The Abel Prize announcement. March 20, 2024

[Music] welcome to the Norwegian Academy of Science and letters here in Oslo Norway and we're here because every year the Academy of Science and letters hands out the arble prize a very prestigious mathematics prize designed to celebrate and reward mathematicians who've made significant contributions to the world of mathematics and today we're going to find out who the 2024 winner is as the president of the Norwegian Academy of Science and letters it's my pleasure and privilege to announce the winner of the OB prize for 2024 the board of the Norwegian Academy of Science and letter has decided to award the Obel prize for 2024 to Michelle tagant for his groundbreaking contributions to probability Theory and functional analysis with outstanding applications in mathematical physics and statistics the AEL priz honors outstanding scientific work in the field of mathematics this year we honor Professor Michelle talag from the C National de scientific Dear Professor tal I would like to extend my sincere congratulations to you on your great achievements through your work on probability Theory and functional analysis the arel prize is a celebration of you as a mathematician and of the whole field of mathematics Norway is a knowledge Nation we depend on Research based knowledge and expertise to understand and resolve the challenges facing Society in this context the Abel prize is important to celebrate excellence in research on behalf of the Norwegian government congratulations as the ab PR laal for 2024 and welcome to ow and welcome to Norway in May the Aral Prize winner is selected by an international group of esteemed mathematicians who draw from the nomination sent in by the mathematics Community to explain exactly why they picked Michelle tagram as the 2024 aror Prize winner we will have a full reading of their salutation by the chair of the board hilga Holden thank you very much mat the the norian Academy of Science and letters Awards the arel prize 2024 to Michelle tagr s naal scientific Paris France for his groundbreaking contributions through probability Theory and function analysis with outstanding applications in mathematical physics and statistics the development of probability theory was originally motivated by problems that arose in the context of gambling or assessing risks it has now become apparent that the thorough understanding of random phenomena is essential in today's world for example random algorithms underpin our weather forecast and large language models in our quest for minorization we must consider effects like the random nature of impurities in crystals thermal fluctuations in electric circuits and decoherence of quantum computers tagr has tackled many fundamental questions arising at the core of our mathematical description of such phenomena one of the threads running through tgr's work is to understand geometric properties of high-dimensional phenomenon and to crystallize this into sharp estimates with broad scopes of applicability this led him to obtain many influential inequalities for instance tgram derived powerful quantitative results to prove the sharp threshold phenomena that often appear in the study of phe transitions in statistical mechanics he also also obtained a useful inequality bounding the quadratic Transportation cost distance between a probability measure and a gaan distribution by the relative entropy much ofr's work concerns the geometry of stochastic processes a classical problem going back to Koro arising for instance when one wants to analyze regularity properties of stochastic processes is to estimate the supremum of a large collection of correlated random variables B building on the work of fique and Dudley ton developed his theory of generic chaining which provides sharp upper and lower bounds on the expectation of suprema oaan processes this illuminated the mysterious connection between the distance function on the underlying index set determined by the covariance of the process and the expectation of its supremum a key result in probability theory is a law of large numbers asserting that the normalized sum of independent random variables converges towards its mean this normalized sum is therefore concentrated using the terminology coined in the early work of milman or self aaging using physics terminology it was gradually realized that concentration is ubiquitous since many random variables defined as function of a large number of independent random variables appear to be close to the mean with high probability ility in an amazing to the force tgram provided quantitative versions of this phenomenon that hold in great generality including the case of discret random variables this result applies to functions of independent variables that are lip sheets with respect to the ukian metric and convex yielding one of the several celebrated toon inequalities it laid the groundwork for a non-asymptotic theory of Independence applicable to high dimensional statistical problems since the work of Edwards and Anderson physicist have been fascinated by the complex Behavior exhibited by disordered systems which describe phenomena like magnetization in the presence of inequalities and more recently also the energy landscape arising in machine learning in 1980 Parisi Nobel Prize in physics 2021 proposed osed an expression for the free energy of one of the simplest models of this type namely the sharington Kirk Patrick model Guera showed rigorously that this formula is an upper bond for the free energy in a groundbreaking article togram proved the complimentary lower bound hence completing the proof of the paresi formula this provided uh the foundation for the development of a mathematical theory of spin glasses and its applications in statistical learning tan also obtained a rich variety of important results in measure Theory and functional analysis to cite only the most recent one he answered a long-standing question by fman and mahaman in the negative by showing that there exist submeasures which are exhaustive but not absolutely continuous with respect to any finite additive measure this fact implies the existence of radically new Boolean algebras tan is an an exceptionally prolific mathematician whose work has transformed probability Theory function analysis and statistics his research is characterized by a desire to understand interesting problems at their most fundamental level building new mathematical theories along the way he disseminated many of his insights in the form of very influential research monographs combining technical virtuosity with deep analytical and geometric insights to construct new powerful tools and answer long-standing hard questions Michelle tan has had and continues to have an enormous impact on mathematics and his applications thank you hga now what struck me about the citation is the mention of how telegram was obsessed with understanding things that they're most fundamental and I believe that was a motto he took very seriously yes a guiding principle throughout his work has been a piece of advice that he got from his PhD adviser Gustav sh sh said that you should simplify the problem as much as possible while still keeping the intrinsic structure and then try to understand it at let level and this is what tan has done throughout his career you break down the problem to the simplest possible terms then you extend expand and make it into a theory and then once you have understood it you write a book about it take the for instance the example of generic chaining tan studied this for 15 years and now he has simplified it so that is says that you can understand it in 10 minutes and of course he wrote a book about it now this approach of understanding things in their most simple context did that affect when he was working in other research areas yes he tried to follow the same principle when they studied spin glasses here there was a formula based on physical intuition approved by paresi but tgram wanted to understand this in mathematical terms so he wanted to have a fully rigorous mathematical proof of the pares formula so now we have two arguments ments for the par Formula One based on physical intuition and another one based on mathematical rigor and of course T wrote two volumes about it always ends by writing a book I think it's fantastic he was able to prove the same thing from physics but by ignoring the physics largely and just looking at it mathematically in fact I'm now going to try and provide some extra context to the research that tgram was doing but I've very much taken their motto to heart and I'm going to try and do that in the situation which is the most simple situation possible while things still technically make [Music] sense the first two concepts I would like to unpack from the citation is the idea of Randomness and having a distribution now Randomness I can simplify down to Rolling a dice something we're all reasonably amiliar with I roll this and I get oh a two one of the six possible values on the dice picked at random now in terms of distributions telegram's work covered a lot of different types of distributions but the most common and arguably the most important is something called the gausian distribution this is a symmetric distribution where you have the most common uh event or value right in the middle and then less likely events and values on either side and this kind of G Ian process comes up all over the place in the world around us the the mass that babies are when they're born the scores people get at school or even the age that people retire at these may seem like they're random values but they're all drawn from aian distribution we can now try and combine both of those together so we can take random values from aian distribution now if I roll two dice at the same time and then add their values all all the values are no longer equally likely in this case I just got an eight and if you look at the distribution of the possible values and How likely they all are it's close to a gan distribution but not quite although the more dice that I roll at once the closer the distribution of these random values gets to a gan distribution so I can simulate picking a random number from the true Gan distribution by taking I'm not going to show you how many an unknown number of dice I'm going to roll them behind this screen and I'm going to tell you that this gausian distribution just gave us a value of oh 30 a value of 30 how many dice do I have what is the largest possible value I might roll if I roll this another 10 times there's a lot of questions you can ask about getting a random number from a gan distribution I can now tell you the second value out is 30 it's going to be bigger this time 30 42 there you are 42 and a lot of telegram's work was taking a random variable like this where it's coming from a distribution and you're getting some values from that random distribution and you're asking questions what's the average value what is the the Supra what's the largest value I would get over a certain amount of time and I can now reveal I was rolling in this case let's have a look here here 6 9 11 11 dice so the average value from that the mean would have been 38 and a half and we mentioned the law of large numbers in the citation and that says if I was to keep rolling these dice or taking random values from a gaan distribution the average value from the results will converge with the true average value of the underlying distribution however life is not always quite as simple as a single Gan distribution in the citation you will have heard mention of higher dimensional spaces and you're thinking well how are we going to get from a probability distribution to that well that's what happens if we start combining a couple of these together I'm going to go even more simple now with just flipping a coin so I've got a yellow heads and a copper colored Tails if I flip one coin our first random value oh is a Tails now that's from a very simple distribution of two possible values but bear with me I'm now going to flip a second coin that will give us oh heads very pleasing that it's different and our final third one is heads so now I'm selecting random values from different distributions and I've ended up with Tails heads Heads that's one of eight possible values that could have come out of that process to understand those eight values and the the way these three different admittedly very simple distributions interact we could label the corners of a cube with those eight different outcomes so I'm now using the x or the across axis as the first coin I flipped I'm using the the y or the back axis for the second one and the Zed the third axis for the third value and so now when I pick these three values from three different distributions what I'm doing is selecting a corner from a cube in three dimensions If instead of being very discrete distributions with just two possible values these could have been continuous distributions they could be Gan they could be something different they don't all have to be the same and each one is in an an orthogonal a new dimension from the others so if I had four five more possible random distributions I'm pulling values from that are all interacting and being combined the resulting space the the total probability distribution that we need to understand to be able to calculate expectations and values is a fairly confusing multi-dimensional object the problem with higher dimensional spaces and shapes is they're kind of Beyond human intuition they're very hard to understand but we do need to understand them just like when I was rolling two dice the height of that distribution or you could argue the area of that part of like a bar chart that represents the probability of getting that value the same thing holds in higher Dimensions we need to understand the volume the hyper volume the some measure of how much there is in different areas of the distribution to work out the probabilities and that is not straightforward so I've got one of my favorite examples of how trying to understand higher dimensional shapes is difficult and we're going to start in two Dimensions we're going to start with a square and I'm going to fill that square with unit circles or unit discs because they're they're filled in you you can think of these as the 2D version of like a solid ball now inside those unit discs I'm going to put the biggest disc possible that will fit it's not very big it's about 0.414 compared to the radius of our unit discs but we can you know use Pythagoras and we can carefully calculate exactly how big that disc is let's go up a dimension we've now got a cube-shaped box and we're going to fill that with eight unit bowls they've all got a radius of one and once again we can see how big a 3D ball can we fit right in the middle and this time that ball from the center is just over 0.7 compared to the unit balls it's a bit bigger which kind of makes sense there's a bit more room to move and you might think as we go up in Dimensions the ball gets a little bit bigger but ultimately remains bounded sadly no if you go up to four Dimensions where you've now got 16 balls the ball right in the middle is as big as the ones around it and in five Dimensions it's bigger and by the time you're in 10 Dimensions that ball inside the unit balls inside the box is bigger than the box and once you're up to 26 Dimensions it's more than twice as big as the Box our intuition doesn't work and this is not a rigorously correct way to say this but in higher dimensions a ball even though it's still the set of all the points that are within one unit radius distance from the central origin the balls become spiky is one way to think of it they become it's a weird distribution of where the content is and there's an area of mathematics called measure Theory which is all about trying to understand these shapes and calculate how much of them is where these coins started as a simple example of how data can be visualized in multiple dimensions and in the citation you heard mention of large language models and when you've got these big data sets they are effectively multi-dimensional objects which we need to understand and telegram's work was revolutionary in understanding these shapes and finding aspects of these high dimensional distribution spaces which were well behaved and then putting bounds on how far away from the mean for example a certain percentage of the total measure of that shape is likely to be because the lore of large numbers still applies to these higher dimensional shapes when we mention the concentration of measures that's a more abstract generalized version of that and telegram's work was fundamental in us having a better grasp to understand these spaces finally I'm going to attempt the most simple explanation in which spin glass makes sense and spin glass is not like glass it's an arrangement of matter that I'm representing with these tokens and on each one I've got an arrow so these are little magnetic magnetic moments within the material and if this was a ferromagnetic material and you apply an external field it would cause all of these to line up and that happens with a spin glass you apply an external field and all the little internal moments line up now a pherom magnetic field they would stay like that even when you take the external field away a paramagnetic material would immediately well almost immediately return to disorder exponentially fast a spin glass however once you remove the field it does start to rearrange these little moments become random but not completely random they're still still interacting and they're interfering with each other and it's not an exponential decay like a paramagnetic substance it's an unusual and difficult to predict Decay and physicists were trying to work out what they could understand about this unusual state of matter things like how much free energy does it have and they came up with some limits and some thresholds but it was tgr's work that was able to put those thresholds on a firm mathematical Foundation taking his approach of simplifying and using Randomness and probability Theory to give an insight and prove what the physicist had discovered so hopefully using counters coins and dice I haven't too oversimplified the incredible work that tegram was doing but I've managed to give you a bit of an Insight a rough context for the phenomenal work they were doing Michelle telegran has been informed they've won the Aral prize and they've very kindly taken some time to answer our questions so Michelle my first question for you is given how prestigious this prize is how did you feel when you first found out you'd won it well that's very interesting because the proper answer is I had no Su there was a total blank in my mind for several seconds if I had heard that the alien Mother Ship had landed in front of the city Hall I don't think I would have been more surprised I had never ever thought that this could happen so it was a shock Pleasant shock of course but very deep shock I'm not over it yet of course as we have already heard today from the citation you've done an incredible range of work across different areas of mathematics and some Physics I'm curious to know though which part of your work are you the most proud of in some sense I could be proud of the things which were the hardest to do but it's a little bit different the first result which I proved which really had an impact uh was uh on gaussian processes and this sort of miraculous result which proves the equivalent of two very different way to look at some object are in fact the same and um when this was done it was such a miraculous result that I think okay it will the miracle will not stop there uh let us dream what in the same direction what could be the structure of other related object and I made a series of conjecture totally out of Fiat fat there was no not a single element of proof or real indication just dream wish let us dream what things be and so that that set up a program and amazingly enough uh two years ago this program was entirely completed with of course the help of many other people so now there is this complete description and all the Miracles turned out to be true uh the universe in that very small Corner turned out to be as simple as it could be so it's it's nice it makes me really happy when I think about that today you are the prize winner but for a long time you've offered prizes cash rewards on your website for people who can solve open problems could you tell us a bit more about those that's a good question because there are many interesting stories Paul OS who was a great mathematician was famous for uh offering cash rewards but what he would do he would offer $15 for an incredibly hard problem well I'm proud to say that in my youth I solved a couple of problems asked by paos but there was no cash reward for these ones in my web page now there are five or six problems for which I offer $1,000 these problems are problem I I thought about them and I couldn't solve them I think they are good problems I hope that now people will look at them one price I offered a significant amount of money was 5,000 $5,000 but it was for a problem which over which I worked intermittently of course for maybe 15 years and I couldn't solve and my feeling was that this problem was absolutely fundamental probably because it dealt with a very basic object and I always had the philosophical belief that it's important to completely understand the very simple object before you try to understand the complicated one it was solved it in 2011 by Rafa latala and vitol bedos and I was so happy to give away $5,000 for that res the proof is the most beautiful proof of mathematics I've ever seen and incidentally it could be it's not the correct proof it's too complicated so one goal I have for young probabilist find a proof of that results that you can explain to your grandmother like fedman would say a proof which the approach get a good idea which makes the proof transparent which is not now and my intuition that this result was Central was correct because it turned out to be the essential Cornerstone to finish the program that I was talking about in the previous question so I'm so happy I gave that price away now five years ago I received the show price which had a 1.2 million uh money coming with it I have donated the entire amount of that money to create a foundation which will distribute price for resulting mathematics in the areas which I have worked and which I understand which goes well beyond probability and this foundation will start in within 10 years to give uh very significant prices in mathematics and the money of the ab price is going to join the man for the show price exactly I'm going to donate the entire amount to the same Foundation so when the first prize comes out you will have a very very interesting story to tell thank you so much for taking the time to join us Michelle and once again congratulations on an incredibly well-deserved prize this year is the 22nd year of the aror prize so we're joined once again by Lisa OS the president of the Norwegian Academy of Science and letters who's going to give us some more background information about about the prize Well the Obel prize was established by the Norwegian parliament in 2002 on the occasion of the 200 years anniversary of Nels Henrik Al's birth norwegian's greatest mathematician throughout the times and the AEL prize exists to honor individuals of course but in a largic context also stimulate interest among mathematicians and society and stimulate young people I would like to thank the abble committee for their excellent work now that we've had the announcement and we know that Michelle won the award what happens next how does he get his award well it is a pleasure for me to welcome you all to the Obel prize award ceremony at the University Ola in Oslo on 21st of May at 2 p.m Norwegian time Michelle Tagan will receive the oble prize from his majesty King har the 5 there will also be a lecturer in honor of the laurates at the University of Oslo on Wednesday 22nd of May registration will soon be available through our website you're most welcome to celebrate this year's ala Laurette with us in [Music] Oslo

Channel: The Abel Prize

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Length: 29min 28sec (1768 seconds)

Published: Wed Mar 20 2024