3/4 Growth in groups and applications
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Formal Metadata
Title 
3/4 Growth in groups and applications

Title of Series  
Number of Parts 
36

Author 

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. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

Content Metadata
Subject Area 
00:00
Variety (linguistics)
Multiplication sign
Analytic set
Proper map
Number
Manysorted logic
Lecture/Conference
Meeting/Interview
Units of measurement
Position operator
Physical system
Condition number
Forcing (mathematics)
Element (mathematics)
Content (media)
Theory
Mortality rate
Line (geometry)
Group theory
Local Group
Entire function
Estimator
Flow separation
Proof theory
Category of being
Number
Arithmetic mean
Computer animation
Equation
Right angle
Annihilator (ring theory)
Inductive reasoning
Directed graph
05:28
Group action
Spacetime
Observational study
Eigenvalues and eigenvectors
Variety (linguistics)
Connectivity (graph theory)
Element (mathematics)
Basis (linear algebra)
Routing
Price index
Limit (category theory)
Sequence
Local Group
Orbit
Arithmetic mean
Plane (geometry)
Vector space
Lecture/Conference
Natural number
Hausdorff dimension
Phase transition
Equation
Inductive reasoning
10:40
Point (geometry)
Observational study
Variety (linguistics)
State of matter
Direction (geometry)
Multiplication sign
Connectivity (graph theory)
Sheaf (mathematics)
Similarity (geometry)
Propositional formula
Insertion loss
Grothendieck topology
Number
Hypothesis
Tangent space
Frequency
Maxima and minima
Latent heat
Manysorted logic
Lecture/Conference
Term (mathematics)
Interface (chemistry)
Saddle point
Position operator
Physical system
Condition number
Covering space
Area
Spacetime
Element (mathematics)
Bound state
Mathematical analysis
Bilinear form
Flow separation
Field extension
Category of being
Arithmetic mean
Turbulenztheorie
Vector space
Hausdorff dimension
Universe (mathematics)
Linearization
Right angle
Figurate number
Family
Resultant
Inductive reasoning
Directed graph
27:30
Point (geometry)
Musical ensemble
Variety (linguistics)
Multiplication sign
1 (number)
Theory
Number
Derivation (linguistics)
Manysorted logic
Meeting/Interview
Subtraction
Condition number
Injektivität
Spacetime
Process (computing)
Independence (probability theory)
Line (geometry)
Tendon
Local Group
Flow separation
Field extension
Radical (chemistry)
Summation
Lattice (order)
Hausdorff dimension
Universe (mathematics)
Mathematical singularity
Equation
Right angle
Matching (graph theory)
37:08
Polar coordinate system
State of matter
Multiplication sign
1 (number)
Parameter (computer programming)
Grothendieck topology
Maxwell's demon
Fraction (mathematics)
Maxima and minima
Manysorted logic
Square number
Physical system
Area
Spacetime
Median
Bilinear form
Orbit
Flow separation
Proof theory
Hausdorff dimension
Order (biology)
Equation
Right angle
Resultant
Sinc function
Point (geometry)
Observational study
Divisor
Civil engineering
Variety (linguistics)
Connectivity (graph theory)
Distance
Trigonometric functions
Number
Gaussian elimination
Frequency
Latent heat
Root
Lecture/Conference
Operator (mathematics)
Lie group
Subtraction
Field (mathematics)
Euklidischer Ring
Units of measurement
Condition number
Eigenvalues and eigenvectors
Forcing (mathematics)
Element (mathematics)
Physical law
Independence (probability theory)
Division (mathematics)
Volume (thermodynamics)
Mortality rate
Line (geometry)
Multilateration
Set (mathematics)
Inversion (music)
Algebraic closure
Inductive reasoning
53:05
Constraint (mathematics)
State of matter
Multiplication sign
Moment (mathematics)
Element (mathematics)
Complete metric space
Pivot element
Functional (mathematics)
Grothendieck topology
Proof theory
Centralizer and normalizer
Arithmetic mean
Manysorted logic
Lecture/Conference
Operator (mathematics)
Order (biology)
Condition number
57:46
Collision
State of matter
Multiplication sign
Element (mathematics)
Price index
Parameter (computer programming)
Pivot element
Number
Voting
Meeting/Interview
Dependent and independent variables
Identical particles
Thermal conductivity
Condition number
00:03
as on and on and on that at this
00:14
stage of of the methods of not for the unit itself will reappear today but we will look
00:20
at the more difficult problem that the including the role of the system was an example but mostly everything say to the devices the prove this In symbols union groups a groups being made groups groups of the tightest affair would say almost too because they to other like I was quite simple almost simple that easy have portions source of Russophile bond indicates that simple so will listings in full force but that guests will that will just serve to make some simplifying assumptions such so what are the main so well a basic technique just a first one was he which will see the sort of work it escaped from some varieties the weak for Helfer pointed out of something would use will prove useful practice skate mainly they mention estimates about them later this initial assessments will lead To what will 1 of the centerpieces of the entire of all proves ever since System tool east of the idea that you could find chloride info basically old fluoride you find containing many elements of Europe all of your life finds itself about the number of times to the 1 year to the work abounded and the store had used for so was to have the store right with many elements of faith in them so authorities have said that the 2 of them in the majors or some conduit thereof but that's so was to have many amendments with the you don't use them for pivoting and induction much like what we did yesterday with "quotation mark looked yesterday but it would whatever it costs to get an positive Friday at any rate no time for a joke involving that I undisputed induction to well actually was a case of Spanish poet in the 15th century said that after being released from the importation engines after several years of torture and that was seen as case of great for me but it might have been the substance lines so I think it would give from about so I love all right so what do we use skate for it's itself used to show that there are many meaning as many as servants of the physical elements of a more intense itself about the number of times the generic In some sense some given sense what I mean by that what just about any given sense but what they mean generic practice outside proper variety for a sobriety of positive contention while Canadian more concrete while saying we want we want development all of forsaken by all all of this on the secondary condition in the sense that not only are most elements for the great maturity of elements of that kind but it's the only way not still have the property is to set aside equation the Battista being a variety of was couldn't so we write an example so this is what America means so for example a generic Jean insulin is relatively simple meaning because this entitlements said to me very very concrete ABCTV for
05:29
example has 2 distinct eigenvalues he finally gave the certifying equation which very simple equation the pistol so you don't decide your generic if you don't have the so escape from some varieties is something that is used to escape from special cases that want to bother about all right to hold the view that while it escaped limits at the bottom of at the space we will look OK Let W B A sub variety a proper so variety in the illfated means for composite indices seems our vector spaces irreducible means all components are of course of course we've got a mention that he's the very mention of each component is lower than the image of interest otherwise somebody would be 1 of the let there be a subject of the prosecution simplifying assumptions made a statement of such starts the orbit remember these your orbit effects this would happen strikes when you multiplied by all elements of faded who might want a modified by the orbit is not contained in so that he sometimes find sequence of phase that will when multiplied by each other than by X will give you something outside of so there he is there it's the subway to escape from the idea that told you know the at least constant and say elements of both the proportion of governments these the off a plane the solve 1 we have that support have escaped from the liver and on the entire Constance here depends only on the the greedy and they mention of that the study is made up of several chunks and indeed agreed to the amended and elimination of full of them and on them all right so as to achieve approved so a felony just sketched approved for a piece of me very special but that's what was the nature of the DS generalize the city was to put just 1 x ingredients just let WB adding years space more not that led the Japanese yesterday to make to be completed is as clear as possible Celso polls look actions as opposed to the prom for old W. of or they mention the basis of induction it was by the about dimensions Zito that you can do yourself so all very well we'll see what happens if WE processes W for all the main while the Cesar case that will not happen in practice because it either you have that's the exists knocking W for all the that Questar that evidently you're not even stuck in this space so didn't even need use your automatically escape from it during the conclusions referral Genie all or what happens or did you have the key to the existing W for some the name has sold well a the SEC generated by a the group generated by eighttenths Jesus a group generated by an innings seized the 6 GX and then Jackson W. and you're assuming that are that is stable and that the action of elements of face and hands of the action of the route generated by the this have to be contained W but if this was his assumption is functioning so there is some way to escape from value however long however rare and then we will
10:38
show that is shortened ,comma
10:41
so his contribution to the assumptions so we can assume that the UW that in the press that he wanted so what we have that we let the pride of the sections of W. G W then they mentioned of Prime and of the so but inductive hypotheses despite our assumption here and we have that's while the slaw dimensional saying that prime something from which we can we can escape inbounded time and many ways for many elements which was but that's all right but this implies the statement here the fact that they check 6 nothing W. crime implies that the formation of the new prime that either each Texas not In W an HGX is ascribed to the pitch here on H no excuse me the strike because hideaway checks is the thing that you're it's knocking solidarity in the 1st thing to have these things basically we is modified with the invention of the so 1 of these plenum systems in that but know that because for every element page that escape we get the minimum tailormade for university that escapes from that a Of course we could have some reputation as a guest house them so evident on right so that's about Tate's whenever reasoning airspace but how did we use linearity ever well we used is the fact that in the section of Alenia Space on another linear space he said in space of northern mentioned for the end of the same off but in fact the that's pretty weak consequence of being in space you can do it you can do something similar for the general we still have that that view reduce the very mention of every component of W. intersection GW the smaller than the mention of diarrhea and the number all the components is founded in terms of their degree and so on of the company said is just basic rights you have the Arizona I the end a variety of it is going to be a lot of people mentioned the 2nd statement it's just secularization of the fact that on the play a cover of the manager of the deepfried I'm going to the most adept in the present .period injured would you have is that if you have some variety of some of the tension and the variety of half of all the degree the enterprise and degree of the degree of intersection for some definition of degree would be opposed the all right but if not to reduce will it's just a slightly more complicated statement but this idea that the number of components of Muslim degree also suggested that depicted on this and other things just get the picture right so this is a variety display state and have another variety and the intersection east of northern mentioned it's enough to make the election works in 3 goes and transferring the induction all right so much crime alright but what makes mention of the nation's most folks from the DVU I'm not around for how many things can even speak of that was you got some special cases would be lower bounds which protect press I wanted where you the Suffield EIS assumed to generate just fatal softened a little hundreds of years have variety of key they were so we have to come in lower than we would like to get the win that the intersects with new bigger than you would expect it to well what would you expect it to be this huge you expect I mean if some something of the federal learned of a certain size recently dimensions he expected side to be about the control of the site citizens here expects this to be the image of the government of the I expect it will be about well you know community about the generates but this will be the grows strong statement will be because have a cable was in the yeah and case Constance and of course the other will be and it will be only partially attained for some turbulence that you have to of owning the other direction also this would be called Proposition 1 because it will be entertained 1 was this achieved for GE almost think he can be seen from a single delivery so the history of the statement and it's developed independent interlaced soul I was always working very well the where when the coach the old about the the stingy my apologies I think you the study all right during the future but the all right some of rights sold out when I was working on these things but at 1st I was for special W. ball as needed and we knew very few now United will prove a result of the sort 1st of foreign UN GE and RCA had to do more and more development techniques for
19:12
there was always much homework and be involved sense that something speak painful lesson peeing on they have proven this sort of statement in full denied the fouryear there are but for a very every specific interface India several they were not In the business they were in the business of even as asking for approval father the justification so on what happened here said .period saddle managed to strengthen with a whistling and are case of reopening tell using both basically it's a combination of tedious and both so both of these things have in their statement very well unlike right so again I would start working was a special case and we will show you that we haven't you we don't use is apparently very special condition that much and consultant this immediate clear this was the case of value when they didn't it's enough to construct I'm not aside from dedicated to its place itself how many times the cuisine of the times told you and canned or what not an injective such that's what I was going to be the return date it not we want them up to my sinks in also in view of the of course insisting that bows and only if the and 2 of the beta decay face some constant why do we won then we would have well injected maps again Endesa kindergarten idea but it's very powerful invective maps of how the property that they make use of the city's same size as the main so that if Kerry already construction it's going to be tool the human side of the image because indicated the Times and the and the other in the media give him something is something that has no moral and that that the the the the size of the an analysis implies previous saying that these books and records the sees a close aide to the case to the 1 already which is exactly what we want because he's still the scene of values which is 1 of the ways meeting all right ,comma legal that settlement hard very well all right so it 1st of all this you wouldn't really really really needed and it is not it's enough if it's about not that if I had to wonder bounded 2 1 in the sense that every animal has supreme interests around the world and with the greater that we can soften that because it allows an exception to wishes with officials enough for every gene you pay the pianist undersized or even death but the intersections section was everything except so on this side so the WCO here the other beauty thank you I haven't even if things to do with that the reminder of where that is Seto areas the severity of the view the purpose of variety this is the future of the means of positive containment and so the fact that the family the twoseater these snuffing the days when there was a onedimensional because it sobriety is going to be contained in and the Union of things of the form just a point everything so would with these guys .period to UK has speed some Pennsylvania many well sold whole going to be fine it's into the fine start here of the reference was sent on his Zito W 1 that is minus 1 tool that Seto times due want everyone knew 1 in various times and so on up tool here are the winners 1 W D minus 1 Jedi mind 1 interests very well but what guarantees the ice will have to choose them will choose answer the map would indeed be almost injected although we do that well what you will see now knowledge will are you going to use University in there and the wonderful thing about varieties meeting saying and reducible space you know we've used to that cases such as this figures to show that the appalling flying Linotype writing you have shown the variety of the whole space and if this is the use of the overall dimensions so it's going to be an exception all right so I assume for simplicity without loss of of generality clean that the parliament they'd introduce contained on so this into because it will be easier to work with the tangent space then so let the tangent space To that view ITV denoted by Gothic W which I will on the the notes in turn by curly W because they don't know how to dropped out of the W I have the Roper across W no
27:31
golf in the audience theory then the tendons spaces To the saddest thing the 1 W 1 the tool wh between various etc I get the call of the attendance and so 1st of all I will have slightly more modest goal before going for injectivity I just want these the mention those lean years faces no the lines into face to be independent to being the defending the 1st of all and they find the ones such as different from that of the so the question was the 1st of 4 east of such a thing come and disputed case for the virology and you know it can be the case because my assumption was the single most important groups the duration of the music business have some idea of what it is and you're not going to happen all right for you should have
29:29
1 right the have but these communities contained In know beforehand because if this were the case then the other the space will be stable and accommodation or some substances I told you was handled and so and so there are the ones have to move meeting minus 1 such that well the 1st thing you independence on the 2nd 1 is that in this and so on so the time of the right image of the full dimensions and very well of course the the ice vests exceeds the Irene Ng but what happens around for the best means to be an independent so the fact that this condition is true it is as if they were saying that a certain equation is not fulfilled all determinant recent Zeidan those in other words the good in the eyes of the ones that live outside of our overriding we have shattered yes shown the varieties not all of the dingy dingy Tennessee because the ice that out there that are made to give you something independent do seized London overcame so far a progressive variety think that is a promise of variety with that of a procession of severity added that such that its values to locate the contain all points of view of Kerry there 1 of 2 main In aid to the over 1 such that starts terrorist all right very well so we have that's the but no what do we have well there's going to university get again because you're going to have what does it mean that stars toppled it means that this modified Jason singular and the origin of the deceased city it's a terminal it's the has known Xena terminal so far 5 seen star we have that the fight at the origin of the trading date indicate entity is announcing he knows he has eaten but the condition of of this sort not having CWA I want to raise I want to be conclude because it's on the not passed without derivative of the 1 . 3 million a year it has the will the derivative work at the most every point in the sense that my definition you have similar derivative on the variety we have here shown that it introduced not in the variety the sobriety leaving the itself a few times something irreducible has sold the variety must be of course difficulty mention what Donovan said he soon seemed ready for all that outside well some variety that is going to have to occur because it just means not writing by having seen in a proper variety moreover the Museo Katie's not all of industry that is exactly what we wanted to see exactly what we were looking for and we that that's the weak and the time this is going to be known as if you have a match between 2 spaces of the same mentioned on the map is something of outsiders for me than the prematch point is going to have the agency and the nation's Ito noted that among the degree because all of these things have been rebounded sums of each other and the variety of the musicians you with on the degree use this point and just about the number of people which have stalled right the mention of Dudley was once so what about they mention of
37:09
W greater than 1 was surprise surprise into an induction they mention he given examples but at the the really has immunity as an example let us have insisted since existing example I can assume that the equals 3 . 0 so now have been to Houston's street like a the Lincoln have in their fight is going to go from crosswise to you I can define these the ways white 1 tool why ego you wonder why want printers he was alright then what do we have we have maybe not that between 4 . 2 east city of about a number of points of his intention for the invention white widely used threedimensional but you will be able to show that comes we can show that fight in various this onedimensional rightly mentioned for 14 of they space during the week because here so took 2 2 2 2 the was so what does it mean that fight all points he go away once the study this means that way Seto you why 1 he won inverters because you will these given so the use of either so why Zito 1 lie on the variety W Of course they mention well what is the conventions numbers independent operations fight and specified that really me the number of equations invested here a number of the amendments servicing of containment but since White because White he of mentioned for this means mention 1 just 4 minutes when he quit for when the state was once in White was quite hot something onedimensional with at the rate of Florida mentioned until the very lucky that it's 1 arm so what we can do now he said we can bounce so that it here PCs Y intersection is copy that almost so we can bound at the intersection of civil right we have gone down from 2 to 1 deficiencies at the Cumberland Island and so far I calls of 20 senior away 1 it is going to be sold me yes subjects "quotation mark society's suspicious said it's always from exceptional said so this set the premonition of why 0 why 1 for why was seen a white 1 not exceptional is going to be what it's not going to be our on right it's going to be the site of the demonization of almost entire said is going to be what's it's going to be not a year over the intersection white square that will be fewer injecting when you have that the printing the image has been much of the mentioned 1 and moreover that this sort of thing applies the mission for not special the intersection where the sort of lacked the political W so and he said most by the argument before aid to the olive 1 to the injured since 1 of our street is so they mention of divided within inches of the dimension of later mention of the so we get the intersection of median 1 of the he said most we figure square root .period I you have 5 also placed CO . 1 the deceased's containing the 2001 as usual and here you have the moxie of a intersection expertise here recently at news value we have shown precisely the except most AU falls of 1 onerous to fight there was no need to draw the line because insistence that very well that was so we have doubts hence the intersection where the basic most aid to the old ones at 2 % these was just they mention of wire were the mention of and this available so I have given you the simplicities of induction the disparities paradigmatic it always goes on display and so the only onedimensional case with it in full and they showed me how to go down from 2 to 1 dimensions and knew that he did that that's how you reading in you their mentions unfortunately the nation's saying again crucial very well so that future result of what they call tied 1 namely leave gives show an upper bound it gives to us and um it tells you that this at a distance of more elements of the intersection of the SIP aims of variety this that have enormously now can we give any sort of
45:07
happy down the law by showing you that there are there are elements of sufficient for well for some special forms gets thanks to the orbits of races here we can show that for what is right this is a centralized surface certain you're going to have that abound any 1880 you're going to have that the they mention of the intersection of the centralized services this initialize remember all the elements that come into it disintegration is a variety so this will be the mention of CO Genie Over the mention of the times a factor that is very small when they grow slowly made mainly in the hands of the health of sentencing even needy selffunded display on rights 148 what I said ,comma satellites much temporarily and growth of excellent flavor my orbit satellites into Gration 4 sets of the last time we have that think intersection which sealed the exit 6 page beacuse she's going to be a success of cheaper these almost all right he was making a variety of La facility closure the severity we they mentioned what well would you would expect the mention of the minus mention of centered on what you doing all is this while in order to get the lower bounteous supply an upper bound to the denominator this this boasts 8 to the so they mentioned it is heating later made mention of but we already have we really know what elimination of sea of GTE's on so you get to that the intersection with just divide made by dates and we get that this is the cleanest claim to they mention all C of the the divided with emission of the times what any of so no 1 was in the state and set specification of Robert sterilized all right but what kind of money you hear the difference there and I felt for all Britain and implements which manages the elements that have anything to say about that because the conditions which so for the relatively simple which the miscellaneous he said the system was the name for hiring distinct eigenvalues that these Greendale their Lisle what we have always have on the main forces we see them make resisting taken volumes the elements that compete with the idea that they on the so that if we fancied recalled that the orders maximum COT commencement Torres so the shows in particular that's in fact for any told us that the system that I softened element of fate you going to have many elements in the Taurus in history the centerpiece of the proof funding has always been so this sort of statement that let me call the statement and from now on the reached already states the Tories would hamper the most so the statement this was already but candidates nothing to you but they prove that the weakest segment and fortunately OK thank you the number of of components of yes if not expect size of what guests to administer this was that we think of the bible for most of the farmers many American speaking I knew going to have this sort of thing that these systems have made things more complicated in the way to and all right so we finally doing this just about enough fine proved that fajita units was less tool but the same EDS this is a modern form so that the media's work for all indeed almost simple In a clear field of areas of the value of generating new trade policies said that can be softened we assume these 2 it's not really necessary there either 8 over 1 of the marinade the 1 placenta all 4 8 to the over 1 the must from this restraints and so not have time to produce less strengthening the guests tell you that the statement was to have almost entirely mentioned that the 1st time you have a that a couple of times more is going to be viewed territory the fact the .period have McManus may be surprised to see the of all right very well
53:06
it's approved on parallelism
53:08
the proof that the last time the before the end of completing and the ringleaders and you could be put before the things that were in this way some of people of the and you're right I either addiction so I assume that the deceased listening to the 1 foresee large and the small so this shows that the 2nd pushed conclusion should holds key to the 1 East almost out of control by the state limit their ceased some seems write to conditions so that centralized consumers someone Torres meaning is consistent with their images some basic caulk India people just like just like last time he shut them up sites defending slaves induces I from this is the deceased's team clearly placed ministering to geographically Lespinasse said this there in order to avoid the sort of troubled that the money was referring to mother plusminus the on button so you know we call this activity feeding the induces vindictive function this when this very well on all right but this was really beginning to many Colorado 1st victory since the 1st case his occasionally gets really lucky like last time state and we do have a pivoting there might reached for a here for dinner there are many elements in the AT and T the dingy minus selfdirected he'll go this coming from the operation and this is course the astronomers think exist for completeness but that's in the hands because since disappeared at some of the subjective we have that parliament strikes early in he said 1 coming from the plus minus the constraints of almost at this address largess is the main the sees a constant to the Forestry Minister of Delta very well at the same time we have that this is contained In something a 35 so we hesitate to the 5 is really big speedily to aforesaid Minnesota and eating contradiction to our assumptions for this moment
57:49
so they say was easy but that's OK speak case these are pessimistic case which turns out not to be the hardest case creating an indication there
57:58
no pivots whatsoever in the entire universe but but but but but but but but but but but but but but but but but but but but but but that's so because there no for everything he anything he feels to be a people that the side there is a condition that cleverness collide and the fire department fails to be passed on to be active this just flights all of them to keep cool and is justified side so it's a failure from the Committee for every side half a year from utilities but then came the same as saying that a towing versus 81 discussed minus so suggest before the failure of conductivity pragmatist and what he tells us get hold of something so to speak His because this means that for every CSI they can get hold of it in a certain sense because that's Sangeeta 17 community to 1031 various such steps and it's not Fabio such that side the and peace in 18 residents say no at this hour the only place where I was use really simplifying assumption that is Soto says that it's not to date indicate will mean that sidekick saying there is no fairness in a simple Of course In the case of the you would have told argument that the same these and you will need to deal with the with the Office of Special K using for the complete and so the the service industry simple as those the centralized server of Of these guys is the poorest itself exists side he claimed salt Lake which terrain which tore there are too many elements and that's because society exciting there is using the name the state because we have monitored all using the it's known people that we have that by the restoring them up there many elements all the the tourists strive to excite investors in the so what we have here there are many distinct alright Securities in but the intersect on the EPA's miners identity so you have a lot of stories all of them all of them are the plaintiff accommodate entity so you've got a lot on a lot of things over the plan and and eroding into the 4th time this number of stormy times states that the number of things we have in each tourists going to school we didn't appear respondents a so get this a very concrete the Sept peace squared the once there it's going so those because the size of the scruffy pick you know that this month the consent of the facilities when not using anything political about that and it's being concrete and so this was to that so the fact that this is bigger than that you could be here this user contradiction to the assumption that they barely grows analysts say almost as large as interesting very well and we we are left with the lastplaced inductive 6 Casey when there are the votes and none .period and because a generates you there must be a for as in Batman there cereals alright "quotation mark next time we will do some remember everything about the reason for this is that on the last day in the last "quotation mark thank you for your interest