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3/4 Trace functions over finite fields
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Title  3/4 Trace functions over finite fields 
Title of Series  Summer school Analytic Number Theory 
Number of Parts  36 
Author 
Kowalski, Emmanuel

License 
CC Attribution 3.0 Unported: 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  10.5446/16419 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
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00:06
and and you can OK so so again I've written down on the blank monster agreement but it is in the making examples that I refused and possibly I want to begin by saying a bit more about the conductor of trace function longdistance Europe representations so I'm giving some samples the precise definition without going too much into the details and explain why we need something a bit more than to information we already know so suppose you have trace function carry much clear so you don't need to assume that it's geometric irreducible anything like that so it's associated to a representation role the submitted legal was intentional and so the complexity is really an environment of the reservation for the 1st election itself you would just stating the number of all possible always stress function OK so we already know too a components of this so we want to measure the complexity and we sold out I said it would take 1st dimension of the that is then there is this singular said so it's a subset of the 1 of the key God which is a finite subset which is an environment of on you have to take this too and this is not enough so what they're going to do is going to tell you what is the definition and then give an example to show that these 2 components which are easy to understand and which are all the obvious environments are not sufficient to get good qualities of the conduct so you need something else which is way we define it it's a sum of contributions corresponding to every single point of an England go this 1 conductor at X of the where this 1 day the ball is an integer negative all so measuring what's called a wide ramifications at a single point so it is sometimes 0 then 1 speaks of tame from it but most of all it so I'm going to give an example that shows that that if you want to have a statement like the remaining is this issue a statement like this to be true we is environment or complexity environment would not suffice to define just the dimension of the officer said In the most suitable this would not be enough and then are you a few examples but not much more are so this is an environment that's quite so indicates and in many applications if you don't know the conductor and need to estimate collect of this would be the the optical control because it really is sold as environment of the representation which is not so accessible from purely dive fontina concrete properties of interest so why do in advance focus so they to resign and so the first one is that if we want the complexity and be some kind of fight so that so what do we want to visit into the kind of mind means we want sets of all such that by the complexity of always less than X and its civil instruments you reducible modular geometric isomorphism Bloomberg on this finite for every so this is a more precise form of what I described in the 1st lecture when I set aside as this complexity as I property at that time and not said what it means to be dramatic adjustable so this is the right thing to do these are the building blocks of all trees functional representations and then you have to dig module dividend as a model because otherwise much applying idea scalar without value Wong in which ultimately the isomorphic to a given but if you do it this way that actually it's a finite set so this would not be true if adjusting the dimension of the was the size of our the single said that we have so if you set they descent of all where sold always dramatically reduce but the dimensions of the plus set said of the report president acts soon would you some of them to mitigate some of some of about this is not final and that simply because it contains all the power can have associated to trace functions retired nuclear F of X where his going onion volatility go so 1 1 does this type of correspondence it's possible to do it in such a way that it's essentially injected from I feel likes to say the words ambitions so I won't go into details and until the boards is that the dimension of the fall of these will always be Wong independently of em and the single boards will always be restricted to infinity In addition to your soul the subject of polynomial is only really strategy that inflated than when we take the exponential Our every value which is not infinite UBS essential intuitively somewhat clear that this is the only bonded consisting of an infected some that is if it is on constant so before the OK and so we are initiating many bring a mail because of the reasonableness and all of them of sent dimension and sent singleserve sodas said is infinite Buchenwald DI property which so we knew something else that in this case are asked to take into account the degree of so what's missing in this case is a clear we bonded area then we have a finite set going with the Peter Bauer but nevertheless find William valid on the degree of effluent is a factor so long this 1 ,comma during the final rules and so on are we are that the swine ,comma talk infinity of such recalled that many couldn't where with the resources to do this will be the degree of effort at this if the degrees .period and in any case it's always at
08:40
most of the so this 1 from the often and stacked over function will always be at was the order of the pull of the function and the single point to industries oppose it introduces degree and was acquitted Jackal plan with the character so and the 2nd example is In some 10 similar that the issue will also be of the same died at this and it is vital for animals with anabolic degrade our but is more concrete in the sense that you might say that OK decide like policies nite but it's not the essence of what you want to do an number surety so this is much more important however this would fail we as a conductor fine just as the sum of the dimensions and as stroll essentially for the same type of reasons so consider can into your kids sponsorship deals from went toward what can or from K and then look at all the plans ,comma care and find the alleged to be repealed thanks to the park so then please you look at the sum S K P which is the good the gal some built with this nuclear extensive usergenerated position of grudge against next game so it's a dissection of the fact that the aquatic Gulf son is the same as the "quotation mark goes on for the real character that this under these conditions you detect the fact of the case power using characters of the cushion who as his standing with his thought about OK there case at cocktails 1 of them will you treated contribution and the other was will give commanders warned us that this is in the Scottish understanding of In 1 goes so what does thousands I mean seems like you of acts of acts where current will be awarded came in and trivial and and under this condition of course we know there are such characters the case of the killings nontrivial so it's not easy not out to find to deduced that attitude take issue trying to being up for the a constant function Wong Westley was intended to people want 1 then this will be making lines for the normalized its OK on the other hand the Associated dimension of typical of the UK plus the set of single is again and I won't last long for the same reason as before it's an example of previous construction resent power so they mention is 1 and is just 1 surety bonds are so you see that an estimated the 1st part the moment but is this does not hold With C1 and C2 replaced by Joseph dimension 1st most you so I would say that we as CEO of replaced by them instantly stumbles look so it's clear that this is geometrically nontrivial but this is not constant as accessories well maybe it's constant which is equal to the budget and not because it's 1 that the it's 1 of the key for icicles 1 and it's 1 of Mexico 0 so this is not published until constant so it is geometrically nonconstant Trevino not isomorphic to trace function 1 so we can apply the 1st by the overnight but this is not and this was not true which will result yeah was there is also some of which was mentioned that there were cries of oratory would say for some reason I but some songs when orchestrations but they are the simplest example funneled warrant monitors about this because in some sense this is 3 years spontaneity 83 the 1 that's most painful no 1 tries to avoid it as much as possible again if you're working with wellknown for its functions windows you would find in a table In a list of examples then that means that the stables only Gore did include information on this 1 ,comma so this is known so let's go back to the examples from so here for energy character they said but he said he's is a polynomial it's it well please thanks to the polar bear that is why on the .period X's etymology although the Boeing was equally tutors who to please so that's dissection of what I did so we see in the future that the cycle toys locally and in some sense but here only depends on the although the pool of efforts .period at it's snorkeling and more precise technical center is sometimes useful and so here are the singing ideals he was full of effort all this time collectors Ozzie so much of the debt if doctors in this sense of all .period in some cases because the nearly fired everywhere so almost 100 thousand 0 I got to summon sounds so singular said was the 1 who became it's terrified at 0 so when Valladolid 0 is 0 and at infinity it's got McCain yeah that said that he does not provide a degree of F than this 1 ,comma .period every single point what is your so I point out the function also to numerous files and folders on family indices also swatted 0 equals 1 at 1 "quotation mark attention is also looking at this issue is satisfied and this action is related to if not bits of the IGC reverted to cast the exponent of the conduct of a new find work you he's always bounded by at most to accept full Prince peoples to 2 Austrian to see the same type of game so the fact that the conductor was excluded bounded by 2 is also a feature of the mummification and in the case office functions he was into the center so this is going to be difficult for for family of other way varieties the the equations of the final wasn't fixed and cables for infinity the FOP large enough that was in beating if you discount the mobile solutions but so they still want to sever the conductor orchestra today maybe tomorrow will have the meaning 1 of the applications I want to discuss solo then going to do now and the remainder of today and tomorrow always give some kind of a survey of some of the applications trying to highlight some of the features of 1 uses the remaining What is this now it measures are presented numbers so sound applications 1st pass from sewing the 1st that I will
17:15
discuss the results while a series of papers who is fully conscious which is where we started working with these objects where we are trying to I understand that correlations with interlocking data so they know what motivating problem but 118 problems not quite exactly where seems completely started but it's something that I would think of motivation is to understand so did model phone and my own tolerance of stadium Missouri so this is a 1st agreement some were but it could be a mass following falls Wall thinking so we have applications in cases all agreed by the Eisenstein's focus field and applications of which this is mentioned some of them we so it's not so important for us whether it's fully coefficients Reagan values but let's assume that you have a guy conform with a character values life again and 1 so normalized so that demeans queries about 1 to from squared is equivalent to a constant admitting that his x so utilizing normalization so we consider souls thinks of prime game which will then tend to infinity and the trace function came the plane we consider sounds like this OK playing which is smaller sums obtained by comparing the nonlethal Renzo and more I think function frequently chance against this for his part on which we view as a function of the final introduced by believes it and the vertical lines is about which is natural because cases fairly model please so we take us was cut off at end of our businesses on which you can sing being between 1 and 2 but resistance cut off and the estimates asserted that the would a little bit shorter Peter minus some small when and it's extremely large so the larynx was it's a computer opted for something that you were just as being in the city and replaces a sum over a half kilograms of sum of all integers ,comma went away and apply known results on freak efficiency nicely progressions to handle this type of Seoul where the insistence that of which in fact selected but the 1st thing to have in mind is the usual picture which missing in region it's as drawing on his survival there wasn't time I've spoken with him and which is the right way of doing this is to avoid an attitude issues and to remaining debt to and than win so In some ways that and we want to obtain demands so little is you who want their finances these objects are collected in a strong sense there should be some consolation in the south of Seoul the roughly bounded on the ranch it is in square killing we section bounded by the conductor so if the lines about the you can expect in good cases to get hold of the the cancelation and and I we try to get some cancelation it is which will show that the power cannot be my dad Iannuzzi where is this kind face on gold cancelation so this can be seen as a challenge and Hong so we think we know something about Taufik phone received we know something about trees functions
22:08
can we actually show that we know something about them all at their obligations a by discussing the scandal as a child and then will see the applications so a basic example that you can having my final 2 examples such as the 1st only needed to show that it's not a purely academic again so a simple example is much of a at cocktail also killings caravan where carries much being nontrivial it said that there should not turn then because the can cancelation that case level and Caroline is known as the same so that's the right lines because of the conductor of the twist is a square and times and his coupon on 2 and so on the site in that case as it currently controls the convexity for the trees to the function come in level I was addicted meaning that if we get our cancelation and then we will get some kind of convicts estimate for this function which are very important in the Member Syrian applications so
23:35
can 1 of the guiding examples we had so let us with no idea what can come out so I don't mention the example of energy Factors example a character 1 it's very well known but due to his estimate of Wilton which is but the 1st example we cannot use it as trying to see whether we can do it so our challenge was to try and do it with the and fossil 1 when Europe's which all which is stumbling block of his own songs I can be sure that this time changes of the universe will be off session random also should be independent of let's say of the provisions of the Bill of functional commented that start of the basic question which is not easy even for is a special case who was formerly challenging Clinton and then many of us would give another example in this this is where we very decay but they're also quickly there are special cases of automorphic farms where you take our fall Aphrodite centuries so let's think doesn't have to do that .period after the he is a real number than alumni for is the sum of their review the power where a musical so strictly speaking faltered equals 0 we're going to that you because this is and knowing this so faults equals the order for aggression subdivide the function and therefore you get the type of standards were looking out for you wanna say clearly K and is a sign of 2 integers mn that is 1 of the amendments to the main he said this would be equals 0 and in general and and also and above all right OK and these are an interesting because they are they give you special uniforms when you're trying to apply continental
26:09
identities to detect crimes and the power of the presence of the IT and the fact that we're going to get uniform estimates expected to some extent but allows you to but it is the size of men and incident of so it get a special meaning of follows but Granger 19 when the coefficients is a case of a man working interest this will be done application later cancel Allstate the results and then see a few words on the pool and but cure highlighting what what is unique about trace functions in this book for improved debate on the relay but this but it's interesting to generate clearances . com In this instant so there is a supposedly from stunt missing we checked it can be taken to benign but it such that for every point .period for every Kmart .period that's a geometric irreducible although this is not strictly speaking necessary but the way so In the Eisenstein case so supposed equals Europe together you have some of the and timescale and analysts simply essentially so if then case constant you're likely to get cancelations some of the UN isn't installation and these cases adds you can't it also doesn't quite work so are we assume that the case is not an additive character you know of any end in the special case this is not insensitive cylindrical steel cases no condition and in the other senses this is the only exception so when I write is you always have to sink so that their representation said to it is not the conditions the pursuant 2 that all you can sing that case not proportional to 1 which is a weaker condition so if I do this decide so we get cancelation so for every you signed on so I write that the implied constantly bans on that assumption vintage depending upon the explicit way so we can actually play games with me to get some cities all society Showtime doubled name speed but I don't want is
29:00
explicitly so in terms of trade it depends on the conduct of the Premier League and it's fun it's not terrible but I don't remember exactly should sign of such again and then respectively we get cancelations and a cancelation is completely uniformed 1 minus 1 explicit so what is striking 1 of the striking saying that his is the complete uniformity of the exponent and again so we always there 1 less independently of the complexity of gave Michael basic jamming of find West paper so To to say that this is efficient means that the sea of cake actual conduct us to be bothered ongoing reconciliation so that seeking Boeing's isn't he I 1 mates and the treaty along with refugees but finds out if you take any of these examples we see explain on Mills a flesh of functions in here or there on your feet off then you can apply this year and for every 1 of these and explanation will be this and the fact that this game is uniform ultimately just fight that the reliable this is just as uniform so remember this gold of the independently of which of the because of fall well known Of course the socalled legs mom was long and the exponent is the same as the best which is known in that case on is also drew passing 1st because skiing and then would generalize this is comedy like this fund is known for some convicts found even for general and would you like is they both so it's also known as a set for the Jewish character of fortress built case could give candidates will turn analysts Maine the and I guess we can interpret values statements in the literature us extremely special cases maybe was slightly different can you but about bottom that essentially as he was known about being tried the newspaper with Bangladesh can be interpreted as giving of this fall for a transform over much of it the switch is missing OK so this is the 1st no they said because we can handle so he also warned that unseeded dependencies on to you when you take as insincere in but it's under control is spread balance so we can use this estimated Fisons centuries interpreting it as special meaning of false together with extra ingredients coming from the body of put it in the middle and collateral countries and identities like his bronze to mark and all summer long .period so and the existence of such that for goldplated for care for its function would plead again as consultants to dramatically reduce civil where is so in that case for some little times we have to exclude a bit more so it cannot the unmotivated his character at times and it got who cares so in that case only clear why we need that for characters because tonight I'm going to write a new was reminded but is a frequent and government because after so just like you know about us all ideas involving uniforms and so on this breaks down when on the functionalism of fugitive and Pyrenees roughly the only entree selections that she was the UK and the game is not so clear but it also is needed but we can do it sounds "quotation mark so some of the love and care of and the and again we get whizzing by constant depending on its side on and on the misstated the welcomed the plan but a mail abound in terms of its criminal of the conductor and then we're going to get something plans 1 minus 1 24 pleasant sight this 0 and excellent game is completely uniform independently of what the trace function signed it only needs to be addressed this not on Yates we passed from total data so the spectrum defined and remains and series which allows us to control the devises sounds like that we should belong so adding an additional under 5 would be it has a problem which we don't know what to do so there is no dependency at all Malaysian Peterson type approximations this is all Tomas forms and forms uniformly if you look at the list of those who were violent Colorado have yet so we get to just decision by 2 so we get 1 was 16th Floor Foster Long and Foster and to get 1 for him "quotation mark sources say some highlights of the portion of not ideas because it's too .period altogether so focusing on whether the trace functions are used so there's a lot of 900 numbers you're involved with a focus on the trees functions solely on the United side so I begin with you and 1 on United side we use amplification and so reviewable SOF was little and we view it as a curse fall Little Farms of weight gain times that of leveled the dimes and what is the point "quotation mark before assumption and we have to find all the space all forms of little PM forms so this is an idea that as the news also about felt like it was because of because scandal Mark caution also being used by the finance they effectively to ratify son results so in some sense here so it's this idea of putting little peonies is very natural a personally but up here we must think about it that's so that's the nature extent they we 125 so use Cruise's form we could use the business from a nineyear lofty case but we would then use quite a lot in some instances attitude causes a formula also about a killer so even joint and devise a functions the function of the One no special features of its weight coefficients we go to full respectful of the including Castille spectral so so we traded by the function fully as a model of the objects and embody a reuse goes following opening although to get Indiana an estimate of for the devise a function which is sharp than known estimating indicates where we applied so easily for the so I basically this is the future so once you do this so there are many complicated steps but you end up for his part so we are led to some exponential sounds appealing apologize if I just sounds which we call coordination sounds of K was expected and now so that I would be in the mountains in PGA Tour history and is tries to I understand the poor nations between not carried bodies were transformed and the transfer of transformed when
38:19
replaced eggs by so chaotic of acts out of the market in act good gate where so chaotic it is the Fourier Transform which I recall is somewhat worldwide clear of why the people of the export of the management of that is why Mexican if so
38:49
wish if you could exclude the value report of I expect you want but it actually works out fine added since distress function as I said extend naturally attention order and then we're going
39:04
to play of the game so if so care was intimately inadmissible that's my assumption sofware transform was we said last week is also geometric your adjustable but we make a change of viable this is also so chaotic still fortress function and it's also geometric irreducible because the change of islands by executive and chaotic human so we are exactly in the situation where we can apply the remaining With K1 is chaotic and K 2 is scared of them and bond is can we applied but this is a sense of are we going to get result as a result and what we need To win its code cancelation however not always dishes if we destroyed cancelation away would be 1 of his problem is that you can't again Michaels the identity and then doesn't cancelations so we are not ready for is just official that this type of men so that they are not they all for small number of diagonal cases where you have no cancelations but with few exceptions which on closer dialog so then what happens is we balance to facts so wish matrixes Denmark come into the argument United signed we don't get a sum of all matrices Imperial to repeat we get a somewhat since this matrices already complicated shapes but which have long feature that they are not antiwhite construction in some sense they do not from the start so far only 1 name from United arguments to the governor has to control I'm not as a structure and actually from direct families but not several that is blood through cracks works and we can that indicated the modification will not reflect like its stable however From the Riemann iPad users that scientists trying to understand those Dunlap original squad cancelations sold meaning in which case the weekend we not applied but only that so and is a constant large enough depending on the on the conduct of care so that I can control so the conduct of transformed because they said it was necessary to conduct of has controlling terms of the quality of care so if any is that not then the only possibility to not get scored consolation was a constant which is an is that this Commission failed positions near your mistake and so we are engaged and that is being fed means that he wonders found to be sold it demand such justified the the correlations son the market is larger than M .period then about the means caveats is for personal to so there exists a modified depending on demand such with this yeah I should say I actually changes stationed here so in the paper would know that language so we we sometimes change the stationing of papers were sometimes be careful if so whether they should be assisted to kill 2 caveats is not entirely clear story I used the same definition as indifferent but anyway so we had this conclusion this is a consequence of the imminent budget so if that is not enough so roughly speaking and must be larger than 3 times was sequestered to square West you want a closer to his conduct of the free transfer more care than a fancy so be must be true and that means about a kilometer dastardly coalition From now on the point of this is that the show that the set about demand so this condition defines several of the gentleman so the set of VanderMark meaning doors for which staff His themselves that's quite a strong future so if you look in item number serial said so sings Western coalition estimates say than typically you might have introduced the idea that it would look like something structure but you would never usually dream to get something that's actually assembled so yeah we get an actual several animal can do again use the castigation of the woes of Egypt will be there very well understood for a long time and using this week insured and that they do not mess up the outcome of so we get a with estimates from Part 1 using this kind of fighting between the manager Mike structure or Genesis and the other way structure of the questions so we use structure of several 1 injured list which is almost the same as if it to consent to a to this is handled from the Germans what so meaning that really extremely few of them can possibly created trouble so dial cases would very skilled and the future of the situation that the issue diagonal cases cannot affects young so nice working 1 I see the workings of originality here we can have a 99 what we need to get beyond the woman's so the 1 in a list of complexity case this is the budgets type exponent which has never been beaten so the new force conveys found to have Bank for the full finals yes
46:14
ended we can see killing this argument where we get where we need to have something that also we would need to understand that the undervaluation of the coalition sounds as gamma varies but in the is coming from said so we we need to get extra consideration from this the families of correlation sums which we can expect but were salvaged from difficult to get so for the moment we haven't really been called the but was very different anyway so that's can of that part of the ideas that come from doing an IPO disease in force Surin 1 so to so builds on this well meaning bond prices sank case so too I understand special meeting your forms recent assurance Our class sold only United side so we need to understand requirements so we use is from identity which is 1 of the we can get very uniform estimate because we use it in a quite complicated way Weizman was amended to remember the days actually was taking 10 demands all are so the parameter in mind antiwar stance Folder Gap between plans you we need to take care the number of someone like intend to infinity was a common this is quick to kill and so we knew that and we need a bit more momentary functions so we need more about the underserved so the settled by demise in some sense so you I just said it's obvious use of gold are entitled to ensure that its at the point of an undue writes of gold million under way some of pigeons to understand that to be important in 1 of the steps so was but the way to go into this so this comes into the body of another man so when we do intelligent uniforms where led to special cases where the demise of the triangular and we need to ensure that they are not too many bad down chopper trying understand that to be the best handled by the candidates and that good OK so I don't and of potential mentioned and a few additional applications but as an example so a kind of a nice example we have assailed by Jemaah so you might as well as the sale but demand is really exist just assume that the Senate but Gomez is trivial something out it can exist and can be quite Indiana not obvious from the definitions relating to give an example actually from Chile experimentally which is opposed to kind that yield on them so it's 1 of the classical example that has been studied by cats used to take a gruesome some at ends square then we dated square mines were so inside the semantic square of the Persimmon songs but with arguments and square so if you use a standard notation of our summons songs as a nucleus of this the S & P minus let's so the fact of taking an amendment that should just the fact of gruesome isn't it and with so he should understand 1 shows that for so for this kid if you look at his coronation sounds was affected was metrics demand this will be of science please so locally no compensation meaning which is 1 of the bad with this season fall in Denmark which arguments and begin to review which family which is seen as 0 Infinity 1 month "quotation mark Denmark in the southern is a set of guidelines contingent to such that they're not going so the single porn 0 in future foreign miners fall from his mother obviously this at a single point of the free transformative presidency so this is not a view but for instance all the work of catalog which allows determine the fridge from the facilities ideas of which responds and this and this is an exercise that it can be used to teach at about 1 so is available so there's something to do with biracial being 1 of the special values full points in the 4th particularly so usually they are not so many matrices which from default .period except when thereby ratio which is a special type so this election not a reason to pull so because a single point for the 4 transformer default .period it's clear that the bed matrices must be invisible this to respect that serial bonds but the fact that conversely In Denmark from 4 points then this With was followed his scale the main square square lines 1 election being is not views so the best course so we have 2 or 3 posted the best is follows from a result of the intriguing clue we have appointed Dodge about 4 and David Ewing and sent us a lot about so that's a kind of this example can the last few minutes well ,comma mentioned some further applications and is still basically the results and your aren't so far the obligations
53:28
of USA so long as they agree distribution of twisted going to twisted cycles so while Britain is quite quickly so this is what we have to get actually use or use as much of a shock following this abuse so papal unmentioned distorted what I was doing the talking please on news that as someone geometrically accessible home audience of a mysterious so we can find more detailed by winning the 1st 2 pages of this so so the idea is really following so we have the fundamental domain where looking at I T 1 replete with looking at the points W. J. P .period just this time ,comma please wear jeans between the war commenced 1 so all this points correspond to point defendant in the name of the golden in F so who prominently corresponding wants and on the issue is how they Chris returned the answer is yes like this is fuels from bombs on the eigenvalues .period but what we do is we twist kind of dissenting measure with twisted wizard trace function key event so well with reporters than so for any the p this complexity involved uniform including any that's a continues his comeback support on the fundamentals remained in England the modules offers he average event I had the nostalgic replete human Swan Industries Inc the this is going to go to 0 so we the case was the case because it is and they should say so geometric here just as you for the consequence of this if you receive a 4 K an examples reduce accounting functions of Commandant of bringing it's a given value so this will take say that instead of taking response only got J. P for all James you can restrict the subsets where objectives is required because you all too because if you all the former R and square .period Cuba's to where is and you will still get the sake institution so you cannot fall this points to light it's a duel of the alternates through for most of Jane by just taking those J which avenues open it's about rescinded means so is genetically quite nice and economic conditions so the abovementioned Venkatesh concept complexity and explained that in some cases this is also equivalent to of conflicts "quotation mark reasonable that's 1 of the application so interesting to apply to put this year and we apply 0 1 for the Fourier transform of which means that in fact is you want just surrendered only 2 of the existence of the free transformed dress functions because right twice before which consumed by audience that's when kitchen so surrendered to rule as some obligations which traveled quite moving down territorial and zeros only minimal information that was from Chinese sources 6 squares functions essentially In the Bank because when you applied the from idea of sums over clients of Christian Sons which Prime arguments which is the reason is that a sum that the example to incident to answer what we also did is this really wasn't the restaurant which is which was used to some of the ideas into the Burnett a paper so yeah the board was we much more streamlined pool for this young man Federline even it's on getting it from the distribution Nagin 1 of 4 discriminatory progressions but with allege that Iceland and even the advantage and improved 309 it's that inside we don't you need the freed only it's not so it's all bank of widening the state line in just statements we can't just almost you just generate colleges ultras functions is to streamline and stronger compared with the number yet it is not but it's only for primary United so it would be a bit more complicated to do for the John Wooden examples for it's a really nasty would say Well you just eyes approving uninteresting new reserves for improving known good results so that's true something that's not normal uh so this is something which is still a work in progress was made the government said so we can handle with powersaving a twisted moment of pride Friedman granted additional characterize values at 1 of so there's a project square times fortune issued model for comfort I also so why is an interesting so well is to rebate the Kaesong highestend series you're going to get Medusa Mankiw will fault which is a which was done in by Matt Young effusive gold and on the other hand so we can handle this and we are way artists we have some we reduced the case where you would of a kind and square so results the 1st part adjusted joke form 2nd element which is a wellknown open problem with powersaving we reduce at Westminster Press function which we all wanted to to obtain and again evened the match on case all proof is much more streamlined excellent is much better so altogether we have some quite nice application listing of this and so was the problem the question of the this something that the you no I mean I just couldn't in the pool of it's useful because you see with coalition and unlocking near Seoul "quotation mark but in fact the estimates of finest meets so you just decompose and applied to every the list In the thanks and
00:00
Point (geometry)
Ramification
Geometry
Complex (psychology)
Polynomial
Building
Divisor
Connectivity (graph theory)
Correspondence (mathematics)
Multiplication sign
Rule of inference
Subset
Power (physics)
Number
Explosion
Group representation
Thermodynamisches System
Strategy game
Lecture/Conference
Stress (mechanics)
Scalar field
Reduction of order
Integer
Monster group
Area
Module (mathematics)
Addition
Focus (optics)
Validity (statistics)
Block (periodic table)
Nuclear space
Sampling (statistics)
Infinity
Mathematical model
Set (mathematics)
Cartesian coordinate system
Functional (mathematics)
Bilinear form
Field extension
Category of being
Summation
Computer animation
Isomorphism
Network topology
Hausdorff dimension
Finite set
Thermal conductivity
Gradient descent
08:39
Logical constant
Group action
Musical ensemble
Euler angles
Multiplication sign
Archaeological field survey
Sheaf (mathematics)
Survival analysis
Mereology
Food energy
Subset
Plane (geometry)
Meeting/Interview
Series (mathematics)
Position operator
Stability theory
Nuclear space
Moment (mathematics)
Infinity
Price index
Functional (mathematics)
Arithmetic mean
Numeral (linguistics)
Hausdorff dimension
Order (biology)
Equation
Normal (geometry)
Cycle (graph theory)
Thermal conductivity
Conformal map
Resultant
Point (geometry)
Variety (linguistics)
Mass
Number
Power (physics)
Frequency
Crosscorrelation
Lecture/Conference
Ideal (ethics)
Integer
Associative property
Field (mathematics)
Spontaneous symmetry breaking
Condition number
Series (mathematics)
Exponentiation
Line (geometry)
Mathematical model
Cartesian coordinate system
Estimator
Field extension
Summation
Network topology
Game theory
Object (grammar)
Table (information)
Family
22:07
Standard deviation
Divisor
Block (periodic table)
Multiplication sign
Real number
Cartesian coordinate system
Functional (mathematics)
Food energy
Grothendieck topology
Automorphism
Estimator
Power (physics)
Summation
Sign (mathematics)
Lecture/Conference
Convex set
Network topology
Order (biology)
Universe (mathematics)
Square number
Daylight saving time
Energy level
Integer
Uniform space
26:07
Point (geometry)
Complex (psychology)
Existence
Greatest element
Euler angles
Decision theory
Multiplication sign
Heat transfer
Mereology
Weight
Power (physics)
Number
Group representation
Frequency
Meeting/Interview
Term (mathematics)
Wellformed formula
Natural number
Selectivity (electronic)
Condition number
Series (mathematics)
Focus (optics)
Spacetime
Chemical equation
Exponentiation
Coordinate system
Incidence algebra
Mathematical model
Set (mathematics)
Cartesian coordinate system
Functional (mathematics)
Bilinear form
Approximation
Estimator
Field extension
Arithmetic mean
Cylinder (geometry)
Network topology
Game theory
Uniform space
Coefficient
Identical particles
Thermal conductivity
Resultant
Spectrum (functional analysis)
Directed graph
38:18
Point (geometry)
Complex (psychology)
Diagonal
Multiplication sign
Heat transfer
Parameter (computer programming)
Number
Explosion
Goodness of fit
Mathematics
Matrix (mathematics)
Crosscorrelation
Lecture/Conference
Term (mathematics)
Oval
Position operator
Condition number
Coalition
Forcing (mathematics)
Chemical equation
Exponentiation
Algebraic structure
Staff (military)
Set (mathematics)
Functional (mathematics)
Estimator
Arithmetic mean
Summation
Order (biology)
Game theory
Family
Thermal conductivity
Identical particles
Resultant
46:13
Point (geometry)
Transformation (genetics)
Parameter (computer programming)
Mereology
Number
Matrix (mathematics)
Crosscorrelation
Positional notation
Lecture/Conference
Square number
Atomic nucleus
Social class
Coalition
Scaling (geometry)
Gamma function
Forcing (mathematics)
Moment (mathematics)
Infinity
Set (mathematics)
Line (geometry)
Cartesian coordinate system
Bilinear form
Functional (mathematics)
Metric tensor
Estimator
Summation
Arithmetic mean
Lattice (order)
Uniform space
Family
Identical particles
Resultant
53:27
Complex (psychology)
State of matter
Multiplication sign
Parameter (computer programming)
Mereology
Dressing (medical)
Subset
Explosion
Square number
Fundamentalbereich
Coalition
Moment (mathematics)
Functional (mathematics)
Bilinear form
Measurement
Fourier transform
Proof theory
Arithmetic mean
Module (mathematics)
Cycle (graph theory)
Arithmetic progression
Resultant
Fundamental theorem of algebra
Directed graph
Point (geometry)
Existence
Distribution (mathematics)
Event horizon
Open set
Number
Prime ideal
Frequency
Goodness of fit
Lecture/Conference
Reduction of order
Condition number
Series (mathematics)
Eigenvalues and eigenvectors
Element (mathematics)
Projective plane
Mathematical model
Incidence algebra
Line (geometry)
Cartesian coordinate system
Estimator
Summation
Dependent and independent variables
Object (grammar)
Matching (graph theory)
Nearring