The work of Terence Tao
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The work of Terence Tao

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33

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CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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2006

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English

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Abstract 
Laudatio on the occasion of the Fields medal award to Terence Tao.

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00:00
With ACGIH alone for the fields man stand down by Charles fear mom and the university UAZ however
00:28
some the work Terrence Tal now it's a great pleasure and the daunting task to prevent the work of Terrence Tao dietary worked on such an incredible variety of problems that I think no 1 in the world except Terry himself has really competent to do
00:47
the presentation and furthermore I have 20 minutes and I feel the imperative to snow no 1 so I have the following idea as have received I'm going to give you the few
01:03
selected being yet and and we what many may consider to be some of the main contributions but I wanted I I want to show you enough to produce all the following reactions OK so method its highest level could make us think what amazing technical power look on my works he might be in despair that actually I think is the cheapest form of mathematics at its highest level but then 1 can think what grand synthesis 1 can think this is obvious how can anyone not have seen this before and perhaps the rarest of all the mathematics from or we're just come from and my goal in this talk is used to produce enough Terry's work so that we will feel all of these reaction now body by snow and no 1 I know perhaps show you a little bit less of the what amazing technical but I think you can take my word for it that many points already curative 3 topics that that I picked up in order to resent Terry's work these these are 3 that that I
02:19
felt together produce the reaction
02:21
that became legal problems nonlinear Schroedinger equations and arithmetic aggressions of let's start with the kick a needle problem it's a problem very dear to my heart the classic version which I think goes back to 1917 deals with regions of the plane imagining ever regional plane and Nederland 1 inside the region and imagine you could turn the needle through a full 360 degrees keeping it entire within sight at all times and how small and can we take and despite initial conjectures the fact is that the area could be taken arbitrarily small that I was a surprise all right now the modern version of the problem is to look at a set
03:08
in higher dimensions around and assume that contains a line segment of length 1 any prescribed direction but of course went to pick the direction you will have to make a clever choice of the transit given such as set called the massacre which said what can we say about the fractal dimension of another many definitions of the fractal dimension and I'm going to concentrate here on the simplest which is the main cost dimension simply you take the set and you covered by bowls of a fixed small radius Delta and US which power Delta it takes to cover by small balls and if it takes Delta to the miners later Our will spend roughly then you say but the cost dimension is better OK so already know enough about that now lead the decay of problems is an important problem and I don't have time to explain why could take my word for it it's related to 40 analysis PTT competent and other things it's essential interesting problem but it's very hard now what is your look like regarding the problem well what it says that indeed massacre which set In and must have been cost even mentioned at least 8 other they then is obviously a function of and it's also a function of time depending on what 1 From tried the best data then it's plausible to guess the data adventure and all right now from the 19 seventies and eighties it was known that the best fractal dimension of 1 of the best which said dimension to West To that's a result Davies and it's With but it's related to the early work event on your part let's see for higher dimensions the classic result is due to Drury dimension is at least and plus 1 over 2 and that's that's related to the work of fish a dog chair and fancy or and that's the way things stood once upon a time but then in the 9th came 2 breakthroughs won by started there and then continued by Tom Wolfe and another again by now a problem know it was related to geometry and then To comminatory in terms of geometry vesicle which sets of small fractal dimensions if they exist have geometric structure which 1 can analyze they they contain bouquets and hair brushes and God knows what they're sticky never might be a big a problem as their realized is related to work done well 2 to work done by Towers related to the the early work a lot and similar ready and this is on on the way to go a Dalrose proof of of December ready and I want to emphasize the fact that the subject Group D and forbidden if you want to work on long list but subject better work very hard and very smart and be prepared to suffer all right now here in particular is the connection with the Torah circuits in the following but fact about a billion groups see on doing for time spent 5 minutes OK so imagine that we have to subsets of the billion group and we suppose that both of them have a pointer I would use pointer glitzy but I noticed that both of these assets 18 have and have at most and points there for you add them together if you if you take all possible sons you may expect to get as many as in square but perhaps a special structure so that there may be far fewer will be met we have a set of Paris G a set of Paris in cross and if we look at some of the some of AT&T's where the Paris belonged to achieve this is this is also with most of that the day were far far fewer some than you might expect if this is true then the theory says that there were also a few were differences that there might be again there might be as many as and square differences because AT&T had and points but in fact the number is bounded by a power of and strictly less than 2 you've gained 113 so the point is to do better than the trivial and square this company Torre fact immediately leads to information about Mexico which sets because simply sliced the best which set fire to a typical parallel Piper planes and AT&T a roughly the intersection classical which set with each of the 2 Piper planes and she is the set of all belong to the best of all right so
08:20
that was a hard and forbidding and now in the 19th 1990 we have in place of the deep theorem we have a little Lemmer by next cats and Terrence shall we have the same assumptions of the during blood instead of gaining power and the 113 over the trivial estimate we a power of and the 1 and relating that and and a slightly a slightly fancier version of a little landmark what 1 gets but 1 gets in quality for the fractal dimension of classical which set which at the time was the best known for it that that was the sharpest result the point that I want to make is that these are the little let's go back the little landmark the little is strictly better than the D and yet the prove fruitful avoids some difficulties he could read proof in an hour it consists of I would say to pieces and each piece has the flavor the my solution With Olympiad problem where owners did that come from it couldn't be simpler and yet after you read this it's not at all clear how how the hell they thought of all right will progress continues and in particular I want mention the till frostbite abatement have Isabel lover and and how and discuss the subject still forbidding it if you want to work in it you have to be prepared to suffer intensely and unfortunately it seems the solution but nevertheless we have an example of mathematics from Marseille and so all over like so I don't know how Terry person feels about this particular result among many it's it's among my favorites all right now now let's look at the next subject he wanna talk back interaction with interaction more of its estimates come from TV by presumably for interaction here's the eye team came from now this is it this is 1 ingredient in in a program to understand certain nonlinearity and understand very deep staff I'm going to suppress the very very difficult technical problems and conceptual problems and I'm going to and simply going to focus on exactly the the interaction more of its estimate itself and what that might hope to do for you OK so let's start with the nonlinear Schroedinger equation there it is he is a parameter plus or minus is an important parameter but once you fix a sign plus or minus and and the power then we have a particular PDU with an initial conditions you was a complex valued function of of
11:22
space so North 3 years in time now behavior of the equation is very crucially influenced by the size of the minus sign causes it to be focusing and at least for certain he would expect better singularity will form and the problem the problems don't understand for instance what singularity looks like on the other hand if if there was a plus sign then the equation is D focusing and and then that means that and on a nanny finite any bounded region of space you grows so small that some users out to infinity and therefore a power a key factor if powers greater than the 1st power will be negligible small and so we we expect the nonlinear term will become less and less important and you will resemble the solution for free shredding so that scattering OK so that's what expect now how can you actually proof that any of this actually happens it's all very nice to say that you expect this that how does it actually happened now before we get to that but just point out too obvious conserved quantities field to form which is the mask and this a combination of the radio interview square and an integral of the the power that the energy notice that when he is fine if that's precisely critical in the office of what the quality controls the potential energy in terms of kinetic energy the fact that the equation is D focusing means that the energy signed here can't see here the media Connecticut and potential terms are connected by plus sign rather than a minus signs of they're both under control thanks to the energy of right so here's how we start prove here's how we start to prove that it in the D. focusing case the solution spread out this is this is more of its estimate of beautiful simple wonderful idea of what happened more of its originally introduced by her a different equation but in the context of this equation take focusing nonlinear shredding and let's take people's 3 to start and here you can see up on the screen what more of its estimates and the point is that the right hand side is controlled what the mess and the energy whereas the lefthand side will blow up if let's say you continues to be about as big as 1 throughout the unit all in space all time of OK so the more of its estimate as it stands rules out the idea that you should stay about as big as 1 Wrexham unit ball for all time that's good that's that starts to tell us that equation focuses but unfortunately it doesn't rule out the idea that the solution should remain about its biggest 1 in a unit centered at a moving point is has infinite propagation speed and therefore you don't really know where the moving point might be enter and so that the weakness of more its estimate now would used up 10 minutes or right now and I'm going to accelerate so if we take the prove more of its estimate then it's based on on looking at the right point idiots and 0 and you check founded by by the right hand side more of its estimate that sort of trivial and then you do a computation which plug in the nonlinear Schroedinger equation and you discover the lefthand side of a more of its estimate before its integrated in time of plus some positive stuff and so therefore the more of its estimate follows trivially from that now this is this is all very nice except that the quantity and 0 well the quantity and 0 gives a special role to the origin and maybe the origin is where your lips maybe you lose far away from the problem and and so after you lives far away from the origin so maybe and 0 is irrelevant and so the solution is just so simple and natural this isn't an example like claim of why didn't people before it's completely obvious the obvious to me that last week we moved we move the quantity and zeros where he lives so if you look at the middle line on the screen you have in place in 0 you have em wide open with the an arbitrary point white playing the role of the origin and that's the analog and 0 with the origin translated wife and then the quantity that start with in trying to derive more of its estimate is away never to those of white where the weight is you square so wherever he was it wherever you live that's where you put the quantity and what could be more sensible and and natural back and so do you obtain the interaction more of its estimate which which you see there before you the left side now no longer picks out the origin as a special point and the right hand side is still controlled by the massive energy and that's all it takes to rule out you think of the order of 1 in 9 in a moving ball now half from this to results on PD is very hard but nevertheless this is at the heart both but the results of the item on so but see what does 1 do with this estimate very quickly for people's 3 D look at the young look at the
17:07
interaction of its estimate and notice that taking all seem to be 1 4th power that means that you need 1 half a derivative where's the energy involves a whole derivative so therefore you don't need a finite energy in order to get some control over the equation it's enough to assume that doing of space short of seed of the energy space for people's 3 now that a big big long hard story that's that's the eve the 1st species of mathematics of firstrate mathematics My God it's hard and suppressing here ideas except the interaction its estimates but in principle of that's an application but I want come now To winter D focusing NLS people's 5 remember that's the critical so that's a particularly important and part equation because it is critical for the energy short time solutions we are easily seen do exist for finite energy global solutions exist for small energy I remind you that the energy what's there and the challenges to prove global existence for large finite energy so notices this is a tremendously difficult thing I'm going to pick up only 1 small I'm sorry 1 small but very important aspect small crucial aspect of the challenge which is which is will grant greater false succeeded in improving improving the well pose less of this equation in the radio case and that is a big part in forbidden by promise but the general case is yet much harder and 1 way you could see that is that in the rate singularities formerly of the origin were jammed in the general case singularity inform anywhere so improved well that's not exactly a statement theorem time and you're my other time when a few more minutes I'm so all right so here work for finite energy initial data we the quintet at D focusing analyst has global solutions provided that sort of space energy is finite and furthermore there exist solutions of the Richard Hill equation which which mimic which which reflect very well the behaviorist find right and so that we use the interaction they use the interaction more of its estimates was cut off that make it much harder many ideas from particular something called induction on energy and a whole lot of ideas that I described here right let's let's say something about arithmetical Russian Prime very quickly Lindsay 1st of all but so perhaps obsolete but you get a sound bite from each from the man himself tomorrow but here goes a little sound like very quickly given let's take cables of 173 you confined aromatic matter regression fact the arbitrarily mental arithmetic progression of like 173 among the prime and in fact if you look at the crimes of tech and for sufficiently large and there are roughly as many such a medic progression as they to be so here's here's the fear of OK now how long do you prove that well To begin with starts with the Great Pyramid incinerator which says that if you have let's say numbers from from 110 m and end a sufficiently enormous you pick up a subset of say 1 1 million of the numbers from 1 and then they will contain a term arithmetical regression and in fact about as many as you think ought to be so that similar here's the variant of theorem for function immediately equivalent to the diversion for sets notice of the reform of the theorem I don't have time to say more but we have a function between 0 and 1 it's defined on the numbers from 1 3 and 4 huge and yet it averages bounded below by let's say 1 1 million and then we can we can control from below an average of a product translated function as you see paint so that's that's function now there are 3 I would say completely different proof of original by Samir ready which on the sound level I would call company probe by 1st Furstenberg which on sound bite loverlike forward product theory and approved by the hour switch on the sound level of called nonlinear 48 analysis now I say completely different to what the 2 were we know how these theories NO Our road not of different groups are not so different at all but for the rest of us that comes as interesting news or right that the work of green synthesizers all the previous work but I'm ready theory in the sense that it holds firms emirate your itself and it uses ideas from the probe both 1st and dollars and my now officially out of that kind of alright alright so idea and the idea of a sound bite level is that we look at a function which is essentially what the effects of of prime and 0 otherwise now by the prime number theorem that has averaged comparable to what I multiplied by a half so the average would be half of cake and so if we could
22:49
apply December ready fearing that it would save the average translated dysfunctional will buy some constant and that would give the number of arithmetical regression but was ever in theory doesn't apply because this function half is not founded on that's the whole point of the day the primes don't have positive density otherwise there would be a lot easier but we know how proven extensions hearing in which the hypothesis that have foundered is replaced by the idea that enough is less than some functions acts some special properties this is called a pseudorandom measured by town and so curator defining conditions for pseudorandom measures the average was 1 and their upper bounds of certain averages products but never mind what OK and of the green here is the green version of similarities theorem which you will not have time to read but this is just like them ready theorem except that instead of bounded found its less than equal to a pseudorandom measure at 1st discussed no time to discuss the troops we we take the functional with it up into 2 pieces in easy piece of the hard things with then what at the average of the products Japan that's a sum of a whole lot of terms in which every time you see a nephew place it either by the easy part by the hard part if you see any easy part then thanks to ideas that go back to Dallas from the reduced peerreviewed control that and that's not somehow that that's not a problem of this leaves us with a single major namely the case would you take the product of all the hard work and it's what you do is that you partition numbers from 1 to end In subset and replace the heart function by average over each of the sets the petition but you prove that that doesn't make any difference a significant difference in the final answer and yet when you've done this averaging about the new year averaged out functions steered out function satisfied officers of the classics and ready fear is that alright approving of those properties of the Of the hard part function fall from the ideas that go back to to 1st of from December ready Here and now green Tower simmering during followed once but applying classic ready and that's all there is to it OK now have the supply
25:17
to the crimes all you have to to do is find attacks which bounced function that you want to get a pseudorandom measure and as such a work comes from the work such a new consumer work gold Goldston yield room for using not so hot analytic number theory let me emphasize that not so hard is very different from not so good it's wonderful and very clever just not so far curious that the analytic number theory that you have to is not so hard all right guests find yes so that that's all I have time for infected yes I went over my my time apologized there is much much more firstrate work is a program to study ways maps by harmonic maps a solution the saturation representation theory joint hits and misses algebra it's very very unusual for an analyst 2 were solved conjecture In algebra OK and finally what next year's somehow I want well that's as it all what
26:22
next how happen that what could possibly follow what you've already done that we look forward to it with the great anticipation thank you