Taming Infinities - Martin Hairer (2017 Fields Medal Symposium)

[Music] good evening everyone thank you for coming and welcome to the public opening of the Fields Medal symposium for this year this is an annual series of events of the field listed by the Fields Institute established in 2012 endorsed by the International mathematical union and purpose of course is to highlight the recent fields medals awarded every four years at the International Congress of mathematicians and this symposium quite clearly we're celebrating the achievements of Martin Herer Fields medalist in 2014 exploring new directions from his work there's a rich scientific program which started this morning at the Institute and continues through the week and on Tuesday evening one of the usual events is an opportunity for students to meet with the medalist in an informal session and and then the program continues after that so it's going to be a very stimulating and interesting week of science at the Institute I of course like to thank the all the organizers for putting together the program and making the arrangements and our own staff s turbines unda fields program team for all of the local arrangements maybe I'll just take a minute or two of your time to tell you say a few words about the Fields Institute the goal of the Fields Institute mandate is to promote mathematical research in the widest sense and to enhance the visibility and impact of mathematics in our society so our major support comes from the Ontario government through the ministry research innovation in science you will hear from the minister a little bit later and from the Government of Canada through our insert' National Sciences and Engineering Research Council of Canada in addition we are greatly appreciative of the support of our nine principal sponsor universities which are the largest universities in Ontario eleven affiliate universities private donors and corporate sponsors and briefly at the Fields Institute we have activities mathematical activities of all sorts of scales from one day two days week-long workshops month-long focus programs and then six months of intensive coherent activity of the Institute we also do summer research experience for undergraduates outreach to mathematics education mathematical applications to problems in industry there's a quite a wide spectrum of activity and you could I encourage you actually to find out more about it if you like by looking at our webpage the Fields Institute was founded in 1992 at the University of Waterloo and moved to the University of Toronto campus in 1995 and if you haven't been to our beautiful building I could see many of you here I recognize certainly have but others perhaps not do please come in and and take a look you might also be interested in there's an e-book which we prepared called the field Institute turns 25 since this is our 25th anniversary it's a book of stories about the Institute's available on our web page now I'd like to recognize the additional sponsors who are specifically for the Fields Medal symposium if that's how it started by collecting together support from interested donors we have and still very much appreciate the Gold level sponsorship with a great Westlife group of insurance companies instrumental in establishing the series and also very pleased now to recognize Elsevier as a gold level sponsor this is now the second year of their gold level sponsorship our civil Silver level sponsor Jim Stewart is a deceased old friend of the Institute former colleague of mine at McMaster author of the most widely adopted calculus textbook ever many of our bronze level sponsors are with us this evening and we gratefully acknowledge all of their support now the next item on the program is to read to you a message that we have from the minister of science Kirsty Duncan she wasn't able to be here this evening but did send a short written message that I'd like just to just to read to you so this this is not me but Kirsty Duncan on behalf of the Government of Canada you can see it's not me please accept my best wishes for a productive 2017 Fields Medal symposium congratulations to the organising committee and everyone here at the fields institute for research and mathematical sciences for putting together such a great program again this year to everyone joining today graduate students mathematicians and scientists thank you for all the work you do to celebrate and enhance mathematics in Canada and around the world I would also like to acknowledge Martyn Herer world leader in stochastic analysis and 2014 Fields Medal winner his research is the driving focus but for this year's symposium as Minister of science one of my key priorities is to remove the participation of women in the STEM fields I think that your guest speaker this evening professor Sylvia Sir fatty from the Koren Institute of Sciences in New York will be an inspiration to many of you in this audience I wish you every success throughout the conference and look forward to the many discoveries your conversations will surely ignite thank you and enjoy the 2017 symposium so Minister Duncan [Applause] my next very pleasant privilege is to introduce the Honorable Reza barati the Minister research innovation in science we just say a very few words dr. Moretti is an award-winning engineer and scientist with expertise in medical physics and radiation safety he was vice president and chief scientist at the radiation safety Institute of Canada for 17 years prior to entering political life he served as the member of rimbor MPP for Richmond Hill since 2007 and as mentioned before government of Ontario is one of the major supporters of fields over the last 25 years since parading please well thank you very much Ian for that very kind introduction ladies and gentlemen it's my great pleasure to join you here tonight for the Fields Medal symposium and to honor the 2014 fields medalist Martin Harry and his distinguished career I would first like to acknowledge two very special people who are sadly no longer with us Miriam is a Hani was a brilliant mathematician who focused on theoretical mathematics it's so unfortunate that we lost such a brilliant mind at such a young age Miriam earned the BSC from Sharif University of Technology in Tehran Iran and the PhD at Harvard and became a professor at the Stanford University in the United States her work and the passion on curved surfaces has made major contributions to geometry maryam received the Fields Medal in 2014 becoming the first woman and the first Iranian to receive this medal understand Fields Institute intends to dedicate the 2018 Fields symposium to her memory and the work which is appropriate I know she will continue to be an inspiration to everyone in mathematics and beyond especially young women interested in pursuing mathematics I offer my heartfelt condolences to family friends and the people here today who knew her we also lost another great mind this year Lotfi a Zadeh who was a giant in mathematics computer science Electrical Engineering and artificial intelligence with his Adi graduated with a BS C from Tehran University in Iran and the PhD from Columbia University in New York and went on to become dean of engineering at Berkeley he is best known for his work as an fuzzy logic both Miriam and the Lotfi his legacy will live on especially among people in this room you you are some of the world's best mathematicians researchers engineers and students including fields medal recipient Martin hare who are we are honoring tonight his genius lies in the question of universality of systems and stochastic analysis which has a great relevance in quantum field theory and statistical mechanics ladies and gentlemen this year marks a 25th anniversary of the Fields Institute and I offer my sincere congratulations to Ian and to the Institute on what you have achieved government of Ontario has been a proud supporter since the Institute's very beginning your work provides a deep and strong foundation for innovation and indeed or education and there has never been a better time to study mathematics before becoming a public servant I was a scientist as you heard an academic and a researcher and the one thing was constant though my career was quite fluent but one thing was constant I saw firsthand how mathematics shapes our understanding of our world and its users are outstanding last March we launched a vector Institute for artificial intelligence in Toronto in order to continue to grow our provinces knowledge-based economy deliver transformative innovation breakthroughs and attract investments and the best talents from around the globe to our province of Ontario last year the Fields Institute began hosting the machine learning advances and applications similar the seminar brings together hundreds of students and researchers and top academics and industrial data scientists in machine learning and I am pleased that the Institute is partnering with the vector Institute for artificial intelligence hosting the seminar collaboration like this among our best and brightest is paramount artificial intelligence will have an impact on all our lives in almost every possible way that is how we diagnose and treat diseases a use and interpret data and the help make better decisions in research and Industry and the mathematics ladies and gentlemen plays such an important role in artificial intelligence advancement virtually every field of research can benefit from mathematics it will pave the way for disruptive technologies that will help us compete and the win in the digital economy and we want students across the province of Ontario to embrace mathematics at an early age to help us get there and that's why our government is providing more math supports for elementary schools to improve math scores and our students we recognize that developing talent in science technology and engineering and maths is essential to building an innovative knowledge-based economy in this province already almost 40,000 students graduate from our universities and colleges in engineering mathematics science and and maths programs every year and we want that number to increase its students and the recent graduates like those here today who will help us address some of the world's most pressing issues like water scarcity global warming and an aging population and I understand that the symposium is hosting is student night with Martin Harry giving a talk called encore entosis atoms and forest fires I'm sure it will be a fascinating talk I also understand there will be free pizza which as we know is essential to solving some of the universe's most complex problems in Ontario ladies and gentlemen we know this is lead to economic growth investments in innovation and the research is investments in economic development and that's why occasionally I refer to my ministry as Ministry of Economic Development for tomorrow and tomorrow begins today so that's why we are committed to supporting Ontario's research and innovation communities we also know mathematics plays an essential and fundamental role in our future prosperity so thank you for showing us the miracle of mathematics and how it can transform our world so best wishes to everyone on a great symposium and again my sincere congratulations to dr. Martin Herod for receiving this prestigious medal so thank you very much Mexico komak rich [Applause] well Thank You Minister Brady for those very useful and timely remarks we always like to hear people say good things about mathematics annexed like to introduce Michelle Regehr the vice president Provost at the University of Toronto professor Regehr is a distinguished scholar and academic leader professor of social work with a cross appointment the Faculty of Law whose practice background includes over 20 years of direct service in forensic Social Work and emergency mental health University of Toronto of course has a special place for us as the host institution of the Fields Institute and now I'd like to ask professor eager to say us a few words thank you very much in and it's a great pleasure to follow Minister Moretti who is a tremendous friend of science in the province but a tremendous friend of the University of Toronto and we're grateful for all he does for us and I'm very pleased to be here today welcoming you on behalf of the University of Toronto to this wonderful event and and congratulate the field's Institute on its 25th anniversary you all probably know that the Fields Institute is a remarkable place and that plays an important role as part of universities of Toronto's extended community but what you might not know is that the history of the Fields Institute at University of Toronto stems back to the 1880s which as I learned from Wikipedia the authoritative source of all information is when John Charles Fields was a student at the University of Toronto and then he returned again in 1902 to teach at the University of Toronto John Charles fields had a vision for mathematical research and scholarship as evidenced by the Fields Medal that he began planning in the 1920s and we are delighted to now at this time be hearing from one of the recent winners so today the Fields Institute is broadly celebrated as the locus of Canada's most intense mathematical research activity and it's certainly a hallmark of excellence for around the world it contributes in so many ways to the rich life at University of Toronto and today's Fields Medal symposium is a perfect example of this it welcomes prominent visitors from around the world including Professor Martin Herer and so this is an obvious reflection of the kinds of contributions that it makes to the university and to our community as Ian indicated it also offers lectures workshops awards courses and gives members of our U of T community a front-row seat some of the world's greatest puzzles and field changing theories in addition the incredible partnerships that the Fields Institute continues to to engage in we have colleagues and students from a vast array of disciplines who collaborate in developing new ideas and new opportunities and the partnerships with local industry and the ongoing partnership with some of our innovation hubs such as U of T's engineering hatchery contribute to the thriving entrepreneurial community and facilitates the incubation of startup companies and so while many view the Fields Institute to be a fortress of mathematical wizardry which it is it's truly a part of the vibrant academic community here at University of Toronto and the City of Toronto itself contributing to extraordinary diverse and dynamic landscape so this is now the sixth year of the fields metal symposium and it's wonderful that it's kicking off with a free open lecture tonight and the symposium is really a reminder of the power and joy of pure curiosity imagination and discovery and at a time where we're talking about research funding and really talking about the importance of basic research this is an incredible example of that we need these reminders more than ever in our turbulent world and in a world that's really looking for quick answers to things bringing together students and scholars and the general public in an evening like this is truly a marvel and allows us to connect with a shared wonder so this really does build our community of scholars and learners both within the university and within this great city and on behalf of the University I extend my congratulations and thanks to Professor Herer and appreciate him coming here today and to the field's Institute and the symposiums organizers I know that you're going to enjoy this evenings lecture because I actually watch part of one of Professor Harris lectures on YouTube the other great source of information in our society thank you very much [Applause] thank you very much professor Regehr for those warm and words and now in the next part of the program my pleasure to to call in Silvia so fatty as educated in France a professor of mathematics at the Koren Institute of mathematical sciences a new york university distinguished highly cited publication record in i'm losing a laptop here in partial differential equations statistical mechanics winner of several prizes the army Pangkor a prize in 2012 and special prize of the French Academy of Sciences in 2013 please professor svaty come and say a few words thank you very much so as you heard we are here this week to celebrate the work of Martin hire in a conference with many distinguished speakers at the Fields Institute and in fact I thought about it it's very rare to have a conference organized in your honor when you're a mathematician in your 40s it's actually only happens when you have to feels bad all you have to realize these nuts I'll tell you a little secret in case there are some non mathematicians in the room having a conference in your honor when you're 60 that's easy happens to a lot of people but when you're 40 much harder you have to have the Fields Medal so as you heard as you probably know the Fields Medal is one of the two most prestigious distinctions in mathematics certainly the most famous journalists tend to be fascinated with it they even like to compare it to the Nobel Prize except that as you know there is no Nobel Prize in mathematics and the Fields Medal doesn't come from Sweden it comes originally from Canada which is why we're here and so it's a big deal for mathematicians for mathematicians who have the the Fields Medal but despite all this glory and fame what is maybe special about Martin is that it hasn't gotten to his head at all I can really tell you that and he's remained essentially the same person the most unpretentious unassuming pleasant in fact the most normal especially among mathematicians and really if you know him you know this he's one of the most pleasant people in the profession we really appreciate him for that so behind this normality there is still something special there is a there's a brilliant mind certainly and there is a maybe you don't know this a genius programmer it's sort of funny to learn that on this side as a hobby you know when he was in his 20 and in his 20s my Martin created the software which is actually the leading sound processing software in the world simply and has sold millions of copies so you know it's not so common that you meet someone like that and I wouldn't be completely surprised if I learned today that he has another life he didn't tell them that I had never heard about so I recall the first time that I met Martin it was a it was in 99 if you remember he was a PhD student at the time and I was a fresh PhD I just graduated and I was visiting the University of Geneva so you may know or now is the time to learn that Martin grew up mainly in Geneva and actually if you speak if you're a native speaker of either French German or English English there must be some here you can go and talk to Martin in that language and you will come out convinced that that is his native tongue it works for French at least I can tell you so I was visiting the University of Geneva and Jean Pierre like Mac who is here somewhere so he awake and that at some point or some she asked me to give him like a sort of private lecture about what I was doing and he told me there's gonna be a student of mine Martin who is going to come and join and so Martin sat in the room I started to explain and he was like clearly listening started asking questions and Brian and at some point I remember thinking wow who is this guy he seems to understand my stuff better than I do myself so I was really really very impressed with Martin at that time and then when yours passed and I heard about all these achievements you know and he's Fame that started to build and build I was really not that surprising fact so let's talk a little bit about Martin's work to conclude one thing that distinguishes Martin other than his wonderful personality is also his way of thinking about stuff and about mathematics and really he has a way of thinking of the problems which I think many of us would agree is that it's kind of orthogonal to what everybody else had been thinking you know like he approaches a problem in a very personal very outside-the-box way and maybe that's what leads him to these wonderful discoveries he really doesn't follow the crowds of the trends and the theory that he has built is really something special it has been compared to a cathedral it has been compared to Lord of the Rings it's really a sort of immense building that's very intricate and yet beautiful it's it's hundreds of pages it resembles nothing else really in mathematics and it has opened up a whole field of study and what this theory does it it's about giving a meaning to equations that usually come from physics and your martin is also very interested in physics I think that at some point in his studies because he was kind of gifted at everything he was hesitating also also with physics and and so he saw these equations that physicists were writing and that didn't make sense to mathematicians because there were some terms in there that were infinite the equations were kind of like infinity equals infinity minus blah which you know and his theory is about making sense of these equations and it really opens up whole possibilities and finally a very satisfactory fact of understanding what these things can mean and and it even seems like some of his experience in sound processing had played a role in his inspiration anyway so tonight I think you will have a chance to hear a little bit about about that about these infinities and how to tame them and so since Martine is also a great exposure you will have I think a good time and you're lucky so so so enjoy this and without further delay [Applause] okay thank you thank you very much for the very kind words and of course I would like to thank the fields Institute for organizing this very pleasant event so the the kind of infinities that I want to talk about today as a was already mentioned so they actually let me try to so first show you historically when how these things came up and how people started to think about them and that actually goes back to the late 20s so that was back in the time when people were building a quantum trying to build a quantum theory of electrodynamics and so that was around the time where you know relatively early on after the discovery of quantum mechanics and well people did you know what you always do in physics which is that you try to make a an educated guess of what your physical theory should look like but then a physical theory sort of you know it comes with various parameters so you know like mass of the electron or speed of light or whatever right so you have a kind of a guess of the form of the theory but typically you don't have an a priori guess for the values of these parameters so what you actually have to do is you have to go out and measure things and so the way you relate the two is that your theory actually helps to make predictions about the measurements that depend on these parameters and so then you measure various experiments and once you've made enough measurements that you've enough to determine the parameters and once you've determined all of the parameters you can make new predictions that then give you you know that's the whole point of the physical theory so then at some point make predictions where you can really predict something new when the theory says something about the world rather than the world something about the theory and so the whole point here is obviously that for this to work your theory needs to have just finitely many parameters that show up in it because otherwise you've never done you know figuring out what the values of the parameters are you would need infinitely many experiments in order to just figure out with your theories and then there's no point and so what happened in that theory of quantum electrodynamics people started to build was that also for certain types of experiments are not going to go into details of the physics and it's not my expertise anyway the the theory in some of them so spits out results not really as numbers but actually as somehow formal power series so there's a constant of physics which is called the fine-structure constant which has you know fixed values so it's an actual number it doesn't come with a unit it's one of the few physical constants that actually are pure numbers it's about 1 over 137 and so the outcomes are expressed as a power series in that numbers so something times some number plus something times alpha plus something times alpha square etc and so what people first did is that instead of computing this whole series because the coefficients were given by some very complicated expressions you know you just compute the first term and you say well that's some sort of approximation to my theory and do you make predictions from that and so that worked reasonably well but it didn't explain some phenomena that you could observe and so then they try to go to the second order and there what happened is that things started to go really badly wrong and so it turns out that when you try to compute the of coefficient in front of the second term in the power series you know UN you have to compute some very complicated integral and then turns out that the answer is just infinity and well you know that's pretty bad all right so I mean if you you know as an undergraduate you learn about all these criterias for series to converge also so here it's not about the sir Syria is not converging it's about the second term in the series already being infinite and so then that's it right games over and so so what do you do so of course you know the first reaction would be well you know the starting point was to make an educated guess so maybe you know you just weren't dedicated enough and you made the wrong gas but of course you know these guys were really smart and so they they did hang on to their gas and so they try to fix it somehow so how do you fix this well so the first fix essentially was something like well whenever you see an infinity you just kind of throw it away you pretend that it's finite and you just take it as a parameter in your theory all right but you do this in a somewhat smart way and the way you do it is well I know as I told you the reason why you have these infinities come showing up is that you have some complicated function that you had to integrate and the integral turn out to be infinite and so then what you do is well you see one of these complicated functions and you know the reason why the integral is infinite is because the function somehow goes to infinity very fast somewhere and so then what you do is you just say well I pretend that this is finite so I just give it a name I know say well so now it's just one of the parameters in my theory and then I can still use that to make predictions and try to determine the parameter right so that's fair game but then of course you you know you start to run into this problem that you know if you do that but every time you hit one of these infinities you just add a new parameter in your theory well you know then you're back with the problem that you end up with the theory that is infinitely many parameters in there it's just not a theory because it will never make any predictions and so then there's no point either and so what you do is you say well if you're lucky then you know maybe you have one of these functions that solve blows up in a certain way and then you pretend that this is finite so you give it a name and then maybe little next time somewhere else in the theory you encounter one of these functions that's not integrable well if it's all looks the same as the first one then you should give it a name but you should actually give it the same name as the first one all right so if it kind of blows up in the same way then you should give that infinity kind of the same name as the other infinity okay and then if you're lucky there's only sort of finitely many types of blow up that show up and so then you're back in a nice situation where you only have finitely many parameters and you have a real physical theory again so you do that it works it works very well but it you know it sort of sounds like a cheat right and so people weren't too happy with that so so here is what what Dirac had to say about this just this is just not sensible mathematics and simple mathematics involves neglecting a quantity when it's small not neglecting it just because it's infinitely big and you don't want it fair enough or here's Fineman it says that the shell game that we play is technically called a normalization but no matter how clever the word it's still what I would call a DP process so so people weren't too happy with this right because it really just sounds very ad hoc it really sounds like a cheat but actually you know first it works extremely well and so then when once something works extremely well you try to sort of rationalize it right so you try to figure out the so way why you know find an explanation of why it works and so eventually so let me try to show you sort of a cool cartoon of the explanation that central is all crystallized right over the years so this so it sort of works in the following way so take any physical theory okay so physical theory is sort of always of the same type which is that you have some cooking recipe that takes is an input D type of experiment for which you want to make a prediction and then it takes as an input two of the parameters of the theory and then it spits out a prediction for that experiment okay so it's you can view it as a map that has solved two types of input the first type is the experiment about which you want to make a prediction and the second type of input is D parameters of the theory and then the output is the prediction for the experiment in question okay so it's a but because it's not given as a map right it's actually given really as a cooking recipe sense it's given as a computational procedure of how to compute this output and so what happened here is that you had a sort of a cooking recipe and then when you followed the recipe at some point you have to do an operation that this proved not allowed right so you have to compute the area under a function which is infinite area something like this and so what you do is you say well okay so I modify my theory in some way in order to guarantee that it spits out something finite and so I could do this by for example saying well here in this case the problem was that you have these functions I kind of blow up and become too big and so the area under them is infinite and that's what caused these problems and so you just somehow chop them off you say well once they become too big then you know you replace it just by some big value or something like this so you make some modification to your theory so that if you want visually it'll still looks more or less the same but it produces finite outcomes all right so you introduce some small parameter so mathematicians always call small parameters Epsilon and then some procedure that you know regular Rises the theory which depends on that parameter and which would have to property that if that parameter is very small then that new theory is sort of you know very close in some sense to the one that you really wanted to explain in the sense that for example you chop it off somewhere very very high on and then of course well that introduces new parameters because well you know you have to say how you actually modified the theory right so there are many different ways of regularizing things so now you end up with all sorts of new parameters that are completely unphysical because you know you just change the thing in some sort of arbitrary way pretty much in order to produce finite outcomes so you have a whole bunch of new sort of unphysical parameters that show up but now you have a theory that of produces finite outcomes but it's still you know you think well you haven't really gained much because what you really want is to send that small parameter you know to zero and when you send it to zero then the things still blow up and you're sort of back in the same problem as before but now the now comes of the genius idea that people had which is to say well you know as you send to zero as you send this small parameter to zero you can if you want you can change the way you parameterize the constants of the theory right so you remember so the theory has a bunch of constants that describe it so that's these physical constants here but they just parameterize if you want the space of theories and now of course you would describe them by numbers but in order to describe that by numbers for you have to specify how you parameterize this thing you have to specify units etc and not somehow irrelevant right so the actual numerical values of these constants were kind of irrelevant the only thing which is really relevant is you know the the set of theories that these constants I'm sure and so what you can do is you can change the parameter ization of your set of theories as the small parameter goes to zero and in many cases it turns out that you can simultaneously find a way of changing the parameterization of your set of theories so that's just to purely somehow it doesn't change the theories it just changes the way you assign a numerical value to a specific theory but it doesn't change the set of theories that you're considering and and so you've changed this parameterization as epsilon goes to 0 and you change it in such a way that there is a limiting theory which actually produces perfectly nice finite values and furthermore in many cases turns out that you can do that in such a way that this limiting theory you know the set of limiting theories that you get is actually the same independently of how you did this regularization in this dropping often and so then you have a you know sort of very reasonable to call that sort of D if you want unique theory which is all described by you know the what you started from at the beginning and you know that works extremely well so for you so the moral of the story here is that in many cases especially someone in physics what matters is somehow more the form of a theory okay rather than you know the specific values of the constants that appear in it okay so what's really important here is of T the set of theories that these constants describe rather than the individual guys so so this guy here talked so he's the person for example showed that the standard model which is the best shot that we have at the moment of the kind of theory of everything it doesn't include gravity but everything else that that theory is really what one calls renormalizable which essentially means that the procedure that I just outlined kind of works and that really works I mean that's basically it's you know sort of basically LHC which you know was built at great cost basically the whole point is to try well try to fault the theory right you can never prove that the Theory's correct them it's not clear philosophically what that even means the only thing that you can possibly prove is that it's wrong and so you somehow you know make larger and larger machines to sort of try to prove that it's wrong and you haven't been able to do it yet okay so it works pretty well so here is the sort of cartoon picture right of so what I meant with this repair metallization right so so think here of each point of this curve as being a theory okay so each point here is a theory and so so here these theories are parametrized by one number okay which is sort of where you are your position on that curve so say that's the theory that corresponds to that number equals zero that's the one that corresponds to the number equal one that's the number equal to okay and now let's think that this is so in the picture that I'll just explain this would be one of these theories which with some finite value of epsilon which is supposed to approximate some kind of limiting theory and now you could end up in a situation that if you keep these numbers here equal then as you make epsilon smaller and smaller something like this happens right but you see what happens here is that if you look at the curve of all of these theories that curve certainly converges to a nice limiting curve which I just drew as a straight line here okay but if you fix one of the values of these parameters and you send your parameter epsilon to zero then here in this case all of these points just sort of collapse to one point and you don't somehow see you know the whole set of theories that you get in the limit so here what you would have to do is you would have to change units as ipsilon goes to zero in such a way that in the new unit sort of c equal one would be somewhere here right so you would have to kind of blow things up in such a way that you you know you really get a parameterization of this curve in the limit and you don't just have everything collapsing to a pole right so that's the kind of thing that you should the kind of cartoon that you should have in mind here or actually this is what I mean in reality it's more like so for example what could happen is that as you make epsilon smaller and smaller and you keep the constancy fixed these points sort of run away to infinity all right and then you would have to sort of pull them back by just making a change of variables all right so that they actually converge to something on the limited curve instead of just running off to infinity so here let me now try to give you a sort of more precise example where you know exactly the same the same sort of thing happens so now this one is more so that's more for the year so if mathematically inclined people in the audience I think there are quite a few mathematicians okay I can do some math so so here remember that okay so a distribution is something which basically just stays takes us input the function and spits out a number in a linear way in the sense that if you give it the sum of two functions it spits out the sum of the two numbers and that's something which has a little bit the same sort of flavor so you should think of the function that you give to it as being something like your experiment and then the number that it spits out as being something like your prediction okay and so one example would be the one that simply you know you give it a function and it spits out the value of the should say at zero another example of distribution which is much more generic is you fix a fixed function which are called eita hat here and now the way your distribution works you give it a function Phi and what it does is it multiplies it with that fixed function and then it computes the area under it under the new function you get in that way and so that gives you a number so that's it spits out it turns out that you can always approximate every distribution by something of that type and so now the question is can we define a distribution which got two parameter family of distribution so that would be these two parameters would be something like the physical constants in my theory if you want okay so now it's a theory of two parameters so can you define something like a distribution which corresponds to some constant divided by absolute value of x minus a constant times this Delta distribution and so here the problem is that you know one of absolute value of x is a function that looks like this which has infinite area here and so the area is infinite in both directions sort of this bit here has infinite area and these two tails here of infinite area as well and so if you just take some fixed function you know nice and smooth function you multiplied by this well the thing you obtain is still going to have infinite area okay and so there you have a problem which is that what does this bit mean right if you interpret it in this way you have exactly this problem of trying to compute an area which is actually infinite so it doesn't make any sense and so now what happens is well we can do the same sort of procedure right so we can somehow chop it off so we say the problem here is that one of X becomes very very big if X is very large and so we just introduce a small parameter which is epsilon and we replace one of X but one over X plus Epsilon and so now it's it never becomes bigger than one over Epsilon and so now the area under this is certainly going to be finite because it never gets bigger than a certain value and so now I have a you know I have a perfectly nice way of defining the distribution a over X plus epsilon minus constant times the Delta function okay so now I have a two parameter family of distributions and so now I can try to send some hype cylon to zero and of course if I just send it to zero I have exactly the same problem again instead I end up with something which is infinite area and so it doesn't converge and turns out on the other hand here is if I define a family of distributions in this way then this actually converges right so I take here any function Chi which is sort of one near zero and then it's zero further away and so in this definition if X is very small these two terms here will cancel each other out because Phi of X is going to be very close to file zero and so you kind of cancels out this divergence e here and so there's absolutely no problem making epsilon very small and converges to a limit it's always going to be the same limit whichever way you approximate this and on the other hand this expression here you can actually rewrite it in this form because this term that I added here is just some number times Phi of 0 so it's actually of the same type as this okay so it's actually really just a change of variables in my two parameter family and here it's sort of clear if you do the calculation that this converges to a nice limit and the limit really doesn't depend on how you regularly things alright so here it's exactly this situation where you you just take a pretty much arbitrary way of fixing things it seems completely arbitrary it gives you different answer for different ways of fixing it but then if you sort of try to remove the fix and just do a repro motorisation the family of limits that you get is always the same whichever way you fix things you always get the same family in the limit and the number of parameters in the families of the same is what you started with so you have this kind of two parameter family okay so now let's switch to a completely different example which comes from stochastics or finance if you want which is the following so so take something like a random walk so a random walk just means you know you stand here you toss a coin if it comes up heads you make a step to the right if it comes up tails you but you can step to the left and you just do that repeatedly so you end up sort of moving right and left sometimes you move further right sometimes you move further the left so wiggle back and forth randomly and now you do that faster and faster so instead of tossing a coin full of every second you toss a coin every millisecond or every microsecond and of course you're going to move so super fast so if you want to get some kind of limiting process you have to make smaller and smaller steps and it turns out that the correct way of actually getting a limit is that if you you know toss your coin every epsilon time you the size of your step should be about square root of that epsilon so that you get something some limiting process which is called the Brownian motion okay so Brownian motion is something like this so now this would be a plot of the position as a function of time so you see you sort of move a little bit say this is right this is left so you move a bit to the right and you move to the left and to the right again but you sort of move in a very very erratic way right so you make these kind of little random motions and so this process is Brownian motion it sort of it's one of the most fundamental if you want random processes we solve the simplest random process that appears in mathematics and so it actually appears in many places you know in real life if you want I mean if you look at you know curves of the stock market locally they kind of look like something like this I think there's a good reason why they look like this is because actually the mechanism that makes stock prices move is you know at least locally sort of for reasonably short time scales is actually very similar to the mechanism that I just described where if investors you know sell or share then there are more shares on the market so the price goes slightly down if they buy shares they're a bit less than the market so the price goes slightly up and so you have this erratic random motion with all these different sort of independent investors develop buy and sell shares and so that's basically why you know the prices of stocks that you see in the newspapers they kind of look like this so this here is a plot of a Brownian motion so it's just one realization of this praana motion and what I do is I'm sort of zooming out more and more and so you see it always somehow looks the same so it's it has this kind of fractal structure there and it looks the same at every scale and I don't know if you can yeah maybe you can see so there's this parabola here so the meaning of this parabola here is that the way you have to zoom out in order to always see the same is exactly the way that keeps that parabola constant alright so it means that if you zoom out by a factor for horizontally you should zoom out by a factor 2 which is square root of 4 vertically okay so that's Brownian motion so now you know so I gave you some sort of intuition of why this random walk is a reasonable model for stock prices but it's actually not that reasonable right because for starters the stock price doesn't turn negative it's always positive and the ensel of the amount by which the stock price changes in response to move these events it's typically kind of proportional to the value of the stock right I mean if the stock price is $100 that it might change by about the dollar you know within the next day if the soft price is only $1 to start with it's obviously not going to change by a dollar I didn't mean it might change by a cent okay so the amount by which the stock price is going to change is typically about proportional to the value of the stock itself so a better model for stock prices is something like that where instead of at every step instead of moving by some random amount after every step you move by a random amount which is proportional to the value of the stock at the moment when you make the move all right so that somehow a more reasonable model for stock price it's now you can ask yourself well since that guy if I make this primary type this parameter epsilon is just some sort of artificial parameter she says I make some kind of discrete model where I assume that things just make a move every epsilon time it shouldn't have much of a meaning so you would really want to send this more parameter to zero so you can ask yourself is this model related to this model in some way right so can I relate the two limits so now if you look at this model here I can write it as in this way right so it says that the the increment the amount by which my stock price changes over this time interval is equal to the value of the stock itself times the amount by which this random walk here changes over the same time interval okay so we have just rewritten this and now if I divide both sides here by the time interval itself then this looks like a derivative right so that's how we teach the relative size of little vertical increment divided by little horizontal increment and then of course the what we always say when we teach undergraduates derivatives is well you know you sort of make some kind of little approximation like this you draw a little tangent you know and then you just make it smaller in small ways and this little interval to zero and in the end you get your derivative right so you'd want to do the same thing here and say that well the derivative of this guy is equal to its value times the derivative of that guy and so you get a differential equation you can solve it so you just get this formula here it turns out that that's the wrong answer in this case okay so if you if you take this model here and you make epsilon very small you don't get this but you actually get this and it's possible to guess that because actually you know the average amount of your the average value of your stop price here in this model doesn't change right because this guy here on average doesn't change so that was my step in my random walk where I have half chance of going right half chance of going left on average I don't move all right so this guy on average doesn't move and so the expected value of my stop price at time after any time should be the same as the expected value at the initial time here so it's true for this model here and so you should between the limit but it's not true for this model here okay and so you can check that if you subtract this term then it becomes true okay so that's more reasonable and actually you can prove that what you get in the limit is actually this rather than this but so now you know what actually went wrong and I mean we had almost a proof that you should get this limit and what went wrong is of course we're not really allowed to differentiate these things that's a whoops so remember this Brownian motion looks like this and so what we said here is we have this differential equation which says that the slope of my asset price is the slope of the Brownian motion times the value of the asset price itself but you know if you look at this curve it doesn't really look like it has a slope anywhere right and it's actually clear because remember in the movie that we saw earlier they spur on the motion it somehow you know it looks always the same under this zooming out procedure but it was dessous mning out procedure that keeps this parabola fixed and so it means that it actually really sort of looks like this parabola horizontally but that parabola had infinite slope right because that the origin was precisely the point where it was vertical and so actually you can show that you know this Brownian motion just doesn't have any slope and you can already also see it by the fact that we move by square root of epsilon every epsilon time and so the speed that which remove would be square lips on the web apps on which also blows up for small values of the time okay and so so that's what goes wrong here is that somehow these slopes become infinite and so you know you write down this differential equation but it's actually it doesn't really have a meaning that differential equation right and so well you formally you can find a solution for it but in this particular case it turns out to actually be the wrong solution so the consequence of this here is that you can get depending so if you take you know you write down the same differential equation and you can interpret it as you know somehow a limit of various discrete equations of the same type as the relation that I wrote down earlier so you could for example interpret the same equation as the limit of a recurrence like that was excellent goes to zero there are various other recurrences that you could write down that at least you know with the same sort of back of the envelope argument should give you the same limit and it turns out here that all of these different approximations they actually all give you different limits but not that many so now the limit that you will get will always be of that type so it will always be of the form this exponential Brownian motion which is the thing that you expect minus some constant times T is just the value of that constant that's going to be different for different approximations and so the so the moral of the story here is that well you know sometimes you have these in probability theory you have these quite singular random objects that appear and you know they come with various natural ways of approximating them and so in many cases what happens is that you know the details of how you approximate these objects do matter so you get somehow different limits by taking different approximations but they don't matter too much in the sense that although you might get different limits for different approach Nations somehow the set of possible limits that you get is quite small so maybe you know you just have so one constant that parameterize is this or something like that and so that's something that appears in many different contexts so one tool of model that probably lists like a lot is called the easing model so the easing model sort of a model for what goes on in a magnet close to the transition at which it loses its magnetization all right so if you take a magnet and you heat it up then at some point it actually loses its magnetization so above a certain temperature the magnet just becomes inert if you want and so the this easing model is some kind of a toy model for what goes on in a magnet where you know what goes on in the magnet is if you want every atom in your magnet works itself kind of like a little magnet and these little magnets they all want to they have a tendency of wanting to line up but then on the other hand temperature has a tendency of you know making them want to just be disordered now right so temperature is basically the same as disorder if you want and so a simple model here is to say well you know I model each of these little atoms by just a number that takes on only the value plus one or minus one so which somehow says either pointer it's a little magnet that points up or it points down and then I say well so my my magnet is now modeled by a whole lattice of atoms okay so that at every lattice point you have either plus 1 or minus 1 with both points up or down at every lattice point and you say well I take one of these configurations at random but the probability with which I pick a configuration is not uniforms so that they don't all have exactly the same probability but I make the probability proportional to this quantity here and the important fact about this quantity is that it becomes big if they are aligned and it's sort of small if they don't align right so if the if two neighboring so this here means two neighbouring positions so if two neighboring guys have the same value so if Sigma X is equal to Sigma Y so if they either both plus one or both minus one then the product is one and if they have opposite values then one is 1 1 is minus 1 so the product is minus 1 okay so in one case I have the exponential of something positive which is bigger in the other case something negative which is small right so you want to have sort of higher probability of being aligned then of not being aligned and there's one parameter here which is essentially the temperature so if that parameter is zero that means you just don't care about this value here and then they're all equally probable and then you just choose for each of them either plot you do a coin toss basically for each of them and in the other case if this value here is infinite which actually corresponds to zero temperatures a temperature of 1 over the value of that parameter so if the value is very large well then it means that with very high probability things should be aligned right and so here are three pictures of what you typically see for different values of that parameter so for high value they want to be aligned and so you typically see so here there are two possible values plus or minus one which is yellow black okay and so here if beta is large then what you would typically see is that things tend to be aligned so either they're mostly yellow or they are mostly black I don't so what you see here is you see something like this which is mostly yellow and then with a bit of blackened inside or sometimes you would see the other way around mostly black with a little bit of yellow inside on the other hand if beta is very small at higher temperature you see something like this right where they are basically more or less you know it looks completely random if you want and then somewhere there's a transition B between these two pictures so somewhere it looks like this or because it looks much more interesting right and so that would correspond to the behavior which is just a D temperature where you actually lose the magnetization right so it's what's called the critical temperature so let me show you so here is a sort of animated version of this right so here would be the situation where you're at high temperature so things tend to be completely disordered so if I now go to low temperatures of looks like this alright so at low temperature now they tend to align right and so you have these regions that form where things are aligned and then they move very slowly and eventually one of the two colours wins and you just see everything in one color and somewhere in between right you have behavior like this so this is what happens so here is a temperature which is very very close to this critical temperature and so here what happens you have you know very interesting behavior where you see you know sometimes rich behavior are those scales and you want to understand what's going on there now in two dimensions this is relatively well understood of course the real world is how three dimensional in three dimensions it's very very poorly understood so actually people have very very little information about this in three dimensions so there are some guesses by physicists there there's very little mathematical information about what happens in three dimensions and so in particular one thing that one conjectures off is that at this critical temperature this toy model actually is a relatively good description of reality in the sense that you know you could come up with many other kind of toy models for this transition and you know they will all have similar features and in particular when you have at the critical temperature for any of these toy models the behavior at very large scales should always be the same okay so there's a kind of universality conjecture which says that there's a very very large class of toy models you can write down there are many many mathematical models you could write down at this they all have different critical temperatures they would you know in their details they would look very different but their behavior on large scale should always be the same when you're at this critical temperature and so then one thing you could ask yourself is is somehow one of these models more natural than all the others right so this model is there's something arbitrary so I arbitrarily somehow choose to put things on a square grid on the lattice I could have changed the lattice I could change the interaction a little bit so there are many many models of this type I could write down they would all be slightly different alright so there's nothing if you want special about this model so you can ask yourself if there's one which is somehow more special than the others and well it turns out that there is so there's so ki so I wrote down here an equation which is the equation that somehow supposed to describe this model I can show you a movie so so here so this here is a movie of an object which is called the free field which is okay so it's somehow a natural mathematical object that describes a number of physical situations it's a continuum so it's an ideal there's an idealized somehow continuum object random continuum object here so this one isn't a good model for this transition so it has a somehow if you look at the behavior here it actually behaves it doesn't quite look like the previous movie that I showed you about the behavior of this easy model at the critical temperature but then it turns out that there is exactly or conjectural II exactly one model which is the one shown in this movie here which has the property that it should be a good model for the behavior of a magnet at this transition in the sense that when you look at it at large scales it should look like the easy model and what you see here you know if you remember the movie I showed you earlier here you see very similar features so you have somehow these relatively large regions of light large regions of dark and they have somehow structure at every scale so you see very similar behavior at these large scales and on small scales this one really behaves like this free field that I showed on the previous animation which is another how very canonical natural mathematical object and so you can write down you can write down one equation that if you want describes a model which has the feature that you know on very large scales it behaves like this model of a magnet at the critical temperature on one on very small scales it behaves like this free field and so there there's again a conjecture that this is somehow the only one it is basically completely uniquely specified basically by the two properties that I just told you so this would be in some sense of more canonical model but then the problem with this equation is that actually it doesn't really have a meaning okay because so in this equation so here this symbol Phi is about bunam denotes the magnetic field at a position in space and time so here this is the derivative in the time Direction this is the derivative the second derivative into space direction and then you have term which is the cube of the field and then you have another term which is some kind of noise term which makes it random right so it creates it generates the randomness that you saw in these movies and now why is this cube term a problem well you see you know you see in these movies but you see this kind of flickering in the movie right and so what this flicker what happens with this flickering here is that if you did increase the resolution of this movement okay so this is of course some sort of discrete approximation because it has somehow the resolution of the screen that I showed the movie on if you were able to increase this resolution more and more and more this flickering would just get worse and worse okay it would actually in some sense flicker more and more very small scales and so there's no way you can actually assign really a value to this field at a fixed point all right so if you try to zoom into a point try to measure the value at a point it just keeps on flickering it just keeps on sort of taking very large positive values then very large negative very large positive very large negative so there's no way of actually defining what you really mean by the value at the point of this field all right so it's a if you increase the resolution that flickering you know it never smooths out it just goes worse and worse the better the resolution and so you know what does it mean you know to take the cube of something which doesn't actually have a value at that point right so it's clearly a problem and so it turns out that actually you know this term if you try to take some kind of approximation of this well this terms of becomes infinitely big and then what you really have to do is you have to make the constant in front of this term infinitely big as well so the two kind of cancel out but they don't they don't cancel out completely because otherwise I wouldn't have written them in the first place there's still something finite that's left over and so you know one thing we would like to do is you know you would want to go from the cell you know why does that actually happen so this is one of the things that you can actually show so you can show that you know if you take basically what this theorem says is that you know you take a movie or the certain resolution for this model here but you see this model depends on one parameter which is the value of this constant and then it says that the higher the resolution the higher you have to choose the value of this constant in order to see something which sort of more or less looks the same in your screen and if you really send the resolution you know 2 to infinity so you sort of try to make a sort of infinite resolution movie then you have to send the value of this constant to infinity as well in order to get some limiting movie on your screen but then you can show that the movie that you get in the limit you want is always the same so it doesn't depend on whether you approximated with the you know pixels that are aligned sort of in a square grid or in a triangular grid or if you you know you can do have various different ways of approximating this equation so there are many many different ways in which you could go about it as you increase the resolution you always see the same object in the limit okay no matter how you try to approximate it so let me give you so in the last two minutes just say one or two words about the way in which you can actually prove things like this okay so you see the problem is you have you know you have something like some sort of very wild kind of object like this and you want to somehow define what you mean by the cube of something like that which doesn't really have a value it any point and so it turns out that well even though this looks extremely wild mm-hmm in some sense you can actually describe it as if it were a nice and smooth function so usually right so usually when we have a actual nice and smooth function what does it mean to beacon of nice and smooth well it means that if you take any point on your curve so well you can approximate it first you can approximate it reasonably well by a straight line right but then you can approximate it even better by a parabola and then you can approximate it even better by a polynomial of degree three etc right so it sort of means that at every point you can actually approximate it very well by a polynomial and there's kind of some small error term so that's what it means to be smooth and obviously that's not the case here all right so this really doesn't look smooth at all so you can't approximate this by a polynomial alone but as I already mentioned you know this guy has the property that at least at small scales it actually looks like that guy which is this free field that I mentioned which is very well understood so the idea is to say well I take you know instead of trying to for example approximate my object by something like a linear function approximated by that guy here all right so I actually replace the polynomials by you know some kind of wild random objects of this type you know in order to approximate the object that I'm really after and it turns out that that actually really works so you can set up a whole theory where you replace polynomials by things that are little not smooth at all don't look like polynomial at all but have you know various properties that are actually very very similar to the properties of polynomials and then you can rebuild sort of the whole all of analysis that we kind of know solve all theories of you know all various function spaces you can have analogs of these function spaces or you know pretty much all the operations that you use in analysis where somewhere at the back there's always this idea that a function is really approximated by a polynomial you can rewrite pretty much all of analysis in this context where you replace polynomials but something completely different and then the point here is to use something different which you understand very well and which approximates the object that you're after and so then you only need to make be able to make sense of this operation of taking the cube for example for these objects that you understand much better and then once you control these then well you know you have a chance of being able to control anything that sort of looks like these objects it turns out that you can you know use this idea then to actually really show that what I just told you is true and so I think I should maybe stop here so I have hope I've given you a little bit of a flavor of this idea that of came really from the back from the 30s and 40s so this idea that you know sometimes you have these mathematical or physical objects that look like they don't really have a real mathematical meaning but there is this procedure this randomization procedure that a priori kind of looks slightly weird it looks like a cheat somehow that allows you to give a meaning to these objects but it's actually very natural and you can really you know work with these I think I should stop here thank you very much [Applause]

Channel: Fields Institute

Views: 9,258

Keywords: Fields Medal, Fields Institute, Elsevier, MaRS, Great West Life, London Life, Canada Life, International Mathematical Union, IMU, Martin Hairer, Kirsty Duncan, Reza Moridi, Cheryl Regehr, Sylvia Serfaty, Lapse Productions, Fields Medal Symposium

Id: Jz63GkM0_eA

Length: 85min 38sec (5138 seconds)

Published: Mon Dec 11 2017