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1/4 Trace functions over finite fields
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Title  1/4 Trace functions over finite fields 
Title of Series  Summer school Analytic Number Theory 
Number of Parts  36 
Author 
Emmanuel, Kowalski

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  10.5446/17022 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
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00:02
so if Hey my
00:11
home and said and became so I'll stop my series of lectures so this is a book infiltrates functions of financial and so this
00:35
will cover a lot of ground but most of it is our joint work With especially inflation and from Commission and their pieces which are strong workforce was 2nd only you go to an ongoing joint work with the sun and the whom and we we and some stuff is in the pudding might put all but 2 people and so I'm going to try and described is to some extent and with the a use of time summit so the mathematics replied bodies the series the without essentially around here ,comma between 50 and 70 years ago by "quotation mark indeed especially cats no more than a few of those community the Ramon I purchases for financial you today and the applications so what we are interesting obligations to a native members you all numbers here in generally but was applications that are going to discuss several United numbers mysterious and what I'm trying to do during his lectures will be to this by all fully that is possible Johnson that you can also apply it to your problems Our in an efficient way so the idea is that we're going to look especially at the reminders of financials and at 1st reading by saying that a black box and then for naturally you will see that the blackbox replied to use it efficiently wants to do a bit of what's inside and all the cost the 4 lectures the goal would be to introduce more and more of what's inside the black box so that the applications become more and more refined and In the end what cases is that 1 which is quicker all right with you but 1 which is our points out where you cannot just say the vision works of cats and apply them immediately we reach all questions but interesting open questions even from the point of view of algebraic geometry where all from sites where numbers so this will come toward the end of but today I'm going to begin by trying to prevent a version of the remembers of 5 steals in an extra original version but has hopefully you very applicable but as a black box and will not what's inside the box to be able to apply its founders so as the 1st can of reference for describing there's a trace function survey from desire to his face function surveyed that the planned fully and I wrote about which is available on my work so the main objective will be to trace functions so we'll be interested in a time peace and so I said over finances what I'm going to describe solely times when the final him but because applications are mostly over Q the 5 years just for the most part just it is clear the prime sales and we are looking at certain functions such a functional missiles and functions conveys function as the final which I have a very underwrite flavors of various functions about structure and the actual precise definition is quite and also I think that by the end of I should be able to give food definition to try and show that it's possible to begin limited like when you study at Norfolk from the beginning the 1st time I saw myself the Peterson formula when there was a better function as you know anything about this functions but I just started a new suchandsuch a place I found some balance and I found something I wanted to congratulate you and you can build not just by knowing that at some point you want to do a bit better what is the best of function but so you don't need to start by studying different equation that strictly to Boston to forms using the Peterson from the sun going to take a little bit the special functions reported to be in and of the examples and at some point you will see that we need to be become more of where these functions come from otherwise we're kind of stuck when applying to but this is and then I opened the box for further and further to the end of the day the food strict rigorous definition of trade book so 1 the examples the basic example by so 1st examples and see attitude and that predicted just at the time here and so can you believe that a lot of thought and care is committed to begin the character of please some of the more ones that in but then netted number series quickly rich scenes where you replace the argument by putting on rational function so it was the same day as the formal performing at 2 of the polynomials with and who's not 0 and they are caught fire and to respond to what we can provide at least for most and we can compute 2 event in computer the government European computing so do others simplistic Cumberland then 1 very quickly pronounced that useful to people to make them he was rejected cactus plants and and these objects magic numbers through the don't arise you just like that we define function and we want to say something about the day come out in a natural way and what 1 is interested in very often maybe most often is not so much value but average values for but the same meaning explanations so usually In applications In the some the first one and then when I encountered problem of something like that .period and square and ranges of being granted ghosts I'm quite quickly also the
09:28
socalled ghost towns of customs best character so as long as 1 is interested but the balance for this so it might be that you want something more refined for aquatic outside the most important application doesn't just want decides he wants sign the argument of the got something but in many applications your abounds and as long as you just said these 2 only wanted about you and I share is 1 of the few cases where the murders of summer's known but here everything I mean I quote from this 1 and tour then has gone through the use of stewardesses were encouraged and then you might want to tweak sometimes you that extra parameters coming they do and wouldn't still be true for but basically also could you realize that these are essentially 1 was the only ones where you can actually computer would exactly so I guess maybe an even more trivial cases just before the prompted some just character Over the course of a book in which case when the characters don't really on there is no doubt that gave a simple formula as for the search for them when you take a stand and 1 of the special functions would die forgetting the other nite is exceptional Jacobson's study is something you have some from but it's not such a nice before and you have basic care is Christian and some so was answering 1 of which was denied by appears to I think this was a notation in question used by which for us it is convenient to normalize by scored of immediately and then this is something which some convertible spoke clearly and of and the interest of X computed you often denoted next cancel christened and son who introduced somewhat implicitly by point when completing the for expansion of series and explicitly and already quite deeply by customer to study orientation just by "quotation mark but it could also and for doing so there for many many many applications that number theory and there is no explicit formula I think what this is all worked in just I want to you the suggested situation now 1 of the important features of this example which this for way beyond the security of all the events of the day Best estimates focus on something new some of the people the nite to start thinking of this as a function of an so important feature is that the substance themselves of and back and back and so it's important to define also 0 which once in 2 months from school and went to suggested the formation come natural special functions and more generally in on families of exponential song ,comma tries by some elements in it he will themselves be nice functions sense which immediately with another modifications of the ruling bodies than just playing around with the 1st example so from the point of view as the facilities like killed 2 exorbitant functions in a fire at a precise science and so on saying that we want to see these as nice function life and that a bit functions are just as nice as the exponential functions and all our functions so I'm just gonna know is the 1st interesting functions are exponential functions and for functions and that when you start going to the SEC's functions bodies are just as nice wants to stop only about them and so on and the step of going from character to the present of such financial center also of this kind so the crucial future that's the reason I put this tells you is that if you look industry functions so whatever they are William examples of some on account provided you restrict to a certain said which of defined by some Canadian registrability on opposition condition then they behave likewise the opera not affect was unified emission and the square the Ottoman energy use Croat cancelation In a generality which is the way way beyond what's history that can give you for just 1 viable characters so far age will be a statement so have assumed the on where the functions ,comma but then this almost a definition what definition will be replaced by again a series of examples and the understatement of the so where speaking without is that you have 700 by geometric data which can be defined in different ways so it could be the observations of on the Russian functions to be viable which it was going to say but I also think it has to fall for this for this but more from arbitrary arrest not just just you to since all you can think of this objective point of view is something called education of Time and it doesn't really matter whether your understanding of the world for most of the victims who will say that the question that something that you can only answer if you really have an idea of what was means the most convenient I find is that the addition of but the easiest to define would be to get all the attention so that's probably what will emerge as the definition as time passes now
18:32
these are due by geometric data and you can see them really has an animal's automorphic forms so you know 1 of the more the form of a number of years already complicated object but with many nice properties or their obligations and subject which is identified as structures with many violence and we will send this to the space of functions the new couldn't over which is the space of functions can differently with so we have some highly complicated objects with proper normalization and is typically defined accountable sets and recent history 5 dimensional vectors 1 dimension of the complex so the assumption is that I am which again think of analogies dimorphic forms or even with additional characters Motorola nonchalant use few think this may be us the being debate all of the 1st values of the character all the firstly coefficients of modules so it's not full information about the model fall if you think of the classical case of due to false you just take it said the 1st Due to the 10 occurred in vital something like this oxygen 1 of the Korean use but that it would like objects and so it was a very important fact that frequently Lucia so from the point of view that we are some complicated to wake objects and then we just take it disadvantages the truth function the trade function only a priori contained some of the information and that is not injected so in particular about infinitely many objects which of the 3 functioned identically so if you just think of the trace function by itself as a function defined European don't tell you that much because it comes from a few too many she's always repetitions and some of these Ailey complicated and those of Somali read complicated so we need a way of measuring the complexity of the trace functions so that we can work with objectives small complexity as a way of saying that we have control on the function we get and the function of information so it's like saying that if you just look at the value at 2 over the dish Okada it's not telling you that much but you know that the conduct of the character is extremely small then giving you a little piece of information so there is an environment which is not quite clear that the United conductor for functions so why is the subject solution representation to 4 and so far we will 2 where the so for every time we would have traced functions which asked the varies so they're very really risky but they come from she's or whatever which of complexity ballad uniform in terms of "quotation mark objects used uniformly by all complexity to end so my presentation and I will write your fault so the smallest complexity of an object his right to free so many analysts it's so these objects on to weather they are at the moment that we call them representations in the cold and Ron trip to Europe will be that result from this from the trees and you know for the complexity of you go and you will see just like you can say a lot about this functionality can use this functions quite efficiently just by knowing that their current formalized as about the future and signed here 1 can say quite a lot about for a functional and apply them quite asec successfully just knowing many examples abound on the complexity and the remaining part of which will come in so far more examples with examples of the complexities McCain and so what 1 and I'm going to say later but with its indepth Frank you can always go back to these examples are usually even with just these examples and a few basic operations you already have lots of interesting features phenomenon and what you need for many applications so the abstract series use food but it's not necessary to already sequestered but so far they the example of the again said that after 10 years something about the complexity of these objects so we take a rational function so now I'm going to take the polynomials in a few banks prime afternoons and so in this series because we are looking at objects which are a little bit like primitive usually character in contrast with just 1 character so the opportunities have to be very careful with the value at all which would be the prime numbers and all integers and just became so we can just say that it's at the point of view Domenici regions went so where F 2 was appalling Westwood as you also have the support we have to say precisely what is the right value the function this is important for us and statements to actually work so precisely so we
25:49
define kilogram to be appealed for then went after it was announced here Monday so in this case from friends don't you because women's article prime just makes sense and 0 otherwise and for much of it getting factors we can take care of the plan is an effort to advance his 0 and F 1 of them is known to all as fulfillment of bigoted characters after we also have to say what happens when we evaluate and that his character and the other 2 4 further cases the archives story that not the truth character and a former venues you have to amend that's the extension of the Jewish character of 0 is kind of 0 0 it is nontrivial but it taste really should be so want you guys equal to 1 and the form of an analyst with so too I said this is technical so united number series to point where different convention for the you just switched the value at this point and with 3 estimates for after switching but for the Dubai statement it's crucial to have the right conventions and harmonization so we are so we have to say what is the complexities in both cases the complexity is about unity and define it but I can tell you this is not going to be funded by the maximum of the descendants of the degree real that was enough to buy it so what can actually be more precise when 1 of the core definition of complexity but for the basic applications you just need to know so what is this useful is going to be useful by saying take To a ratio of the work efficiently animals which are fixed and then you take the viable pondering to infinity and their fallen after face to conduct of whatever you I was a transcontinental Kennedy balloting independently of the execution it is the best example the 2nd examples I thought themselves so that they can do is to take care of it and the kids here from Shalom Oliver tools comments were explored both to expand in please besides thank you such that the product equal to that of clear exit and that's when it is known to all and when it is equal to 0 again was to be careful referring to the right it's minus 1 To all divided by P 2 the latest I think anyway so it's important that instead of when it is equal to 0 so there is a way of computer his values right so I call returned more Nova please cancel so it is incumbent upon With want the ideas from outside coincided with the case an equal Michael Stewart and correct that the next election this this is something which OK so far I think to strangers on the revival of ensemble on songs that tell spoke about when explaining how 1 and socalled Type 3 sons all the time and the functioning of the questions from the the no I can write to dissuade so he and his role but in the end is a matter of vitamins and equals you wouldn't have to use a 2nd phone line where anything is nonzero then the excited to be here yeah all everything is different from my functions would just but I mean I always forget that you some including as during defender the 1st and I'm you write it correctly then the equal 0 case is is given by the informing I am not sure if the where we don't want to get away from the front of the that's 1 of the lines Florida farmers well yes and no because I mean flies equal to twice which assigned the right definition of resources from minus 1 was put of the times the sum for equal to 2 of which I didn't do so I just anywhere from 2 thousand people wanting to distance but follicle to tune you should have yeah writes for young OK so this is the fact that functions and this is due to the and I think that this is indeed a defined by the following 1st what kind of man another fact about the complexity so I can tell you about the complexity much but the
33:05
outcome of the qualities of the complexity which research cities after the this is something that I might forget to say now it's a little bit this is kind of the same direction but it comes at a good point because you the strength of his remarks will be here are so precise about the complexity of important so the 1st it's a little bit like heights In the sense that when you look at the state of trace functions with the normalization which I will discuss later With complexity bounded by a constant is the financing so the loss of trees fortunes just like the set of additional characters June but when ,comma told usually characters finance said and this is something of which so this is the wrong way of saying it because we need to be more information before we can state is precisely but roughly speaking the set of care complexity will act this finite out to treat your modifications so this is the right to U.S. statement but it gives an ideology of what is content and so as you maybe noise we should they denied a conductor as defined by the events and some like the same statement is true for the outcome of the forms 1 of the few the case and the 2nd remark is nationwide this is really deep are amazing fact about the complexity so 1 has audio 1st way of showing that the complexity is interesting ways defined it as a provided that the LNG regional is bounded by the complexity so when complexity what should measure welcome together a function 1 way that function can be complicated is by taking large various values and you were saying that the His mother by complexity and here but I didn't say which sets so is the complexity of him off my clearly is founded by 1 was a constant depending only on so let's applied this fact sir results batteries functions with ballad complexity and the fact that the shooting is bounded by the complexity whatever it so prone to immediately is that there exists constant they are known to such that the IPO instruments and the founded by a for playing at all not because they divided by it's a likened to a divided by sport of speed this season we from the very powerful fishermen's so it ain't was equal to then this would be focus this sounds which comes directly from knowing that it's actress functions with this type of property and so it's not surprising then to see that to prove this the disastrous function in the sense that mentioning civilian lauded the remark was off 5 years In nontrivial bonded complexity printed by the way defined by a series of precise definition of the all of them in the the past 3 years we've got complexities of the smallest complexity will be 1 and the interesting question 2 to find the smallest complexity of another function is not the object reflections and action of sordid that about a million miles we always play on about we trace functions which complexity roughly P which side we know objects presentation with complexity refugee restrictions here and what is the actual special so this is just knowing that an object like a focus of the music for selection without knowing anything else so this about the information that may be in 10 years the progression of France functions and this is a I you would find in the 1st 2 pages of this hypothetical much of the gotcha and you need need to know anything about baseball or type of Publications among you just functions so it's bound by complexity complexity policy and then you applied so this is somewhat similar to the fact that the functions became 1 of the Code of Texas In a very rough and manage its image stupefied again so not only go back to after the population to more examples to shoot a variety of type of things that can arise so it has a general class of trees functions which opened by accounting solutions of putting on equation so 1st example would be the effects on constant watch on me all at once it could be rational functions will be fruitful promoting you counts for a given and how many X there are in innocent people With this effort that people want you just kind so this is on you it's only bounded by did you because that is not constant could be 0 includes from frequently but it is something so this is a trace function and its complexity abounded In terms on the degree of so tend to pay his train function is still clear again to enacted constant bounded by the degree of and therefore always your polynomial means that 6 grand which is an integral part in mail then you get something which complexity independently of people this case of course it's not about to compute the number of shares were squared equal any person
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and the last example which is again just as a special case of something generally issue trying count not the solutions to understand what level finally equation but not all of the family of and the best example is supposed really interesting examples from leaders of community can take care of him to be case minus disclosing normalized in just west of the number of solutions a square of y squared equals explains warned it's minus and so this is done in such a way that is going to go for an equal 1 as you whatsoever and so is a trace function but I have to do so as to define the scope of the movement conscience so it's it's clear can have divided by school it's a trace function with complexity bounded past periods and you can basically replaces family to kill by almost any time you have want to buy a variety
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depending on 1 Parliament that you can't think of 1 of 2 technical but we we want them to go well I mean you might have to strike the right well for that about points and also refuses to take a surface and if he does that make streams this would only be the right thing to do just that dimensional community or whatever but for families of killed this week except for the doing the right thing and about points this would be the right this would always almost always response so design gives you some insight and so 1 of the things you I you can see is that once you have this on just these these examples in statement which is valid for any trace function which it will statement I will immediately give you many many different types of statement which very often adding numbers who would otherwise be we have treated 1 by 1 we may be different other methods and what's examples of some of the things we do this for me and shareware we are statements which are vying for every trace functions would that to be up front at 1st but once you have a list of examples and will virtually nothing of substance focus now I can say during the nite but right because didn't fall for such a big so I think I'll keep the examples the Commission the way so good was saying that I'm going to state the American I cannot quite stable just with using only the words that would introduce a bill to introduce this 1 extra public trace functions which have abandoned efforts in concrete ways in the statement at least applies almost immediately equated version of Verdi's estimates for representing 1 of the few definition because them so because the extra what I'm going to introduce should be condition infrastructure will be always satisfy for the example 1 which is the classical Twitchell something viable so this is essential the accuser also welcomed you can't trust people deserve to me once and that the really essential work is that of 2 McCain Swoopes a prime and we take a warning came to trace functions will With complexity is at most wants to respect so I could "quotation mark you want equal to sit up to half of that is the complexity of Reno would again because it's not such a meaningful number with meaningfully is being being obvious so you have to say that sales to 2 equals 10 thousand yuan equals insolvent and has already lots and lots of such things as the examples above all show but now we knew assumption on K1 K2 so the state would be a question 90 but for that we we need some assumption to reassume K1 came to what is called geometrically reduced so we have to explain the what is going after 1 you can take care information that in case 1 always so in case 1 if you have to 192 balls of this time still wanted to monitor because fuel was meant to be getting people's whichever way then it it is always then they ought to concern so the Warren and associated to representation of the holder devotees underlined objects which article 1 analyst who solely for 1 and all to hell not to mention the national thing so it's another piece of information to explain and for any case along with the case of former rival character sons will again be quite you would be then are the U.S. Code consolation when we call a clear when state so we somewhere PK want entity to buy half so it's a new
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product and we get scold cancelation implied constant depending only on the complex and 1 can compute
47:09
that the upper bound can the streets he want granted With a definition was taken so this is Uniform Code cancelation as long as you some and as long as he wants to about I'm still in uniform in searches for way that you don't need to us wants provided they could go a little bit respect and still get extremely good that's the 1st 2 months and could the 2nd part sources the examples the bald the Roanoke right but that's that's the condition for this is that to all its original woman so what what would this will reveal that once we have the 2nd version would this reveals that there can only be 1 along with small complexity and residual response so there a gap between the complexities of interest that's what action was look at the 2nd stage weapons in Europe that we all want geometrically isomorphic to to because the 1st thing is we don't know what that means but 1st consequences which we might be able to falsify meaning that preferred defies the condition of course when case for so that 1st information is that 1 disproportional cater to full sound as far as it's really promotional everywhere so including possible but status was the fact that using implicitly current definition of the values at every then not all of them but cuisine known meaning that the sale of killing at will and well of course it's out 5 times the sum of motors of Q 1 of its square so we cannot expect cancelations and in fact what we have is more like an asymptotic from long distance and fight and spirit so in about K1 K2 executive and up to and which is against the constitution and the same power as in to about 10 minutes OK so long we understand it bit better these conditions mean and we have to to handle these conditions checking them and so on and this is the next general Bible assertion of the model for the post of finance so it's not quite equivalent because we still working with just 5 years and some of his own viable but it's it's basically awesome following a seasonending upon them and other consequences of Dylan's work which are modified to meet with the decision of the trend OK now yet the current phase of the peace plan this show exactly like products of traditional cartilage operative might being primitive you might have to change at about points something primitive but is a unique underlined and primitive the man who is the wife of yes but then such images irreducible it will be tied sky to different components you only allowed to applies if you multiply tied by complex a moment want you consider to 1st function that's you measure the adjustable but if you much by 2 when you have different ways of doing it but it always is to conduct some kind and trains and this is not that otherwise would have the cold soaking 1 equal to get to you either here or in this case and in the state of the land of the small and I thought he was not I believe that you mentioned irreducible then this is through an otherwise this happens is going to happen with situations where the complex now it is something that that Our down legitimate their magically isomorphic but our key 1 is equal to the 2 discount but then this 1 also to would be extremely so the deal voluntary so using these values but very powerful insulin into a small enough that essentially the product is isn't good because what is attributable McLeish said that but so the train bound for this sounds like a lot of extra to conjugate vaccine so trade rebounding times I getting thinking on board and we noticed that most students so you see that the fuel complexities to realize that this is a meaningless in given information so there's no contradiction OK so what our does it mean to be dramatically reduced chairman the disability and isomorphism yes right now that's so that the environment is an environment of so objects 1 or 2 there really meaningful of 2 sides morphism class in particular the 1 . 2 OK so what does it say whether these conditions remains so here we have 2 entirely and began to wear these functions ,comma so it can trace functions Thomas 7 objects this implies there exists only on his soul or Israel Mofaz and so who I want to respond dimensional but so it's a highdimensional doesn't assure you can think of me as a complex vector spaces but the writing on the right thing to do would be energy trip to Spain but it's only find the way distinguish the complex that spends of some finite dimensions such that the care of is the trip also on elements for being here a month depending on an acting on the science of space the end of the the year and rise and show which I don't want to go into right now but OK so that you can you know 2nd information so there is some objective so always really amount from something called the decision depends on the tour highdimensional of GM fundamental spends so if you have a representation and you can ask that the rotation reducible Anderson what want so precisely that exists in almost served I want to know and to say that all is geometrically irreducible we know that there is no that's not exist summons for his own people and there does not exist W inside not able to not here stable under direction of the circuit OK I'm I'm still trying to keep on my my example become difficult what is there were no longer exist such so we can more precisely so I can't tell you what is possible and that Illinois and is the just to get out of the workload of the here so it's good that depends on your tourism offices
56:44
and almost everyone is together for about the same thing here surrounded by closure of these functions the model the small group of functions is under way so it's not really something of such a complicated objects that quite misuse of the point of view and often does need to know we were there but investors will but so we have this condition and the fall in particular so simple that the dimension of his mom so this at once in case 1 in the 1st example so many title value is not the rational function at the human so 1 has to know that this is the case that this is the case then it is definition you don't need to anything the use of dimension came from Switzerland 0 you're looking for property of space that surrounds it down so it is automatically Georgia next on geometry isomorphic soulful wander into cheers you want and what to do and to tool geometrically isomorphic differently if there exists only in your mouth isomorphism the isomorphism from the womb to the tomb of send them a trend as a union of such that look after registered 1 critically so for every 1 to do whatever it is useful diagram this has been the action for every gene does so can match couldn't by I'm in the 1st instance to begin with what you should think of the moment as what has to be Georgia is among the nation's trade functions of that would you should think of and we'll see more refined versions of but now speaking becoming all and all I'm starting to get it all the time but since an organizer but Anderson so I want to show all this means the fails 180 form of estimates so just by knowing the surety apply it was enough to recover instruments for 1 of the characters look and out of the performance from its full simplicity that's most interesting from Kenya petitions to articles to soak in that case would take care of that 2 consignments to adopt a Sunday is clearly an act of sex and violence with conventional and you want to you .period there is no way to France so irreverent long 0 element which is thinks that could and toward confusion and Carol elections in of Iraq's perfection berths for some considerable In the course 4 K to take the function 1 file which you can see as kills you if you want so it is a trestle China and by example long and by what they just said so complexity of 1 is bounded and inventory of P and complexity of the problem solving and they are sudden geometrically irreducible because of rank Foreman said and therefore acknowledged that stopped it examples again "quotation mark so we deduce from a former stated that the instruments are respond In his government is bounded by code kidneys absolute implied consent Our provided distressing on a geometrically Asimov so where are all 1 to unintentional Dimitri so can it be that agrement appears more supposedly I'm then what happens that means in particular by case too we've they want to reduce some of 1 Republican to K 2 K 2 is a constant so K 1 would be a constant and as soon as you have won an unknown 0 defunct destruction can be companies whose fault is the Council would have to be but OK so as long as there are a few of these 2 elements I think that K1 please not it was presented to so we get kind of quantitative form of the system without having to to know anything at all except that books like could also clear to started trying to move on the examples of tried to stop doing this and that comes up your songs were and see where we get stock and what would be more information and on the same scale I also want you had was that it was something that anyone would want to get to rescue argument room so I don't want right yet so this this will be no problem I will see you so trees functions only we have the 1st example but they also have a rich formalism and particularly replace eggs by 10 times next fall and 0 the amount of the fee and get another 1st function or you can replaced by 1 of our work 6 and then you want is the function of and then this would often Beatrice function and conditions but a full can of standard Randall examples distributors function and the complexity will be bounded solely in terms of complexity of cable companies can this is highly nontrivial soldiers will require there work of cats on the city of conclusions but the success there is no time to exclude the theories of his yet because it's not dramatically reduced as you can see so 1 of the consequences already of what I said is that for small complexity the estimates of what I'm compatible so the fact that this cancelation if you take equal to 1 just a few data to equal to a want the sum of 1 square meters square is essentially playing this is equivalent to a be forceful complexity so 1 Creek Weill said to more than 1 quick way to if you're given function you know it's a trace function you want to know whether to mitigate this on contradicting square is the main square close to peace that is
1:06:21
dramatically and we'll see also find you can check that Christensen's optimistic that disabilities will with Minsk office from some to conclude intersection please just by tighter sausages from on Monday that is the have end of this notice that summer the plan so we know they existed the composition 2 sons of adjustable pieces whatever little technical trick but sessions from it might not be easy to find so in principle you find them by taking a product it's likely that I cannot more this is sort of space but it's I mean that in future many visitors we have highdimensional space but some of his intention in game interesting many basis vectors so it's out there she conquered all of them to determined the the composition so you cannot computed because you have extra information it can be very also has a number of the open part we still are fighting freeze invites applications is because we have something we don't know that estimate the adjustable and we don't know exactly what so it's 1 of the fundamental problems that can arise in the future
00:00
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Thermal expansion
Cartesian coordinate system
Functional (mathematics)
Estimator
Number theory
Object (grammar)
Family
18:31
Polymorphism (materials science)
Complex (psychology)
Musical ensemble
Multiplication sign
Equaliser (mathematics)
Prime number
Mereology
Proper map
Group representation
Maxima and minima
Analogy
Product (category theory)
Moment (mathematics)
Infinity
Functional (mathematics)
Bilinear form
Category of being
Vector space
Hausdorff dimension
Module (mathematics)
Normal (geometry)
Right angle
Uniform space
Thermal conductivity
Resultant
Point (geometry)
Polynomial
Presentation of a group
Divisor
Number
Prime ideal
Thermodynamisches System
Lecture/Conference
Term (mathematics)
Operator (mathematics)
Harmonic analysis
Function space
Integer
Contrast (vision)
Rational function
Series (mathematics)
Addition
Algebraic structure
Mathematical model
Set (mathematics)
Line (geometry)
Cartesian coordinate system
Field extension
Network topology
Object (grammar)
Coefficient
33:04
Point (geometry)
Logical constant
Complex (psychology)
Group action
Musical ensemble
Presentation of a group
State of matter
Variety (linguistics)
INTEGRAL
Direction (geometry)
Multiplication sign
Insertion loss
Mereology
Event horizon
Wave packet
Number
Frequency
Lecture/Conference
Term (mathematics)
Energy level
Selectivity (electronic)
Rational function
Social class
Series (mathematics)
Addition
Focus (optics)
Reflection (mathematics)
Counting
Set (mathematics)
Functional (mathematics)
Bilinear form
Field extension
Category of being
Arithmetic mean
Shooting method
Network topology
Noise
Equation
Normal (geometry)
Right angle
Object (grammar)
Arithmetic progression
Family
Resultant
41:47
Classical physics
Point (geometry)
Complex (psychology)
Focus (optics)
Product (category theory)
State of matter
Multiplication sign
Functional (mathematics)
Number
Estimator
Group representation
Lecture/Conference
Dependent and independent variables
Right angle
Object (grammar)
Family
Subtraction
Condition number
47:09
Complex (psychology)
Group action
Rotation
State of matter
Direction (geometry)
Decision theory
Multiplication sign
Parameter (computer programming)
Mereology
Food energy
Expected value
Group representation
Conjugacy class
Oval
Square number
Ranking
Diagram
Social class
Physical system
Product (category theory)
Spacetime
Closed set
Moment (mathematics)
Functional (mathematics)
Bilinear form
Discounts and allowances
Category of being
Arithmetic mean
Isomorphism
Vector space
Hausdorff dimension
Phase transition
Right angle
Uniform space
Identical particles
Directed graph
Point (geometry)
Geometry
Metre
Perfect group
Connectivity (graph theory)
Student's ttest
Distance
Theory
Wave packet
Workload
Morphismus
Thermodynamisches System
Lecture/Conference
Term (mathematics)
Operator (mathematics)
Reduction of order
Antiderivative
Rational function
Condition number
Standard deviation
Scaling (geometry)
Element (mathematics)
Mathematical model
Local Group
Estimator
Residual (numerical analysis)
Formal power series
Summation
Algebraic closure
Network topology
Dependent and independent variables
Hydraulic motor
Object (grammar)
Matching (graph theory)
1:06:19
Spacetime
Product (category theory)
Manysorted logic
Lecture/Conference
Basis <Mathematik>
Game theory
Cartesian coordinate system
Mereology
Freezing
Estimator
Number