1/4 Motivic periods and the cosmic Galois group
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1/4 Motivic periods and the cosmic Galois group

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Part Number 
01

Number of Parts 
04

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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. 
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Abstract 
In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights. Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties. Many Feynman amplitudes in quantum field theories are of this type. P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'. One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power. Topics to be addressed will include: 1) A Galois theory of periods, multiple zeta values. 2) Parametric representation of Feyman integrals and their mixed Hodge structures. 3) Operads and the principle of small graphs. 4) The cosmic Galois group: results, counterexamples and conjectures.

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around and the and in the end of the movement who were an there is a need for a the end the the war the ground and it is not just about so thank you for coming I appreciate that as expected there is a mixed audience of mathematicians and physicists so His 1st lecture will just consist of an extended introduction with lots of examples and unexplained for mathematicians order Fineman into go it is and why it's of interest in number theory and I will try to win the secondhalf illustrates for the benefit of physicists what this Daloa ,comma Gallagher does and what a very powerful tool in understanding amplitudes also appreciate that you might not want to listen to me droning on fatality so this lecture you'll see a thing that's gonna happen of course so this is just an introduction and examples so the the goal it is to define and study a guy was very balls finance In a completely rigorous manner be no nothing conjecture about the on and I should really mentioned my intellectual debt so that inspiration the story behind this is quite a long 1 but on a bit started off with calculations due to "quotation mark Western crime the nineties and think 95 who found out and not for the 2 values which all the fire later as altitudes in Marsalis fight the 4th the sudden physical therapy calculated from OK some of the devices a certain numbers and in a few years earlier this year the horror and felt independently and that's exactly the same time 89 I'm had developed the theory of the movement of limited fundamental group he 1 minus 3 points and the sort of philosophy that emerges from this is that these numbers not positive values are intimately related to a certain group called into gavel all of a sudden category called makes the motives of the Z which at the time didn't exist and all those conjecture at that time I so what happened next is the cavity In and around 1998 said well we have on other in physics related to numbers in mathematics that are related to the Gallo group well could there be and he coined the term cosmic so that it can do vehicles make Gallo group that's that summer acts on amplitudes and corresponds to this uh this group here which read which was conjectured to be underlying the structure not positive so he made some vague statements but he she and invented the work of McDonald's another 1 contrary contributions due to consider 1998 who suggested I counting points graphite the services of a finite field which I Wednesday March and that it's proved to be however a useful tool in in trying to understand the structure of and I should say mention work of honor :colon and mentored mockery who in 2004 and 5 I wrote a paper in which they constructed what they called the operation voters ,comma because it's different from what I'm going to define they constructed a cosmic Golonka I'm on it is related to the renormalization group so I'm not gonna say anything about this underscores the tool is just so that you are aware that there is a phrase out there cosmic our group in the literature I don't know of any connection with 1 2 I would say anything about this and then there an important contribution To develop and Brosnan In 2003 I which started to cast some doubt on whether opportunity in this theory was in fact not positive and is a tool that yeah so they founded graphite the surfaces foreign general
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type that was when you take a look at these things come points of a finite fields of and cut the services defined from confinement grasp you get pretty much anything possible but their examples as you mentioned was physically convenient unrealistic so the corresponded to cross which would add never occur in any quantum field theory this is followed of the road the following year in local history was not few cases that so that no until very recently there not infinite families now at the beginning of the end of the it is began they all come to their results on later in the 2nd half they computed numerically many examples of an cottage harmony but a convincing number and they found that they agreed up to a very high precision with multiple victories show that numerical fit policy of holding these prove that they improve the
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acceptance on a handful of cases that the aptitude was actually more positive and it is verified numerically to high accuracy the board has ,comma thought all come to that later this is of being sketchy about want to mention everything the term that came before and I'm not important contribution is of a paper by the offer In and ,comma and in 2006 here I'm defined what they called the motives all of a sudden family of cross in this series for primitive Cross and in this particular theory 5 1 4 dimensions and and there's been a huge amount of recent work the I'm not gonna say very much about a torn and frayed time is shorter than mine the my expected I will mention names my collaborators under reach Doran and Panza on August nights and candidates the on enormously influenced the way I think about about this but the the bottom line of cookware lot body of work is that we now on that there exists amplitudes and fight the 4th in this particular theory which are in fact not support expected not to be multiples the values so this initials of correspondence to doesn't really work and in fact this story is even more complicated not because not all the most positive values actually occurred as ends in so it would seem that this whole project this and that it
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was found in the during the coup yeah I think that's a false man but you have to teach an when as an Indian enforceable so the initial analogy the inspired to call his cousin got a group has fallen apart because of genes in the last much more complicated than not for the 2 bodies and and Fox on the story much more and could than anybody imagined and by the point of this costs is to say that in fact despite its complexity this idea because got group polls some sense and that's what I want to explain so in this cause and we will define I an affluent group scheme this is slightly United will really be several front groups can for the introduction this will be an unfunded schemes the cue to seek legal group and Associates 2 A. I am certain families all Simon amplitudes the extremely general by the on the the the the sudden
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would reverse more restrictions so these attitudes nonfactor Vol Beaver general depend on opportune asses an arbitary mentor cities such families a fireman amplitudes Associates a on metallic period and those rights are a subset subsided and change aren't you and so on we will be able to speak of and the Gallo conjugates the energy of all right and passes memento for any element in this group so far from actually get something needed
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something new so remark used the word Motiva Inc ,comma In hommage to gardening because all of these ideas really come from 1 of that is playing a role in this and the theory will be entirely um rigorous and Bobina conjectures except to database and conjectures the restaurant so as part of a special case of this theory physicists and we will retrieve the notion all also the symbol amplitude so this is the common industry industry recently and 100 physics but the symbols on a defined for very particular class of aptitudes which happened people and women defied the material .period is will be holds everything underlined the symbolism is a much weaker notion than noted .period so we will achieve this and be at this stage and often Out of this theory as a side effect will also pop out the renormalization group and in particular the conference hall thought about the process of normalization and the furniture John explained that the this will come a free although the precise connection between the actual renormalization group and the cosmic out group remains to be worked out and a 3rd the consequence of this theory which is what I want to talk about today Is that implies a very subtle I'm recursive structure on attitudes I find it gives phenomenal constraints on what you expect to find as opportunities but in particular has a practical application to study from here said these were the most beautiful and I don't know and so the wellunderstood deserve some opportunities and it was forcibly animals and my understanding is there's been a huge progress in understanding the into grants organizing integrin structure for an acquittal animals for which it wasn't they partly out to write that down but the actual integration we actually integrate over some domain is still completely open from our wanted the importance the law laws of the land was posted on a lot jumping ahead and yes no it's OK for free that just over it that they will be a quotient of well define a representation of the target group of a category of realizations I work with a cat curator market attitude of realizations and that has a group but every time I give myself some accounts must pay geometry the group action that will be the correct 1 what would she would is expected to be the correct 1 if there wasn't a bit Catskill mixed motives it should give exactly the same amount it's a way to get around all the conjecture you go now asserting his motives or I will explain all of it the 1st is Britain with suggestions strength Figueroa Europe can happen because it is a scientist at the system 0 the both invented and destroyed because the gullible
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all and this is now finding grows so I will consider I'm scalar scale quantum field theory In Ireland in Euclidean space so to the deed 1st the new ideas and new number even number of spacetime dimensions and so the final graf time will be connected graph later on I might make them not connected with the today they'll connected so graph G it is on a set of vertices a set of intelligence and Cedric some alleges for the years and set of vertices the G R I intelligence cities up has a vertices not necessary distant I'm very often these edges be labeled and security wall yes absolutely and we can have a that and then we will have some external hostages Richard called dialects offices so we have crossed and topological data finance this kinematics data and 1 a particle masks and he's so Tiberio number call every In June and 2 a and incoming momentum if you are buying which is a vector In the oddity would is the number of spacetime dimensions for every external subject to amend and conservation this is a dishonorable incoming Amanda that is 0 and and sometimes only to specify which masses are going to be 0 and which aren't my course which memento for allowed to take to be 0 and arts and so to do this it's convenient to draw massive lines massive edges that is to say those with anything on equal to 0 as African lions because it is an example of phonograph me maybe I should have President Soeharto version Q 2 Q 3 and Q 1 and this means that we have masses and an end to which will be nonzero but the mass of the 3rd particle here is 0 and momentum conservation tells us that Q warned prosecutors Q 3 I was in the form of of a dozen motto is in the formerly room square of Vermont 1 area times I will always replaced and if we have a graph with several incoming momenta Q 1 Q 2 optic you and we can always replace it with a single external Lake which carries the some of the because if you like this defines an equivalence relations crossed but this is I'm missing something funny tedious but it will be important for later on time for once went to nominate we can specify the set of pages with promising the set of vanishing In masses and mentor some says all the thinking each is gruff comes with the data of which momentum masses of nonzero and if such family will define a maturity period so what I mean is that we will view of the Fineman aptitude for some about to define all I G which the function of these masses and the
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momentum as a function so here we we should say we mean function it can be Multivalued in general that means it's a function on some choice of universal coverage of some space and maybe it made me well illdefined so many Internet everywhere so when I write the fine manageable sometimes and I will not assume that convergence beer and a formal expression Soviet function on the complex points for all of us and I find some variety of em change Annex V for its use of his right the set of masses and Momenta where the moment the masses earned a 1 1 0 for all In Multan the animal and the Momenta will be enough find the space no 1 0 for lies not in the queue archaic and boasts "quotation mark subjects school subjects to 2 madam conservation conditions so this is sort space on which you want to view and the firemen aptitude is it's the 3 conversions will have singularities this year I'm not certain here although and yes this is we specify sunset of edges for which the masses vanished some sort of matter then vanished likewise the moment but I understand how to do this is to take a limit as a mascot 0 was a momentum going to vote for a tricky that's a question I want to dress a tall and so on to specify who 0 news announcer and some of this is that the Euclidean regions so what it means to be an Euclidean Appleton and costly space Is this the set real means we restrict of real points of this variety we would focus on how to define the employment opportunities cemented to jump in and define the firemen amplitude directly in Parliament exposed which is very oldfashioned and on younger sisters so maybe not so familiar with additional that the derivation from the momentum space representation is an old hand the trust me that it it works and it always will so this is the sort of 1960's presentation of fine amplitude and involves grafting good food so they aren't spanning tree to enclosed the 1 million take is some grass Armed with K connected components which of the and it's the eyes a tree and it has no no lieutenant such facts the vertices of these trees cover all the virtues of a graph the vertices of G is the disjoint union Of the verdict said to reach team of the Union of the verses of the teams and then on some terminology to internal PG Of the graph we associate what physicists call issuing a parameter offered the from those who define a couple of foreign nationals chief of the
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annual growth going on here is often called the 1st semantic phenomenal and it is Britain side G because the science moreover over all spanning 1 trees and the graph and then for every such tree you take the products some of it is not in that tree about and so this is 49 In addition parameters and house coefficients and Z spanning 1 tree the team is expanding wanted something versatility to equals T 1 this is span spanning 1 tree according to the definition of yeah the firm and then now I the 2nd among them and that I'm going to about foray she Q is not going to depend on the Momenta it's a song Over now spanning 2 trees all right at morning at this time finding into trees 13 and to take the products of the edges not in the union industries after the at times 51 squared where and cue T 1 was sick of the minus Q 2 2 it is the total momentum that is incoming the comes into 31 on the square inches of new difficult to say and if Wendy dimensions to Q I has components Q 1 attitude he did say that the squares Euclidean are you In an Instant fact which which actually give rise to all its arithmetic coming out on the field there is the fact that these polynomials here and here have is to coefficient and that's the fact it's almost never used in physics and so they have entered the fact that this has integer coefficients the very important because it gives all the arithmetic on and it's almost never used example so do this graph again 3 the spanning 1 trees all 1 2 3 2 and 2 1 3 and so the 1st semantic for nominal kickoff for a nominal is the sum of the compliment the products of the compliments of the edges in each country that is just after 3 it's just for 1 offered to and the spanning to trees all times as 1 find this isolated verdicts here this verdict here and 3 arms and then there's the text message and so on for Figes Q is equal to so we take the momentum the total momentum going into into 1 industry so that citizens text that's that's just Q 1 and then we take the products foliage is not in the span inducements offered to Of course if I the momentum and into the other train you get to keep his Q 3 but because momentum conservation on just the same thing as if you want Harry gets on the 3 squared off on offer to the last roughly here Q 2 squared I for 1 half this because it's a very concrete .period anonymous coming from grass and some remarks which is the is always homogeneous finance degree there is HG which is the and better number 1st but the number of the grass In the standarddefinition sister and I will think all this the number a number of Intergraph just the dimension of H 1 an anthology as a function of the this time homogeneous as well the degree 1 more we turned finally we let the find a 3rd polynomial the local side I don't think this is standard terminology to call this excitement and suited sigh and commentary will be the 2nd Monday polynomial the sum over all In internal alleges mask word of the age times offered modest but but
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so in this example creates a Florida passed on enough along the south to this other thing we want to use it on the other side of the office yeah I said this in in yes in the office and to attend on his another financial which will be constructed out of these plan onions 3 so this is the fireman integral in Parliament to form um yesterday the rights Ng for the number of edges of any damages and across the time being and then the farm integral RIG 4 of em Q and it also depends on the dimension I really don't bring a fixer mentioned for most most the time it's gonna here too integral saying and you did where on and Q it is 1 0 over the 1st someone's appalling Arnold to the show times but engineers a powerful Ng minus edge to overturned this is what we want to take the dimension to be even to avoid having squared everywhere times Omega the Sigma 1 on coming to and FirstEnergy some energy is some particles 1 to minus 1 the I Alpha Phi the Mets and made off with the and remind Sigmund it was some locusts and projective space of the mentioned Angie minus 1 Israel .period so the corner simplex the real quoted simplex the men go right here but the Sporting News stressing what is the 1 10 projected Hornets it's the region where the alpha energies to offer by and radio and monitored this summer marks or so of course I should say this this and maybe an illdefined and could may diverge most the time it will diverge badly at this summer remarks and is the underground Ermengarde and here is homogeneous degrees 0 that's a small calculation we we know what the degrees of society and fired told you here and plug into the Formula One and onehalf to check that it's homogeneous degree 0 in the office and once you've made the remark that indeed the integral doesn't make sense the projected court you don't like that you can always restricted fine shot by setting 1 of the officers to 1 for example and you get a standard and School of Law the and minus 1 or something another markers that and the amplitudes In a general I need not necessarily scalar quantum field theory a much more complicated but they can be expressed In parametric formed using similar into girls but on ways enumerators then you might as well be some sort of polynomials and Alpha's the coefficients insomnia some Clifford algebra or something and the point is that the the the
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geometry of the firemen integral I will not depend on the the new writers so I expect and hope that everything that I say can be extended to more general quantum field but for now and it's not much of a restriction to consider the scope for this region yeah but you be we arrived in you for all of us I'm not sure I mean I'm not sure I assume you mean that Sony drive this doesn't doesn't exponential and um that use this chemical is gonna come here but this is not regularized and in any way that's if you take that if you take the that the momentum space definition of refinement integral and you do the trick I need to the army and momentum schools using Gaston former forecasters could you get a form of writers is hold yet that OK so that that's how you part C of E the miners something on and then that's that's when you pass to regalia so if you work in ah find space you get collectors say With the 2 something and then you can do 1 more interview integral integrate out it's a minus proper level times and under his homogeneous you integrate Outlander which will spit out the scanner of function and you get a projected this is 1 1 stage further than the normal are fine and you're thinking of the the parametric in school the dealt the delta function and unremarked that maybe not all mathematicians aware of is that almost all on the predictions fall and collider experiments the FAA obtained and from computing such quantities and that's why the calculation of financial this isn't the enormous enormous industry so perhaps default breakages give Simpson's 1st examples of opportunities and try to convince you that there did you get interesting quantities come from so here some time so the random selection of examples from literature I'm sort 1 what are the sort of things you can get the wanted Graf would be this triangle Graf reexamine earlier With some choice of Macedon lamented doesn't really matter which other examples would be polygons like this remain a mentor this is indeed equals 4 dimensions then all these families of firemen to growth I'm it turns out is always expressible and tentative functions and the logarithm general right an Israeli 1 of tax because minus log 1 1 6 and the dialog the 2 X "quotation mark thanks to the case the case squared so this is an that he was the 1st to the service but there is and there is that it will provide 70 Geffen don't know what is explains so you need subsidies to function to describe all the fun amplitudes at 1 and where the arguments of these dialogs among rhythms while there will be some complicated the governments of Eastern complicated and algebraic functions after square root 2009 all of the masses and the percentage mask Western this I'm sorry so that's really wanted 1 on the
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2 groups gets more understand on 1 example that's been massively studied in recent years has gone long history is the sunrise diagram again with musical 3 possible masses and unwanted momentum coming in so we give you the growth for stressful fun you queue he squared off for 1 2 of the 3 and this is a very long history bed the upshot is that since this is families of financials given that dialog with them and the most recent work on this is due to Adams governor invites you so in their references as the full history this and family of interest I'm but the amount is that that the general to .period grounds general trade advantage it is our duty is not known so what I mean not known I mean is that it some function which doesn't have a description In terms of many mathematical objects and there is a vast array of examples which can be expressed in terms of .period logarithms not reporting over them for the literature for searches full such calculations behind and the other interesting examples among the 1st for myself at least 1 such as these family of grass the so here we need to was 2 dimensions this so and edges so these was talked about recently by David protests in this very room not long ago they take you could not so moments I'm a emits 2 2 draw the external momentum and it would have little small line that illustrated 0 and all the masses but equal to 1 of the Baptist 2nd these lines 1 there is not L can be anything yet that was to have attended tuneups and then to the twostory stocks because we don't we're stuck I'm not knobs just a different from the example so figure on equal to 2 and so on so here that that point on oxide doesn't depend on doesn't ban thing now it's songs the also of life times PCI G against formative years pledges assigned 1 of the we are fighters so that the 0 locus of these on polynomials defined interesting surfaces that have been studied From mountain of politics in some cases and here hearing some examples of the corresponding opportunity I too is the integral former attitude of I B 2 1 and I B 3 I think that was awarded to the report is 3 times the director L. function for the unique surgery character March 3 which is which is nontrivial offices 3 times some car and and square I'm hands we start again once the numbers are the floor equal 7 times each of 3 at a value I think beyond
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that they're not so these assigning numbers which are very interesting the number of arrests I'm in fact there are variations on this graph France honors and awards relevant to the study of such prominent roles which experimentally mainly In some cases conclude deals a whole array of special values all of El functions for more forms at at someone uniforms for the price of 2 the comments of the rest of the so wonderful continuing the main example of emissions missionary have a 10 minute break coffee so this is this class of Gross's Caldwell divergent precisely because the gunman the effect it will produce a poll that but this notation that is right by G to be what the scum factor was the remains of this integral and it is on G "quotation mark square and so this is a number if it converges harmonic convergence on and only the for every subgraphs the number of loops as for the number of ads is a strictly bigger than twice number every strict subgraphs answers this means that the big offers primitive has no subdivisions and so an example the example of a graph which is all divergent as this 1 so now because there no masses and no external momenta would play a role I can drop in the pictures so this is certainly not divergent it has foragers and tuneups but it is not primitive because it has the subgraphs 3 4 contained in it which has 2 edges and 1 and 2 times 1 is too she said violates this inequality and what is a common for this this is what is quantities relevant but in this case we get I say we get a real number functioned anymore diskette number so why these quantities irrelevant for physics and because they give renormalization scheme independence contributions to the beat functions of history you can always say also the 1 I write fivefold that means in graph verdict on this simply that this the growth has no vertices Paulson degree they didn't equal to 5 so the valency each vertex most 4 OK so here example the leaders of the calculations originally due to borders Yukon which which started off this whole business so wanted there's no 1 question this graph minus opportunities 1 the number again and 3 lives there is a single example all of a primitive log divergent graph which is the wheel with 3 spokes and its amplitude 6 times 6 each 3 and at Fordham says a single example which is the wheel with false books I guess 20 the 2 5 a 5 the several examples I mean but 1 of the most interesting ones but there there are couple of others and this gives us 6 Zeiger 3 squared that's the that's the square of this fund amplitude and really understand why that's the case Ahmad 6 lives there's more examples but the most interesting 1 is on this graph which again was completed experiments the way back when and that only rather recently has been approved on the and so many others I'm opportunities some
1:00:41
complicated is 27 Over 5 the differ ,comma 3 please 45 before 2 2 5 times at 3 minus 2 6 1 over 20 years each and so these technicians were 1st at the borders ,comma numerically up to a certain degree and the Desmond spectacular progress In recent years ,comma too much harder border and proving the actual quantities rigorously by the time the tension but the 2 different bodies that there is a lot of national the but yes 44 with respect to right this is a it has 6 edges so it's 6 lamentable for 5 dimension into groups and the government has 60 tons Yukon computer that way it's it's terrible so you use different techniques momentum space so you use a cycle the gate about expects technique and you expand in terms of certain um give up the Mules and Men accelerated convergence as a whole industry of situation which John to compute numerical but that's succeeded because there now and algorithms that do this using the parliamentary representation cases accused millions and so on polish that a different approach using single value Moscow put ovens so now many of these can be done much more efficient and exactly all families are nothing new 1 he is yeah these all these alters them then and there are examples now up to up to much harder for the so there's been a spectacular progress in recent years from but I'm further illustrations of today out of just that these examples OK so the 1st observation is that there are 1 so there is a need there were no infinite families none but now we know that there are infinitely Serena an infinite families now some which explicit and some of which it is approved by General fans which almost positive ideas from what was it about you is this nested song and the 1st defined by in the 18th century on but now we know this is an extremely long story about which I regretfully will say nothing there we no longer expect that at some time and it expects all these amplitudes demand for that is that there is no longer the case and we do not expect more of the 2 bodies in general so the question is what not so fond of another known examples teachers to convention and where you have an evaluation of the firemen integral and something like an on some way all varying a variation of this definition we could agree to unity and standards transcendence conjectures predicts that is not possible to write it for more drastically and the examples you can myself motivation as lilies I'm proved that the armed with cruise the graph officers modular so the piece of the commodities at the mountain 1 of and so that's that's favor system much more so catastrophic this yeah but it is said the same Islamic state media this is history community but done and wonderful on changes changes the UN the the title number completely so the question is then and 1 out in the same general Was there left to say In contagion advantages at all given the viruses the examples and I the 1st thing 1 might toy with his sons of invariants and weekend variants all of them not for the 2 bodies like the weight so the way to the multiple value is the sum of all of the for the arguments and so that 7 of the weights on on these examples this is shorter trying to compete if we can announced some way so and if economy here the weight and another corner here twice the number minus the way is it is 0 I and here's this candidature examples will be more here the way to 2 3 according to this definition is 3 and twice 2 times the monastery's 3 I'm here the way is he defined as 5 and this constitutes is to transform I'm 5 new to 6 and this quantity 7 at times this the grower has weights this month was devoted wait 8 and it's minus 3 please 9 so already we see that even looking at Avery crews have environs like the weight we have some sort of Upper Bound which is twice the number minus 3 but in these examples here the bounds not attained and we say that these examples have wage dropped so receded down even understanding awaits is a tricky business but I think I should say the reaction done with Freddy well distance the setting so the weight is the 1st on hand or something he limited government so the prototype for an inspiration for this entire subject is the following experiments try to complete the go action on fight the 4th of the apertures which we know in 4 3 and Exeter those which are and infeasible close friends entities so In the end 2nd lecture next week and I will define a hearing
1:08:31
falls inadvertent ,comma Mattel Inc periods it's an invisible admitted I wanted them anyway so we have a ring the school P M sub page for explain next time and it comes with the following structures it comes with a period homomorphism from these elements of this trying to actual complex numbers then it comes with and action also are fine group scheme Q the others called so this group will act on this train and it has other structures and in particular an increasing weight filtration so in general the waiters on infiltration but in in the examples of another look at the most positive values it will happen that the way to the grading and we can speak about the way so the small abusive terminology but through 1 about to say we land in a substrates where the way a grating and so I use the word weight as if it were greater than in general wanted remember that it is not he noted the nature the finer next time it will be here I can cafeterias of realization yes but that that will be on display in and what a Gallo theory of and the Jews means and I'm not to want to explain how it would explain a lot of the structure that see so I think this disease examples of very striking and what the motivation for constructing so examples that things ,comma material multiples he values to define a few years ago again they actually they actually live in a sobering of another ring of periods which injects into this 1 but I'm going to identify them with image this arms but the addressing thing that drastic I'm so the objects corresponding to most of the devotees and the period it is Vermont wasn't a case of unwelcome with group well we can define the god of conjugates balls the announcement in this range so this will be constructed next week go ,comma this will be the new combinations of the G where is in rational point to this group and that for every In fact every comments will generate a finite dimensional a dimensional representation that I will call the Sox side of this group and we will get a representation wrote From this asset group about which we know very little but to very concrete could the matrices so the idea is that you want to replace actual numbers with representations of a group of this idea should resonate with physicists where I'm ready replacing objects with representations of groups so it turns out that we know how to computers a lot about these representations in the case of multiple for the so Salemme and give give examples with us in the case of some simple positive earnings so the 1st example is Clemens so the Gallo ,comma good and even the the justice was itself the this case on the queue it's a matter to land and if so how does the the Group G act on this rhetoric the value what just multiplies it by some number the 2 and so that means that there exists there exists a homomorphism From G to GM which called Amanda and so on every even the 2 values get scaled by land to its white so this is kind of a bit interesting 1 To get something more interesting we should look at all the devotees nowadays the at the double conjugates former space founded by Warner ideology divided 3 did not twodimensional representation times g calls is on all the divide you it Land achieves the same landed before To the waves lost the company
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Energy Services thank you so there this Gallagher with a delicate if you like takes an auditor value scales it but it can also add a rational number 2 sets are transformed and Xymenes means of regular representation which in this basis strategy 2 awards clearly and G and this basis will not to the matrix 1 S turn press 1 landed the 2 entries .period G so we should think of an odd and even the 2 values a onedimensional optimization and all the divide is a twodimensional art and science excesses of his of a function from G 2 Q it's function from 132 . 5 9 that is a function from I student this 1 is a function From Klickitat Q and and this thing is a homomorphism this is is a is almost present in the additive groups redirect much active group 2 these functions on GM and this fall to a group of men in this group a matches properties of era this is 1 didn't work the 2 and 1 not I have 0 yeah visitors who want to blow the can year the fight the idea that used to train useful in many contexts considerate to the actually arrived at the Florida Museum thank you hold these days are different from the I'm so that we can take that we can take tens product of representation is renowned for the 1st 2 examples time tells us how to With the Gallo Konjic it's all of I am product of an old Zito value time and even the dividing wall solar Gallo ,comma visavis its sounds from the 2 previous formerly it's too to carry press 1 the same thing us what occurred on trumpets 1 the Zita today but as follows from multiplying the 2 previous examples together so I'm particularly Gallo ,comma gets golf and analysts example theater 3 the 2 tiers all up and into the vector space the 2 to times and what you can take 2 to 3 times at 2 and there's not much of the group which sends it too therefore the 1st to really interesting example it is the 2 3 5 and you have to trust me on this but there's a will there's a way to complete this this the Gallo ,comma and it's it's a big subject to take for a long time to explain and we won't need everything but the the upshot is but the gambler conjugates power and the number itself z 2 3 and 1 this is not a and the other reason is nontrivial she follows from the fact that the divided nonzero John yes this is foolish enough because there's appeared homomorphism in can so now we get a representation on this vector space strategy order very times In this basis L and the eh minus basis we can write it as a matrix 1 0 0 but and on cute None of the gates this is best 3 this is minus 5 times S 5 students in competition to be done to obtain as minus5 and there's some new man from GE and severity not from GQ to Q which recalled S 3 5 such that this thing forms that is is a homomorphism of groups so what work the Yosef Fink yes 1 goes to 1 of the more likely it is while the leaders of the universal right this is drowned maybe I'd like to to a lot the of the horrors of legislative I'm not tossed about in my left on top them yeah we can discuss with this guy or all or the convention and a young young adherents OK and then in the With regard to him 1 in the 3rd lecture In the 2nd general tried to set this Bob said always has a rigorous meaning and so on works and the furniture I will define a on the period the energies to the limited all g in particular for example for energy on all the time we looking at here so primitive an overload divergent and fight fall but it will be much more general enough but was looking at this case now so we will get on an analogous
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construction IMG In the same times such that it's period is the final attitude toward which by these assumptions converged as I said earlier and the key observation which is the whole purpose of this series of talks is that the substrate sponge why the matter from it is time is managed and some precise and but the story cited more complicated than the 1st since is actually closed under this Goshen and so the goal is to make his precise in some
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sense ,comma Qatar's original dream was that the on the day of the fun out there that there should be a good bet farmland students and it should be the same group as the 1 acting on what that since we have all these counter examples in this complications that can't be cured but the the point is then there is such a group backed opportunites but now's not talking actually preserves the space is no reason why should take amplitude and take its gobble conjugal which should still be opportunity and that's it is not the case and is an absolutely astonishing fact and and I will not try to explain to you why it's such an astonishing fact on these examples so this this fact has extraordinary and predictive power for the opportunity so the purposes of this store was ready to try to motivate his entire construction with with the example Munich so as a and to test out the validity of these ideas let's take these attitudes which are computed is as what and it but it'll end belittle and everywhere that's reasonable because there's standards transcendence conjecture from what the values I suggests that the on the period map on multiple alignment not for the devices His injected and I so that's not madam does not request a prairie so that is assuming transcendence conjecture friends of these amended following experiments and so then so or whatever said he took his vast data of amplitudes renowned In before the 4 3 INI computed the El Gallo conjugates of all of them and he made following conjecture based on that numerical evidence which is that the periods I'm sorry that the the vector space span by into the capital for G 8 has above primitive Arevalo divergent and for the 4 theory what is clear is closed under go action section so this is not yet it will be written up in a paper with and from the Council of belief made many contributions so let's assume this conjecture through another thought experiment that's go through the consequences of this conjecture Safonov let's just assume that the conjecture is true and draw consequences From last and that's assuming that in all the examples of it and look at that we know but the amplitude is a multiple value some facts would their theorems that will guarantee this In the Union for the families of examples so that's that's not much too soon and that's assuming in these examples that we can predict the weight and that's also the case I have effect with conscience and which we explaining how to predict the wait list amount agrees OK so arms company so we need senator synergies samples machinery noted opportunism opposite ideas and we know the weights small much so it is a table here is a table runner boards How can have ask residents here is a table of modest positive ideas up to wait 8 to the want to these are the numbers which expects 2 finalists low degrees coming out as amplitude and fly 4th
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and here right basis for the space of Matera Kansas City's cannot wait some weight 0 this is the number 1 and like long there's nothing In waits to this he committed to nothing else she desert metallic 3 nothing else might fall the Atlantic false and nothing else and wait 5 dissident 5 incident Inc 3 times the tentative it too my 6 6 and intimate Terek 3 squared In May 7 the 7 maturity 5 the intellect to it's not like 3 this is committed 4 I'm weight aides we get z to material 5 3 this interesting the foyer at 3 5 but this is the a after assent and the deleted 5 the tentative at 3 some intelligent interrogated the top then 3 squared to it's 6 5 detected 3 and the 1st submitted it not for the device that is not a product of things which came before 4 3 the yearago net so we have all the stages of numbers and we have and conjecture and we have a graph lets on what is this put the pieces together as soon as conjecture His a time 1st of all that is no graphs His amplitude it is a not presented too the can't because because we know something about the way the world crosscountry graph with 2 and a half years or it could correspond to 3 graft which has a drop in the way but there is only 1 Friedrich Crawford said so there can be a graph His period is a 2 2 the man with the the begotten conjectures says that's on the knowledge of the law I'm not saying I just assume under the weight today I know that this court has weighed 3 so and that and that no grass in between that the 1 often 3 across this so there's no space to have attitude it cannot easy to tune in In this theory and that implies that there are no numbers whose conjugates of the 2 to can ever this implies the number is heat and easy to times heated to can never any new border so we take our table amended and we strikeouts having struck out the 2 2 the crunching conjecture tells us that this number cannot this number can't occur and this number cannot occur and so on I would like example so example and wait 5 years compared to 5 so that afforded comforted but your fault we know we have a graph here Reno it's waiters 5 a priority the amplitude since assuming it's not some opposite the value should be an inspectors space slipped 5 so it's not a rational mobsters 2 5 plus a rational multiple receipt of 3 the so what its congregants B by the examples I worked up the bosses beforehand the conduits would be the number 1 beta Zita Matinicus too and the amplitude self but we know the conjecture implies that's the conjugates must correspond to grass but there is no Graf was .period is heated to so forces Peter equal to 0 said tells us that
1:36:04
the amplitude and cannot be an opportunity combination that has to the amount was defiance this is going and is I waigel cities will handle all of this is that you know have is so I'm which is is tricky so we can understand on when Grosz were dropped and can in and some additional funds of cases and we can guarantee that there is 1 drop so the ,comma tutorial criteria we can just look at his guy see immediately that it has to have 1 waitress but predicting 2 or 3 or more weight jobs his rhetoric and there's no there's no general recipe will you give me a graph and I can ride on the way to do that this not abound there's not a band of about it's always to H minus 3 and I'm and sometimes a bankers none of the Army you go .period yet so I'm assuming Henderson assuming but um and there also yet the triangular bound using I will come back and forth lectures so and is in horse theory does there's not 1 of so that you think 1 is it can carry big job but that's the Krasnodar sponsored an austerity know that the become more G H N of the of 80 of us smooth variety has which to and to end there's a bizonal abound as well you have to justify that so in some sense there is a lower bound Our astronomical said was sorry the president would your quality of course was they've been something that gives a language which means that a beyond a certain point I'm beyond a certain point the only way to get easy to 2 years but you have you have to do some of the there no problem the use of and what many pictures from the authors of course you know have who always the issue was out of respect to the the yeah and I don't argue is not satisfactory level forward but yeah that that there the await something extremely subtle and very far months and has no no recipe that predicts that their son some interference that tell you from the ,comma talks of a graph that doesn't oneway job but the twoway job sisters out of reach mountains questions OK so not continues to likewise by staring at them at the examples there is no no grass corresponds to the 2 Fawzi to 6 because this is not end and some owners of it's 6 so list of this example here some of 6 live graphs await 9 and that there they're going to be more interesting opportunities because of With this destruction structure this gogo structure I'm but that's the way job case is going to be weighed 8 I'm gonna something interesting so this graph and and so a priority attitude In this basis is going to be after the 2 3 easy to 5 Peter 0 2 3 5 gamma the debate plus delta the 2 3 credited to and aggression projectors also we would establish that there should be no to economy of this graph that goes out I always looked at this this to be out and this should be out but we are allowed to have the congregants of any possible in a combination From tacticians accurate idea can be amongst the numbers 1 G 2 3 2 2 5 on the attitude itself I'm but that's OK because these these numbers do occur as previous opportunities they want answer this dissuade conjecture Wait wait thing is that the the collection should it should die respects the wishes respect number registered the Hanson to of final commences university now 1 finally the city's do these jokers opportunity 1 more 1 fine example I'm quite 11 this would correspond to a 7th which until a few years ago was totally beyond what was computable forcibly computable so as 7 of Moscow's Intergraph which involved the following periods is a basis of limited not positive ideas wait 11 times this is wrong the peak years wanted to the land and 9 officials however that this is the basis of all telecoms and wait 11 and is 9 dimensional and the Cochin conjecture tells us we're not that we don't see the tutus and seek to force the 2 sixes at previous craft to these numbers should not be shunned offers opportunities and so will we do now as we take a I'm a 7 Graf so I should say these extraordinary difficult to computers aptitudes and pick a random 7 aircraft "quotation mark peace overnight and you computers opportunity finds after a huge amount of work the they can be written and this the basis of what the devices and you get some professions out sir and you in the United that these the confidence of all this fall quantities vanish in accordance with the correction conjecture but then the gotten better to go beyond that it it says that if 1 compares the Cochin on these numbers you could certainly find something await 8 and
1:45:13
so you have to 9 in the space spanned by the a period of 5 of bandages and fight to fulfill and if you look at what the correction conjecture tells you for the reasons Gallo action conjectures his cattle action conjecture implies that some constraints on these numbers In fact implies precisely that 3 thousand 24 . 0 5 ratio should be exactly equal to the ratio of the 2 3 5 2 2 8 over there so 27 of 5 Over the 2 6 1 20 and you can check the guts indeed correct so however this illustrates the discussion projected give some extraordinary extraordinary subtle constraints on the possible opportunity which which up to now more than ever consider the cottoncandy you take this quantity computer collection for the 211 is primitive it's not going to give anything so the only thing so we're looking for things that will be the confidence of the form as he 2 5 3 was 2 8 and this the 2nd 1 in this basis was chosen in such a way that it's gobble conjugate listen to 5 times 2 3 services and that's the only ones that occur the 3rd 1 in the 4th 1 and so indeed the collection so gavel action since the 3rd 1 in the 4th 1 2 2 5 3 and edit respectively and visit 5 3 in 2 8 have to occur in a combination that has previously occurred in the quantum field theory because this isn't this week combination of numbers here the only way it becomes an intifada for theories of the 2 5 3 injured 8 always occur with in Wisconsin proportionality and so you have a new deal 62 calculation wants finds that give you some information the computer from reporter 5 by this conjecture so I find it's actually astonishing that and that's not knowing something in Enloe degrees and gives you constraints to all orders in alterations so but there was these calculations motivated story so the about 250 examples the rejects for area and where all nuts and pounds of this type the condition up to 11 years the constraints as I hope I explained this correction puts on a possible opportunities actually gets stronger as you go higher and as a report increases normally it's the opposite ends physics you can just do things of right know noted .period receive something worth getting more and more powerful as the award increases and indeed on the dimension of the space of amplitudes aren't exactly 4 which sits inside the match was based members of these .period 11 because 9 so the physics is choosing a very special subclass of numbers and that carcinomas is entirely predicted by this this go actions 5 dimensional but but the point was that army here we had a on a threedimensional space of possible opportunity to 2 5 3 2 2 5 2 the tape there's another graft the gives you 2 5 2 2 3 it's 6 loops and you only get a twodimensional space out of the full possible 3 dimensions I forgot to say that I not .period inventions and minimal .period so the fact that that that explains on yes I should have said that more clearly here did you have this I committed a line here I should have said that at 6 minutes you only get a 2 out of 3 dimensional space I'm not will propagate opt for all highly reporters in see occurring here a constraint and things he did he has the image it was none whatsoever because the army and generally don't get not for the 2 bodies so and for such numbers this conjecture would be even stronger because it will predict and if you get a much bigger space of possible periods and you'll be enough In a very small the suspects is a racetrack example due to shed some pounds aware and you have a and order song so you have In my minus 1 to some power here crimes the spaces on songs that I mentioned hundreds of belief and the the conjecture does it has to sit in on in vector space of dimension it's a tiny fraction of that I don't know what precise numbers of insurgents from 402 5 or something so if the numbers are not what was it about is this becomes a stronger and stronger production because you want a bigger boxes possible period and and so the boss of the remaining lectures will be to prove a version of a weaker variants all this conjecture so the much weaker than that and they will be fight for opportunist imminent as well this is of at yes so that the poor the problem will be that when you the gavel confident over graph defied the 4th the way if you do the mathematics you expects Congress with high over to put in 5 to 5 the opposite he can get involves contracting edges in the graph which would increase the degrees Of the vertices so could just be that on this conjecture is true because I but because the amplitudes in Hideaki verses know more complicated than the ones with 4 overtures editors soup after about this foreign picture that could be this conjecture goes wrong and in hundreds but yeah that's exactly precisely that the statement of weakness is not clear what whether the conjecture is true on this 1 reason why my name is known projected I'm nervous about that time moreover is a brave man than I am and the eminences is this phenomenal but we don't know yet whether such a strong statement can be and very briefly that there was a plan for the for the sequel in the 2nd match I will be just pure mathematics I will talk about maturity periods I'll try to do thing in the election will be completely independent from this 1 the 3rd election I will talk about and profit motives this amount break John relating to profit assesses and will be totally independent from the previous lecture and 4th lecture put everything together phone calls Miguel groups and a small profit theft
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and he