups why you basically did it for a split group so according to and then you have the the classification and then so he had a whole bunch and and if you looked at them you could see that some more they were they were related to representation of the story I societies associated a parabolic group of rank one and they they were somehow related to a representation you take the reductive you have a parabolic group you take the reductive sub group and is which is a rank one and then you so you have the you throw away the rank one part and you have basically and some kind of l function associated to the other the automatic form on this sub reductive sub group so what you have is you have Eisen you have you have all the products that are attached to a representation of a group and so all your products are these are these are there's a series that number theorists love and that's what you want you want you want to swim so he had these year and if reach group they or you had a large list of groups and so that already suggests something maybe and then you begin to fall you could say you can formulate this this you can see somehow where this where this is coming from you can see how to formulate it as as a representation associated to a naught of morphic form and a representation a particular representation and what I call the L group before L series and so there you were in that you start to make a guess you have this in general and now I've got for particularly reductive group I have a an order product with an analytic continuation associated to a rip a definite representation but you're thinking oh maybe it works in general so once you have that once you have something that might work in general you think about how you're going to prove it but this must have been extremely exciting it was yeah and and and what what about did you continue the class in the Russian or or Turkish or did you no sorry I tell ya I suddenly I get I gave up the boat and as I said the Russian teacher who is a sweet woman and I think she liked me because I was an industrial student but is it fair to say then that your discovery comes out of well you were extremely exhausted you let your shoulders down you play and you have some evidence and you have a discovery is that if there is so this comes through your exhaustion and and you're just playing with mathematics for a while yeah that's that that is fascinating when when did you get this if you like epiphany where you saw the connection with the with the the the arcane conjecture and so on I'm going there was a Christmas vacation in the appropriate year 1966-67 yeah and this was before you sent the letter to Andre no the letter is a it's an accident I wouldn't I wouldn't have sent them that normally sent a letter 200 a the point is that he I went to a lecture books my turn and they went to the same lecture and we both arrived early I knew him but not particularly well so we both arrived early and the door was closed and we couldn't go in so he was standing there in front of the door I was standing here in front of the door and he wasn't saying anything and I thought I should say something so I started to talk about this business and then he didn't understand anything of course and probably he behaved it you would behave under those circumstances you else is young fellow he's talking to told me so I just assumed he went away so he write in your letter oh he never read the letter I know but he added written Alps it printed out yet it typed your letter type that's right yeah this must have been a wonderful Christmas well no I mean I don't good to discover it but you're young you don't take these things so you do tell a story where you had this well just an epiphany where you saw the connection you were standing in in fine hold it at the seminar room looking out the window yeah can you explain tell us about that moment I was there I lived about we had 40 we had four children and we lived about a few hundred yards away from you know Princeton yes yes no wonder you know Bank Street a about yes you know University Place yes I do you grabbed University Place you cross the principal road and then you go down a small Lane we were the third House on the Left okay and so I can easily walk to that was not the new math building but the old okay okay you know the old math yes it's a it's a shame but it used to be much really beautiful yeah so I had an office I had an office so you go in the front door the president the garden was behind you okay and you went and I had a small office on the right and then there was a seminar room when they left so I would normally you know what I work in the seminar room because the blackboard was bigger and so I was standing there thinking and looking out the window as one does what I'm thinking and and then I could see it so that's where it was and that's why it was because that was there one evening probably well leaving my wife at home to put the children to bed that must been have been a marvelous moment it was happy I could tell you that yeah but this is not the only moment you scribed where you're definitely not sitting by your desk and working your at another occasion you tell that you're walking from here to there and suddenly you see something is that a pattern of Urist yours is that how you find things I have suddenly seen something very seldom in my life so I don't drink one can speak of a pattern yeah we call me talked about this prank are a moment you know Frank arrived and he entered the bus yet and then all of a sudden he saw their solution to something he had been thinking about for months earlier but had put aside and then all of a sudden he saw the solution so I when I read about this story about you and you saw the connection with the art in conjecture that you were doing you know which was contained in the paper that you or the letter Europe to buy in other words I wasn't searching I didn't I'd know I did it I would stumble across the another billion class field perhaps it's time that you actually tell us what the Langlands Prague the program is all about just in broad brushstrokes the person whom you talk it seems to me I I think we discussed this a few minutes ago yes we did we do it again I wish at Frank over here because after many years I thought I would like to learn the geometric theory and Franklin has many articles on the geometric theory and I read them but then what scared they're quite pretty and and he explains them clearly explained situation clearly but so I wish but when I read it I was troubled by two things first of all they seem to be dealing with statements that they couldn't prove and they just couldn't prove at all there wasn't that they couldn't prove them in general they couldn't prove them in really specific cases but this is the geometric the geometry but but but basically the Langlands program as such what it what is it about what what does it entail all right actually I think I wrote so to speak something explaining that later so so you want to ask we sort of know what the we know what the quadratic reciprocity is right that two things that appear to be quite different are the same now so and we also know that you know after they one can be fine Zeta functions or L functions if probably better and you can define them over a finite field and you can also define them if you have a global field you can take a product of the ones of a finite field and you get some kind of l function associated to a variety or even two if you like in particular the degree of ecology and then variety and so you assume so presumably once there's a basic problem for arithmetic for any kind of estimation and it's number of solutions of the offending equations are these elf functions that you could have formally associate and a one and a half plane associate to the kamala G or you know to given the degree of an algebraic aynd of curve over a number field there there so the problem both presumably if you could deal with these then you can somehow other sometime or rather do more things about the estimation number of solution and nature of solutions and so I don't think well they're known as any a very clear idea except in very specific cases what you could do with a knowledge of these al functions and global L functions but they're there so you will want to prove there I've analytic continuation so the only reasonable way on the base of evidence is that they will be equal to other morphing L functions all right now from the point of view of the variety and the quality of the variety you have a golden dick formulated I don't know to what extent he he actually had a complete theory I don't think you had an the notion of a motif and a motif was it has certain multiplicative properties and and so on so you had a whole family of functions which people behaved in a natural front or manner and you wanted to prove that they could be analytically continued but he got to a managed to associate a group in other words these these motifs were associated to representation of a group who nature had to be established that outside it was there so you may not ever know its nature but you should be able to find out its relation to other groups all right now on the other hand what you would like normally would to establish the analytic properties of these things to find out to break dramatically you associate them to something that is defined analytically because a lot of morphic L functions basically at least have analytic continuation there's some question about it right because you could do it for if they're associated to g ln and it's in the standard representation if you and that's the theorem of jockey and gautama but but but and then so you need to do two things which which are more or less makes namely for another morphing form associated with general group you need to show that that oughta morphic form it's really sits on g ln and push it towards PLM and and then you make then you define the l function so it's not just inaudible form but an automotive form that can be pushed towards GL n now if you can so that needs you to think well somehow inaudible form is associated to a representation of a group which which has to be defined in other words a structure in a collection of all art of of your forms you could pass them for one the digit associated with G you can if you you know it isn't true that if you're associated with the G you can pass them to another group G prime if G goes to G Prime this is this so-called L group whatever and and you have to push it forward but if you have this motion and you can push you can say the autumn working for him here is equal to one over here so the l function is the same so if this one take over here is g ln well then you know by exactly the language that you can handle it so if you have this kind of so if you have a way of passing whenever you have no very formal one group to pass to other groups and they appropriate form formalism then you can handle the analytic continuation this is what you call funky reality yes yes yeah this passing like that so that means i mean if you could pass like that that really means if you think of what it means it means that you can describe it by representations of a group so so and this is the same thing this is something similar is happening over on the algebraic geometric side and there's another group defined a similar way not to let the growth beginning and its motifs so but when you have the two then you can do all the analytic continuation you want and give what you get is of course something for your great-grandchildren to discover but but tell me I mean the Langlands program as such I mean as you described and very not a real program I don't know but but is it is it related in in a way I mean philosophically if you like to the so called air lung in program that Klein introduced because I don't think so because he what what client explains itself of course is that you should study geometries by looking at the groups that makes the invariant groups of the things but climbs describe it's also that this was sort of a light motif for his thinking that did but well yeah I would hesitate to use the word program but I think that probably this light motif is it is right and in other you have these two somewhat surprising structures on both sides right groups and thin groups moving from groups groups as well with it one goes to another and and and and you have an under one side which is arithmetic and the other side which is analytic or geometric depending upon your point of view so yeah you move around and you know that everything can go to gln and so and with GL and you have this one example of it have an euler products and you can analytically continue it continue but i mean if you listen to what I'm saying I think this being out of a void or in devoid of yeah but but it could be explained why is it important this seems like a very naive question and it is why is it so important analytically or meromorphic Lee continue this L function why is that so so crucial why is that so crucial yeah that's a good question and why is it so crucial to know anything about the zeta function I when you go you know you go to estimate and a number of solutions and things like that I mean what do you do with the information you have about the zeta function and what do you would you do if you had all the possible information so do you have an answer no I don't I think okay is it that we haven't worked with it sure of course we know that the classical set of function tells you something about the you know prime numbers distribution of prime numbers is that right now of course the deletion a thing does something with the arithmetic progressions and so you get that kind of information but it's clear that that's where people are hoping for but you get as well why do they want if God Only Knows oh so so what is pushed so to speak by preconceptions in a certain sense and you're trapped but and the way you think mathematics should work ok but but is it the the other question I have is that in you had the classical class field theory I mean yeah and and of course that can be explained or or it is about you know a billion extension of the of the number fields and and you know how Prime's sort of the split that cetera cetera cetera and your program also called the Langdon's program the generalization is often described as now you look at non abelian extensions of the of the number fields or a function fields over to find out well you know according to if you if you have very around then every number fields means you're dealing with the question is with an equation with a finite number of solutions you're not dealing with the curve or variety so I mean the but there's very de functions means that you're which which are also can be put in there meaning that you're dealing with something about a one-dimensional variety of two-dimensional variety of three dimensional right and so on but you ask me what do you understand the implications of all that no one is speculating I wanted hoping to prove rather small things in what is potentially an enormous thing may be completely irrelevant but that's what the analytic number theory is Andrew Wiles yes he said and very openly said that proving this modularity conjecture I mean he was very much inspired by your philosophy put it this way you have to do that very very much so could could you expand on that he used kind of we're speaking about a specific kind of theory right our theorem and that the specific theorem of this sort that he knew that he needed happens to be available by some sort of happenstance system so just just a curiosity this the particular case of this general theory that I'm describing that is necessary for Wiles is there okay okay but but he said I mean that for him the longest probable sort of a guiding light the guiding to what you should look for I mean he well that may be so you have to us that's the case for me I guess in some sense but maybe you have to ask Wiles but he specifically meant because I understood it to mean simply that he had this one case which was enough for the fair mafia okay I mean just one case of an ever of an order product that you can manage okay because it was related to an order product someplace else but while states it rather interestingly in in one of the earlier interviews here that is something like to the lines that the taniyama-shimura conjecture had to be resolved so it's a problem here what he says it's a problem in the middle of mathematics you can't just go around it and forget about it you could go around firma and just forget about it and that's the end of it but this problem was completely central and it would one day would be solved so it's I don't know what the glue show more associated the modularity ammonia Larry theorem that that was proved in order to prove which led to thermal and and he he gives this sentiment that this part of the Langlands correspondence was of such a central importance that that almost lent him courage or he this this would have to be solved and and is is that something so you you propose a theory of mathematics that in some sense is it's rather encompassing and it's not a particular thing and it's a structural thing it's that well I think one is looking for a structural thing with a lot all of whose the particular instances are of interest I think something like that I'd be there's so much that one can't do that I hesitate to answer your your your question but yeah I mean you like you have the one structure and the one from the offending equation which is sort of dated in one from Arabic forms with one inaudible conforms you have a lot of and it has the structure of its own and so you have a lot of information about the elf functions here and they move back here and that more or less usually that does what you want I think but I'm not a specialist about those things let's move on I mean this so-called fundamental lemma that was proved Bongo got the Fields Medal for it in 2010 and actually Time magazine said that this was one of the most important but there is sure but but the point is that this I heard that this because the lack of a proof of the of the of the fundamental lemma kept the theory the lianas program back 20 years or something like that because okay no no okay the fundamental lemma is is inserted to deal with specific kind of technical question I this is not a it's not a good example but but I want to try to explain something I suppose you you have something like s a group SLN and you have su n I thought you know about after miles theory or something you know about the representations of SU n I doesn't basically the standard finite dimensional representations of this group now if you look at the SLM situation what happens is so the SL n has more representation than su n do su n it is a for certain form of it it has less than SL m but in addition over an SL and you have so SL n has a law is a non compact group it has a lot of representation but in particular it has some things which are very much like those unless you in which the characters are basically the same you know the character Minister yes you towards each of you i invade of - each of the - I overeat is and now let's go over to SL and it has well Harishchandra theory I mean we're doing phrase SL true that it's previous to Harris changing you you have corresponding representations in other words in its whole theory of representations of semi simple groups or simply or reductive groups and therefore in the theory of autumn or reforms and therefore the whole theory there's what happens is that for SL to those things where there's only one I mean you know this one su - there's one representation H each dimension but so each one has basically something corresponding over for SL - so called discrete series but the discrete series it has at each end it has - it's just this one thing which is only one hand for the unitary group becomes - an SL - all right now but these two are for all practical purposes the same they're just two pieces now this what this fundamental lemma of course because then you what you have to do if you're worrying about the trace formula you want you want some part which is a really useful part for say sl2 and that's the part where you put these two together as so they look like su 2 and then there's a supplementary part where you have to take account of the fact and if they're not really they don't occur with the same amount duplicity so we have this extra stuff so if you wanted to do the to handle the trace blown right you have to say you want to compare you want to say su 2 is more or less like SL 2 so you compare the trace formula on the true but there's extra bit over here it's causing you trouble so that is so and the reason is that somehow one representation here breaks up in the two here and and some of it is really so to speak it doesn't have much to do with things it's a it's just there you just take the difference of the characters rather than the sum and you have if you're going to use the trace for anything else you have to end you have to understand that that's part that you don't really want all right and so there's something the theory of endoscopy and what so called fundamental lemma it was a fundamental lemma in the context of this specialized theory which you introduced for this special only for this special fact the factor that things which sort of speak should more should be the same can sometimes differ so you have to what you do is you treat them as though they were the same put you together and then you take the difference and then you have to look and treat those differences separately they look like something coming from the torus itself the this circle group it sits in there so it's a technical necessity namely if you want to compare the representation of two groups you use the trace formula and but this stuff this extra stuff you have to get it out you have to set it aside and treat it separately so you can compare what's left and then and so what matters is what it's just to understand but what you can't compare on its own and that means you have to so you have to understand the differences so you have to look at the just at the circle group which is which is all that matters and then and for that you need to fundamental lemma so the fundamental happen it's a fundamental level for these technical purposes and all that don't believe what you are what you read it was hard to prove but it's not the sort of big fundamental fundamental lemma takes care of exactly that problem I mean yes the whole theory part we have the rather complex but it takes the taste yeah but you say also after the fundamental lemma and you say that for for the fact or reality thing which is the important part of the language program in the this fact or reality question that that the key use you surmise that the key to making progress is the trace formula in the in the stable form of Arthur could we could you explain something about why that's it why do you think the fortress formula is so crucial yeah well so what do you want what do you want to show you want to show that I can transfer everything to to jail and basically okay no you let's put it some of that differently you want to show that I can move automotive form from one group to another okay so that means somehow you do want to trace form you compare the two trace formulas right because you you you say something that of course from the reality is such an important with the most important language program and to make progress you say that you think that the crucial tool is going to be the trace formula in in the in the our third source table yeah or and and and I I just want to know why is the trace formula is going to be so important what you want to do is you see if you have g and g prime you want to be able to move things from the group g to the group g prime and you want to be able in take nur to handle the elf function because they want to be able to move them to jail in right you say and I say these things work at the level of the L group but let's just work with jail Anderson okay sure so we don't have to worry about her so how are you gonna do it to what you want to do is somehow you want to you can pretend you know you can plan you you in principle you know in other words you say here is this this group every time I have a homomorphism of the group really the L group one end to the other then I have a transfer representations alright so that means that every representation its obtained by transfer it sort of speak there's a natural transfer because you could see this when you see when you see the distribution so to speak of conjugacy classes so so what should happen what would you do right you would so there so to speak a smallest place a smallest group where it sits and then it's propagated to the other groups yeah so for example so yeah there's one group do you want to understand it right so you say well here's a smaller group so there has to be the contribution of those things which sort of sit in the bigger group at that smaller group so one away went away went away you do it you do it all wrong you look at the trace formula here it'll be the trace formula here and they cancel in other words coming it comes from one place you look to see what it what it can sit cancel something and you go along and along the wrong when you know you understand it and then ultimately the real building block is those things on the big group which come from the trivial group mmm all right so and then you saw you that sort of speak the last stage is to analyze those so you can look and see well how do I take the small group and I want to send it to the big group and I just have to look I take the trace formula up here it cancels everything I know they on these it just canceled everything I said it should be made up of pieces and he should come from some smaller group so I just say well this comes from the smaller group this comes from the smaller group this comes and then I have to be careful because you know they could come from a bigger group and a smaller group and has to be careful I don't count it twice so I said this was it should be so they should be equal I have to have a clear view of the combinatorics I so I have a clear view of the common everything comes from a smaller group song that comes from two things a smaller group some of it comes three and so on and this depends upon the image group but so to show that this is really true I just show that the some of that the trace formula gives the same up here as that's worth something collection from the various groups this is pretty vague but I think your principle it's not so bad okay okay that's how it works in fact but it you know up until now it are very low yeah yeah yeah but that's that's very interesting to hear yes so that is that the four from so of your investigation III mean that at the forefront of artists investigation I think if you want to hear the definitive or what's available along these lines ask Arthur but you're thinking in a more differential geometric line well I was the three I've been thinking about the geometric theory and the geometric theory is not the trace formula idea metric theory is this basically this yang-mills theory yeah and that was a recent really that is a recent release of yours so we can write out ready to release but it's not it can be found on the web yeah exactly well I guess we have to round up with the interview more or less but but before we do that I mean you have this characterization of mathematicians sometimes you call them typically problem-solvers and then you have some that are you know the form theories mathematical theories how would you describe yourself are you a theory builder or are you a problem solver are you something in between well I I think it's the wires I'm anything at all it's probably the first right I see you take advice prepare my theorem is the problem solved yes yes and in that vein because he was and you see early in your career you said that you admired Groton Dick and Harry Chandra a lot yes and and was that because you saw them as let's say fantastic theory builders was that what impressed you with them or certain you're right they both of them yes sir so perhaps before we conclude completely the interview it might be interesting to hear whether you have some non mathematical private passions or interests of some sort music literature language or poetry or I don't have passions but you know it is true that one would like to take a look at other things you know history is fascinating you know modern history ancient history the world's the Earth's history universities history these things are fascinating and the shame to go through and so life and and not been some time on them contemplating on that but not to spend some time contemplating on that yeah or something mean not everything of course but just to think about it a little bit well on behalf of the Newton mathematical society and the European mathematical society and the two of us I'd like to thank you for a very interesting interview and again congratulate you with a diabla price well thank you for the invitation you

Channel: The Abel Prize

Views: 24,405

Keywords: Robert Langlands, Abel, Abel Prize, The Abel Prize, Math, Mathematics, Christian Skau, Bjørn Ian Dundas, Bjørn Dundas, Langlands program, Langlands interview, interview, math interview, mathematician interview, Abel interview, beauty of mathematics, Langlands programme, trace formula, reductive groups, Artin Conjecture, L-function, Reductive groups, Fundamental Lemma

Id: 49anoCJOJ9I

Length: 72min 33sec (4353 seconds)

Published: Wed Dec 11 2019