Cocktail of Mathematics with Cedric Villani

[Music] good evening people so mathematics yes is a lot of things and tonight we're in for a cocktail of [Music] mathematics this of course is uh the kind of battle that you don't want to have your cocktail in because it's impossible to keep it in the battle here what is mathematics about first one has to remember that mathematics is a science and mathematics in such studies the world in which we live it is about the examination and the understanding of the rules of the world just one special thing about mathematics is it never interacts with the world directly if you are a physicist you make experiments if you're a mathematician you don't make experiments you just work on the mathematical formulas which are reflection of the mathematical reality there is one poetry which i like to invoke when speaking about nature of mathematics this is the lady of shalott by alfred lortensen this is the illustration about this poem and in there the poor lady which is there has been a victim of a curse by some evil magician it's a legend from the arterial times and she cannot see the real world directly only through the mirror this is the mirror the curse she abides by until one day comes lancelot and then he's so pretty so handsome that she looks at him directly and then the curse is on she dies it's tragic it's very tragic yes and her body drifts through the on the waters here and then many people have wondered if there is a special hidden meaning to this poetry and because the author did not explain as usual we can imagine whatever we like i like to imagine it's an allegory of the mathematician when she mixes reality and the reflection then she dies she has to only study the reflection and these are the abstract formulas so if you're a mathematician you study the world but through the formulas mathematics is indeed abstract by definition it may deal with very concrete objects but through the abstract reflection for instance the water here on which the corpse of the lady of charlotte is drifting you can feel it you can touch it you can make experiments on it but you can also try to write the equations of the flow and this is a mathematical adventure which started in the middle of the 18th century and this is an abstract equation that can describe any flow provided it has the right physical characteristics in this case incompressibility [Music] many other examples arise about abstract interesting phenomena whose study becomes an abstract problem of interest in itself and which are inspired by reality and which then can be applied to many many real problems if mathematics is often thought to be just abstract with no applications this is a sure mistake we'll see examples of applications coming from mathematics very often not wanted i mean not imagined by their authors of the theories another misconception about mathematics is that it is all well known of course what we study in high school is has been found long time ago but this is a misconception and there are many many open problems which are being solved from time to time around the world each year there are more than 100 000 new theorems being proven which is a huge quantity of course making it impossible for anybody to know the whole of the discipline or even a significant portion of it if you ask a mathematician which is the most important problem that he would like to solve you will get an answer which may depend from mathematician to mathematician some will tell you that this problem is so important that that other problem is so important and the answer may vary but if you ask them which is the most famous mathematical problem certainly they will answer is the riemann hypothesis here i put some big name words deeper scientific mystery about time but it is really a deep scientific mystery so who was riemann one of the most extraordinary mathematicians of all times more sexually scientists of all times and his hypothesis so it's something that people believe to be true but it's not proven yet is easy to explain okay don't uh here there is a bit less contrast than i expected but never mind the riemann hypothesis is about a certain function a function is like a curve you know depending on the parameter this is the definition of the function of riemann called the zeta function just it's a function d prime defined on complex numbers so you could imagine as a function defined on the plane imagine that to each position on the plane we associate one complex number so it's like another couple of numbers and you ask when is it that this function becomes equal to zero it's known that it is equal to zero for when we are at point minus two minus four minus six etcetera minus eight minus ten whatever but we know there are other points where it becomes zeros and remain conjecture that all these other points are on a single line all on the line vertical line then you would say okay let's compute them and check but there are infinitely many okay let's compute many of them as as soon as computers were made even from the 50s people started to compute the zeros of the riemann function and they found that they were all on the same line and when the computers became more and more powerful it was checked for more and more zeros now it has been checked for thousands of billions of them and still it's not proven because the mathematicians they are the extreme guys you know the really hardcore guys even if we have one billion clues we don't consider it proven and that's the only shield of all of knowledge of man's knowledge in which we are so tough in usual life if you have a prediction and it's realized thousand times it will be more than enough to accept it as a valid theory but in mathematics one billion clues is not enough you want the complete proof which makes absolutely no chance for error now why do we care about this human hypothesis well for many reasons in particular it's related to the distribution of prime numbers prime numbers are these elementary numbers like 2 3 5 etc 7 such that every number is a product of these prime numbers in a unique way these numbers we know they have many properties but their distribution where they pop out and so on is not fully understood if the riemann hypothesis is true it means that in a certain way these numbers are randomly distributed from a distance let's say a few years ago uh a british journal fantasized that if this hypothesis was solved it could bring disaster for internet it was one of the prizes put for reward by a math institute for a million dollar and it's very weird why would that be a disaster if we know something more now the reasoning you can reconstruct is the following because it's related to prime numbers and prime numbers are were extremely useful to code things encryption security of transactions online and so on was based on prime numbers for a long time the deduction from the journalists was that if we solve this problem it would be easy for pirates to crack the codes of course if you understood what i said before you see that this guy hasn't understood anything you know if you want to crack the code you have to find the product of prime factor so you have to find the prime numbers behind the key what good is it to know that the prime numbers are randomly distributed if you want to find them it doesn't it's not very useful you know not very useful but anyway we don't need this kind of sensationalism the important about the important about the the riemann hypothesis is that it gives some beautiful insight into the structure of a basic of a basic object the prime numbers also it's related to other fields of science for instance there is a very famous connection very weird that we don't understand between the prime of the distribution of prime numbers statistical distributions of prime numbers and the statistics of the energy levels of a random atom it's a connection that was observed and never explained this shows you the number of consequences that could come from this understanding riemann was remarkable not only from this hypothesis by the way he also left his mark in geometry and now we call romanian geometry the basic non-euclidean geometry which is what happens when you deal with species which is not flat but curved let's say distorted the earth for instance we know is not flat it was shown long ago so if you do geometry on the earth it cannot be the flat geometry that we learn in school how do you handle this riemann it was in the middle of the 19th century showed us how to handle this in the world of riemann units of length may change from place to place so what has this length here maybe has that length here and so on and you still have a notion of shortest path between one point to another even though these are not straight lines for instance if you want to go up here from the bottom from down you will go through something which is not a straight line but that maybe is the most economic path the one that consumes the least energy similarly if you fly i don't know from paris to new york from copenhagen to new york if you look on the map the trajectory of the plane it will not be like this it will be something like this it will be a curved path how to understand all these curved paths i mean in geometry the lines will understand them line has simple properties and so on and we study them when we are children but a path that can be any kind of distorted path there must be some way to understand them mathematics is not just description it's classifying and understanding and riemann and his master gauss showed us how to classify these shortest paths or judaism curves thanks to the notion of curvature so there is this curvature is something that can be positive or negative or zero zero is like flat and if the curvature is negative the geodesics will spread apart they will become more and more distant from each other very fast what if the curvature is positive they will not separate so fast here is an example of a romanian geometry this one is called the hyperbolic disk hyperbolic space it was discovered in the 19th century beginning in this artist's view every fish has the same length just the units of length change from place to place and if you are a fish here making your way here the universe appears infinite you approach and units of length become smaller and smaller and smaller and smaller and if you draw a triangle in this using shortest path here is how the triangle looks like and if you measure the distance between these two points it becomes extremely large very very very large as time goes notice that the sum of the angles of the triangle here is less than 180 degrees the triangle is said to be skinny on the contrary if you draw a triangle judaisic triangle on a sphere the triangle will look fat and see what i was saying this is positive curvature this is negative curvature a positive curvature it's geodesics they separate rather slowly take these two geodesics here they cross and they will cross again on the opposite point have we ever seen in the plane two lines crossing once and then crossing another time of course not if two lines cross once they will never cross again but in the world of positive curvature it's very likely that two lines which cross once will cross a second time they don't get separated as much as the lines notice that the sum of the angles is more than 180 degrees in fact this is three right angles so 270 degrees and when i do this in high school always i ask the students try to imagine which is the maximum sum that you can make for a triangle on the sphere and usually after some time they manage to find that you can go as to as much as 540 degrees for the sum of the angles of a triangle so much more than 180 degrees and that's typical of positive curvature so the rules of the geometry are changed depending whether you are in positive or negative curvature this gives two beautiful results and so on it's also very inspiring and let us see some examples of artistic realizations which were inspired from curvature so where can that be this is a picture is taken from a science museum in dublin and if you go there be sure to visit the hyperbolic coral reef museum it's not guaranteed that you will meet the lady who is here enjoying in the in the museum but you will see these beautiful shapes which look like it's coral but it's knitting made with a simple rule negative curvature constant negative curvature the contrary of a sphere if you want it looks like it's imitation of real life but it's made from mathematical rule and if we draw a triangle on this and measure the sum of the angles it will be less than 180 degrees and we can understand very well the geometry on this it is also negative curvature constant negative curvature here in this pseudosphere which was exhibited on display a few years ago in the exhibition at the foundation cartier for contemporary art in paris and it is still constant negative curvature which is the geometry of each of these units is this kind of accordion which represents the solution of some problem of mathematical physics now look this this this look like such different objects but if you take a piece of them they will all look the same constant negative curvature and this is enough to determine it completely we see on this example what i was explaining mathematics is about ordering and see that sometimes the same equation is underlying several very different objects so maybe it can be inspiring but also it is not just something that you say once and which becomes the eternal truth as in all of science people will use it to develop it and to build on this more and more and to be more and more complex and the work of freeman was expanded and developed by people after him like this gentleman grigorio ricci who was italian geometer from the beginning of the 20th century he developed this notion into what is now called the richie curvature which tells you uh in which are the directions in which the curvature can be positive or negative in a sense and it then left the field of pure mathematics when einstein discovered through conversations with his mathematical friend grossman mathematician friend that the richie curvature was the key tool for developing for expressing the theory of general relativity like this r here constant for either richie or riemann but that's the thing that they that they developed and you see at first it's just for mathematical curiosity and then one day by accident somebody discovers it's at the basis of the representation of space and time and then wait more time and gosh it arrives in your car when the gps there is a problem about some measurement about space and time and it has to be done using information that grows at enormous speed satellites and so forth so you need to use a generativity so for this you need to use the curvature the ritchie curvature for this you need to use the riemann curvature so in the gps there is a little bit of einstein in riemann and many other engineers that have worked on it yes yes if you open it you will see the remains and so on and uh is the same very very often see it took like 150 years before there was the idea in the brain of the mathematician and the moment where it is in the technology of daily life very often this is the case [Music] now riemann was also very much excited about understanding some of the geometric patterns of the fluids like for instance so-called singularities if you look or fluids or or light and so on it can be this kind of cusp in your cup it can be these lines of discontinuity of physical parameters when you have a supersonic plane called shock waves can be also the shock waves which are around the kalashnikov whatever whatever this is not a very pleasant image but look at the beautiful shapes here and here in the physical parameter and riemann wanted to understand this and develop the first mathematical theory about this in fact riemann was interested in everything and if you look the list of concepts which are named after women you just become very dizzy wow wow is this possible all the more that this guy was dead when he was my age you know this makes yes yes this makes a remand a very special romantic figure you know somebody who understands everything before in his short life and so on actually once i met a very famous rock singer who told me that from time to time she goes meditating on the grave of women for the inspiration you know showing that sometimes the life of great scientists can be an inspiration for the artists because it's all about imagination you know i like that and i too like graves you see this is ludwig boltzmann one of my heroes he died in 1906 why he committed suicide let's not for talk about this anyway 2600 years later i was invited as one of the main specialists in the world of his equation i was invited to a big conference celebrating him in vienna i knew he was buried there i knew it was famous i knew that there was the famous entropy formula on the grave it's a formula which expresses the fact that the discrepancy between our macroscopic world and the microscopic nature of matter with the atoms and molecules the discrepancy between these two scales lead to a wealth of phenomena related to what is called the disorder of a situation of a gas the entropy which tells you whether it is easy or difficult to reconstruct the state of some gas for instance which you observe i knew this i had worked on his equation for more than 10 years so it was natural that i visit him you know he was like family so i arrived in the cemetery and i see that the cemetery is huge and i know there are these famous people they are not just bullsmen there are famous musicians and so on and i am desperate how will i find the guy and then there were some tourists around so i asked them do you know if there is a map of the cemetery where i can find the people no no map who are you looking for borsmann ah he answers the guy aha s is k log w that was the answer the entropy formula you see here i uh like to say that this formula certainly is more important than the most famous formulas that we know that like einstein's equation is mc square or so on because this is not just about special relations on microscopic quantum mechanics and so on it is something that we encounter every day of our life the statistical nature of the world and in this anecdote it was like if that formula whose power i could appreciate was transformed into a kind of password between people in the know in the cemetery what did we know both the guy and i about then the nature of boltzmann's discovery and the entropy well burstman and maxwell around 1865 1870 they developed the idea according to which gas is made of many many very very tiny particles with crazy velocity crazy speed this was very daring rest a little bit and think about the air around us it's very quiet right now imagine that in this in every portion of this air there are billions of billions of particles moving in all directions with crazy speed bumping into you at each single second of your life billions of billions of them and so on no no don't don't try to do the particle maybe it's just a trick for uh for establishing contact with the neighbor okay okay now here is what the view let's take this model with many particles and these are the velocities so back from time to time they bump into each other and then the velocities are changed like a huge billiard game how are we going to model this and what are we going to learn mathematically from this this was the challenge of boltzmann and the others and they understood that it was just impossible to look at the positions of all the particles and the velocities so instead they look at the statistics of the of the particles of their velocities this is maxwell by the way and they focus on this curve rather than on the particles individually now it becomes statistics and they knew they could take advantage if it is statistics of the large numbers you know if you want to handle complicated things one thing then another then a third then still a fourth if you have many many problems to deal with it becomes a burden if you have more and more problems but if you think statistically the more is better like if i take a coin and i uh send it up in the air and see whether it's head or tails it's impossible to predict but if i do this one million times i know that roughly speaking half a million times it will be head half a million times it will be tail so when you think statistically large numbers are an advantage and the problem of large numbers starts around 1700 with jacques bernoullis in switzerland around 1730 abram de moivre makes this extraordinarily important discovery the gaussian law with this kind of bell-shaped curve equation like this which he understands arises in the fluctuations of the this problem of header tails of course if you send the the coin roughly half of the time it will be head half of the time it will be tased but there is some error is it a large error or small error well very often could be a small error and sometimes a large error and the statistics are given by this curve very often it will be a small error but a few times it will be a large error and this is the tails of the distribution and this profile is typical de moivre can be claimed both by the french and the english that's an advantage he was a french seeking refuge from you know religious persecution because he was a huguenot as it was as it was said he was protestant let's skip before for this time we'll do it another time maybe in 1810 laplace proves that de mois intuition is right but also that it applies to all random problems not just the problem of the coin tossing but many many problems in which you have a lot of uh a lot of experiments for instance if you do a poll the error will be like given by a gaussian or if you measure the fluctuations of the ocean it will be like gaussian and so on and so forth this was also exported to social sciences by catelyn the middle of the 19th century and galton who studied many you know biological beings and so on and human populations statistically also marveled on the role of this gaussian curve i know of scarcely anything he says so apt to impress the imagination as the wonderful form of cosmic order expressed by the law of yes cosmic order thank you good cheers the law of frequency of error the law would have been personified by the greeks and deified if they had known of it it rains with serenity and complete self-effacement amidst the wildest confusion it is the supreme law of unreason that's well said back to boltzmann this is young boltzmann between young boss man and old boss man there is uh the arrow of time and it's related to the entropy the law that entropy increases disorder increases and so forth so he was working on fundamental problems and boltzmann has this formula which tells about the disorder the more that the state of the system will be fuzzy the more it will be distributed the more it will be the entropy will be high a lot of disorder and here's a formula for computing the disorder of a distribution of particles and boltzmann shows that you can study the entropy the disorder of a gas and he proves that this disorder is always increasing and that this governs the behavior of a gas let us give an example suppose you take a big box and you put particles in one half of the box and this is the void okay what will happen if you let the gas free to do whatever it wants well we know particles we go all around the box and we like to think that the void sucks the particles in however boltzmann tells us you know the important thing is that the gas wants to have a higher disorder higher entropy and that's why it will spread all around because it's more disordered so he replaces the idea of a force by the idea of the rule of statistics here is an analogy imagine you have children coming out of a class at the beginning maybe they are near the door of the classroom and this is where they can enjoy and so on during the free time but pause between the two classes and after a few minutes they will be just all around you know spread all around not because they like to go in places which are void but rather because they all do things which are unrelated to each other two guys going to fight in this way that they've been going play that way etc so they are acting like particles which are chaotic and uh act as if they are or independent of each other almost boltzmann was using these beautiful objects yes yes yes yes partial differential equations to make his claim and to prove his contributions people like me are specialized in understanding these things or part of them and understanding how the solutions of these equations behave each of them represents a physical natural phenomenon for instance that one is the boltzmann equation which tells you about the evolution statistical evolution of a gas that one is the velocity of equation which tells you for instance about the evolution of a galaxy statistical evolution of the stars in the galaxy this is a schrodinger equation the basis of quantum mechanics this is the euler equation the simplest model for food mechanics these are the compressible navistox equations of fundamental importance each day they are solved on big computers around the world to predict the weather that they will be tomorrow or one week from now and so on they're also used enormously in the hollywood industry for movies for special effects because you know it's much more efficient to use mathematical equations to reproduce special effects with water than to actually do the special effects with some water and so forth etcetera so all these equations have meaning and you can write a whole book on any of them bossman was using this one to predict the behavior of the gas it's like one molecule of gas you cannot predict what it will do but all the molecules together statistically because there are so many you can predict with great accuracy what will be the evolution of the statistics to take an analogy it's impossible to predict if you are look at an individual you cannot predict when he will have babies when she will leave the country or whatever but if you look at the whole population you can predict with very good accuracy over the years if you know the natality rate the death rate the immigration rate you can predict the evolution of the demographics of the whole country it's the same with the molecules boltzmann discovered that this entropy goes up and he showed how you compute the increase of entropy there are there is some nasty formula that is there and you can ask many questions when you understand this for instance what will the gas eventually do in the long run if i leave the gas alone in the box maybe there is some preferred state of this of the gas that we like maybe it's the most disordered of all states so let's go for the maximum entropy and see what is the preferred distribution of the gas and what we find again it comes the supreme law of unreason that galton was speaking of the law of errors it is the equilibrium distribution of velocities in the gas around us the if you pick up a molecule statistics of molecules around us the velocity distributions will be gaussian and this is not only mathematical theorem but you can check this with the measurements extremely precisely so this is the kind of thing that i worked on for a long time study about the increase of entropy i worked on the conjecture by some italian researcher about the rate at which the entropy increases and so forth many works i did with italian collaborators on the theme how fast does the entropy increase and so on the bustman equation nowadays is used for many things let me skip a lot of this but just mention that the theories of boltzmann were inspiration for many people including einstein and smooshowski in their theory of branion motion or code shannon in his theory of communication which we use all the time when we send messages and so forth and also it is used by engineers for instance to understand how the gas flows in a motor in an engine of some car and so forth so once again a huge gap time gap between the time it was invented and the time it was applied in our daily life like more than 100 years sometimes however there can be situation there are situations in which the application comes very near the invention or sometimes the invention is motivated by the application wow it's very apparent in the work of leonid cantorovich another of my heroes so kantorovich was a russian scientist he worked on many problems from very pure to very applied like atomic bomb or taxi fare he was one of the pioneers of numeric approximation he was a very very clever man and very dedicated always seeking the public interest and never his own his masterpiece his work in 39 and he's improved in 59 is the best uses of economic resources a theory of planification which earned him the nobel prize in 1975 in economics kantorovich is one of the heroes of a very weird novel which i warmly recommend called red plenty by francis perford studying the rise and fall of the soviet economy from the point of view of the mathematical idea which is planification some of the heroes are imaginary some are real ones so you see kantorovich in here you see khrushchev and so on and very well explain how in the old days in the 50s some very serious economists thought that the soviet economy would become the first in the world because it was the only one that could achieve massive planification which presumably can give better result as a whole and how it collapsed how it collapsed partly because if you want to planify it's not just the mathematics it's also the men and women who are doing the economy and they will not act like the planific like the the the plan hopes they will do spontaneously kantorovich's life was completely changed transformed by this plywood yeah plywood is not very sexy but it's a an invention of the father of nobel you put various kinds of wood together and you press them so there is hard and soft wood whatever and there was this plant of plywood that needed to improve the results for the next plan whatever and they had the idea to ask advice from kantorovich he was very young but already recognized as a super mathematician he had his professor position at age 22. you know he was a very gifted guy and he told him like professor contour of it we have this problem soft wood hard wood some comes from this forest some from that forest we have a machine like this that can handle so much of this and that that machine this other machine so there are so many possibilities how should we handle organize the production to make it as efficient as possible and korovich contour rich had no idea but he thought about it and understood that by putting the problem in abstract formulation he could make an impact and that was the birth of the mathematical theory of linear programming the basic idea of linear programming is simple if in a problem which has several unknowns in this case two unknowns you can put all the constraints in the form of linear inequalities like the admissible solutions of parameters have to be on one side of this line and that line and that other line line is for linear constraint then in the end the admissible solutions are part of a polytope as they say and one of these extreme points is the best if it's just three equations you can solve it by hand i remember i had exercises like that in high school and was finding this so amusing but now in real applications there can be hundreds of lines hundreds of variables it can be thousands it can be hundreds of thousands in real applications how do you do well you need a mathematical theory for that and that's what kantorovich developed kantorovich found tools to solve the problem and also realized that it made it possible with these tools to solve an old problem set by gaspar monge at the end of the 18th century called the optimal allocation problem it's an economic problems problem too imagine you have something which is produced at locations x1 x2 x3 x4 and you want to send it to some sites where it will be sold for instance or transform y1 y2 y3 y4 how would you make the pairing between start point and final point in such a way that the total transport cost is minimized this can be anything maybe here you extract things from the ground you transform it whatever but should you send this here and this here or this here and this there it doesn't cost the same for the shipping and controv for instance prove that you can transform this theorem into this problem into another problem which would be like imagine there is some guy who offers to do the transport by himself and he says okay i will buy the good from you at this location and send it back to you at the final locations and we'll just discuss about finding the good selling price and the good buying price and these two prices will be set in such a way that the difference of prices between start point and final point is never more than the actual transport cost so that i'm just being honest that's what the guy would say now the mathematical theorem of contour which says that if the guy does what is best for his profit maximize the total profit it will be equivalent for you to minimize the transport it sounds like no big deal this kind of theorem but it was strictly forbidden in those days to tell about this in public this seemed to be like a justification of capitalistic theory or whatever it seemed to go against the uh marxist idea that price is determined by work and not by how much something is wanted and it was strictly forbidden to say this in public in fact for developing a rational theory of economics it is very clear that the likely fate of kantorovich would have been death if it had not been for the fact that he was so important for other strategic problems like atomic bomb nowadays nowadays linear programming is used for any many kind of problems that arise in economics as you can see in those examples which i took from a tutorial and all these problems they are can all be formulated in the same mathematical abstract way as a optimization problem and linear constraints and you have many programs that can solve this and of course before you make the programs you have to the software you have to understand how it works and which is a mathematical theory and this is a very very successful theory it's used in particular to set up the prices of public transport or whatever okay so so far so good i hope but we saw various theories right with common points and huge differences we started with the riemann geometry and the fat triangles for instance of positive curvature then we move to the boltzmann theory and the disorder and what it implies for the gas and kantorovich and his optimal economic theory and so forth but now we remember that here this is about cocktails and i will make a cocktail of these three theories and that's exactly what happened to me around 2000 me and others we discovered that these theories even though they deal with different problems different techniques different applications they could all be mixed together and be combined to get a new theory in each of this field it was not obtained by rational thinking as most of the time it comes through chance and random encounters you know life of a researcher is never like it's logical and you think and let's do this and this and this and see what happens it's much more tricky here let me borrow let me borrow a image from a cool science blog this is how some people think science goes maybe you ask a question you think it's a challenge you read you think and ha solution nobel prize whatever in fact anybody who has done science knows it's more like that wonder accept challenge read science ah somebody already did it what the hell do math some instrument breaks whatever amazing results turn out to be crap and then you start again what the hell is going on thinking makes sense no sense oh that's good yeah this makes sense ah they figured this out 50 years ago now we are back and forth and so on and after some time wow something comes out this is it's quite like that's the way it is except that to this you have to add another extremely important ingredient these are the encounters with the other researchers so imagine this figure but like many of these uh many of these units tied together and this is more like like it's and sometimes you meet this guy what the hell oh this is yes it wow in my case two encounters were critical first with felix otto my german collaborator met him in france and went to visit him in santa barbara 99. and then john lot in berkeley this is a beautiful berkeley campus politically speaking this is the point of all the place of all america which is the most to the left so not bad maybe and it's always sunny it's great university and so on this is the not so great math building see how ugly you know it's famous the evans hall so ugly and when you are there you discover it's not just ugly it's not functional because people don't meet in this it's like organization in layer so if you are in this layer never meet the people of the layer below and so forth it's a disaster and there is no common space so people don't meet so you are here i was invited as a visiting a visiting professor visiting associate professor sitting in my office and i knew there were all these great mathematicians around but not a chance to speak to them what to do you know my my duties were very limited there was no course i had to give no administration all i had to do was attend the lunch once a week and uh speak to people you know and prepare a lecture during the whole semester just one okay and for this i was paid the three times my french salary and in spite of this i was not so happy because you know sitting and what's going on until this guy arrives he was invited by some institution nearby uphill from there and he talks oh hi my name is lot i'm a geometer i read your work with felix otto about entropy and transport it's great and we are going to use it to do this and this what what the hell is this guy talking about and it was the start of collaboration we collaborated for years and we had big hit with the start of some new theory and what did we do what did we develop building on a previous work let's call it the lazy gas experiment it's a new point of view on the curvature i will end with this mention that of course it is based also on the works of others these are the people who really were in the development of this my work with felix otto cardero is a a french and argentinian guy mccann is a canadian guy schmuckenschriger is a german guy strom is a german guy lot is american i am french even though this is an italian surname let me explain you now why the three stories i told you are all linked together and how we can understand this and see this let's ask whether the curvature is positive or negative and let's try and ask it in a way that bossman would understand if he was to come out of the grave so imagine there is this gas sitting in a certain configuration initially and i asked the gas to change configuration and this is the other configuration maybe there are some regions of low or high density for the gas i don't let the gas do what it pleases i ask the gas this is your final destination and the gas will obey it has to but the gas is lazy so it will choose a trajectory which has the minimum energy which costs the minimal energy because it's a it's a gas that wants to be as economic as possible now all during this process from initial configuration to final configuration measure the entropy of the gas the disorder and plot the curve as a function of time if the curve is always like this concave it means that the curvature has to be positive and you see here a relation between geometry that is positive curvature economics that is the most economical way to move and statistics because you are looking at the entropy how do we understand this let me try to explain this picture is like positive curvature right see the geodesics first they separate and then they start again a bit like you remember on the sphere the lines were crossing once then separating and then crossing a second time it's a bit like this this picture is typical of positive curvature if the curvature is positive in between the gas will be more spread then because the separation between the trajectories is large a gas which is spread is also a gas which is more disordered remember the gas spreading in the box and its disorder increasing so the disorder will typically be higher in the mean time than it is initially or finally in this experiment and that's the result in this curve so you see here how the three stories are related and this is a new way to understand the curvature it was actually the one of the key ingredients in the thick book which i wrote after our works with lot i think is something like thousand pages you know with big nice fat formulas and this was also a way to solve some problems in geometry that nobody knew how to solve before because the reasoning would go through statistics and so forth if you are just geometer if i may say how would you imagine this why talk about the gas and the entropy it looks crazy but then by accident some guys finds it this is by accident it's do a lot to chance encounters and that's one reason why we know it's so important to have places for fostering unpredictable interaction the institute which i run which i have been running for five years now it's an old institute and situary poincare yes the guy in the the uh the being in the movie before the start was talking about that institute right uh it is devoted to inviting visitors from everywhere so that they can meet so that they can interact so that they can have new ideas who knows which it will be we have different themes and so on but we never impose any program to visitors and chance encounters most of the time will give new ideas that nobody would have predicted that's all thank you very much you

Channel: Science & Cocktails

Views: 2,293

Keywords: science and cocktails, science communication, popular science, Denmark, Christiania, science lectures, Science & Cocktails, science talks

Id: bQY4o_8zu8w

Length: 57min 58sec (3478 seconds)

Published: Mon Dec 28 2020