6th HLF – Laureate Lectures: Random perturbations of Euclidean Geometry

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6th HLF – Laureate Lectures: Random perturbations of Euclidean Geometry
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Wendelin Werner: "Random perturbations of Euclidean Geometry" I will survey (in a non-technical way) some recent mathematical developments dealing with the following questions originating from theoretical physics: What are the natural random fluctuations away from Euclidean geometry, and what properties do they have? This video is also available on another stream: https://hitsmediaweb.h-its.org/Mediasite/Play/e055b61979b541cc8b3debc6466db1a51d?autoStart=false&popout=true The opinions expressed in this video do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the video.

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good morning everybody my name is Helga Holden I'm the secretary general of the International mathematical Union and the IMU is handing out the Fields Medal and the neverland award and it gives me great pleasure that there in this opening session or today we have one fields medalist and one Neverland winner so the first talk will be presented by bendlin Burnham who got the Fields Medal in 2006 as a first probably list he's working in probability theory more particular with random processes and he is a professor at ETH in Zurich so when Dean please the search is yours [Applause] thank you Helga it's a great pleasure to be back here I mean in Heidelberg or close to Heidelberg again for a second time for this forum last time I came I said you know it felt a bit strange to come back to Heidelberg last time I was here was you know as a young student like like you coming after the night train arriving with the backpack at the Heidelberg station in in the morning and trying to find a place to get a breakfast and I remember that for the first time in my life at those at this time the only place that was open was a luxury hotel next to the in the center of town that was already open and so we entered and and somehow we had the the luxury breakfast that happened to be not very expensive surprisingly and so and then I felt like this was really a different world of you know these fancy hotels and things and now when you arrive as a laureate you know you you're offered a black limousine you know to take you from the airport and you arrive at a you know crazy hotel and it's always very dangerous because you you you sort of have to you know first yourself to not get used to that yeah okay so my the the other thing I might also repeat from five year for five years ago is that you know when when you are given the opportunity to speak in front of you know a crowd of very clever young brains who are going to build the future in their respective countries it feels it's having a responsibility it's a big burden really feel you know who am i that you know to give you no advice or say something that might be interesting or relevant for for all of you you know in some sense the the most important thing in science is to be you know first of all to like it to do it because you like it not to where you're told about the prizes and you know we are here this laureate thing is about the people who got prizes and we are sort of here to tell you somehow well if you do science you know the you know the worst possible reason at least for me is to in order to get to price one day because that's not the certainly not the goal is only would lead to frustration or you know and the other thing is like you know sometimes science works well you know we we take we try something out we have ideas we test them sometimes it works but most of the time it doesn't right so otherwise you know if everything would be like always you know a piece of cake and every we succeed and everything there would be no you no merit in it somehow and and so one has to also just have you know accept the fact that sometimes you try things you have ideas I saw it but then sometimes works there's a part of luck in it you know it's not just you and sometimes it's a it doesn't but nevertheless we still like the the path of trying to invent or create new things I also what I was preparing today's intervention I was thinking like you know maybe it's me getting older but sometimes you know in today's news in the last four or five years I felt like you know the world I used to grow up in seems to change you know the number of things we you know values or that we felt you know was you know being shared more and more sort of stopped being eroded here and there and and okay so but in some sense the the I think it makes it even more important that you know science has you know there are certain ethical values about you know truth and and also sharing and international aspects and all these things we have to you know keep firm on all these things and don't you know forget to reaffirm it at any occasions so I'm sorry if I do it pontification all thanks to you young people but I just wanted to say that that's one of the you know nice aspects of doing science to be able to you know live in a community where mostly these values are will be in clearly shared okay so now I understand this is the you know very often in conferences when we go scientific conferences there take place in during
an entire week and the Friday morning talk is the maybe not the you know I understand you had the Bavarian evening yesterday and so one shouldn't be one has to be gentle you know the the brain has to be warmed up again and and anyway I you know we can't give you you know technical talk when you're on the mathematics side of things I'm not going to teach you you know give you a lecture about introduction something with definitions and so it's a bit of an acrobatic you know thing to try to stimulate your imagination to get you some feeling about some nice things in mathematics maybe and without actually doing any mathematics you know here and in some sense I'm fortunate because I work on a on a field which is actually quite amenable to such exercises at least I think because it has to do with randomness and which type of randomness where one can maybe try to see some pictures so you know I can show you pictures and rather than write down slides with definitions and so that will be my goal today too basically so the guiding principle will be to ask ourselves some seemingly naive you know were natural looking questions that in some sense the type of question you might you know if you're on a train ride and you start having your mind wander around and you see you know you know you might actually end up asking yourself this type of questions and and then I'll try to illustrate the fact that you know look try to answer understanding these questions well first of all when you're asking type of questions that are have to do with physics world the the rule of thumb was that physicists asked and solve that question before you right so they have they tell you they give you some answer they have some you know there are very very clever people and but nevertheless of mathematicians you know we maybe we sometimes we have some some some things to say and so I try to illustrate this with some you know tell you about some recent themes that mathematicians are working on and so the general question you could ask is and again I'm this is just a ok I was mentioning that some of them physically before this is like a populist version of the question right so I'm not claiming that has anything really to do with with physics but you might you know ask yourself things like you know we learn about Newton's law we know the Sun we learned the earth and the gravitational rules and things like that and you say yeah but you know what is this is it really so that you know the mass are really given and that the force is exactly you know you know that this this exchange of information between the Sun and the earth is absolutely you know a deterministic thing that is absolutely precise the earth knows exactly the mass of the Sun the Sun knows exactly in the mass of the earth and the the the the the forces are exactly given by this rule you say well but maybe there might be some little fluctuations maybe you know this piece of information that goes back and forth you know about the measuring distances and things like that might that might be a bit you know randomized there might be some little fluctuations that are inherent to some fact that you know the there is this exchange of information between these two things so the natural question is somehow to say well you know what are the natural ways you start with the Euclidean world your Euclidean 3-dimensional world and you ask yourself what are the most natural ways in which you can try to make little two Asians away from Euclidean geometry okay so you try to ask you know what is the natural way in which the distances between points can be you know fluctuate away from the actual Euclidean distance in the plane in three dimensions and of course if you want to do that it has to be somehow consistent right so the distances it's not just the distance between two points that you're trying to let vary but it's the whole space okay so it's the whole notion of the whole function that to any pair of points associate their distance you know how could that be and this big thing that you are going to create has still to be some sort of metric space um you know consistent way to measure sort of distances and if you asked this question in in physics this type of questions has been around for a long time because in some sense it goes down to you know the elementary questions about interactions you know forces sort of you know field theory is sort of the the things that are behind and so in the 70s and 80s this was really look like one of the main themes in theoretical physics I would say and one of the the names associated to that is not random metrics because that doesn't sound very physical the the name would be quantum gravity that sounds more like something but if you're talking the mathematician if you say random metrics that looks like much more stimulating thing anyway so so now I want to I I want to basically out the other thing is so maybe we can can switch the screen to my what what I see on my laptop the other thing is I always
feel my rule of thumb on Friday mornings is the first talk should not be BMO talk for the reasons I mentioned before having to do with the Thursday night and as you know mathematicians have a strange relation to computers in the sense that some pure mathematician will insist on the fact that you know when they are in their office they want to have their picture taken with a pen and a piece of paper just to insist on the fact that the the way they work is really has to do with it you know their own brain and the computer has to be and actually I myself in my office have the computer on the tiny uncomfortable desk in one little corner not on my own desk because I think that you know if the computers or the iPhone is too close to you you spend time surfing and doing nothing rather than actually thinking and as a result of it I have big back problems as some of you know because I keep always sitting on this uncomfortable chair in the corner surfing on the internet anyway so so I'm going to use a you know I know there's no blackboard and nevertheless I'm going to use this you know a little program and of course we are you know we try to be ethical in ok so what I'm using just for you to know is a program by written by Denis ah who's a very top symplectic geometry who didn't like the programs now that were you know available on the Internet in order to write things directly on the on the on the screen and he wrote this software called kernel X like journal with an X in front and just for you to know that this is you know sometimes mathematicians do that as well but writing some some software that seems to be good okay so I'm going to use this right so I'm going to you know this is going to be my blackboard and I have some couple of pictures you know hidden underneath that I'll try to show you and there will be no you know the usual font format of choice okay so now I have to start my actual scientific part otherwise there will be no scientific part and so the first thing you learn when you are looking at trying to understand random objects your first may be for me my first big friend that I encountered in my math studies was this object called Brownian motion and as you probably all know Brownian motion some sense is the random trajectory right of a particle where you know informally you might try to understand this random evolution of the particle by saying that at any time it chooses a random sort of a direction in an isotropic way and it forgets where it's you know it has any you know it's memoryless so it's a bit like the the flight of a you know ER as we know insects are very clever and they have lots of things but if you if you look at them for far away you might say well you know a little fly has no brain has no memory just flies around completely like crazy and at each time chooses a direction you know at random and goes in that direction but and then immediately you end up with you know something yeah but if you choose direction all the time then the path can't be differentiable you know there's no something goes wrong with that thing but still what we learn is that this Brownian motion exists right so that there is this sort of a you know random so I draw it here in the plane you know this random trajectory that you can define in this way and that this has you know that the the that this random trajectory is actually a random continuous function right and we know that because of the isotropy of Brownian motion you know then it will be related to the laplacian it will be related to you know potential theory in a harmonic function harmonic functions you know are exactly I mean I talked about the Newton potential you know one over R is the nice harmonic function in in in three-dimensional space so you know it's related to all these zoo of objects so there's just one little comment I want to stress about Brownian
motion which is that so this is a
trajectory so horizontally you have time and vertically you have the the fluctuation right so you go up and down like like crazy like this and the the thing I want to emphasize is that as I told you this function is not going to be differentiable anywhere and it's very easy to see that this you know if you if you would try to draw this with your pen the actual Brownian motion the pen would go out of you know the out of ink instantaneously because the you know the length somehow this red path would be is infinite right sort of the it's oscillating very well it's it's sort of it's a continuous curve but it has this property that it will be you know have infinite length immediately so in some sense what you have in this random function is that these sort of random impulses to up or down you know the this function would not be you know you have infinitely many impulses up and down and somehow they compensate right in order to still create a continuous curve right so it's a little bit like you look at all these little impulses up and down you take the sum of their absolute values that would be infinite but somehow when you look at the gap of time T there's all these plus impulses and the minus impulses that happen before because you look at them in a time ordered way they sort of average out and then you see some slight difference between the pluses and the minuses which is exactly the the height of your your Brownian motion at a given time so in all these continuous random objects it's essential that somewhere you have some cancellation of infinities right so you have infinitely many plus impulses I mean - and they cancel out somehow in some sort of magic way so they're canceling out phenomenon as you know you can see see it in that case as some version of the so-called central limit theorem or something like that okay so that's one dimension wrong motion I just want to emphasize that you know if you draw the same thing in two dimensions so now that's the path of the 2-dimensional Brown you know of a of an ant you know walking at random in the in the plane then this these are the set of points that have we visited and you see that this is a fairly rough type picture it looks pretty messy it looks almost fat right so it looks almost plane feeling not quite but so it is a very fractal a very strange fractal curve and you know the Brownian motion is doing things in some sense that you never would have expected any continuous function to be able to do right so in some sense what you're doing the philosophy you'll you take a random function the natural random function is going to be this Brownian motion and then you end up with a function of a type that you have never encountered before so for instance just to illustrate this on the Brownian path there's a result by log L which happened to be my former PhD supervisor a result from world well a couple of you know many so three decades ago something like that that says that you take the trajectory of a brown motion the plane up to time one so on a finite time interval so for the mathematicians they know that it's a continuous function therefore this blue thing is a compact set and that there will be exceptional times on the path where that the Brownian motion will have visited an uncountable number of times right so it's already not so easy to to guess that you know the brown okay of course of the brown rush can do a loop like that there will be a double point and then maybe you know it's already you have to be pretty clever in order to come back again to visit this double point once more trouble triple point and what the Brownian motion does is that they ourselves point that it will actually have visited infinitely many times during a finite time interval and actually there will be special points that the brown motion will have visit uncountably many times all right so and if I ask you to you to you know give me an example of such a function that has infinitely many points in the plane that it has visited you know it spent this uncountable number of times visiting those points you will not be able to you know it's it's not something you can construct it's a little bit the philosophy of you know you certainly know this you know the standard things about you take a number at random in zero one you know on the interval 0 1 then you know it it will be like in any chaotic decomposition the number of the different digits will be you know were fairly well you know spread and if I ask you to you know show give me any example of such a number that has that property you have a very hard time to find it so right it's the tip example that says you know if you take something at random it will have always a certain property but if I ask you to give me one example of such a such an object then you have a hard time and that would be a you know a general theme about what we're going to say okay so that's Brownian motion and that was just a little warm-up because I want to now to go to move to a second step which is this idea that imagine that you are looking for instead of a
random path so you can see the random path here the Brownian motion you can see it as a random function from the time axis into all right at each time you associate the height of the value of the brown image in our to or r2 or r3 but let's just look now at function real valued functions now in these there's a sort of field theory in general the idea is that what you want is to you have space it's given and you want to find a random function that at to each point and the plane associates something so you know if you are doing fancy physics the object you want to you you want to start to each point in the in space you want to associate a random object that would be in a li group okay something like that but let's look at here we're going to look at sort of a million stuff so commutative stuff so here we just want to look at to any point in space you want to associate a random number so it's exactly the same story as before except that instead of having the time axis as a parameter space you replace the time axis by space right so you want to have a random function from R to or R three into all and of course you don't want any random function you want to have some constraints so you want this idea that this function should be something like continues like two neighboring points you know the values are to never be there's this constraint that this function has to be you know you know holding together somehow like the Brownian motion of this random continuous function okay so the way to think about it - you know stimulate your imagination say you know instead of looking at the fluctuation of a violin string and you say the you know the violin string is straight and the fluctuations away from equilibrium of the violin string will be this Brownian motion type fluctuations now instead of that you have a tambourine skin that would be the two-dimensional version the flat tambourine skin attached at the boundary to be value zero you know the height is zero and then you look at sort of the fluctuations there what is the natural way in which the sampling skin could fluctuate away from the flat surface and then what you expect to see is something that would look like a random mountain landscape right so you're instead of looking at this one-dimensional curve you just now have a real you know like in Switzerland and now you know you have some real mountains when you look out of the window and you have the real topographic map that would describe it and now you ask the question what is you know is the what is the analog to Brownian motion for that question and then the answer is surprising because it is even worse you know I told you choose a random function in one dimension you know there's Brownian motion thing at random and what you end up is a random function that is really weird but it's still a random function continuous function now what happens when you try to define what this analogous object in this random mountain will be the first surprise is that you get this random object exists but it's not a random mountain right it's you know it's a little bit you choose the function at random well it's not a function so what is it so this object will be the idea you can do a little simulation and right
so if you do it on on a grid and you try to you know look at what happens and now
I okay I should zoom in a bit less and
bit more okay
so what you see is that this random mountain that you're going to see you know it will have you know spikes that are created in both directions a bit everywhere so you have these sort of plus and when you let the grid go to 0 these points will become you know higher and higher so in some sense your random mounting is getting crazy it's plus minus infinity everywhere the mountain itself not it's derivative but that's just the you know the the random mountains you would see when you try to do the little simulation ok so what's going to happen is that the object that you are going to get in the limit right there's natural this random mountain will be in some sense intuitively an object that is plus and minus infinity everywhere in some sort of dense way so you will not be able to say what the value of the height of the mountain is at a given point that will not work but there are things you can do namely you will be able to say something
ok if I take I give myself you know some area like that so I you know I cut out a little area here here and I look at and I ask the question what is the mean height of the mountain on that area on that piece of land right i sample this random mountain and ask what is the mean height of this mountain on this piece of land of course it doesn't make sense per se because the mountain itself is not defined I told you but what happens is that the random object that you can still define is this mean value right because if you integrate these plus and minus infinity spikes on this area a piece of area what happens is that you have these plus and minus infinity is that I cancel somehow and on this piece of land you will have an actual mean height so for those you know this for those who of you who studied math and remember maybe you know generalized functions or Schwarz distributions or these type of things what this is telling you is that you can integrate any test function somehow you know smooth test function and say what is the mean value of this random object on you know here the indicator function of the over set and but you can't define the object point twice so that's exactly a random of schwarz distribution all right so it's an object that say it tells you you can you can do these type of objects but of course they these there is you know the these random plus and minus impulses that you are you know summing up they are not independent right what happens here and what happens there they are sort of you know there's this web of constraints that tells you that there is an interaction between what you see here and what you see there okay so you have this sort of fairly mysterious object and the in in in the somehow in the physically to assure the general philosophy was okay this object is just a random generalized function so it's geometry you can forget about it but we are interested in what so call these correlation functions which are you know just some you know some some numbers that you can still read out of this random object but you don't want to look at the random mountain itself anyway okay now so this Gaussian free field that's the name of this random mountain if you are in two or higher dimensions it's this random completely crazy generalized function which has plus and minus infinity is everywhere that cancel and that you can sort of average out okay so now this is sort of the this Couser if we feel is sort of the you know zero building block of field theory in physics right so for physicists that would be like the trivial you know that's sort of it's it's a little bit like form for us you know if you're working field theory this object here is like the zero for us if you're doing number theory right that's just the the most trivial object I just want to emphasize just one thing you might have heard about you know that there one of the big questions is has to be with young Neil's theory constructor young meals field like that here the idea would be a little bit the same you know the goal would be to try to and that has sort of motivations coming from understanding the standard models in physics and things like that where basically the question is to say well we don't want to define a random function the natural random function from all three into all but we're now the values of the random function takes place in a lee group so you can think of matrix matrices in su something might fall in mathematicians and now the constraints will be that to neighboring things by two neighboring instead they will still have to be you know very close together nearby but the interaction because of the fact that matrices are not commutative you know that there is some there's a little modification you know in the the way the the constraint that to neighboring things have to be actually close together that have to do with the fact that you know if you if you look at the mode if you go up here along a loop and you come back and you you know integrate all these little modifications that you have done and your height somehow but now the height doesn't make any sense because it's in some abstract non commutative things then there will be some constraints there right so what I mean is that the sort of natural attempt to try to generalize this Gaussian free field in the case where you try to say that you know instead of being real valued and commutative because R is commutative you are taking the values in something just a slightly bit more elaborate then you end up you know in one of the main open questions of concept free mathematics which is to build this young meals feel ok now let's come back to our initial questions which have to do with with the idea that we want to find the natural way to define a metric in the dimensional space so what I'm going to do now is for simplicity and also because that's the topic where we can actually show things work in two dimensions so we are in the Euclidean planar geometry two dimensions two dimensions nice because as we know our big friend Riemann is there and complex analysis is there there's a lot of structure that has to do with conformal transformations in two dimensions that make that you know in some sense it's a very very rich structure to work in two dimensions so we work we want to find the natural ways to distort the Euclidean metric now here is a an idea that dates back to again physicists and I think one big name would be Polyakov physicist in Princeton in the early 80s and but there's a big Russian school a big French school a big British it's like a big big big story and the basic idea would be to say well the natural way to do is first of all instead of trying to look at the metrics the distances maybe we should you know view as you know if you distort space somehow then maybe you can change the notion of area right rather than measuring distances you measure areas so the idea would be you know what would be the natural perturbations are away from the Euclidean area right so so you want to distort somehow your your planner domain in such a way that you know now any given domain that is here will have instead of having the Euclidean area will have another random area and Polyakov said well the natural way to do that I know then that the physical way is just I know the random area would be exponential of gamma times this Gaussian free field times DX dy that would be the natural way to do that okay so the density of points near given at a given points will be a multiple of the exponential of the Gaussian free field now what you learn is that the Gaussian free field being a generalized function at school you learn you are not supposed to take the exponential of a generalized function that's strictly forbidden doesn't make any sense because it means you are multiplying these things with themselves and that's does you have to make sense however because of the natural special structure of this Gaussian free field it made this possible by some a regularization procedure to make sense of that so the idea would be that the places where the Gaussian free field is plus infinity will get a lot of mass and where it's minus infinity will be zero roughly speaking but this will be in the some sort of fractal everywhere time since and in this way you can define this random area for reasons that are I cannot explain here one very special value of gamma turns out to be important I mean first of all you can do that only when this when gamma is not too large when gamma is too large then some of these plus infinity e things you will know we're having a way to to to make sense of this object but you can do that and now what I want to just to show you is so there's one special value of gamma that turns out to be you know it's nice to have a little bit of numerology right so of course I haven't told you what the gas Rafi field is so right so here's the the good value of gamma that is the natural fluctuations of you know the away from the Euclidean metric or education area thing that would be exactly that object for that one particular value of gamma now time is running out so as planned so I'm going
to now just show you further you know
pictures just to illustrate this thing
so this are pictures I'm going to show you I stole from there's a one of a very bright young colleagues maybe it's not that young anymore but younger colleagues younger than me less all the me who who's a chase Miller in Cambridge I'm going to show you is webpage in a moment when you google him time Jason Miller math otherwise you end up with some other Jason Willis you don't want to see and you this is just you know somehow the the density you choose at random some points the way to understanding according to this area measure they are thrown like that and so basically the the places where you have lots of red and blue points are those where the these are you know the areas sort of there's a lot of area there and the sparser places are those where you have less area and then you can you know because it has to do with complex analysis we like circle packings you know so just to illustrate the fact that there is complex analysis behind and that you can make nice fun pictures like that but this is the type of thing you would end up with if you try to do something like you know you have the sphere you have the random metric of the sphere you sample your points at random according to this random thing this is the type of object you have now big developments in mathematics in the last ten years and the probability side is basically about the understanding of these structures saying that these random areas correspond to random actually really random metrics and understanding those so that's a nice picture here what you
see is for this nice random metric I'm giving you the shape of the balls right so each the boundary of the red thing here you know the everything that is covered here is the number is the set of points that you can reach that are a distance less than one from the center right and you have distorted now instead of a round ball you have this sort of weird structures like that and you see there are little holes there you know that the balls have little holes because there are little pockets of resistance where it's very difficult to enter there so one of the big results I mean one of the elementary results is that the fractal dimension of the metric space you define the random matrix space you define like this is not - it's full okay all right so that somehow the the immediately when you try to you know make these fluctuation you have these fractal pockets of you know resistance high you know places where it's difficult to you to to go to so here's
another picture where again it sort of illustrates that on these random object then you can try to define random curves Naturals random space-filling curves are one of the tools that two of the main players which are called Jason Miller and Scott Sheffield in this in this direction you know used in order to understand these these structures right and that's another one that is not
directly related interests indirectly related but just to show you that sometimes randomness can you know it so it's always the general question I remember that when I was a student there was always the question about you know is in a keys music art you know it's a random generated music actually a piece of art okay so here I like to think that this is maybe more compelling evidence that indeed you know sometimes randomly created objects can be you know you can print it out and put in your and look like a piece of art and pretend it is which it isn't but it looks quite interesting okay that's another one of
okay that's one of one of my papers with the Scott and Jason but and just to
conclude as we have heard the Internet is very important nowadays and the future is there and that's where so if you go on Jason's webpage which is here then you go home and you have images here and then you have all these
pictures I stole you can recognize you
know you have lots of little
illustrations of all these what these
random quantum gravity here then you
have all these metric balls here you have this thing that is called you know
like simulations of things that look
like diffusion limited aggregations on these random geometries and structures like that so you have lots of fun mathematical objects and these are not just curiosities they are actually you
know the objects that you have to use in
order to understand these you know random geometric you know fluctuations
away from the Euclidean metric and with that I will stop so that I'll be that my chair doesn't have to stop me and so I would like to - I hope I gave you a little glimpse of some of the topics that we like - that you know some of the people working in probability theory are interested in these days and I remember I just conclude it's this you know the train arriving yesterday night I reread again you know in order to put my brain in the right frame of mind about you know what my responsibility was or addressing you people I reread this essay by Max Weber which is called I think probably in English would be scientists as a profession or something like that this is proof so he has about two two options Wissenschaft has been off and doesn't shaft and political or something like that which is a politic and he was a sociologist and his talks were given in Munich exactly 101 years ago something like that and he describes the differing academic world of Germany in the US and so you hear all the time that everything is changing and you know all these things have completely transformed and so but you read this and you feel well not much has changed actually right sort of that and he was giving this lecture in you know to no dealings of like you in some sense younger people and and yes so he was insisting on you know the the fact yeah all the ethical aspects of how one has to behave when we are given the right to think you know and to do science and and yes just one comment is he already points out that one of the key points for scientists and good science is freedom and not only freedom and it's just one way because I know there will be something about how to do good science a bit later and I just want to already peek a little bit in maybe some of what the you know the general thing is that nowadays we have to you know make grant proposals right every one of us and when you do a grand couples of course it's fine because you have to you know the money has to be allocated to people who propose to do something but I consistently write my grant proposals just saying I'll do my best dot okay I'm not going to tell you exactly what the plan is what you know the road map and you know there's all these things that you have to fill in you know where the road map you know in March in two years time I will be switching from that question to that question while the previous one will be solved and things like that and I think especially in mathematics you know if of course if you have a big lab experimental lab that's relevant because you need to know who you are but in certain parts of mathematics is really not it's counterproductive right and so it's important that we you know of course we need the money so we have to fill in these forms but collectively as a community we we do it but we don't accept it right in the sense we don't integrate it as being part of setting up our agenda and and that's just one one part of that you should one should really you know when I say I'll do my best I mean it right I I work until 2:00 a.m. on Sunday on Saturday night and you know I work hard right much harder than you know that's part of the game of being a scientist that you have to you know it's difficult and accept that that in the report you can't write it yeah you can't say yeah I really worked hard it didn't work right but that's normally that should be the standard thing and as a result because you want to be deliver in the end people don't really take the risk of working on the difficult questions because you are too afraid of failing if you don't you know if you don't write your papers you don't get a position and then you're out so you have to go safe right and that's just the so the idea is my only piece of advice is it's okay that you work on some safe problems but keep a substantial proportion of your time to dream about the things that are you know the really big things because they're really big things they are might you know there might be much more in within reach than what might expect at first you shouldn't one should be afraid of the very big questions okay so I expect that the young mills problem you know will be solved - right in the next ten years that somebody will connect it modulo if only you know people try okay so thank you very much and thank you very much friend limb for a great talk with lots of inspiration and some very good piece of advice for the young researchers so we are running a little bit late but I would like to have at least one question from the audience and as you can see we have a different system here in so called Catchpole contains a microphone I'm not going to throw it randomly out in the audience and whoever catches it has to ask a question but if somebody wants I'm going to try and hit somebody well it would have been nice if you'd been more in the front hello yeah I have a really silly question do you know what a gamma is big enough yes you say that at some point which is
basically in two dimensions the cutoff is gamma equal to 2 gamma is equal to 2 so if gamma is to you know gammas smooth it's just you know what ok I'm talking in front of you know raghu Varadhan who you know large deviations he's here and it's just a elementary last deviations estimate for Brownian motion that tells you you know if you try to take exponential gamma something that you have a martingale and you know it's it's very elementary textbook things that tell you that indeed you know the cutoff is at a certain value of gamma that happens to be gamma equal to 2 2 so you know square root of 1/3 as well it's a bit you have a bit of room of margin to you to till you get there but that's that's basically there are lots of I mean of course the other values of gamma are interesting too and there's lots of numerology going on and interestingly the numerology just is you know some of the numerology the physicists got it first and then it stimulated whole area of pure mathematics which has to do with representations theory of infinite dimension Lee algebras for virasoro algebra for so you heard about these things where you know one classifies certainly and these numbers you know this 8/3 here is related to some special representations of you know something has is going on there so I'm not going to go you know say big words but it's related to some you know structure or mathematical stories but yeah but intuitively it's pretty clear right sort of you you want to have this plus and minus infinity that sort of compensate if you take exponential gamma thing then if gamma is too large then the whole distribution about the explanation of gamma thing will somehow it will tend to be concentrated around the the point where gamma is I mean the the the field is really much more infinite than the other point somehow right so somehow the whole metric space will collapse into one point that's somehow what what what's happening you know that all the area is there and the rest has nothing that's basically what happens when gamma is too large okay thank you very much I don't think we will have time for more questions so if you can pass the cash box down here so we are ready for the next one and I just I would like to apologize for you know being here and you know the rule is to interact with all you people and you know I arrived yesterday late and I'll have to leave this afternoon it's just that you know one of the sad aspects of life is a sometimes we don't say no enough to invitation so I have been giving four talks this week you know that was a miss planning and therefore just so my apologies for not being able to interact and play the real game of this form well we are very happy that and Lynne took time to come here so let's give him a big hand [Applause] [Music] you


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