Add to Watchlist
The dichotomy between structure and randomness, arithmetic progressions, and the primes
51 views
Citation of segment
Embed Code
Formal Metadata
Title  The dichotomy between structure and randomness, arithmetic progressions, and the primes 
Title of Series  International Congress of Mathematicians, Madrid 2006 
Number of Parts  33 
Author 
Tao, Terence

License 
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. 
DOI  10.5446/15977 
Publisher  Instituto de Ciencias Matemáticas (ICMAT) 
Release Date  2006 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (lowcomplexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédis theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 
Keywords 
Szemerédis theorem ergodic theory graph theory hypergraph theory arithmetic combinatorics arithmetic progressions prime numbers 
Series
Annotations
Transcript
00:00
I they regard start now It's really a very great honor for me to be introducing by Terrence and stuff all of those working there Congress although it had quite a bit about said Terry Yesterday I still find myself but rather a lot But try and that quick
00:35
As I can so as not to steal time From the itself charges I think not so much a very great mathematician is a matter of phenomena may break extraordinary PARIS them His born in 1975 16 years later found graduating from lenders University of South Australia and That was 1 year After he published his 1st book which follows A book about Lim Problems in Australia not solves aluminum I'd like to know that has just been raised by Oxford University Press and if you hurry you might never get a copy here this Congress I 1st met him at my age gets in France and was introduced and that maybe we could talk man Some points that I can make it Became visited me in my office and I mentioned that was He said it was an anchor care type problems as well as paper program of hybrid I understand quite tough it proceeded to give me the most incredible exposition which explained what all the key ideas The underlying idea is really quite simple will intuition behind right there Um So I realized I was dealing a rather extraordinary person that stage she was 22 23 a it still extremely young he's not published whatever a hundred pages and the use of parties there is a lot of them extremely along I'm tired it Harry also finds time publish more books that these 2 more than 1 combined the CUP stolen 1 here than the pipeline will come out of this Maybe it was as well and to close what I want just illustrate Uh extraordinary and modern neighbors but their statistic He's got an incredible went ahead and I did experiment with Google might touch the world Terrible just terrible to see where he would appear And you perhaps more extraordinary then typed in time Pound plus perhaps you'd like to refine rules which refined my search and Terrence comes 11 good considering competition around right Watson enough about Because got to work on talk budget things about the told voters have not won in the is Time The new fields meant that it was also ready down the Internet services Lot as a special deals medalist lecture for house Another 1 more thing to say you here The other thing is that he's changed his title like now hold the dichotomy between structure When Rams toe Thank you remote liquid warming urged members of the region I did not talk
04:07
Quite a lot but it's actually a host of along ethnic Russians the prime you know there's almost no the section title would be The violent talk about the What's truth Well I I want focused more on the theme that That is in the economy structure and because This seems a systemic in both the register with Michael on this In actually Has come up in many other areas Analysis In fact was in here worked in the economy and different people call the different things but I think it's something that needs to be popular with more so this book talking about OK Firms So soap His Very general which so many your bags at once Now talk about the the crimes that he and on sales many of his Basic problem the comes up will look in analysis He's like which is that he was studied Do something will object class and it is given an arbitrary object and you have to say something about But the bomb that it lives in very high dimensional space maybe intervention space and that makes it and have structure so examples that yet settled As a subset of and points Got embassies a functional and values may be the system and degrees of freedom and head I don't think the that this is a twominute committees objects And most of them don't have any use for structure 5 Athletics 4 because of nationality you It Such a process in search Pete comes up because things at his compactness that haven't yet found don't was converge I don't in subsequent A controlling interest more and more water like big 1 nor was it doesn't buy another no end Objects of because have had no but are not look complexity OK but In all these situations This Amazing moment that gets around it Even big hide states where they live He Goes on That The space of opt also interesting objects of space Is actually often a lot more They often brutal dimensional Compaq all of a sudden the tactics classified and this will in Pete becomes a especially you have all she will methods vary Good at taking generic objects and smoothing them out and replaced with a much more structured objects that would keep open but the baby here because of my analysis and was the most dramatic example of how now The Richie flow method takes about 3 metric Performs through extremely structure this concentration back Beyond that's another technique that allows you to get around the fact that Bonuses compact in the dimensions but if you choose like apology to choose the right Backers of expects subsequent antics extract out of by keeping be important part Of course solution and the rubbish and then all these wonderful structure income which works which taken out before we got up to And and and and and give structure of this they submitted letter of wickets drop structure But the structure of the games that I persistent structure And Will this added structure The year subsets added group structure and last but not least because the signal processing The vector applied mathematics So they know what ages you have a big noisy signal region which is to take steps huge Stacy compressive cleared up place But something much Lord complexity but still does the same thing So So what will be common half white and you take a big dimensions of space summer compressors Motivational 1 so I think all this the really coming from the economy became 1 of structure that aside color convention slide in green structured and includes amended Still a among all of Jewish being objects They want studies on 1 hand it's obvious below structure algebraic Richard objects or structured means is it's hot Quite sure that His some examples A list of all functions the periodic was smooth lowfrequency of all grass will complexity of classic described status We so be that exists Compaq orbits Functions Contact PDD analog Ways sold and then The other extreme Updates Modest structure of public structure have they behave abandoned And objects so take functions but the periodic predictable remanded very high frequency for a raft of across the essence of cigarette rather than the system is not makes And of Punches turning themselves become a both himself and beat you get there Radiation waste that The case Rather than that they could hear so it defines previously hot that you not definitions of what associations offices Yet idea What did What I mean got this sort of A sidebyside comparison of so what Nomi's would have been precisely that The report was on greenhouse Restructured mobile would include surrender At the expense the lost 2 rows of not only do have the contrast between these 2 concepts techniques used To understand other different so structured objects ones which you want algebraic methods exact qualities identities used and what the Duma to capture in contrast with abandon you want to use
11:57
Was a probability or analysis estimates the identity of the was sacked that out of this debate So what what why is this Economy ways to to notions useful Well It stems from this 1 respect that All these to comport extreme cases of objects structured of random objects demand and not usually doesn't do anything if you're doing Doing some sort of average Random incentive and evidence or and so on Ladies of statistics averaging in a goalless coalition that the optic they ran off nomadic led and the sort of waste The source of this exhibited in ergodic theory stockholders At the moment but I but general affinity your system will functions which makes things then we becomes 0 0 constant very simple lots of things impede his admission that modified equation By solution by adding noise noise but not actor concentrating at any given point if some space undergoes more something then It also doesn't do very much equation income choice accounting limits if you haven't yet The craft make little rendered graft All above the big aircraft and the statistics calmly triangles on any of that Things in that draft that will be basically proportional subset of of the office to see big got and so they can learn about neck It is we can offer The White House and this result was a very very easy to prove Regional morning Alice's method and back all of these things you can just get by a cash Schwartz quality not so much the which they more of a skill might have fairly simple results so because of this you like to know given any object is a grand structured because because we know we can go in and scientists and just focus on structure and In general use of child is typical conjecture that have Natural objects and the small be structure The After what you don't see any more structure then people start conjecture surrender So this comes up with some conjecture comes up all over the place but it usually incredibly hard to prove these considered projector to acute established something to 2 Mystic is randomly is thing is that we don't have soldier and just to examples So we have rejected that we had no idea the triumphs are not render because some of the almost or The structure of the they behave on and we want to more be modified but after local obstructions we expect after After all that to be taking cab no more structure became and the ways to make a mail Conjecture formal bidding added But it too was conjecture The number theory officers and we can forgive for PDD very highly nonlinear PDE we expect that even if you have to tell Somehow the solution should should become very random time except for the fact that it has told a conservation laws that the is obvious Structures that Should diffuse although all configurations which consistent with so that the public just Mechanics rivers Citizens completely a very difficult problem even more difficult as negative Oregon article which has this feature that that you that you must somehow it's exploit of bad diffusion process but we don't know how to actually make that work and then Complexity theory that you want somehow tree evidence that still that But still behave ran and the risk is to expand the grass support As an example and presumably also The because PPP PPP you have to somehow quite understand this is very well and possibly hazard for that when we put 2 elected by supporters of the the may possibly be the people like that that much most of what I do for us Anyway So the fantastic Oracle method would take any object tried and tested as is a random structure that would really be be fantastic We can't do that usually on that It's amazing waiting to buy for much weaker principles that always true and always known when they do they become very very useful just So much for Prince was here So again I'm very general specialized number theory support enough for sale The 1st thing is that What we can't told what these structured as a random I can tell you that It did not Where they of a structure He has structured you you yet coalition component which is structured so Hold on those they did you are not structured system randomness incited still example Here you distribution process function regular distributed if often blame distribution you wish Large Oracle some quick officials meanwhile moves some some generalizations published by austere distribution and was The sort of office tied Benwood what is talk green and & Sons of be the work of often Dallas the Mideast is also based on the idea of a much more generalized similarly In got a of a systems of justice is refusing to make function which is not Madam The kind both of shifted it has to be because the presence of nagging function which were structured object to things go on and she just solution is not Reading dispersant 0 it's because you that bounced trapping all Maybe that orders some coffee some some coefficient some basic away with lot so potatoes peas the will look like this
19:34
And they they use different muscles of for it at the prospect of A similar although For Become very The In any case you can actually use hostel for him us to do something bad The Center for Public Health It was haven't economy he can then innovative and then you get some really useful to get a structure that are often any object and A generic object structure for random become Into a structured and amended And that's really useful because Because Structured optics Amenable to what type of analysis attack using algebra and dormitories and bad things you can attack using public analysis and sold Structure New School for techniques to You understand General Which we would have been accessible subsidy And this This is really Quito along a lot of things for example to the price so The Haganah many examples of such structure of discuss 3 classes But the most movie 1 is especially that That is operator The 6 of a joint over space Then they can can decompose especially in the discrete spectrum which is almost periodic reiterated the view the old South of Macon Compaq and and mixing up which is continuous or continuous But of to the decomposition of the repair decomposition analysis decompose any Any object to a smooth part which is sort of course scales and the hike by What is the position of things and then there was the famous isn't ready regularity photograph of Baker complex You can't work It decomposes The funny thing is you have a the graphs function as you 1 and splitters thing Decomposition But I got this is really of decomposing into a local practitioners of the summed this way Between classes and then you have some some regular grass that between classes And the petition drive structure and the and the having assists from random songs statistics but so these out The restructured often economy And get on and so we have a job functions It's structured done not That Give it to random done it falls through and that so structured You take out And because smaller And now you look again As This To remained a and is done otherwise Research about Rating And then you you you you get reading PDD may episodes of held what equation in order to get the car decomposition Agassi argue that the charges of possession projectors Monday's out Alto version structure that OK and the very useful because what position and universe you end things going port your new book a structure that what you such a part of the 2 other principles Sunnis were too so I don't want to have structure but as a bitter structure As often amazing anomaly Brewster a lotta structure so Close rigidity of the good name for this summer It is is is almost A completely structure and He can repair Clean it up And he make a quick structure of this If somebody could give us almost wasn't edition of says Wait Examples your your your back and said they want the time so you want and that is the closest thing you can actually get That means that you actually closed much much more I should be closed And then you go initially by a group or a convex set A progression of or some sort combination of all of these things so the famous in the refinement the extent and the flood of transition impede year because my thing is have a smile had a spell on that but if you can't minimize Version of the my energy and you have And you getting close was limited then You know it was is a compact Have A sequence of Almost minimizes of it and get close and closer to actually summary action analyzes as well and that is This is the way you often constructed minimize the problem of function If extract compact and effective as have mentioned by was cut yesterday so You feel for have to Defined the compact properly in order to get a personal but when you do you can't have that men in and besides So distant from the that Get a humongous graft function of the Internet And you want to verify property owners but you can't you do what is just the whole Graf But just Random peace and and revise the property But you you want them locally that you is focus of Almost The sort of folks satisfied then you can often That is not always has served Modified A little bit adults Thaksin edges The naked satisfied and this is how you probably has so these reasons often observers also are actually quite deep usually are yet to use structure feels a lot of trouble A lot of poppy testing Oates used a object and the epoxy structure it has restructured Don't have restructured You can cost So want it has restructured often described updates I'm very explicit I just 140 of algebraic Geometric fashion you can study holes of all Hodgman Jarmusch Some very simple examples But generated in Group begin the basis and now you your last year but was costefficient sold While the interest emissions she faces major he diagonalize like former can also take it and I still more On Was a simple example and more advanced ones PD Most famous ones Assault on telephones can be described in many many ways But some very nice but regulatory methods using to give only because of the loss of really really pretty and very structured formalist exactly described and then
27:12
As an example which I think a brighter crown talk about Today Softening An ergodic theory Setting current year because restructure component of All of any donor assistance is force school cactus actor and those because I had no by the new systems which have which arise from group's Newport new groups on dimensional would be groups so popular that so that those are often also were difficult and they're not The other thing and was often great analytic this is 1 of a much larger break But it's proposal bravery general things that's actually a concrete example So For example which Illustrates A lot of these principles wants most of the year The Chris Ford essentially sample it's amazing Watson ready so State We take any subset of images of positive density so you take a 1 per cent of its didn't find but Well it's easy to find that visited me said Dick Sydenham's Munson and for some large you look at the The proportion of The numbers in a it's if this was from below The offer so yet to take what 1 person and 18 % no matter what you do you you Inevitably We are forced to have a Which contains arbitrary Progression amazing it's really quite an amazing is of you get to choose the set him amid given any pattern of example if you want to have to eliminate old city pairs of adjacent numbers And what very easy to choose a set Which avoids What is known as even numbers But you cannot As long as large as the city only what about A lot of no matter how the move around you cannot you cannot destroy Progression he cannot be removed completely West and that's that's really quite an amazing not And this Is buried at 1 1 1 example of 1 of its speeches by seeing how prove that the people proved a selfless on hundreds of ready for 5 contour method 1st proved it again using it and got agreement is very influential in public and Dallas Use of Korea and that there having been accused sent by spoken also another walked out Support on so The most famous is hot add a short list is because of yesterday's arbitrator on to be very structured a structure say few removed veranda And actually more Mojo is a combination of both and structures as progression Advances the progression progress have progress different reasons and so that you cannot Hope were once by You must Set into the chief wants 1st before you can do and before you could so what structured me here Structure is not easy to define and that actually hottest proving what Ikeda depends approved emergency crews and each of It is easy in 1 area and difficult enough And the was around a what it would mean that we took all those 3 proves that he was a 4th for and easy each 1 easy declared passion together anyway so What a structure needs a good example structure pure except the more was 100 Set against you want said More generally was actually So when I was in such that Ritu times and is very close Image that's a set which A person with what period Who was shot and the more interesting examples The quadratic except when it ends That seems to have been square wantonly upon a rational official disclosed of these assets Describe algebraically And However The thing that Is that if you know a lot of progression said then you can often tell that the next government and that is what we sampled you take a provisional lead and the 1st to more and you know that the of a mobile and the same type of thing is almost truthful whether credits at the 1st 2 numbers The systems the progression said infected by half the time The about what you the quadratic about Set You take a progression of war and the 1st to be The The 1st step and we were a lot 8 before a lot and this is what is this because what the fuck quadratic formula Jim describe 3 points in the interplay OK so Because of that coalition that never a few elements of a few Points in your The progression get the rest prefer free that that's really cause of white pressure purity It could be a long at pressure because of the coalition and the college course our brick structure such as fuel and diesel and generalized Computer City and Below what captured portly groups which I not talk about a thing Brinawa talk about it this afternoon the bus structure on on the other extreme And so What And so the basic example is action man as you take each number And you will be decided by Would start But he died but each John each number and becomes 1 foot in cent Independent Men upset more generally Take says mistake behave and because certain college coalitions more example figures said over shifts Can you If you ran expect density of the intersection to be after the product of densities Although some this is there But still not earlier esophagus misses a twopoint coalition often need point coalition and drop but That's the type of thing You want and you have enough to and then you can start Then it's all man enough that can use public and basically The mystery elements All of said Progression The next guy Be event The next that was worth it Independent of all The events The abuse but and The polio But a long progression in your this should naively because the product
34:49
All the people in the region but few Poseidon seal of the quality of so So why we expect of the provisions in the manner that is A little tonight but that But what does what you expect so so in this case A version of course No by structure by the absence of structure by independents coalition Did that to scholarship completely different course progression and then And then we have hybrids commission vote So This is where most sets for the continued a completely random structure that they happen to be Bits of So that we can take to lend structured so We think even numbers structure that Neurotic And wouldn't even numbers to random thoughts of about 2 % The public even numbers at random will the 50 artifacts And then you just 2 random sample even numbers That new demand more structured but above More generally take a cigarette The model of a single circuit of the petition Additionally images destruction pieces and each piece choose band so you might take place 3 % even numbers What is in the odd numbers random And the things that structured progressions and you know that Do I said What's structures progression Utica random subset of structure Matters will continue proportional number Progression to the extent that more general that's The sum which ability problem earlier And so what's structures progressions than anything any random will also have rich as well and so that this because these hybrids progression so in this case the progressions come from a combination of outbreak structure Which which is Initial Canada progression and discoloration Which was refined to Still keep some questions and then the key thing that makes Things work is a structure and depending on the proof that use somebody's professor but we've got structure is different but it's always present And basically The said Think it big dent says that you can always find a large which Which Assad fully surrender but its suitor and set up a structure so example in the because all the hours The a statement a big dance that confined the progression of structures inside except where civil on and what have I done Because Because of what I'm saying is that every that contains high and hybrids as a progression why you accept pressure and be so structured follow phone Economy and because to do so Like this But said It's not doesn't being like some coalition Big Bend The reason is because it college in construction that means Depends proof of the things it some some progression of your cost basis When density This is bigger than normal the other bird doesn't looking for militias of decorum it as always The economy underlying the buckets of tempered versions of these that each proof digs at least call out to talk about what he projects not talk about my own thing green which is a talk about what Dead subset images but Much much faster said But the specific the price of the conjectures about what To aggression Beautiful conjecture according to my conjecture says that A set of Take positive images some of reciprocal diverges then
39:12
The new contained Truong progression so that's much stronger race and it would also like to hear We have no idea of proof that it said At he felt again and another for now what do we have the results for the Anyway But we can use so It is that before And we just result Is not surprising based on the conjectures we have on price we we do believe that the price being randomly after what except for local structure that the affect most or all of the above obvious structures created you expect are randomly members of the size of the country was rejected This The conjecture example would that Leads us to believe private true quite good but it is true as part of its why this is the biggest dick At you Underlies quite a few of the year Police In most remote price so that it and if you ran after should be between aggression but any difficulty but The things we can't verify that we don't know where we still don't know Rented friends we do know 1 thing what are the ones was ousted is a useful we do not I paid from Iast which is a weaker notions Demand essentially the group of prime Except when they obviously small And they became obvious enough smoke coming from local structure Things Some of the wall and the work optional 1 half assembly where huge so the reserve major arts with things of things a bit Said But It was not present in my Knox than we do have sit around and this this is basically a sound and but got off And this is enough do to get some of this conjecture You can get The 1st of the 3 in the price from back and this is done by the corporate 927 But unfortunately it's now well understood that stories is actually Fans Well it it it is for this is a whole haricots around and for years he wrote the list and It's good enough to drop
41:40
3 that even for prisoner of war
41:43
It's not enough examples of event said Which do not have the right number progression of quadratic critics mission example so this is not enough
41:56
Well The increased for Friday or something to talk about How you can boost this To pile pressure from the circle but that does extend but it's not trivial OK so the farms Densities you as for the further funerals of prime But Density and the number so you can't him directly What will they would exploit was that you know the prime spots They Things in a different sense that they don't respect to another set of almost So this Well Almost Kansas had defied the world The way we have to use it as an IQ set of a weight function a measure that But you should think of Almost crimes Numbers virtue prime numbers only state 10 practice at most and those numbers Who was known for time doubt that those guys count because And that they do do behave like we expect The price But they are still after accounted for The obvious reasons but not right and that is Thanks a lot The aimed toward making best starts basically young The primes to meet Extracted the parts of the and expected But since such as the Super tossing that Was accused practice because of example Europe and private Story to lots of fired up to root and then after that what's left of the pride of the runway doing to me steps And It's easy to see what happens meat or what the tumult of but want get more And your I would be doing many many done little changes many many of those who remain a mail small sets Big to greater private party control But if its officers earlier they had been told I'm not screwed event and the 100 than you can still see control on what you have left I will not let me do tricks Essentially can stop What the earlier and still understand exactly what what users but then you don't get plans and we almost saw surely crimes which a cofounder in than and 1 of 100 and have at most 1 wunderkind backed up anyway So the crimes are not case over the last dance In the images Yet The they against almost Prime Besides of almost crimes and a lot of The old define almost Was the most wanted man factors the private density nothing but want to set up effective in almost OK so the way we use that we managed to prove an extension of the Which of course villages of that So not just any death of images agents of all these 2 grand images so But 2 colleges take a representative subset of this case almost tracts of this sort of your focus group which is the 1st of the visitor But even represent the whole images then They need was transfers any fear of proof images should come out As counterpart For this represent a sample is become a part of my bed Dickerson random civilians The things that you're still have of drew on Russia and This is why we get progression of practice because the crimes are dense almost and all of us to abandon by the here and Thank you can see we we'll watch is nothing but a price I could even take
46:07
I did Right they wants in the front and I still which so in in some sense From this I was really a big chief of the reason why I want reasons why the result If desired Versions of crimes as a hopeless you lose price by doing so that we don't actually you very much crimes hit a new low of progression Because we Particularly we have this doesn't settle the question of funds around the world and refuses to avoid a question instead It den subset of something of and that's enough to give all we do have A subsidy program actually tried It plans to act Get bending talk about that and time before updates of the latest took a little bit of a Harley could just as relatives so we used we used this image as 1 ingredients and recombined Chewelah facts so still Again with based on charges that Take dead subset of a random said It could its NEW random itself An intractable what action which now its but that it might not be but nevertheless Anything which you take any death It turns out It It's a large bill which which is to rent It's about a mail receipt of what upset of the whole said Pursuant subset of Dent said But it seems Sample logical back because of injuries Dance concerts Our Siemens sources that had sued Budget And what of privacy shortly but does that anyway If you believe that for some of these the primes the plans contained A large subset which which amounts to random embittered this U.S. something debts and then we have a negligible yourself that says that It gives you an upset What would be a big university very fast Postal container portion what subjected to go assumptions that that Basically pursuance of should contain a fraction of the expected fraction of all 1st big said What have you done Those results not because simmering that then said in a lot of aggression and then This negativity without that I need to a subset of Odense Foster progression that I just told you that There But the prime contained The death of the a registered so that these 2 things ought to be ingredients in an argument but But they fall into the framework of discussing discussed earlier Still the naked abilities Not so difficult it's a just like Schwartz On Yet you Carter Schwartz And you are the main difficulty in proving that set of adaptation of that That is basically what Akashi Schwartz All and defining what to do and that the other difficulty Washington Dallas So Watts So OK so they they have The harder part was structure and destruction Of becomes the economy that Take it and some of us Pursuant to the far almost That's still thought through and then it turns out that actually colleagues with New song structures gone but What They the key of that he said Box structures said that you call it a day And this This summer what what what what drives so the funds will not surrender Because You know Collated bias very structures such as a progression 1 1 of the structures systems like appeared said was a critics said those of sort things that our goal is the dance structures set that up walking 0 and and that's really And This is it that dance is key because accusing instructed that the the that would dent in conclusion which is from the vessel the 2 applies in reduced the band said using the sparse Have Almost the truth because the actual details about meant that the So I just to reward But then I working nowadays 2 years ago And so I just mentioned show you that sets pressure of the density of water progression of price pressure that they don't exactly how many they are listed in the remaining but more Quantitatively Did you ask how progression of lens case other amongst the problems up to some level and And river bank What is the asked can be more precise question was The fact that there are few attitudes A lower back but it's not the shot dead shot so the next question is I thought I had shared you really can't finessed anymore more so than in the in previous year We took the prime on babies man and restructure we don't know That He case Progression but depending what history and you'll find a different number progression and getting what exact count of progressed he have to know How many times are found and That sort of Linux focus Can't say that I get an ingenious way to start using the year of the other leaders of the I mentioned relied mostly on the 1st 2 games of the 1st 3 of his negligible ability the economy and the structure of the wheels of his rigidity and costefficient and those Up to give you deeper information As a What more quantitative information So the prototype of this is this Was and hope that that Your account progression 3 . 5 must end The answer is actually quit it over So instead of the normal person week period the presidency wearable again So Square log tube and the 3 us and we want after that but Local The crimes of honor with 1 exception which which increases the number of all the Ferguson We could order because 2 numbers Progression expected orders well And what got lucky we can't that even so you have to make adjustments Will want to obstructions would 3 modified girls You put all those working together this product Which is the number 1 . 3 something I think it will be looking beanstalk and so on A victory in the precise count of 0 how fast it goes depends on removing officers that we know 0 even on unconditional no hypotheses the way people use these hiding Commander and the U.S. didn't start and the hundreds of comedy shows you that if you want to establish a result you yet a corporate crimes The local history what's going on They jocks circle And all you need to do is to show that On my Knox that A partner local you want you want a who is the political crimes More than you can 1 neglect them and then your local trips and local churches of work would a product of the Well The info analysis we destruction of its completely class that Characters exponential your not surrender of years surrender call it with a character But needs of a large work and We need to do is to contemplate between students structures that you have to show the prime don't they don't call a minor heart attack Show that the crimes don't call it was to And 2 guys and there's a merciful across the Chautauqua head You do that End What does that and you get this thing which is right so it's much stronger than significant pressure over 3 harmony So More recently than I would to do the same thing for other crimes which started provision for I hope eventually we will look look said we get Andy Near the primacy in which Which involves Except for those that only 1 parameter They have a go at least 2 and apprenticed averaging Neglect We see parts so it wasn't so bad Really annoying caviar could we could do things are different from conjecture legal but conditional on things But we can't do that we can Can account for the cubes How iPads the crimes of the civil service of dimension of war again so the lovable What the description of this so and this is much more an extension of the hiding the the and a previous so we we we generalize So common that replaced Food aid with a more Hi of the dose uniformity And basically the key point is that You wanna show that crimes After the meeting Local obstructions which you always should be kept out of stronger than for And Not not strong enough to do things account twins In prime Sokol but did not account these motor premiered at the reversal of for the Fabrizio autocratic was replaced Point and step aside to practice on And So The point is that ensure analysis structure of its characters The highwater case structure of its more complicated than they have recently declassified at least For the year And a getaway for progress in the war Former 1 groups have what no sequences coming from a cautionary importantly group and that is There is a natural organization all characters flows coming home from work Taurus and and is still want to have classification which is not trivial but we need to do is show that The prime don't Please What did they do call it with them We a job show what what I that but if know sequences don't come from local obstructions expect the tribes Of course example expect all I could do it because critics said much earlier You sector crimes the purported front on the track because of the exception should be the density of access Modules Local structures which led to the about and that can be done by A refined version that although that sort also revealed so that each of these 3 . required to pay his work 3 papers And the social critic of before But But anyway Mike became can now get of progression for And soon hopefully other practices OK The book what war among them think you have