 You are director of the research at CNRS and you have been at the IHES in the early 1980s. Last month, on the 16th of May, you turned 60 and the IHES is organizing a conference in your honor next week to celebrate your birthday. And also, this is one of the four conferences for events celebrating 60th anniversary of the creation of the IHES by Motshan in 1958. So, Auffer, I will ask you a few questions about mostly mathematics and how you work in mathematics, how you see them. And my first question is very simple. What is your first mathematical memory? So, maybe when I was a child, I was interested in elementary questions, but I started seeing that from books that were not taught yet in school so I knew about ... so I cannot probably date it, but I know that around age nine, 1967, there was a game that the students played about kind of drawing a figure in one line. So, this is essentially the Euler's Seven Bridges problem, but it is, for general things, I mean, you have some points and some lines connecting them and you have ... it's connected, but you have to pass every ... to have one line that covers each segment one. So, then this ... so there is a criterion in terms of parity of the number of edges. So, anyway, I found the proof of this, or I thought about it around age nine, and this was the first non-trivial scene that I can remember. Of course, I was interested in other things. I mean, I figured out like how to ... you can exponentiate to fractional powers by taking roots. And then I imagined how to do it with reals by approximations. I somehow figured out how to do some basic analysis of some ... But when I was ... I mean, I was in elementary school, but later I attended actually when courses for talented children, so I was taught things around age. Yes, so that introduces my next question. During elementary school and middle school ... How did you learn mathematics and is there a teacher or a person that you remember as especially interesting, influencing, inspiring? No, but I ... now I cannot remember exactly, but there was ... when I was around 11, I met a professor from the United States called Professor Vernick who was a distant relative of my family, and he asked me some questions. Maybe there was the game of Niem or something. Anyway, I could solve some certain problems, so then he was quite enthusiastic and he sent me several books for my bar mitzvah when I was around 13. I still have those books with some dedication. So I wrote to him letters about what I was interested in. And so I met some mistakes in English in the letter. But in any case, he sent me something about number theory and group theory and he wrote a book on elliptic geometry and other. And so I ... and of course I studied ... like sometimes I talked with some ... like the brother of my ... the friends in school was in high school and he was ... he liked ... they knew about complex numbers and the relation is science questions so he explained it to me from his notebook. So also I like ... we bought ... like there was those sham series books about various things, some of those volumes I bought and I knew certain things from ... like about advanced calculus and some other ... So these were essentially the first books, mathematical books that you studied. And more generally, what was your relation with books in your childhood? Were you reading a lot and mostly mathematical books or books of other types as well? This I don't remember. There were books of other ... no, I remember, certainly there were many books of other kinds with ... because it was not like now where there are a lot of sources of information at that time. But of course I don't have those books anymore. I don't remember, but certainly there were books also about science in general, like about the inventions and ... there was one book telling about people who invented ... I don't know if it was a book or part of a book who rediscovered things like some guy in the 19th century, some Russian old man who figured out calculus and then he saw someone and told him that this is not a book that actually it was invented by Leibniz and Newton and then he thought about it and then he died. Anyway, there were some stories about this, of course, about all the inventions like the steam engine. I don't remember now the ... but there were other books about stories on ... not about ... but fiction maybe not so much. No, there were some fiction or ... I remember that I ... And today, what is your relation with books? Do you read a lot? But now because of the internet, it kind of becomes atomized into looking at things on the web and it's difficult to reconstruct one. But I also have many books and I read some parts of them as they are. So sometimes it used to be that I went to bookstores and if there was some book that had some part that kind of is attractive, I would buy it and it was not only about ... it was on many subjects but usually not about fiction. There are several fiction things, but not necessarily. But certainly I did read some fiction. There were books of Amos Oz and Israeli author and I read some and my mother sent me one of them and I was at the top of like my mic. But I read other ones and there was other book ... If we come back to mathematical books, in retrospect, which mathematical book was most influential that marked you the most among the first books that you read? What is the most important book that you read? This is not ... Also when I took the exams for the first year of university without attending the courses but by speaking with the professor and there were several books that in particular from the library that I ... You couldn't sing it out. No, there were several such books like about the causes of university like about algebra and all. I remember the author, some of them were in Hebrew like people like on ... like some guy called Meisler wrote on calculus and someone on ... forgot now the author on ... But then there were ... like there was a book of Rotman on group theory that I read at that time. Or I read, I mean looked at something, but I knew enough to pass the exams. So it's ... and I read also about some analysis things. I don't remember now ... I remember that I ... it's fair to say, it's not clear that I would be interested in algebra. I was also interested in things in other ... Professor Vernick in particular sent me the treatise of Hobson on functions of real variables and I've written them in the 20s and I did started to study things from there. And he told me to know certain things in the causes of university like the Reese Fisher theory and people were impressed. I knew the name because it's like I am all okay. But this is just the beginning. But what influenced me when I was a student at Tel Aviv University, I attended more advanced lectures in particular by Moises on the total algebraic geometry and then Marie or Marca Shaffs about algebraic topology and also Gottlieb about algebraic topology and Professor Sternberg related to his book with Sternberg and Gilmin. So actually, those advanced things were very important for shaping my interest and then I was invited to Harvard. But the books are kind of more ... Welcome to that. Yes, it was not ... So a different question. Gottlieb said that when as a child he discovered the circle and he was fascinated by its beauty and also the extreme simplicity of its definition. So is there a geometrical or algebraic object that you discovered? Have you had a similar experience? Something which you found so beautiful and inspiring or perhaps not? No, it was not like this. No, but this is not ... In geometry, for example, Euclidean geometry, did you find a special interesting figure or in algebra or something that fascinated you? I don't know. I remember I had a grandfather who was a teacher ... Actually, he was a teacher of ... Not of how to ... I don't know how to say it in English. He was inspector of teaching how to do manual things. Anyway, but he also knew some geometry. So I asked him how to construct a regular pentagon and then at some point I read in these books as I mentioned before about differential integral calculus and I asked him what it is and he didn't quite answer me because he thought I'm too young to understand it. But this is ... I don't know if I answered the question. I think it's fine. So when did you decide to be a professional mathematician? Was it very early or at a later stage? You were at Harvard or even before? No, at Harvard I already ... You had already decided? Yes, but before I think ... Already I took those advanced courses ... It was already in maths mainly and it was not in ... So my mother accounts that I stopped my piano lessons. I took from about age 7 to age 10. I don't remember the age and I stopped them because I thought that I want to do math instead. But I cannot tell you the exact date. But certainly I learned for three years and then I was more interested in thinking about math. Yes. When did you discover Burbaki and EGA? And how did you assimilate Burbaki and EGA? No, but when I studied the courses at Harvard I looked at ... So this is maybe not in the first year I certainly studied things like ... I remember like heart stones, residues and duality it looks like and then Altman, Kleinman ... But it's already very advanced. No, no, but I didn't stop from EGA. I didn't start from EGA. The point is that there were courses like Ironaka. Ironaka said in his course that the EGA is like a dictionary and no one reads a dictionary. But still I needed, for example, writing things for my thesis I needed to reference it. So I used to make on sheets of paper a list of small remarks and go to the library and verify the references like EGA, such and such. So I certainly used this system of EGA and EGA references quite well when in the fourth year I already looked at various things like even what you wrote in perfect complexes I used in some places in my thesis. This was a late stage. I figured I found it out and then it was very convenient for me to use. And there were syncs on the algebraic geometry like the paper of El Quique or Anzylion. So I discovered it. I mean, I used maybe the end of my third year. I remember I was working on one chapter in my thesis and I went by bus to the library in Baryland to find this paper and to figure out the reference. So I used some serums on page 568 and just I put this. Anyway, but the Boba Key and EGA was not the primary thing the point is that I went back to EGA. I understand. So my next question is that you looked into some passages some chapters of Boba Key or EGA and then you read them carefully. Did you find any mistakes? In Boba Key or in EGA? No, in EGA several times I sent emails about virus command and even Boba Key. In 1984 I sent a letter to Boba Key or two letters about some something. But later on I found several things like in the Parthenonological Algebra there is some sign question. I don't remember now exactly but it's they have some but in later times I did a little more in the new commutative algebra thing and also in some exercise and even in the old part of the valuation there was some strange exercise. Maybe they didn't think about what they were writing. It was and sometimes it's not a mistake but surprisingly they didn't put certain facts like when you have like a rank one and zillion valuation and the gala group when you complete it doesn't change. Don't put it explicitly, it is well known and I had to use it in some places so in fact I wrote down the proof in some places but it was not strangely it was not there but there were also some mistakes but not usually. Did you keep a record of these mistakes? No, those things are very scattered. Sometimes it is in some emails but sometimes or letters but sometimes it's just not that I put on the margin in some copies. In 1981 you received the Erdos Prize for outstanding results in algebraic geometry, algebraic graph theory, presentation theory and partial differential equations but you had met Erdos earlier. In fact it was in December 1973 when you were 15 years old and it was I think on the occasion the lecture that Erdos gave at Tel Aviv University. You attended the lecture and then afterwards you talked to him and correspondence followed. Do you remember what it was about? No, but my mother found some documents in... But do you remember the kind of topics? She found some documents. These great mathematics, what it analyses, do you remember what it was about? I remember that I wrote to him something about what I was interested in but he sent me some problems. My mother kept some record. I mean she found it in this file that his daughter found in his home after his death and also some letters that he sent to me maybe they were kept in another place but now I don't remember. She told me some time ago that it was in his handwriting. Now I don't... There were elementary questions. I think she sent you... She tried to send you a copy of some of those. Yeah, but she didn't succeed actually. Yeah, so I don't... It was like platinum type questions. It was not... In any case Erdos never did things like the Golden Dicks style of the Braque geometry. It was contrary to his taste. It was just singular. You don't have a lot of theory. Although he developed things in the directions of some subjects like set theory did advance some kind of complicated combinatorical things and in fact there are many things are named after him. He did an analytic number theory I think but not the algebraic part. But actually that brings me to my next question. You have worked and you are a known expert in many domains of mathematics. Yes. Combinatoric, discrete math, analysis, representation theory. So why eventually did you choose algebraic geometry as your preferred working area? What especially attracted you there? No, but I started already in my thesis to do algebraic geometry. Why? Why did you choose that? You might have proved very deep conjectures in partial differential equation. Yes, but the partial differential equation part is just a subject of the minor thesis. It was given to me by Professor Stern. I was supposed to read a paper of Hormander in 1978 and this was about the existence of wave operators in scattering theory and I studied from some time and went to some libraries and papers and books and so I succeeded to actually improve a certain technical thing with estimates of some two-fifths. I don't remember. So this was, then I told it to Stern but actually this was not written. Later I wrote a letter to Hormander about this fact and he said, well, it is in my book on partial differential. At the same time he wrote this book or in any case around 1978 where the letter was in 1982. So in any case, I discovered, so this is my contribution, maybe they heard about it from Sternberg but it was not the main subject, just the minor thesis which also in France they used to give such things for people who do test their tab but they had to do it orally. At Harvard they had to actually write down something but I didn't succeed to write it down. I wrote the introduction, then I had some files of pages about this. I didn't succeed to complete it in time because I also had to present my mantises and a lot of work at that time to type and so on and so forth. Now the other subjects like I worked with Svigalil and Tel Aviv University asked me some questions but this was kind of not the main subject, I solved some problems. There is a famous paper about super concentrators which just uses elementary things. But eventually... It's not that I'm expert in studies, I work for some time and also in representation theory I worked in the early 80s with Joseph and Viktor Katz so there are several papers. I continued. Sometimes I return to the subject but I'm not... Of course the Tel Aviv World Shift is related to the Kazdanlustik conjecture so in this way I continued. But you know, Grotendig said that he shifted to algebraic geometry, arithmetic because he has worked and did deep things in functional analysis. And then when he said that he discovered the beauty of arithmetic geometry a new world to him, a new landscape and it was so much enjoyable. So in your case I think you enjoyed many things and maybe a slight preference for algebraic geometry or very big preference. At first maybe you see the courses that I had in Tel Aviv which were the most advanced ones that were not part of the first degree but like the algebraic geometry so I first thought I will do something related to this but then at how about the main emphasis was algebraic geometry because of course they had, at that time they were the world center in algebraic geometry with Ironaka, Emanford, Griffiths of course they were also major and advisor was in number theory so and Kazdanl came there and he gave a seminar on Drinfeldwerk in the end of my second year maybe and then in his office which was also full of religious books he was also religious. Then there were questions about the talcomology, Drinfeldwerk considered in a special case and by hand constructed talcomology for a variety. At that time I also went to a course of art on the talcomology and MIT with all the fundamental things. So I, and I found so while waiting for a bus from Artin's course I had this idea on the golden equation on the Brouwer group how to solve it and so this was one of the chapters in my thesis. So at that time of course the Brouwer group gives you H2 and then you can try to but then I found out that like analyzing what happens in rigid analytic variety you try by hand to calculate so knowing about H2 some of the exact sequences that appear in the analysis you can try to make sense so I worked. Of course it was working by hand without trying to define by hand what the talcomorphisms there were no, of course the papers of Renault didn't appear at the time so I had to prove by hand all kind of basic things and I think I succeeded to do at least some basic things at least but this was so in any case my interest in talcomology comes from that time it was easier to work on things like the Brouwer group so I wrote my, I had another result about purity in dimension 3 so this was my thesis and then I was interested in of course the question of comological purity I found the way by blowing after doing dimension 2 so this was something that from time to time I I looked for this exposure of Arty in SGA 419 we had some rather non-trivial tricky arguments about 360 doing the fine comological dimension I sorted it from time to time finally it inspired me to find the results of my work that in 2005 about, so in any case the question of algebraic geometry from that time in 1976 about the talcomology this was the important thing that I returned to so I was mainly interested in for some period it was of course without ideas one cannot work on it but still it's so I was to ask you later about what is your most memorable experience in math but I guess perhaps the discovery on Brouwer is among the most memorable experiences so now the different type question if you look at living and dead mathematician so which one you think has marked or influenced you most no, you see I discuss with many people a different period so it kind of changes the emphasis is on when and where and so it's it's and the subject changes because I think that no it's very I never met Glotendick of course I just knew from reading but there's certainly the professor at Harvard influenced me but there was many of them also Maser himself gave something and I did something like I observed that using kind of probability on the wrong model the C there this so Glotendick was influential yes yes no I like they are related but sometimes the people but you cut me I said about Maser that he gave some course or something about elliptic curves and L function any kind of observe that this Manning constant is integral and in fact people know about it because I told so I did so there are influence I'm saying from many professor at Harvard and several colleagues then from in Israel there were less but then I came to the IHS and there were several people that like Katz and and as with and Friedland there I was and then there were courses like in 82 Seusslin gave course and this was part of what I did was very influential influence by Seusslin and those people in the KTV seminar at that time so they are for every subject there is a different influence and so for example Thomason came in the early 80s early 90s and then I was reading his paper and found a way to apply what he did for purity and talcomergy in 1992 and I talked about in 1994 so this but this influence just this part this period so it's not and then and I cannot say you I hope I'm not cutting you but you said you didn't meet Gotendic of course but I think you met Seusslin right? Seusslin was where I went to his courses many times so most of his courses in since I was in the IHS in 82 so in the spring of 82 and how would you describe these courses and was it influential, was it important or yeah but this was kind of important for the community that did algebraic geometry in number theory because Seusslin was also giving some in the beginning of the talk he was giving some survey of the seminars this week so everyone had to be had to go there to be informed about the right thing and to speak with the others and so on so yeah so it was one could not miss this this and of course it stopped just when Wiles did the formalis theorem it stopped he didn't give a course on formalis theorem maybe people didn't know quite in any case he I think in 93 he gave the so I followed most of the courses except found that I don't have the course from 85-86 I don't know if he gave it in Paris or in other place but it's and also tits I went to tits for a few time I went to Alencon and and how about Deline? okay I didn't mean of course it was very important but he was a IHS just I met him in 1980 there was a small intersection yeah there was no but certainly he told me about Vale II in 1980 and then I found few days after I found the proof of the purity of intersection coromology using the stuff that he did in Vale II and this was in July 1980 and then of course there were many discussions at lunch about things so I have some of the still some pieces of papers from from lunch but it was also discussing with many other people but he used to yes there was a lot of information but now I don't remember Vivid League all the so he left I think in 1984 and I came to but he was the one that because of him I was invited to the IHS in the first place my visit was extended gradually and then it became a CNRS visitor long-term visitor so yes I think he was but I'm saying it is only for this result some related things on the monodromy filtration and other commuting several things that I did in 81 I told him and of course I told him when he was at Harvard I met with him and I told him about what I can do in talcology at that time and he was very so it was yes so I remember this was but I I somehow work I see I there were people like Katz that continued to come to the IHS many times and I discussed with them much more because of the many years I but also he didn't come back so often he came back in 85 but when it was a few years in the early 80s I did discuss a lot but afterwards did you keep a correspondence with him or just at some places just from time to time not the regular correspondence no there was no he tried to invite me to the this was in the 80s then I did not there was no essentially very little I have some file about it of course there are some things but it's just and of course I came here in 2000 maybe I I don't remember if I was maybe he well I have a file with various things but it doesn't mean it's not about it's about inviting me to about going to Princeton then there was one at one point I sent him an email about his work related to what I mean this paper he wrote around 2012 and if finally sent me after some time what he wrote but it was I don't remember that there was a lot of email maybe and of course I gave a talk at the conference in his own or with other people so actually there was and of course I have some letters like when he visited he wrote me by some letter in 2013 about some questions so there are some documents but it's not really a big maybe we change topics for a little while and may I ask you questions what did you start to say and change topics for a little while and may I ask you some questions of maybe philosophical nature so when you do mathematics what is your feeling do you think you are discovering things or you are constructing things there are some people who have very strong thoughts about that I don't invent I don't construct I just discover what is God given somehow no but certainly there is no there is in the presentation in the way there are many variants of doing something like you can do algebraic geometry in the very language and you can then do some foundations in a more advanced way like you can do old fashioned category or use some people who do infinity categories now so of course I think that some of it is the style of different people there could be a huge realm of possibilities and we only use small part of it for various reasons in particular because it would take too much time to exploit all of them but there are many possibilities to present certain things in different styles and it can look different or dependent people education and so on so there are some people who used very style of geometry because they were not comfortable with skins but now like people who use Grotendick style sometimes they are confronted with people who work now using much more elaborate machinery but I understand that there are several ways of describing the mathematical objects but do you think you are maybe a builder you construct things or you discover things which are certainly I discover properties there is a question whether there is an objective mathematical reality in nature what we know from logic that some questions are not decided but like the continuum hypothesis is it really true or false in nature that could be different opinions and whether what about the set of all sets if the sets exist and why the set doesn't exist so we have to avoid such questions we just work on certain things like if you work like analytic number till we don't have to worry about the set of all sets but still for the philosophical question or whether the mathematical object really exists in nature I'm not sure but still you certainly discover properties of the integers like when you have the law of quadratic reciprocity or when the Greek discovered the Pythagoras or any other things that they did it certainly discovered those things I mean clearly those geometrical things somehow exist in nature or at least the usual nature and then also many other concrete things about various functions certainly the Riemann Zeta function is I don't know it's so you discover properties of the Riemann Zeta function on the other end if you construct some theory of let us say model categories for doing some homotopy theory in some context and you have different people they have different versions the question is if they invent it or discover it but well they decide to put some action in some category as finite limits or co-limits and some guy can say well we don't need it we have a weaker one so do we really invent it or discover so I'm not sure I think that you work on some formalism and there are different ways to do higher categories are they invented or discovered of course it's related to other people kind of compete with each other so they like things to be different but even if they are the same sometimes they are sometimes things which are different sometimes they are the same sometimes they are the same they are considered to be different and now there is no but I think that some fundamental things are discovered but there are some formal theories which are more like invented and there are different ways to do it but in some sense they are also discovered because maybe all those ways are written in some book and you have to this question of inventing, discovering constructing is also linked with a very good question a philosophical question the nature of mathematical objects so roughly speaking there are two viewpoints right the platonic and the materialistic so what is your view is it both or one of them so the platonic view is that mathematical objects exist independently of any physics independent of Big Bang independent of anything and others would say well no the mathematical objects in fact exist in the brain of mathematicians for example in your brain or in books in libraries in electronic documents so just atoms or gluons or what not what is your view? Well there is some consistency because we know that the theorems that are proved correctly are sometimes you have other proofs but you don't find contradictions unless someone makes mistake so this holds for the main body of mathematics although there are some cases where there are some obscure points but at least for centuries it continues to build in some consistency to be applied to physics so this doesn't mean that the platonic viewpoint can be accepted completely because of the logical problems about set theory that I mentioned like if you believe that sets exist and what about the set of all sets and in the usual foundation we need set theory but there are people who found who worked on different versions of the foundations of course you can just restrict set theory to like type theory of rustic and white just you have natural numbers of power sets and power of power sets and you can just ask whether this really exists in some platonic sense in particular then this would mean that the continuum hypothesis is a true or false in some platonic world and then you can have different views whether it makes sense you can find in the literature some views and there are people who write some books on philosophy of maths but I think that those kind of debates are really very usually relevant to what mathematicians do because the problems that they face I have to do with more the content and the sociology of certain domains where the foundation of things like this are not important relative to the huge amount of variance in the methods and the details that you have to study you cannot even study deeply all the possible approaches in different subjects so the but on the other end you can find material about the philosophical view and in fact there is more than two possibilities there are more than two possibilities I think I can cite properly the authors that wrote about it I don't have the language I don't remember the language and content of those discussions but there are some authors that looked at it it's just not really I don't think it's really so relevant but it's certainly there is a view point the formal view point but I'm not completely in favor of this because there are certain things that you do like when you discuss for example working with different kind of functions like you have different domains of mathematics like you have continuous differential algebraic geometry but then you have like analytic geometry of different kinds and you have different kind of functions like over convergence and so now when you have functions you can compose multiply add differentiate but you don't specify exactly the situations in which this is possible sometimes you can discover a new kind of functions like you have rigid analytic geometry now there are different ways to do it but you can find slightly different type of rings let us say you have over convergent and then maybe some other anyway we know that certain things develop in stages like you have different kind of fontan rings with different kind of over convergence so before certain things were discovered you cannot really make a list of all the possibilities but still there are some manipulations like I had some experience working some version of Elkika approximation you can make some proof with certain that should work for different kinds of functions but you cannot explain formally to a computer what you mean by this kind of intuitive operational knowledge or skill I mean you cannot it's not completely formal in the sense you can give it to a computer the people who do formal proof some of think about computer verifying the proof they have some idea like having some function and operating with them and they can see that it works in different context like in differential or certain condition of course you have to verify that you can do certain things but it's it's not possible to formalize this kind of thinking which is natural for people and even can use it in proofs by saying that this is clear that how to do some things so in fact the proof is not really formal it just uses some kind of human abilities which are not formalized and so this is yeah this is about not whether a mathematical object yeah but I'm still but anyway there is this formalized viewpoint that you just deduce assertions like in Sermelo-Frankl system but still it is not a complete representation of all mathematics because you can do things which are like studying other systems so in fact people think that there is no so I've seen some paper of what is his name there was this Huzel maybe he wrote in some in some but he did he did some algebraic and analytic geometry but he wrote somewhere that there are no complete foundations of mathematics in some place anyway I'm not so you cannot have a straight division or yeah it's more complex you know situation is more complex and just the dichotomy between yes no but it is also possible that because knowledge expands in different subjects we don't know what will happen like we see like in the archive the number of papers grows exponentially maybe at some point it will be so complicated that mathematicians could not agree with themselves about what is true and then there will be different it will not be one subject but just because of limitation and ability to and even now there are subjects where some parts are not so clear or not so that leads me to a question which is somehow related when you do mathematics yes how do you work you see there are people who think about mathematics while walking people who just think about mathematics and they are in bed for example closed eyes and think of things and people like Rothenlich who was at his table and taking notes writing so how do you do mathematics do you work do you take notes? No it has to work to remember certain things but I now a lot of the work because of the technique that uses computers like I answer email or work with some people or write some comments but then this can take most of the time because I split the task to smaller ones and then I have to check references and so on so then most of the time it's just devoted to actual if I want to work then I have some tasks that isn't complete so in fact now I'm just working on writing emails and stuff like this I'm not working on what I wanted to do but we'll come to this email question leader but if you suppose you don't have all these tasks but you want to just do mathematics to think so what is your preferred way you just walk and think I walk no but sometimes I have ideas and I use to write them on briefly or not on some so in older times I wrote more to myself in this way but then it becomes it became influenced by the technology so now sometimes I write to people some emails or scans and so on then I write to myself but it is in some notebook or in the margin of some then it is difficult to collect all the information because it is done in different times I cannot remember what I did on the same thing sometimes I can reconstruct in sometimes notes because time progresses things are more are more atomized and difficult to reconstruct and then I can I can still work like so before I give a talk I have some notes and I work out various many details and it's but it's usually there is some idea how to do solve something at some point like this like recently on this question of week functoriality of community so I found some blowing up observation that anyway while walking so this kind of was again it allowed me to give this to a person to give a talk in fact I already told them that I will give this talk before anyway I found something it was using my talk in Berkeley and so on so I there is some kind of inspiration but then you have to read things and think about different and as I said most of the time because I work with various people just to correct and I delay something so then if I want to go back to some joint papers and work on it will take all the time just to work on different versions so it's really because of this practical thing it's kind of not the but also it changes with time it's really because of different subjects different time and each time it's not really the same method the same same technology the same people and the same means of communication the same and sometimes the subject was simpler like when there are only the papers of Thomason on Algebraic Aethery so there are two main papers it was much much much easier than today where you cannot really read all the but since there are people who manage to work on the now I don't know exactly what they do because I know the so you so people from all around the world write you to to consult you to ask you questions and you are very scrupulous you answer everybody and with a very precise answer and usually it comes in 400 scan documents and I observe that there are no crossing out it's just very neat or very few crossings out so do you make a first draft or does it come all armed from your brains somehow it's already in your brain and you just write and it flows naturally it depends on the case because sometimes I simplified and so that I can write we already thought about something and I kind of simplified as usual you skip some difficult technical thing I mean you work you can discuss it more informally and you can communicate an idea in two or three pages and then you can send it so it's kind of more attractive to do it because the other one will get it fast and so I I but sometimes I have some I I don't succeed to work on something in time sometimes I come back to it and sometimes I write so sometimes I have to draft also in email some long email sometimes I use to work for days or even more on some email and make some changes and I had some computer problem that got some assistance then I I more in more recent years I use scans because it's more but I I remember that I worked with there were two long emails in 2007-2008 pseudo reductive groups to Brian Conrad and those were important for my work and I did work for a long time on those emails I mean it was maybe a long time it's not just one day but it was after so many revisions yeah there were many changes and so I was printing out and correcting things and so usually I did this for then there were other things like well different questions some things that I did like comments on SGA3 there was also emails or scans but do you think you devote maybe now that you devote too much time to this tasks answering emails well actually actually more time for yourself and actually you count the number of emails you see it's not so much in a week or so but it's not that I write every day and less than one it's not there are people who maybe write many emails every day but maybe not in business or I don't know but I yeah so the the point is that certain thing can look simple but because of the time management delaying various things sometimes there is a conference there is this to do this to do so what happens is that I somehow I can fill the day without doing much and still I can fill it with various things then the next day is the same and so on and then it gets delayed and there are other questions and then I cannot you see so in fact I don't do enough of this I think that I could maybe be more efficient because I already know the ideas so if I have two or three things I know the ideas I could just but it's not like this so it's but sometimes it's like this and then it is what you said that I write those things without mistake but sometimes it's not and then okay it's it depends on the time and then I there is some psychological processes that are all the time kind of which input I get and this if it is about the same subject or not and but then I think maybe I can say that still you enjoy it you enjoy answering but sometimes it contributes to my own work or it's about joint works with other people so it's really and that maybe brings me to the next question so you have helped and still are helping many people many students and young students for example I remember that seminar on your work in 2006 and 2010 2008 yes and so you have helped older students like myself for example so have you ever considered teaching because you are very gentle with students and I think you enjoy it and also the students enjoy it immensely but this is just you keep you look at isolated circumstances as I said it's not regular teaching for a semester and so on I don't see how I can teach and at the same time work like for example when I work with Ramero on this very long project complicated because of this work so I I cannot work on teaching at the same time or I cannot work seriously like if it's every other day then I can maybe answer some other emails the thing that I did at the other time but I cannot maybe for teaching it will take me depends what kind of teaching what I do is seminars on what I do from time to time the B University a little bit advancing but somehow it didn't turn up they did not succeed so well I mean it was algebraic like algebraic geometry following Archonsburg there was some formal thing about shift theory that I tried to do and it was too slow and the student didn't like it they wanted me to explain about etalcomology or well I don't know the it was so I think that you need to have a good balance I mean to know how to teach is different and to know to actually interact in seminars and and also the subject change even if I want to teach advancing and those things change rather rapidly so I cannot teach I mean if I don't know certain things now do well enough I cannot well I can teach some things but it so anyway you enjoyed being at CNRS I guess and so it is a question why did you choose CNRS to stay in France so you might have been invited to many places first in Israel and then in the States in many places so why did you choose to stay in France? I'm not so sure because when I was here it was natural they proposed me in 84 to apply to CNRS I'm as I said I was all the time busy with small things with going to seminars and writing letters at that time it was in written form and like to remarks on some other people papers and so or other things but in any case I was since I had all the documents here kind of complicated to move because that if you have a lot of paper documents part of them are still in Israel it's kind of not very attractive I don't think that there were actually any formal because in any case if I don't want to teach there were not so many possibilities it's only CNRS or maybe the Bison Institute in Israel yes but I don't think they would I'm not so sure there was really a real options except for the CNRS essentially in any case I I made this application and I was given a post rouge in 84 and actually a year later I was given a permanent position but actually my salary in the first year was higher than the one because in December they start in a low level and then I gradually get far but it is it's kind of so in the first years the CNRS was not kind of very liberal they formally had to write some reports but they didn't care much about it and then later they become more and more demanding and it become more complicated but still I got my promotion to DR2 in 95 when it was still relatively easy I could write something by hand and it was not took one day more so I was typing a few things and then for DR1 I needed a lot of help to do this more paperwork yes and of course now I was told to get DR2 you need habilitation so on you need much more work to do the same degree for the same things are changing you know just leaving the Ecole Normale I was admitted at CNRS I had done nothing so it has changed a lot anyway a different question maybe I'm not sure you will like it have you ever made a mathematical mistake because you know people know that you are such a high level no no but there are mistakes and there are even embarrassing mistakes like in my paper of course there are small mistakes in the proof of the Cohen-Gabbert theorem there is a small community of algebra and it is repeated in the volume of the hysterics and if it's out the point is that because of time pressure I didn't check things which I already know and even with other things there are still a few mistakes so but this maybe can be blamed on but still in the original paper there was a mistake and in other like John Pele with some Indian people there was some I had a version and actually there was some remark about curves like it was about the index what was it the brow obstruction for about the modular space of stable vector but anyway there is some remark about the case of genus which is actually not correct in one case and I don't know how I so the reason mistake that I did not someone wrote so usually what the theorems that I claim are correct but one point Colliot-Ellen wrote in one of his paper in the early 90s that I announced the proof of purity for the bra group and in fact it was not correct there was no such proof even I was supposed to talk in the seminar and then I changed the last moment I sent the subject to something that I could prove and later I published even a note in about this not on the bra group in purity that just wasn't this lecture which is easy to write down then later I gave a talk in a bubble talk about in 2004 about the results that I could prove so recently Ceznavic used a young mathematician to solve the problem using perfectoids and so anyway this was not on my name but someone said that I could prove something in fact I could not prove it's so anyway it's as I said there are some misprints or some mathematical mistakes in some places it's not formal proof checking by computer as recently made some progress considerable or not I'm not so sure but some progress so do you think that one day a computer program will replace GABER certification of signs in homological hundreds when you certify that this diagram has a minus sign then people believe you no no this I did a long time ago you see I did it with Brian Conrad in the book on residual duality on growth and duality and phase change which was being arch on residuality and correctings of particular sign questions and he didn't quite understand what I told him but finally he wrote some compliment or a letter to the book that explained the sign then I there was a sign in Takashi site or paper in 2003 about the spectral sequence for vanishings I forgot what it was anyway it dates to some time and then more recently I didn't have the time to go into the sign that August and his students were looking at they explained to me August and some new proof of by you yes but there was also a new one also by August maybe a new one I think August worked on the Pika Levchets recently yeah but in recent years and I was not able to go into it so in fact this idea about checking sign it goes back to the maybe early 2000 and I so there were not so many cases of this and I know the basic I know a system of conventions that I I used to think about so I could it's it becomes more and more complicated because the objects themselves which are studied are more complicated and so one is cannot really make sense of I mean if you use some very fancy things with all kind of motivic things that people now do that all kind of things go into the definition you cannot really check in the same way without knowing the like give it a little bit of Vodsky motives and different I'm not really sure but I think that the thing of August and Lashin is more classical you have to think a lot on this and I did succeed to send him some remarks on his log geometry because some of it came too late because he was for his book I mean so the science thing is not I think that if you want the computer to do it you have to put too many because the text are not formal so you have to somehow manage some knowledge which is not completely formalized and get meaningful comments on so I don't think a computer as I explained a computer can work very blindly and not it cannot use intelligent analogies and so on between situations so there are six operations in etalcomology and the signs which occur there you think maybe it's too remote to put data in a computer No but people have succeeded to actually have formal proofs of different results but the point is they also use different systems and the question is now if you do it in one system can you go to another one in fact the system is not equivalent in a proof flower and then how do you know that what the computer does corresponds to the problem that you want to solve intuitively so you have to check this is how do you really check that you formulate kind of the so you have some problem like you have actual papers that people wrote now several decades ago which are not completely formalized but then you have to translate it to some formal thing in a computer and the computer will give you some information on the sign but how do you know that what you give to the computer is really a representation of the problem you can just maybe someone could just put something that tells the computer to give a certain answer I mean you need some way to check that you don't cheat when putting I mean and this for this you need the question to formalize this but then you have an infinite because if you formalize this and you have a computer working on proving that this is that you didn't cheat and how do you know you don't cheat in this then you have kind of an infinite layer so you cannot really it's difficult philosophically to understand how you really how you really formalize things because somehow you have some people who work on this and of course you can think that because they have some pressure to have results and so on because of grants and promotion and so on so maybe they kind of hide some things okay and then you have to check that they don't when they tell you in some language you don't understand what they do maybe they are hiding some subtleties so how do you really check it because you always so you have this thing in they speak to you about things you don't really know if you have to know it you don't have time maybe to study other subjects so you need to trust someone if you need to I mean if you don't trust then you cannot okay so it's not clear I mean you some people think that having a human understandable proof is the best and kind of I would say that GABER certified proof is the best anyway you proved many conjectures important conjectures and you also suggested many problems and developments and imposed many questions but is there any specific conjecture of yours or who that you would very much like to be proven or fulfilled is there any special result conjecture that you would like to see proven maybe I don't want to go to very recent things you see I gave talks in recent years some of it is not yet written about proving certain things I think I can also prove some of what I conjectured like the related this work of Cezna which I mentioned but I know that they also I'm not anyway I have some methods to sometimes to answer partially or certain questions or sometimes I succeed and yeah so I usually I sometimes have complicated plans how to prove certain things like in Piatik Hodge theory it was based on of course Fulton's Purity but in the special kind of nice things and there was some art some thing about homology of O plus in nice cases I could prove vanishing I mean this is not written some part of the discussion with Ramero so the idea was to combine this plus some special result in kind of logarithmic situation and some the young type alteration and to do Fulton's style of course what didn't exist yet and people have all kind of approaches and I mean kind of something with complicated so I thought I could make some progress by combining several pieces and I only wrote some part of this input and so and then Schultz's work was led to much faster progress and he did not using his perfect so this approach still has some elements which could be still be interesting so I'm saying that it is not that sometimes I have some ideas how to do certain things sometimes I give talks on it or not but it's it's not that I hope to prove something that I don't I mean except that it used to be like before I could prove the all the problems of et al homology I hope that this like looking at this paper of Artein would allow me to prove some but this was already accomplished but now I've done specific you see I know how to to to do the point that certain things that are not completely worked out like about algebra extending the theory of pseudo reductive groups like doing certain things on the compactification and central extension I have some idea how to do certain things comparing like formal external formal groups algebra groups in some sense I it's not in fact some of it is already announced in the Wolfach report and so in some I want to kind of work out some things that they did before but it is not and so there are some plausible things that one can of course there are always some new things which occur and it's not it's not really something completely open problem that I don't know how to attack this is not a good idea also there are several ongoing projects and then I from time to time I have new ideas on some of it like what recently I talked on in Berkeley and so and then I continue like to so just incidentally you have among the ongoing project this is long ongoing project with Ramiro so how is it okay so now I think that it's turned into doing Andres so the idea to do Andres perfected by Anker Lehmann slightly more general integral refined integral form and then you have several application community of algebra so some of it will be maybe in the manuscript like direct some conjecture for a regular and some application and then some maybe more sophisticated things like some derived variants and so I'm not sure so depends on which time Ramiro wants to finish the manuscript so it's the problem is that doing it there are also things which are left over from before like before in the book we had some condition until there is floods then we generalize it for doing in perfect geometry we work with more general so in any case we have like checking that the thing technically the certain thing that we do like work under condition B instead of M so I'm not so sure if the part is already read maybe one is to still complete it with certain things so I don't know how much time it will take but still in terms of there are some new relatively new thing community of algebra but then it is since the work is coming takes him a long time so I think well I'm not sure you're not sure how long it will take no I'm not sure because it can it can get into some like there were several years just because of formalism of of stocks and the scent and so on that was kind of getting getting into okay so let me change topics so but still on conjectures so it's about the hodge and the tit conjectures yes so Grotendieg I think wrote some plays that he thought that there should be either both correct or both false yes so we'd like to know what is your feeling about them no so I think that okay this is related to question of logic that we alluded to before about incompleteness and so on so in theory it could be that some statements some interesting statements are not provable in certain formal system so the logician said there is always maybe a stronger formal system that you can decide so it is not clear if maybe the Riemann some of the problems like the hodge conjecture maybe they are not decidable in the system that we use and so we will never have we will always use do some implications and so I think that but I believe yes in any case I think that some I tend to say that the Riemann hypothesis will be proved at some point but for see and it is possible also things like the Beersfinnert and Dier conjecture after some technology suffice the advance probably people will be able to cause you have to work it's and then and this is the most likely of the clay problems that could be and the Riemann hypothesis is less maybe less likely then the hodge conjecture is maybe it's on the side where I don't know and the Tate conjecture is related but I don't know if I can place it the point that some progress in special cases is and there are some you see there are some also some surprising developments in motives and so on like the recently proved constructivity conjecture and some other techniques maybe it can lead to more so somehow maybe somehow you will have some and there's some differential are not related to the problem and so it's possible somehow it constructs algebraic cycles this is a Europe's work I'm not familiar so much with it but it seems that if you can construct non-trivial cycles in some context maybe there is a hope to prove more but I don't know I think it's it's not also the old conjecture could even be not true actually so it's not it's not just that I know that it's I think for the best venerant and dark conjecture everyone think it should be true but for the old conjecture maybe and how about the gotten the standard conjecture of left side style for example you have to construct cycles of course because as you mentioned in a Europe's work there are some mysterious algebraic cycles improving the conservative conjecture so do you think that someday it's possible do you believe in the standard conjecture and in those involving constructing non-trivial cycles but anyway it's closely related to dodge conjecture in character 6.0 let us say how could you prove them in characteristic piece you see then it gets into discussing those some recent things I don't want to okay we don't want to say opinion on to finish on some more a more general question so what are your dearest mathematical dreams the thing that you would like to see proven achieved independent of your your own work or work of people around you so what are your most dearest dreams so every mathematician has some dreams no but you see I now don't work on dreams I work on things that I can do like in this project on almost string theory perfectoids different kinds of perfectoids different kind of rigid analytic formulas and applications to different kinds of problems that sometimes I lectured on in previous years and so I want to develop those things that I did in previous years and different techniques including like topo-storatic technique like the oriented product but now you have different originality things but related to homology and maybe okay so you have different I know the technique in other words I'm working for myself I'm working on things that continue what I did and I think there is hope to succeed in some and the point is that there are some people that work with more complicated techniques that I like topological or shadowmology I don't know but there is like when you work on questions in periodic theory there is some room a lot of room to do certain things but it is like continuing previous threads it's not and so I don't but I don't have any because there was the Dugan town and there wasn't like this maybe I know but the question I cannot answer the question is not you said independently of my own work I cannot say exactly because because I don't know enough like if you ask me about people who work on automorphic forms what they cannot really because you are not sufficient so I think that so for example the successful completion of the Langlands program but no it's really even but there are more and more sophisticated things that geometric Langlands really gets so complicated like what Gatesbury does I really don't know if it is my taste now to do those things because of this you see it depends on if I really want to study the machinery that he uses then okay you know maybe if one lives long enough then for several centuries one can study other subjects but then there will be even more because the other people that compete so it goes exponentially so if Gatesbury does those things now and suppose you live for another 100 years then even if you will be able to know what is done now we will not be able to know what so it's not clear what will happen in mass because really no one will be able to know all the all I mean it's okay so maybe actually some people think that it will somehow it will decay I mean some mathematics will after enough time humanity will not be able to manipulate them I mean and already like the number of years is for graduate students like maybe now in some places 5 years it used to be 4 or 3 I don't know kind of more and more difficult of course you cannot expect people will study like 50 years this is not possible so at some point there would be you can see it cannot continue for a million years or so I mean how can people know so I don't know so maybe it's kind of a singularity in the human experience there is some kind and after maybe people will be doing only things that are useful in technology or maybe finally if technology is sufficiently advanced and some terrorist groups will be able to destroy mankind so there is some limit maybe we will not that's a good point to leave it here thank you very much