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2/2 The orbital circle method and applications, toral eigenfuctions and their nodal sets
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Title  2/2 The orbital circle method and applications, toral eigenfuctions and their nodal sets 
Title of Series  Summer school Analytic Number Theory 
Number of Parts  36 
Author 
Bourgain, Jean

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  10.5446/16999 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
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Transcript
00:02
Hey a top and I had hired Webb will they want to work they want to but basically concluding that the subject there was still talking about we the 1 more question that maybe of some interest to the riches In another question the related to work on their frontend properties of global cynical the but although this this 1 is obviously at different nature and it has to do with the Duke ticket distribution :colon in it's a positive thing that attitudes so just as a reminder that there would fix and then a positive discrimination promote positive this clean which a strong fundamentals that say we think the square feet then we can consider the ballot quadratic forms with the intent of coefficients with that given this came and and the collection of this quadratic falls there can be partitions in a finite number of classes from the Glasgow where the the equivalence of Toulouse quadratic forms that simply means that 1 can be transformed into the other 1 venison the consolation the so no 1 can associated tool in each class the fundamental geodesic it's it's only depending on the glass which is the coastal desert in the unit tangent space over the years of the model of which is connecting Alpha Maine is being this manuscript will be too old to work do it's it's it's continent in the world that can take care of reduced which would imply that we have a purely periodic continued fraction
02:24
so Duke exterior tells us that a locally normalized it we're looking at the measures the acting members understood 6 and we average than all the closet then there's the goes to infinity this is going to converge weeks thereafter the measure of the national measures on the unit tangent space this is Duke still in what this is all those amazing scene which is an experience they have been in journalism is that somehow there seems to be some discrepancies between people the number student people doing the good acutely so from Duke student 1 can infect their life the statement the individual peace the world most of them because what that statement and price which is really statement about the totality of the city's of this of these measures associated to the fundamental Joe exisiting that effect for most fundamental to the success with that we were getting ticket distribution so this can because they I had these questions I was asking this question to accuses it of course is through and then just at the number number of years and we don't know about such a sink and then eventually realize that this is the kind of thing you pulled the same way as assertion on the exceptions sequence the room which so if you have you looking at the victim measures associated to the egg functions and then you have convergence also except for in sequence this is what's known Boston in fact plant can be quite precise 1 can even make Magnuson a quantitative the being the sudden precise weight I would come back to that immediately a in well it is basically in the then there are a consequence of all the previous stealing and because 1 has this new Killen in explicit quantitative fall 1 can call for some results about what it means most fundamental this which are quite precise and I would come back on that in a moment but that merely call you that there is a conjecture that in fact when the closet he's not so the largest possible if we have closed which is and of things at home then 1 in 1 expects that there in fact all just visits with will began taking distributed the following the the standout cast member formula what it appears he will believe in this conjecture and then will looking because that is what we're going to do at that events in other words fundamental to the 6 which are not well distributed then we have to look at it for quadratic feuds another fundamental solution here epsilon which is quite small In this would also be a patent in constructions that on regarding this conjecture that that is a killing of ballpark which tells you that the conjecture is true if the classical this is an example proper that the classical is smaller than some sufficiently small balls office over the which is a result which is cool when using was police techniques but in fact it turns out that Duke still in place the statement which is just a little bit weaker than that of any without going to this display of sophisticated technology the Weddle diesel is fundamental not fundamental 1 Molly market is that the evennumbered years of age 1 1 can't quite pull down but we can only get it with with class groups which are less than beautiful "quotation mark basic for the and get bit so early
06:42
gelding consoling that Joe basics there was a problem I don't know whether this problem was evident on but the mostly from call told felt his questions along a sewer looking at doing it's the fundamentals of the 6 which are not would distributed and so what we can for instance asked it is whether we have infinitely many of these features which does not get into the coast state stayed within the region away from the test in other worlds that are often 0 9 and as well in fact that what is in the literature of and infected it's noted papal of people select which is called the basic local job physics is a similar question whether the of were asking the fundamental geodesic could be listed committee which means that the underlying the quadratic investment has a continued fraction which is spend on so actually it is that paper on this apocalypse eventually what happens in the papers at the start studying mark of peoples and sings like that but is in fact a disservice problem as it would appeal may prolong or finding square feet that figure to these developments along the ah finds his way so it goes back to the papers so what is it what is here is half of what we pull thing the 1 papers is posted there is another paper which is in the making so do the 1st page of the paper which is posted there is is that the answer is yes and in fact we have the following statement for any exit on there is a compact region wide In its on the set the or positive fundamental discriminants with the following properties were 1st of all if we're taking the indeed and we're looking at the the classes gamma the fundamental Judaism's Gamla which they inside that regionwide where we have plenty of them in fact we have as many of them as the the classical but the ball 1 at another statement is that this said these quite large because a number of elements in the Lisbon going to be Hoffman is at tool for these 2 statements of Oberst possible basically the best possible that the reason is the following on 1 hand and what we prove is that no matter what is the you with taking the Gamez that means that the fundamental to the success of which the gamma remains In such a compact region wide Weldon members of the statement's is bounded by age should be the size of the closet will be made upset before the going to infinity well that is going well basically the epsilon can be the made there a small as we want to do so Ed what you see after losing the 1st of all if HB small abandoned some balls sufficiently society further the is going to be yacht if HP is going to be smaller than than some of the power of the then the city is going to be empty so that comes close to the the result of popup sold 1 way of being not key distributed of course is to stay with it wouldn't set such a compact regions and in fact the pool of that result it's exactly what I was referring to the before is that it can be deduced from a quantitative offer of Duke student and affect what we're using some explicit results explicit estimates that appeared in the papers of the president and on 1 usually estimates in terms of ecstasy Sobolev norms etc. that would then composers to pull that so that state and this statement is best possible exactly because of this release it is that the what happens is that no matter how we choose the otherwise we're going solely with this it was monsters so no matter how which was widely confinement that sit on it so that this property holds so that statement is this possible know what evolved that 2nd statement whether a
11:38
should go back to war would be you the cast member followup if 1 takes the sum of rock Epson on the a central dealers and fix this thing is well understood on the other hand it is much much harder to understand the sum of HD for the most antics In the past and say that the Sonics Jesus is 1 of the papers on that and also there is a papal by Bay will be about the same time well they studied this problem and get the results of which are unconditional as long as it's Monday is quite small gets but to work at 2 peaks of sentiment that there has been a more much more recent World nobody better way but in any case it is sinful smaller on the on the other hand what holding what is the conjecture is that he would take a sample of the Agency for less than meets the should be even not mistaken conjecture to behave like exclude square of something like that and in any case it
12:42
means that it will looking so basically as a set value we should look at this came in and sold which the the classical piece of extremist seismic scale to be no it we believe Jesus and adopting than animals such as training and has to be there has to be smaller than an infected the powerhouse of prospective so this this can get things close
13:06
to work what we can expect no what this work has to do with the that we are it's well how we put users Lang the fundamental geodesic well if we're so we're we're looking again like it did before long designer the problem at the cynical generated by the spectacle of the natives makers here wears baseball bat capital a it just to ensure that we're are going to have bombed the partial cautions and the Coventry the corresponding fundamental corresponding geodesic is going to be no line the so the of which is that the president that he hazardous came in and which is plays gamma square minus 4 where gamma and doing a lot of floatation confusion here but so damn this is that there are 2 ways to matrix In a dome Celtic in order to get so the problem is to get fundamental once so basically want to have this disk will be so the problem is to find elements a gamma of that fall as many as possible but for which the players Gamla the square of place of government is fully square feet this is basically this debate persona ,comma reciprocal Joe visits there these that led to the same issue about getting square for use in the mark of the press which is of course much more child but here's what would we can't produce this problem in a those simpler way because to get their plays Denmark the square minus 4 square feet what it amounts to is to have his gamma president is gamma minus 2 square feet no held we do that well you have a statement here but what is really going on is that we're getting we're making sure that there's gonna press 2 and gamma minus 2 don't have small prime device and small means some power off and so I really do the the statement this statement follows from another statement is that the number of Gamez in G 8 which are the bound to make ends a ball of ages and well that number is going to be an the Apollo to build that it from what they discussed before now but if we really quiet plays gamma press 2 and plays gamma minus 2 not to have planned the rise of less than end to Epsilon and the question is how that something is going to be that's going to depend on the level of distribution of the sequence while there we still will have a lower bound of that fall which means that the set of places is going to be at least and at the Apollo to note that a miners want another Western aid is getting Larchwood a ghost 1 so we're going to have an ball 1 minus and sending but either way what you see there if the number of places With the pulp of the is bigger than any other poll 1 Maine something so let's call it eat up time and we we normal over that's a State plays down the Bristol workplace gamma minus 2 don't have time devices which are those which adolescent and yet well even the steps along is going to be larger than 1 that will be speaking time we can of course come group that it square feet so the whole problem there is the following is that we want to have a level distribution which does not the general when the aid goes to infinity because on 1 hand they want to be close he is to do so I have to take a lot but as I told you 1 of these drawbacks of determined another comment which is used in saving is that we don't really know he eventually how this has announced the region is going to depend on the on the the alphabet we're starting from sold just by using that the problem is that you take bailout just to get this to built up a close to 1 you may very well lose level distribution so need to have another device they to take care of that and that that's what makes this thing not completely Solanki said this this this is hopeful that the story a there is another element that I didn't put it down because it's not the papers not completely certain is that in fact we have a similar deal and for the reciprocal joined it's so a gain that you have the statement that there is a set the office positive fundamental this came in and the witches the same large and with the property of the that followed the indeed we have at least 1 L the basic Joe this it's a which is low lying in the sense that it's contained in that in that the wide and well what his wife basically the reason why we have to we have to take we have to go high high enough and so in some sense this it's an answer to the question questions costs that was asking at the time so basically this is mall is what it was planning to say and that in the previous took the could not conclude no they don't have quite time enough time to work 2 to speak about the deal the topic and what they will do is just to a select few if you were aspects that over what they wanted to discuss in the in the next and we need to have no surrounded the next subject the time the number to elect acute problems and that's that
19:32
that has to do with the told a
19:35
conventions and it there's a lot to say they'll but basically they want to restrict myself to tool to aspects of this story but 1 hour moment inequalities in the office told 18 functions and then you the 1 that is so the problem of the number of the coal and steel and and the the the number of normal domains so the what is interesting about the spectral clearly the the Flateau said and perhaps the people working his earlier this most people may not consider this as that exciting but say say as a harmonic analyst and like the subject because it's kind of at the interface of of people to well you see on 1 hand the French and given by explicitly the nomadic put enormous the complaint to do some kind of explicit analysis on that without having to go to said Michael local titles national William basically don't see anything anymore so this is completing the cable which doesn't mean that the problems of Cleveland and then because we are talking about special special putting almost because the frequencies they correspond collectors .period feels that there is of course also number fuel In the desert the 3rd reason why this is perhaps increasing the SME steers correspondence between phenomena that happens fold the demands reflect also sent the out not statement not published in the sense that it's important to keep the goes flat agreed to limit the committed between users we was features so hand this features for the bidding for the flat and on the other hand you have phenomena which of course not pool and that the conjecture of India India at East India admitted capable case and there is some kind of similarity between these phenomena something which is not that well explained that in any case and that's not the world about that built aspect what they want to discuss the power of certain features of the distinctive features of the flat ghosts and these features are not I'm not typical of a typical in the sense that it was looking at all the many false we we don't have said for instance holders he a situation is quite different so let us start with the classical moment inequalities it is well known that the control of the old Eggen Fengshan sales Sutherland is always quilt of the control of the tool is going to give up control on the people in the last line of this size bombs on their in the world basically you don't have to worry too much about this response on 1 hand you have the behavior and infinity the industry can be grows Infinity is going to give you the man is 1 or 2 for Infiniti bond and then there isn't the pollination but it's not an anthropologist with it's of these wastes interpolation where the critical exponent here is to was 1 of the men is 1 soulful twodimensional manifolds decreased 6 so there is a compilation of proved and exponent in between that exploded into these Bonsall best possible and easily seemed to be shuffled his feet however folders filled the flat was 1 expects
23:12
that in these conjectures are also the conjecture that 1 would make safe for the dimension magic hyperbolic many falls so they that the breaking point the studio will be managed to win the conjecture is that you have a uniform control of European on due to oneonone please less introduced demons to the twofold decrease for the to it should be be for all people life and then the evolved to deal with the main to 1 has these but an attitude you will be Manus to 1 hazardous virgins which is planned to no this conjectures very hot and is basically open that there has been some progress electoral booked on 1st of all there is the result of which camps close to such inequality although but it is still significantly weaker in the sense that will rule was in effect along the 2 units and that inequality well a show on the these simplistic to plead the cost was another story may say workable that in a moment but for the acoustical complete these facilities quite significant and will 1 can obtain 1 get the results will be the bond between Duke was 1 of the main is 1 note that this is almost a perfect inequality Jamil inequality for explicit photos of explicitly knowledge and affection people out of so it's doubtful many examples in harmonic analysis where you can pull something in the without going to an interpolation argument for an even more so this is 1 of instances and then have gallon the Saigon statements what 1 can pool was at the 2nd statement disapproval for the sufficiently large and that depends on some work on on this subject and also on the on the 1st day against what is the most interesting is the pool of that fill statement which is a consequence Of all the general harmonic analysis
25:22
still in which I think the sits in the middle of a few months ago and which somehow he calls needed to be Copeland conjecture know what this thing says well it's 1 of these what is nice about the statement is that for once it's not quite cable harmonic analysis statement which is clean In this sense that you get exactly the right exponent and that I've bonds fall almost no advance so this is a very old very general statements there's no special going is that we're taking a complex malt type of Silverstein of the with a positive 2nd with a positive difference again for them and for the year we don't have positive it we have a number of the 2nd fundamental form that is a version of that but requests suitable the modifications and in this form it is not correct then will looking at simply were taking a small built with taking all functions which leaves which fully transform relief we've been dealt neighborhood of this type of service and then as people like to do in that field is still the composed this shell in a tangential plates which are going to be so that the sickness of the plate is built up and the size of this is going to be like skilled health scalability and the main as onedimensional and then built in the early dimensions and we can make a corresponding decomposition of F but they can't fully transformed respecting the before it comes from 2 debates and taking for this for illustrations that all of them function better it is being altered by it's done so then you have an inequality which is which is of the amazing is that you can balance the LDP normal F. the the blade the square functions here that didn't do not all European almost all of the death of the it do for inspections still there you can do that for me to about manuscripts along with epsilon can be taken between the small and that the school for the equals to deep was 1 of the main 1 and therefore picture there's something missing he appears to be to stick to the bigger than tool and then what happens this is automatic consequence by interpolation start completed the Buchanan dopamine but you can't open it a 1 will have the obvious bound but which this is what they care about is speakers to the present demands on the obvious bouncing between between that the and infinite no that exponent is best possible what 1 expects that this is an extremely hard problem is that in the subcritical regime you don't need adult mentors at any wait at peak was too deep as 1 of the men is 1 you will need something it's it's not clean bound to know what is remarkable about these inequalities that although it is a general fully analytical inequality its main interest are really their frontend applications and that in some sense what you're getting the past statements which by medical and for which before a there there was only not medically proven known at all all they are simply knew so this has an amazing however say has amazing unaffiliated implications which is 1 of the reasons why somehow in the past they were scan of a fatal believing in such a conjecture because it really needs to scenes which are explicit statements about said of Fontaine the equations minimal solution of the company questions which I don't know I didn't know how to pull for using some of the vise functions and in a way that the results of the 1st result is just a manifestation of the stick opening TOM if you take France's Jeannie equals tweet so the exponent here is going to be followed so what is known is that fall below the key dimension flat us that you get the control of the unfolding on a bailout that would give some to Lomond factor lined up to give some shouldn't be there but anyway the pool that is using some kind of enigmatic some kind of easy you vise of considerations towns of that these are really facts from analysis and which faces at a very interesting question is that the 1 would really like to know a
30:16
half 1 control finances and this is something I've been working on in recent months but what you get none of the electric it were not going to that is very limited reminiscent of another inequalities that there be no doubt also mean value human somehow what they started 1 is whether Levine and others mean value during this distinction analysis so they have a clear analysis conjecture that put in place which is to open in general would he hasn't quite remarkable work on it but it's only sold K Street not for larger so it could be that in fact what is lacking is a very gentle harmonic analysis principle yet occasionally getting a certain amount of funding consequences of that which which although no I didn't
31:02
want to work but don't want to talk to much about the twodimensional dollars for the twodimensional tellers of courses of different because very few matters points but what it comes out is that the old inequality Sigmund Koch about bonding the former made it 2 1 on that is still there indeed that the best 1 novels and in fact In the world of all flat delight the speak only uniformed Balton Bacon functions we have this is a load of creative saying that it's all we have which is while the shameful but that's how things are so what Elliott said couple of years ago we tied to study and this this problem of whether it's expected savings including mentions the what 1 expects is that there basically you will control uniformly all the moments and you will 1st stop looking at the 6th Small so the typical saying is that you would stop trying to con solutions of equations P want to speak to the speedy because before this before it could be 6 With beyond the lattice points along the given circle and whether they have done some work with Colombian and that which was some say well with it and we got some sayings but altogether was well below disappointing and so we tried basically using we could think about which could only limited success so what I'm saying is that what's the other surprising is that the common at all approach leads to results we don't know how to to get in any other way so there is no on cable results that using the incidence geometry on 1 hand and then also we looked at the so go false analytical approach you end up with problems analytic modeling because online so ridiculous so this is the kind of problem would say well the time is not ripe because you wait until people understand mobile dental politicos material really worth doing if you spend some time during anyone wonderful sense right so then we can eventually convinced of so that we got an interesting result so in any case it's kind of interesting so what eventually is state of the situation is that using the incidence genetic approach we can get the control on 6 known which is not what you'd like to like to put a constantly of that something which is better than where what we know how to get a ballet although means and that this basically is a consequence of the simulated told you know there are stronger incidents that conjectures in particular those of socalled elBashir unit distance conjecture of and has been quite a bit of progress on that but it's still open Elbogen medicines conjectural tell you that if you haven't points than the number of Union distances longer in 1 visit to stronger the eldest distance conjecture then you can remove the 112 would includes then about having control for all the where we can pool which for most seats no foremost is missing another really exciting because for most East while the number of lattice points is going to be on view only going to be this quilt of poverty of so you like to look at situations which have a little bit later where you have Molitor said particularly like to take for the small number in there well then we but basically what we like to do from the beginning is going to do something with elliptical what we managed to do was to pull that most small numbers and estimate that to enter gypsum so here capital and may be confusing capital N is the number of lattice .period space was multiplicity of spirits so we get that kind of inequality peaceful most small numbers and on Cingular's this is so exciting but say the technique which is behind it may be of some interest in some of you and what is the
35:16
nature of their technique is the following a Salt like a set you go bald follows laid down basically all looking at the equations a P 1 can to between the because of a given .period baby so what you're getting him of the equations the in the fall of unknowns so you can eliminate toward them then you get that kind of which is a gene is 1 girl if we assume that it hasn't signed integer .period who can we can take technology then getting the needed go and then we know from the general Julie that we can control b the number offers of Integral Solutions by an an exponential believing you would rank his bonded goes well with than that and Fulton Street well in that at the time people were convinced that things will no longer what happens here is that we have certain specific family so more people tend to believe may be that within this fixed extremities this is the expense of the bank made uniform involved but even despite what the recent advances in body gallons on with you don't we don't know that so we will not willing to assume too much we still want to do this and so what we did is the following we made a lot
36:48
say more than standard conjectures saved then stuff conjecture sings about the lanky and so what we assume when we don't have to assume 1 American involvement we assume generates annual so strong that boasts an abundance that conjecture that is not so much of the time and in any case so basically the only hold on whether 1 way standout way to get a hold on the line is exactly told this this conjecture that it would tell you that it is although vanishing when the cold analytic Frank of the corresponding there has and functions which is associated community caliph at the point was 1 and that is an object that has been studied and an explicit formula for
37:44
that which is a duty to work to very very explicit fall and it allows you to vault the analytic bank that a certain expressions which 1 can component and then you will see that you have so that the main problem is the 2nd time here so this 2nd Bell is the cell wait that sounded the capital excessive bonhomie to hear a so what you get here is so weighted sum of quantities which I call a set piece the Interstate said the III I'll while basically certain characters sums know that the problem with this saying is the following a just estimate this character sense of getting something exquisite been so if you put curious quilt of they're going to have something here which is going to be size well abound you get the scramble to whatever abound on be the bond on the being exponents and peaks and doesn't unions so what has to be exploited here is a double cancelation so we need to exploit the cancelation from the summation aches and also from the summation and there's nobody who knows how to do that presently so since we can't do that we have to do an average and so people have done that starting from the work of Goldfield a and then he's belonged considers exponential moments as needed if you want to control the number of lettuce was because of exponent of the bank but all these arguments are nice families now basically I made a statement for almost moot for almost all small numbers so what you're getting eventually when you look at the underlying elliptic curves what you're getting is a family which is not the enigmatic all this is a common at all family it's a little bit like what I was talking with before is that you have objects which are purely common at all but what is nice is that when you take the user's explicit form that you only care about the behavior of not speed for each individual piece so you do a reduction of his family will be so most of the work we had to do was to show that this common at family when you reduce it will be in effect behaves like like a nice family and that the then basically once you have a nice family but you can start using a new can start using exponential sounds exponential where Dr. Sun estimates although financial future a basically a tactile sense estimates on that when you rarely hear of it In a and saved my sanity that's all it goes over sp speed and now basically yeah you always have the double cancelation as long as a Jane event is not constant and this is the standards and so there we could make something work so I think this baby offered amenities observation that in fact you only care about the behavior for individual these kind of interesting that may also another situations this is something new something that works that have not been done before
41:20
now the the 2nd thing I want to discuss there is another problem which may be of interest to some of you which is what happens with columns mobile domain the deal and that being the case say the twodimensional flat the soda the general feeling of coal and tells you this is always a debt that and state has most and noble domains but then for plentiful plan of the domain Saiful panel member amendments is a pleasant go also there's a better result but players which consequences all the public on inequality which gets you that none of this year 0 . 6 9 1 except vessel character remember that number and while this is not the best possible inspector conjectural bound to wouldbe tools by 5 from evolved recently there has been some very small improvements on that have some results and then not something that is standard of where we go down a little bit it's basically microscopic improvement but say it may be the argument this is the most interesting part because what is exploited here with his exploited they have a certain stability over the the socalled public trying inequality which is underlined that and then also Sam all prison notable Baltic packing densities of disks and I think maybe this is the 1st application that this is a result that was booked on it for years ago DeSoto blind to their several papers which tells you that if you back to the success of years between 2 bonds say ANB we could ratio so be over which is sufficiently controlled then the density counted more than the density to get for the so that those and such as opening the way people care about that at the time but this has an application so at these are local bounced balanced accounts using intelligence there is a result of that is that there are constructions the same goes back to 0 2 hands maybe the annual showed that you can have the 4 of its last show a valued customer construction dollars situation only 2 modeled it's so know what they want to talk about is that In a subject that has been world of popular all over the recent years which is the Mogollon conjecture Sobel Vermont Schmitz there's you know that the asymptotic distribution of mobile domains of Celtic 90 false is universal and is this character that the conditions here no this is there's nothing pool there but noted that the expectations for the number of mobile domain sustained Kansas more than what you get so they can of interesting to title standards no is what happens is that OK so this is Celtic many false we're looking flat although not Celtic but there is this is barely principle that the expectation 1 expects that say whatever happens to fall for a conventions in In the Celtic Casey when you haven't had hyperbolic manifolds is going to be reflected by degenerate behavior of a functions on the completing the demand for and so practical ,comma dollars so these are many beliefs and 1 should settle at the different issues here so the 1st issue is whether there is really something like that and know that the mating behavior of 4 of the following fall on a conventions France's view take care the CIA but you can't take random spherical harmonics 2nd count their mobile domains is there leading any distribution when the stating that he goes to infinity so this is Paul as albums sold in a very nice paper with some equally nice follower in volleys the circuit allowed the existence the systems and joint will go the people of some American Eagle Whitman have filled a liberal this techniques in any case what happens with this technique is actually call for that there is some convergence toward him distribution but it doesn't tell you anything about what is needed in in distribution so at this point what is legally so rigorously known is that this that this average expected number is going to be bombed saved by . 2 2 would set which is significantly smaller than what you get from players but still well it's still about 4 times as big as what it should be on the other
46:16
hand that there has been a matter studies by various people and business studies they kind of predicts the number but not quite so that the tenant seemed always to believe that the indeed somewhere that this expectation is stonewalled the what is described by the collision model that it's not exactly that in fact there are no obvious units means there are flaws in the justification why this number should be should correspond to send to some declination principally because of the features of the different they know what they want to talk about it something slightly different basically no 1 liked to say that the indeed there behavior of being touted to the floor for Celtic in terms of Celtic dynamics said it had been hyperbolic selfless is going to be similar so there will be adding to the problem how can you determine mystically and implement this random them way madam In well that that's not that's not so easy In the last summer potential developments there uh no and so in some sense you could try to justify that a buy extending the the general belief say that in this hyperbolic world French stand to have a goal should behavior so we can we could build on that and make strong assumption that somehow what possibly justify to behave like an underweight models but what it terms old so there are some results which means there is a result in this direction which is not clear that which is a condition was the a nation's most should maybe not condition but in any case it a certainly not directly out of aren't to will go mostly but there's just something on paper is that the you don't have these link we phenomenon the number of mobile domains is going to infinity and in fact the results of more precise in they are they are basically the same may be the most interesting part of techniques which involved in this it's and on and on previous statement that it wouldn't say is directly related the government to make public well what the town's is that you can pull over the communist out the flat and after statements fear which are bolted from the the keys the 1st statement is that the fall in inner circle full density EV of taking a conventions any way you want not talking about random a conventional weapons doing is taking a convention say coefficients that you choose to be flat sold stake we take the all the same to be 2 1 so this is it becoming used the example of what happens is that you will have in the limit for the number of mobile domains the same isotopic as in on the way home if you want to be a communistic result whether you have the same statement inserted under certain assumptions on being particularly if it ends in a sequence of images a full of which the number of Prime factors remains bounded you will have the same phenomenon also so what we trying to mimic Gaussian distributions and
49:53
well basically there are 2 members the related to sings which going to that in the 1st East a phenomenon of making distribution of lattice points also so you looking at me like response on the Circle in the Square of AT and you looking at the and angle distribution of these letters bonds and you have an estimate on the discrepancy which is known gets you something and it basically has to do with the distribution of the Gaussian crimes in sector
50:33
now and other input that basically ignore it you want to work and to pull something about Gaussian behavior so you will have to estimate moments when all knowledge of the so what is important is to control additive relations in the frequency when 0 well well well beyond it being in the bonds will really want to have that I behavior so what comes out of this is 11th in the in the 2nd statement said that the mystic statement he said In fact much more should be told that say as a consequence as a consequence of the of the Sept space what 1 has a uniform bonds on additively nations and groups of all the tank so what happens is that the if we control the number of prime factors then we have a good thing our friend practice we have a global and then we can use this is the work of it was shaken by an Schmitz uh which will tell you that if we say the number of solutions all the unit over unity questions they want equal something solutions with that of the general is bonded explicitly in terms of all the of L and the Bank of the gold in a explicit way in fact 1 even expects to have battled bonds that would give the even better results without having to assume almost anything on the number of contractor's all but so far 1 doesn't have his uniformity in particular would like to know that the number of solutions ghosts say subject soup exponentially in the event that it is not known the bombs exponents no anyway I don't have a duty to question here so you would have to reduce it to community equations in this form so you will have an extra factor here which is the end which is animal blood sponsor this is an extremely strong statement because it basically tell you that you don't have that among the general directions In some Holyoke and put these things together Paul Whitson were mostly soft woods depth that you will have a distribution and that it was an idea of calls you can pull the Candlewood becoming sick implementation of these underweight models not the advantage of this is that unlike you in these experiments which will then made his various people even want to check it is unwise model No we have we have some construction that does not require you to generate to abandon coefficients and was never really clear to me how the general plan petitions anyway but let's say he had faced with don't have to to do that we can just a cold cough to be 1 the so censored on how to operate in a computer basically they had to rely on some help on what well this is not really a work that I asked the business during the women's gamble Houston what'shisname his name Michael McGee they did some subplots and well this postal ballots the square fuel square feet this distorts that don't
53:47
much so this is something that is maybe that'll visible goes Umali which is above the level said so you're talking about the black Elliot where if he is going to be less than onefifth of its interest in them the White with his small anyway this is not to say
54:05
then folded In the sort of forgot what other animals here single day the first
54:13
one square feet the 2nd 1 is not a square
54:17
face and holders of different behavior but what is quite interesting to me it's something
54:21
else you can do account for all of this noble domains and what the victims of is that basically the the number of mobile domains is pretty much in the ball game called The Book of Knowledge make it would be the all of this will mean you have you have situations where the 3rd it cosponsor a quite a quite well so here for less than 1 0 0 0 0 6 which is exactly did the constantly want in any case the number of fuel getting here out very well below the deterministic Singh we are getting from Tom Brady and all conjecture played Bond up pretty much in the game of Baltimore mission it and in fact these experiments there are situations where this let bonds are not so well can distributed so what they should do what is ask him at some point to to make you maybe also to 2 2 "quotation mark the discrepancies of the distribution of probably 1 no 1 will see that these numbers are even more and correspondence when discrepancies insist on getting quite quite small this amendment should stop you and you go to the 2nd goal more with the support we would use the book few question 1 yes the original the origin of the for information the tour conducted the Government of the Union of his generation the only reason originally right note no we just look at the policy Of course because of the nature of these relatively small numbers we don't have too many for most of them the number of the 2 of you know Iowa well above that because state if you assume something which is a little bit better than what 1 or 2 pools in terms of the defendants on the bank of the group it wouldn't be unusual anymore so this is not this is not really so much of an issue the question was involved with all of you pull something theoretical but then you may start asking when this this stuff really kicks in and the pulpit politics in a very low level which is kind of is with what they wanted to check his halfhour 1 has to go to see something that and presumably these things are at their wits end labeled them what you do with with random proficient because of cable looking at this monarchs and the great fissions with the good results of Hamlet or something frequencies and don't see how you generate hundreds of random questions this is just so you need to use some sort of random Solomon wonders does it so that this section of your career you in hallways the norms of the United which is a lot more than so the Arabian wise there should be some transition between small medium and large fees at holding leg wires going just on the number of it is good that and then you see basically whatever comes out it's what you get from checking what happens on zeal so you just look at the contribution of the major comes on the new it said this is it ,comma maybe this is not but still the chance that this is it this quarter but this is confirmed for lunch it is a lateral thinking that he met with the union idea what's going on so that announced the easing of natural EPA some 80 government native because many of you have not just where he have remained in the area and I don't know the Donovan experience all belonging to tickets so what's the question is what can we have some of the the love of the game give me the outline and yet well this is a list of 8 7 0 Of course there is nothing that sign this is so the very modest thing would be to so the 1st thing I like to show is that there is some that infinity bond then the clearly not OK so even that is not well understood so 1 they hope to have a ball on game days and in the Mennonite make cases is not known so what is known is again of wellorganized wasn't there but not not apologetic question which is how the list try to get something on paper for your 6 not this has been recently issued the chief value in some some welcome also can and sold each of which shall be based on mutual visits restrictions of the functions of the
59:44
way to connect to PC arms on the whole manifold and then to announce basic restrictions of some developments there that eventually led to a statement with you that it's not that they are not pull basic almost nothing is proved not even apologetic on admitted and indeed make case welcoming some of these things out call conjecture that the ball was I'm not necessarily the quality this is not me OK carriers also you have something to do with the loss of the use of well what using as a statement and using statement Indiana spoke of which is is modern and need which is based on a lot of people buying these days which the peloton encounters John all of which is the absolute to accept this year the cell I don't know about the ineffectiveness of and Constance what matters for me is that there is an explicit not only effective but explicit depends on them and that that is really that size and you get a depends on the length of the relations which is not too much of an issue what is mobile initial here depends on the length of his depends on the bank arch exponent in the bank and 1 effect believes that expansion sinks ineffective there but that will not affect the Bonson terms of depends on the so I think that this is this year its rises the level of his thinking about the insides of original resolution would you only have already submitted his memory ,comma relations the article did you want him on some provisions of operation so that number is yeah right it was not made think there you know what is enough to have a much more modest you and you probably don't have to to go to use system of this absolute such systems to deal with that because after all the talking about enigmatic about indigenous substance I was so mad about the students killing about the numbers knowledge of like none of this will groups of finite stated gulps of all the its symbolic thank sending the complex numbers so there is absolutely no assumption about city and still the course of this multiplicity that 1 imagine that it has approved possible that doesn't go with these machines so distinguished large Heights small said vessel that that's the whole process which when there is a table addiction that you can assume that these these guys you can find a model state agencies is kind of logic that general principle is that you can always that the face the problem equipment with the problem in out in front of a like number we don't clearly specifying what is the height of the number and then 1 has to stop distinguished so you would imagine that has to be proven that does not use them good that this is not I don't know the details of the argument that should have told you that right away is that this is what I had done all the question without getting to the point I never had that and if used to work at the liability has this here so when I used to work in the managers of the citizens of the region that interest over 1 of only you can hear the tho there well I should have told me about the danger would amended that and it was 1 of the leaders of the 2 years since he was generated from instead of what they said and so this this week we have a case of 1st of all it turns out that all these days they sit in the Senate In the 1 thousand of them uh so you can write the best equation and then there's parliament demanded there and this number is so it's 1 Palomita which is expressed in terms of baby remember P 1 and to press between across B and so we have distribution there which is all the while distribution but it turns out that when you look at not be somehow control that this distribution not the becomes basically the uniform distribution that that is what is going on and then once you have that then the saying is completely stand up because you can mean there is a staple cuts in the bulletin that studies that can't of questions so when you doubled cancelation and so we have so that's the next 1 so we have
1:05:05
this exponential sent here the unique we can't exploit the cancelation signing the but we can exploit the consolation that when artists are mentioned in some in some manifold unit be provided with women on constant invariant this is or a neutron constant and to bits of debris I knew that the differences in the demanded that answers to some of the lowest the authority he is the head of the Roman soldier was also wounded during 1 of their initial valve the men only begins good when you use OK said for what it's deputy pages out just to show that these these distribution is good not be which is not included and of course is not perfect like most is perfect but sings out good enough that it can implement the uniforms with different things and the presence at the moment I'm going to vote and years of age and so did the opposite what none of his business many of you go back to work to this this other tables that don't expose people which that that this made in any moment what they exploit that little defended yes sold so well with the man is only the Rosenberg the very well and sometimes can horizontal settled yet yet but we know who is unprecedented for such things as the leaders of the work that he survivors to work with mutilated in the newsroom In some cases could be something of movie about and that man method whose his of conjecture so there we would fix the elliptical and start playing to make an estimate of the analytic drank for a given Atletico by using a double cancelation that involve social the mothers well I think we didn't know about such results but may be maybe you know that there are such things so in principle there may be such things what you expected on my 2 questions of the day but I don't know who is the it was referred to the level of taking with them With cuts and he didn't seem to want to for him I didn't seem to have so much alike some kind of horizontal Saturday at the wedding which is which is not so easily in but as far as I know all this all people Goldfarb woman his the use the 2nd average averaging Parliament to the use of them and then there was that's much easier than using the modified to fix the model the the Senate and
00:00
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02:23
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11:38
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13:05
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19:35
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25:22
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30:59
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36:45
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41:18
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49:52
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53:45
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59:42
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