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2/4 Growth in groups and applications
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Title  2/4 Growth in groups and applications 
Title of Series  Summer school Analytic Number Theory 
Number of Parts  36 
Author 
Helfgott, Harald

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  10.5446/17046 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
Series
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Transcript
00:02
so if the you can and so as
00:14
say said yesterday and today we will start with something that is of historical importance the sum for that theorem on but instead of proving that and then using that to prove statement about Russian groups oppose show you how it's close related and in fact essentially equivalent to a growth in the offline groups of various nobody wants a simple because of the simple growth with very nice small groups and then they will show you how to prove growth in that group so sampling theory and its relation to someone groups it was historically important played a role in a simple influence along but will see how really there right wrote heroes insolvable groups not so much this so what you see some progress here so overfunded feels here to several people who live in the names of the bank at so Kwun Yum 21 so it says that for any a subsidiary of see if the star that is not tool large it has fewer than he repeated 1 intercepted them and you have that either 810 say meaning the for products of 2 others of fate is much larger than they are a plus 80 is much larger than they what they mean by much larger and larger by unexploded 1 conservative where there aren't any pence on units so for our refiners to that the fact that for a fine until you have such growth with such strong grows it's actually quite so but I wouldn't I'm not sure whether surprising might we are familiar with is fine but it's not from statements on so many years put here compared to what it means a financier and which we not talk about but basically when people were standing and the people are still studying which sets Rowan the addition state then we are nowhere close to being able to make growth statements of the sort at all but the unification of field you allow an either or statement I dare you have grows by addition or be modification then you are able to give rest their power cell disorder statement will be familiarity to a few of you at out because it was already known for hours have been known for a long time said it's an note so Freston was already known overall for that matter receiver derided his Eddie From a while ago the absence of timidity has been arbitrarily put they always forget varieties of seller but many of emphasize that if the you could say elephants be favoring it feels as mourners so they should be easier if not fact if he's quite a lot harder than our because if has a useful topology all proves that I know of that where Nova our user topology some sense and anxiety and the with if it is harder clause so of course in the streets topologies valid at not and also that the sprawl over peters out to generalize pretty much to all fields IOUs after putting conditions to make sure that they not a subfield if has 709 figures have feels that this is 50 people disk you then have to put in the condition that they not be very close to us have failed since very well so rather than proved the sum for the CIA and I will state the main intermediate results 2 water system for that ceremony will show you how to into follows actually from a statement about crossing groups said we will prove without using the theory so I recently CIA and if you wish for the lack of a better name tho the thing if it is to be found in the garden the beach opinionated says led x so that and the straight away is just as good for all intents and purposes if you have 2 subsystems so it's infected with a bit more generally even 1 of 50 another 150 star then Parliament 4 Y X whereby for 80 I mean people stayed 4 times as far away 6 plus 2 ways where they 6 it's going to be number of Parliament's is going to be at least half of the main all the exwife of course it's XY Cosby near the peak and that would be absurd to so in effects and peak and some 3 assumptions assume I'm excited because in the structure and some existing very well all right so what easier find broken hosted connected to all of these parliament DVDs so 1st of all let me tell you what's the modern perspective in my view is you should see these not as a statement about field or about 1 group it's a statement about 1 group acting on another world so all of this is really
07:03
about 1 so which Robertson which 1 here you have to worlds the multiplicative group and the additive probe the multiplicative rope absentee at the group the other way around that's basically administrative law the new Masisi health you can put all of these back in the framework of just 1 object if you wish
07:45
I really think this is the most natural thing so many media Fine wrote always feeling and hallways and defined this rebuttal from formations and the line so it's yesterday right it just makes sense and it's a very this a very nice example I'm always forced to look for a substitute a substitute for simple women talking about them simple wrote a song group it's Americanized small example of some of our group and any castle need Dr. be devoted several so there's rescue much of the maximum support and several so you don't need many of you are familiar with service that you don't need to you know what they mean U.S. needs it's another that these general terms from
08:45
which these are paradigmatic examples so either wall and that potent groups and Nunavut and ropes are essentially more complicated versions of the same and the basic ideas are already here to the maximum you people to several so it's the there were matrices which 1 of logo and the Ford Taurus he had the maximum tolerance while you're on the kind of Torres would be the least not the mass murderers is a maximum tolerance but any maximum tolerance is confident to so announcement Austin written in this way if you change the basis Miller told last time centralized server I would remind you that nation the centralizers CGD of liniment GE is simply to the state of all ages that community so on the it's a simple fact that can be checked quickly 4 the agreement do not in plusminus In general the right conditions for you to be revelers simple meaning having distinct eigenvalues but for here I'll see GEC maximum purpose In minutes early this 1 but conduit all right so we will look at 2 options I promise so the interception of unit sales which is just a group operation so that it corresponds stool addition if you wish see clearly from here and there is the action of the on you Hardesty Avenue by communication so this first to modification is the items all right these are the
11:34
actions of those studies Selinda reminded quickly what we saw last time namely the August license here sir efficient port utilization brief reminder of so we have a group acting on the set and then amend the legs of the set and you have a and B in key then the intersection of the Swiss stabilize effects were sterilized I remind you of prefect's discussed the centerfold no group state that 6 6 so suddenly so effects we have a large intersection she's a generalized service sterilized because we're doing it's not for subgroups assistant at these traditional but 1st sets as this will be the star 1 and societal also the decision sterilize reason to LA and as we sold yesterday this is really quite easy to people so France's knew that many consequences out of these quite easily said here I suppose that was due would prove which of our Hans but using pigeonhole essentially but you can also get it immediately as a consequence for which realize so let's it's the simplest this kind of consequence so how several of have a subset of of the denied playing that the intersection of the universe we stage he's at least 80 over arms where are it's a number of Costa offstage intersecting it's the holy approving I want to invest application of the licensee defined next to begin middleaged Sergey Arts on GH by that was publications God you just said X to be page which is the minimum that is seen as an element of and so you have that a nursery intersection they subtly saved his life 18 states from the service your orbit indeed and he something other than the that all right knowledge has worked with the specific groups cinema the devious nothing you'll be in a potent subgroups supplies with Jesse the quotient let being a substance he you with playing I assume that a snort containing because minus Q so there are at least some major which oddity .period applied from testing is 1 all Data NYTimes . 1 while throughout this process because putting in the minds of we just have these from the letter X we made but not until then a intersection you it's at least a over by a so you're going to have many elements it a cells might have no union potent elements and no elements in your favorite stories however I was it's uh just a squares the and say will have many Mapleton elements and perhaps even more surprising lead will also have many enemies in your favorite tourist the company here face
16:52
transplantation for even 1 incident ,comma all sorry I couldn't I seemed I hope my handwriting is better on the boards antifamily note yeah it's definitely but for which the Lord as for any jurors of the port of the form the whilst the effects of these case so for any exteammate which stalking you and that's enough for years forgets that minors over the enough that it's not being you all right so how was approved let a be various EU on which EC squared you because he's going investors by it's going to be at least 80 over a pint of raising hopes so conceded the action of the UN itself by conjugation the most interesting action as it was saying yesterday so by differences have 447 lies you have done a university study laser effects is going to be at least well but what is established affects the centralized current injection liquor division which just a centralized which is what we define victory so this is going to be at least a by it's of course is is your midway the seeds g Eckstein so it's really part of the controversy class I know now I realize that the choke is hiding in a box very well I alright knowledge as a finding the they you'll definition loosen in the importance of so what we already know that and this is because we knew that that was easy I was quite easy assist no 1 but now we have to work as they need it harder for the rest so we find in the I believe a university the effects so so as it was saying a effects the sorbet by communication this has a simpler than it is clear they affect 16 tourists said some change in the cardinality 1 will multiply the mineral Parliament we also have that Eurex exchanges East and animal wastes containing is said to contain a form because some of the additional forces things in nature and their innocence and it's also continue because it's something the bracket because he's so an of the form Y X the 16 versus wintry continued so the effects as community at most 84 intersection you but no played the 2nd that even easier have all resemble aces 0 applied To the action G to years later communication and then we have that 84 In this section you because you here from the stabilizer off the identity and you although the Cedar appointing Indian mother you so this is going to be the 2 5 because we are applying the squeeze before itself a and B. thanks to the 5 ATX and ATX is nothing other than firefighter it's a texts were excused you it is seen as the suspects accused you seen as the minimum all human you so AX with affair hominy decent animals have just by fact that said that ease by but it will last and conclude if he basically stayed over 84 but 1 which is what we want in cases sold making your head spin a little gift remembers so you are the things that I just have 1 of their own but on the hour when we play was brackets where basically using the fact that in the Rossini safe so it's all communities so that's why the bracket of 12 x white singers waiters ease of that for it's like a thick and tire fade just because he's so for pie of the saying his consists of looking up the 7th soap I have these since our but entire these matrices 190 view has something else CNA pieties these meters was still this making sure and what I said before that this could also be proven by pigeonholed was his interest in media because being holy fighting here but this single singlefigure follows so I'm there has to be at last age it is the most popular 1 there were the ones contains most elements of it has to contain at least a or are elements of of of and now if you take a fixed element of that class and posted held by all the other elements of the end of last year at elements of that and it's clear that there are at least a year such distinct things but I essentially redoing the proof of its sort of trivial on all right and what have we win doing here we were trying just to starting from when they were trying to isolate as many Unionport anonymous as we could and that's about as many as we can and we're entertained that there are more than that but that's as much as we could possibly hope for you have ace in different colors classes but it goes out that you can guarantee that the most enticing class the 1 where Iris 1 is normally known as popular in India the gross amount is going it's quiet here and the here so that the Saudis and the sort of statement we don't assume anything about how slowly rapidly grows
24:45
madam them among the more slowly grows summer the statement he's so effective at 5 1 in Ireland then you have a lot and a lot of elements in the section of the spread and the maximum tolerance very well the Our sorry it's not contained where I think it some variation in Cuba very well on right now he gets to the meeting so we have already liberated all starting from an 18 out isolates large subsets 1 made out of the Mapleton matrices In a square and 1 made out all stay on the made dismissals in a square no show how starting from 2 such states 1 of the animators is 1 of the new potent we can and get a load of growth Indian important effect almost as much as we could solve for the are fun different groups of think you a potent twofold the announcement favorite ones that have to be that can write that way and just say it's was headed for the years 1 containing yuan 1 containing and we make the leadership soft assumptions they use including those prescreening team safely frequently not the case with 1 of the keys guest adding the descent some say they'd entities 1980 used and the use of the identity of the grand 8 the square in EU To the 6 recently 1 I mean EU aid he says foreignsponsored almost accepted the sum for the CIA and found schools said it's almost has played to uses some products here from there because if somebody gives you a setting nifty star as its nifty years construct matrices having here the elements of the setting of the star has their arms and it this is here having the on their elements of the city nifty after a year's supply when you did psychedelia made the statement precisely fight has more space and more talks you'll get accepted the forms sum for the same year saving a few ladies and their ages but no need to do that and so on as usual we need peace here just in case any UAP gets bigger than which would be absurd seeding union this may very well so how do you put 100 proved the new we're not you going to use of some folks here missing clients prefer this passage equivalent the simplicity of history facing all right so How alone to do things well I support is the part of me but I think it's in the interest but that must have heard all right has approved golf so a people late is nicer because it's clear that a snow the faint article 87 variable ,comma and would call a people if the functions site area the that goes from a you team could you given by preview of the it's taken told you all the affair meaning you will also keep a clean so you call a aid the and a in you upheaval if this function is injected now we are going to have several cases so 1 honestly :colon superior but it's like a medicine it needs met people at an emblem of the world if you have a favorite sister pivotal ready in EU then could have automatically been knows that definition of vindictive mouthpiece that wall or find it sets said up such that the credibility of the domestic was a reality of the enemy so the content of the sequestered at you proceed which he's AU times all right so and so they take case which do not take the acting action would see actually conjugation yes I the statement that for the trend by aides to be here for you so 80 squared which he's a subset of the accepting money you make give us Canadian so this lives in India by the way as to make the tallest living so I this is going to be at least right because this is contained in that and this is he going to be equal to a to that they're very easy so alright but what if there is no pivoting you are aware that there is no pivoting you want to add that this is the case for I was sitting in the car this case it was the lenient the 2nd 1 the upheaval so In other words injectivity always fails so we saved you know what TV and leads to a key tool collide for a new East indicated he fails there so because there is no appeal for area there there will be 2 distinct areas that collide so for every continues there exists because of fears that existed you won 1 utility to the St. colliding for but all right but you also have an insistence dissipated beat sat there but as about the about as subtle as a linear
33:15
equation 1 variable which is exactly what to phase of this that finally accepted you
33:22
also have that for any of these things Pearson you have they can collide only for 1 day at most 1 so exercise but you seem be few undecideds Juscelino equation 1 variable which must have at most 1 solution provided instantly view or not and that is the immigration has serious to me solution that happens if new 1 because he tool and the glamorous conditions more like all right so the sense on the Net :colon maybe a number of conditions in the way prosecuting noting Eucharist for then put England ahead then you going to have that the sum of the total number of deletions of land in the words you you you hit the and 2 sets of views all AU and sorrow I think it's maybe this is slowly creeping dyslexia are AUA TV ads that have made this a year
35:10
very well thank you but with out all right so here you have today on terms they that it was given by None leasing parents so far any to any pair of another pair going to have happening is 1 solution but there aren't that many of those I am going to have 200 to have the contribution of the distinct areas right now right away I knew her reason why I was saying this is not the hardest days since for every you'll do going to have a k a he's going to have the bigger than you 18 because in case we are commanding the trivial collisions and there by assumption and the pivot so there are more than there is at least no 1 interviewed information and from this and that we have that this implies that's the few times lately it's greater than scored the fuel it's just that the large and again in the sort of problem of the pros from large subsets and easier in general than the problems for small subsets so In on
36:51
behalf of the country prophesy because it will be fulfilled in next but in the next couple but what they mean by knowing that behind well off when you have a problem says are these large that cases leading member series where this is exactly what it you can do what tall and 75 cases and the hard cases impossible but here it's Acer casing the sort of problems it's all fun deals was to attack it was a bit of pigeonholing Bush's strides and that's exactly what we are going to do so really there because of the size of seeing said this implied that defy the ban .period here is starting to smell victory so Tuesday a this most puberty so they a search that cave face minimal then they definition the meanwhile what would Mississippi so the the contributions and then you know you have no more than average because you're the worst maker she shots To keep everybody happy you have that of this will be at least AU squared in discredited and so all preclusion gives that feet tried a new 18 because of this and that you're going to have that this is at least 1 half of the harmonic mean last P and and tea produced at least 1 of them in its but not quite stunned not quite done why it will be done if anywhere in a year a you but the mind and the fact will win the or fall it minus the matter it what if what happens if it's something all in all rights you would say
39:42
faces in the EU it's sort of reachable but it will not be precisely because it is other people so what I mean so because these are the people this the Commission announced you have to Letterman said distinctly collide phrase then the money is going to be the same for Mexico because of what they said before in the 51 were to you would have known it will be the chance it will they cannot collide of unsold the following month will be indicative of sum up from the to U.S. well and this is this is injective and you can see these quite simply you what this in this way sexually just have a major escalation because then you are multiplying homemade takes you multiplied subprime left are 1 on the 1 hand and by our tool and the other 1 hitting the pollution and wonder of wonders it's something entity right very well In generally for deletion of big matrix proved to have the root system but it's also sort of trees OK so it's injected so what are you going to do with it so we like to apply injective maps because even change the size of things unknown level and to use the fact that the civilian so since the civilian continuing to be able to finalize his goal step unfolding good will see what happened so this seems make of the nation T 1 of you off the sudden just writing down definition of FY a here and he had this nation of side he said quotient of the distinct supplied the 1 you played toting the coalition so this is going to be the 1 you and the Sibelius I switch the order of TNT 1 very well army and look so any before was something unreachable right but the face but this is seen here such unreachable because whether we knew when we say that this collides foray we mean that you want the 1 8 he it was a new tool to clear right so 1 a sequel to an insurance is going to be our old friend because wanted to you when you tool all our old friends face a stranger that Marulanda tried but we're trying to reach this is between various the what so the very fact that these knots I would guys not the people at a with that's our analysis to express it in a way into himself since we know the team was the was rather we can apply something to namely the smile and get something we know we apply to taken up would only gets and winner so this is just a interest it to so here did so weak because even write them down just by applying them up this unfolding love and using the fight and has not committed to meeting due this month goes to migrates to the inside and makes a very nice all right this is good because now we know flight will play the EU Heatley is contained in a while all of these things things that are no longer the staring you would make the acceptance of the 69 the area due to the very well and as I was saying we really loving it he loves because they don't change the size of said along this so you have 18 AU for use by containment greater or equal to the left side so that the ceased because he wanted to recent dictated the sequel tool police and the city's recesses said whose size we managed to lob from below this just as a window I wasn't a pivotal managed to express woman about from analysis of this my base and as we note recently 1 but it were distributed to life and the while it's still would say this is the case suggests that we managed to introduce several concerns 1 of being this technique of unfolding right in palladium all right the
46:43
existing that since you met none in this is the 3rd case so that there are some people it's a not luck none of them I knew you wouldn't do that so in particular that there exists the event's unknown people since a newly expanded into cleaning while used 1 of the advantages of doing very simple some minus a indicated by assumption generates the Minifield entities it's a you generate so there exists unimpeded Plainview such that there exists D I need you for weeks Jason Peter Clinton so this this idea that the swingin only if you have a said that this neither NT nor does it occupied in terror groups and you have the acid of generator for the group the confined and the set and this at a narrator such that the narrator takes you out of this and this is great because turnout at 70 all the years that it allows you to use induction in the usual induction is the same insist that you have the narrator 1 of the state of India said so what happens then because GE said Beaver the AGE new Crescent he continued so what now so clearly like the element to 80 it's a pivotal swing to allow us to move things are going to get grows but we cannot say that we don't know where the 80's GA is not an EU aso O'Sullivan EU but remember Asan under yes from the fact that some say is that the people we can get our hands on to it In terms this unfolding because aides none of this is that appeal that it satisfies the equation involving all the things we know that was over there and this is precisely what we will do the unit so because these are the people 2 distinct you won t well through to to collide at a and millions sold as fresh fighting 1 t cool no applied to fight th beauty so this window is going to be large because society a senior active because Jason a people this is going to be the 1 you theology TO they almost as before the tool a few of the deal In variance and this in no way is committed to meeting so you end up having tea all of while precisely this thing over here and this will be TOO few to university 1 story before so we get to 1 you think you think of screwing burst into a room and sharing the the 2 of you to you very well and just as before we use the fact that siphoned injected we get less than 18 square in the USA nothing magical about 206 by the way we discover the constants that writing tool 6 takes place writing C 1 and C 2 sizable small so this is at least as we just showed all of these things using sneaky and EU cities at least 5 years teams now these bogus India TV doesn't change my tennis and USA .period so this is a group of people and the scene of the pro but even those who seems to be this time it's really important step when you're doing that you you apply stepping induction century and although you can see people inductive process and then the corresponds to the end of the induction a corresponds to the beginning of the debates case induction season really that it says here that you don't need a prefixed ordering for induction toward using a set of generators but well undersecretary will think we have that city some of these gives right away sum for that's the presummit the CIA and not in order to get gross newfangled we just have to apply this to their was a memo withdrew before guaranteeing I said AU UN lasted 18 the following will be delayed DVDs and USC 4 times before a continent's universes and identically under a
53:15
not contained in a maximum tolerance could well in maximum authority with the approval of the most groups the boat crossing then either a total of 57 not optimal but hates concrete and takes as much trouble as ratings he said 7 East at least square root of play Wednesday it is hadn't quantitatively optimal the most you can hope for safe will in you know couple of steps sees a power here and for aid to the 57 1 players 10 speed and most statement through here a poll of Hughes containing 100 thank you for mentioning number 2 at the tool to the something the other day which makes numbers and really really small very good guess at the numbers the mention of the ones get from this sort of thing after applied sort of argument too many times and many other arguments with so the BC over G word he's just some GE may need not use since there exists each Norton security we have HGH University in various and yet it seems that bracket so this is what they always mean byproduct of HGH interest this year because some breakfast injured so hence we have that 84 euros that empty and we a supply and not them play dilemma so obtained a you he cleaned and we applied so before fool he told intersection under new applied from Congress all right so this is really quantitatively best they said that this once by a phase of Site of the state to the excellent if what happens what will happen if I if they were a very small by a fate has to do with this consisted of of the identity and you will not have a problem in the offing group really you will discover problem in unit put and growth or what he said the same a problem in the group in region ropes and must be now grossly begin groups is actually you would say all wrote it's that simple and that the notices simple it's a very nice constrained sort of group strongly growth in 8th will be easier to deal with and this is slightly more complicated are fine but in fact now so yes the active case is the 1 that got so that was studied 1st at least 94 minute Oryx and is still being studied but less still known about them that is known about the ethnic groups and we still don't know what went to learn the when will we can guarantee growth by a power India a Namibian Indonesian rule below getting closer but but and this is pretty much the situation in general but the degree of resistance so slowly groups morally will always morally before doing something slightly more opening meaning of any overarching generalization muscle groups are like the ethnic and Neil potent groups and I am concluding here I like you would Heathcliff key so Namibian groups this is a classical these hot we don't know when grows by a powerhouse in many cases it does not on so sometimes the neurosurgeon group the cities of itself and that makes sense sort of of Austria worry because Nathan Roberston chameleon and means DH is not usually quently so sinks the world more rapidly UN rights and they think and these things here now and I am tomorrow while not tomorrow but next week will see a more complicated the group which simple but almost simple isn't too if and I hope to give you the manatee there and we will we would then Indiana would take a brief look at the other completely or cylinders for the final namely the symmetry group the question was made to have different ideas this summer all of this morning there yeah so good question was here Mutua doing the and for that's the only thing they gave the strongest results mainly out there you have dealt a good tool but you have several sums further right now it seems somewhere around 1 of our 10 for foreign aid in no bigger than scored defeat and fray bidders would just be a syndicated statetostate and stays investing
00:00
Entropy
Group action
Solvable group
State of matter
Variety (linguistics)
Multiplication sign
Water vapor
Analytic set
Perspective (visual)
Theory
Number
Power (physics)
Manysorted logic
Lecture/Conference
Field (mathematics)
Units of measurement
Condition number
Physical system
Addition
Product (category theory)
Theory of relativity
Validity (statistics)
Sampling (statistics)
Theory
Algebraic structure
Group theory
Local Group
Field extension
Proof theory
Number
Summation
Computer animation
Network topology
Figurate number
Arithmetic progression
Resultant
06:59
Flow separation
Maxima and minima
Multiplication
Lecture/Conference
Physical law
Right angle
Object (grammar)
Line (geometry)
Local Group
Substitute good
08:43
Group action
Observational study
State of matter
Multiplication sign
Decision theory
Group theory
Mass
Number
Maxima and minima
Centralizer and normalizer
Latent heat
Matrix (mathematics)
Lecture/Conference
Square number
Units of measurement
Condition number
Stability theory
Addition
Statistical hypothesis testing
Process (computing)
Element (mathematics)
Basis (linear algebra)
Set (mathematics)
Cartesian coordinate system
Local Group
Subgroup
Orbit
Arithmetic mean
Universe (mathematics)
Intercept theorem
16:46
Group action
Collision
State of matter
Multiplication sign
1 (number)
Sheaf (mathematics)
Mereology
Subset
Maxima and minima
Mathematics
Matrix (mathematics)
Conjugacy class
Manysorted logic
Meeting/Interview
Square number
Cuboid
Social class
Stability theory
Injektivität
Area
Product (category theory)
Spacetime
Structural load
Bilinear form
Functional (mathematics)
Proof theory
Arithmetic mean
Graph coloring
Lattice (order)
Right angle
Identical particles
Bounded variation
Ocean current
Metre
Observational study
Centralizer and normalizer
Pi
Lecture/Conference
Natural number
Subtraction
Addition
PoissonKlammer
Forcing (mathematics)
Element (mathematics)
Content (media)
Division (mathematics)
Incidence algebra
Set (mathematics)
Pivot element
Local Group
Equivalence relation
Summation
Universe (mathematics)
Fiber bundle
Gradient descent
33:13
Addition
Summation
Lecture/Conference
Large eddy simulation
Phase transition
Equation
Total S.A.
Set (mathematics)
Recursive language
Number
Condition number
35:08
Area
Series (mathematics)
Collision
Multiplication sign
Pivot element
Subset
Arithmetic mean
Manysorted logic
Lecture/Conference
Term (mathematics)
Harmonic analysis
Right angle
Sinc function
39:39
Logical constant
Sequel
State of matter
Multiplication sign
Correspondence (mathematics)
Event horizon
Subset
Root system
Matrix (mathematics)
Manysorted logic
Lecture/Conference
Meeting/Interview
Term (mathematics)
Knot
Square number
Energy level
Units of measurement
Area
Injektivität
Coalition
Process (computing)
Generating set of a group
Expression
Element (mathematics)
Mathematical analysis
Variance
Set (mathematics)
Local Group
Summation
Network topology
Order (biology)
Quotient
Universe (mathematics)
Equation
Right angle
Inductive reasoning
53:14
State of matter
Multiplication sign
1 (number)
Parameter (computer programming)
Rule of inference
Grothendieck topology
Number
Power (physics)
Maxima and minima
Root
Manysorted logic
Lecture/Conference
Square number
Units of measurement
Prisoner's dilemma
PoissonKlammer
Horizon
Total S.A.
Mortality rate
Local Group
Field extension
Summation
Cylinder (geometry)
Arithmetic mean
Symmetry group
Computer animation
Phase transition
Right angle
Identical particles
Resultant
1:00:02
Lecture/Conference