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

Title of Series  
Part Number 
04

Number of Parts 
04

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Publisher 
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.

00:00
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06:10
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14:30
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30:09
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37:58
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48:05
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56:03
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1:10:07
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1:13:22
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1:23:22
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1:33:21
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1:53:06
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2:03:01
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2:19:59
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00:04
what do you think that the group would do the eye "quotation mark we got the hang of it and go so in some reminders last time so we have a connecting flight to grow with N 2 years of age to I cannot think masses moment cucumber and the growth in of new and this is the son of spanning trees and the product of the shrine of parameters not that spanning tree and then started to find his opponent oxide depends on Q this also planning to introduce take the product of the not despite the victory then take a moment through 1 of those trees this matter which and some factors involved masses and undecided From this week concocted the and the fundamental goal the cheap and twitches controversial factor which and for the sake of Allmendinger Q what take you it was born society To the dimension of space time to some of the books and minds and just the times and I dancing contain projected and undermines was based and was called "quotation mark simplex which is where all the China prompted From this redefined of Florida and it's a reminder that wanted the codes but the new people from all 1 the committee and became a case study of energy is I who want and you might want to die of there is Wilkins converge diverge Wilson exactly when and from redefined the services yes yes the services excuse which is and the XP's Union X side you where this is just the vanishing of the 1st "quotation mark on what is homogeneous and this is the vanishing looks of the 2nd in 2nd .period on the family the services but we're going think of it for the time being it's just for the fun of particular choices masses and Memento and so this is contained in the minus 1 important point but in the case was he has masses of momentum and then I'm going to put aside the costs of Sergeant Severin portend and this and that stupid case
06:11
that if if you In if the this costing here it is the energy is strictly bigger than HT dearer to them than this factor is gonna Council With respect to him and not gonna get it in the denominator and fight we can consider a simple geometry in which we just have this perception and not the other way and I'm not going to consider the case that would mess up the notations wanted the slightly better and that the vanishing of
06:45
the standard OK so an example just to arguments some many examples and cost I think too quick way to apply and managers thank you so massive lines depicted with a a double and the combined the growth in just this 1 must understand 1 thing but and so we had the notion last time motives except grass so these are the ones where something bad happens in the geometry the strict Maddox of graft in his case on the semantics of the subgraphs such that by the when you couldn't hedge either the number goes down or when you can edge it no longer connects to all the mementos and contains all masses so that is too strict Maddox of grossness case this 1 because if you cut any edge to sample it will not longer connected examined hiccup 1 then it will no longer contain a massive lines 1 right this is also these multiservice tell us that something happens when certain corners good 0 this tells us that went off 1 because of the 2 equals more the polynomial vanishes likewise for this 1 "quotation mark water well I yeah I do this is all the politics of growth defines a subset of the the edges of the variables and animal and and it's not the ones Montegut tells us that there's something bad in the on happening when those variables could but motorcade I defined last time it's quarter technical the definition of a sometime before profit depends on the kinematics animosities and yeah it it's like the subtle definitions yeah we won it at this time frame the focus of the picture effective so we the last time I called the vanishing of hypertensive called and this is the grass surface X G we also mustn't forget Telegraph the surface it's going to last summer's going around but it's it's plans for strong and and and so there something not not normal crossing situation going on in precisely the Malta look at the site were those variables this is the demand of integration it all also but you've only if all matters and yes were the only movement except Graf because of the whole thing itself it was all Father of the year 1 officer was not a point of the projective space you can't take there was no way that you can't take all of 0 protecting people so you any instance treatment in that case the notice of a bigger holes of the whole graft we on instance treatment program and in other words it is fully offers go to 0 that defines appointment it this is I know of no answer because OK because an overall and so we've we've already taken into account as again a factor in the firemen integral In the 1st lecture and there a dozen overall divergent rewarded care so that's when you pass projected corners you already I'm getting regularizing the owner the Oval divers in this case yes so there's a lot of time we I do this and up along the motorcade for this reason I'm in this case it looked like this we blow up the the 3 .period and get something like and will this has done the devices do you want to do 3 times D 1 3 Due to the area and this is what I call find .period as defined canonical way the mojitos and and the space the ground surface of sentiment looks like and it's called water and I'll be on wide but the strict transformed the Hopis surfaces downstairs that's the picture we have of this book that's what the denominators organs and all of the possible singularities underground and the boundary of the domain of integration so in general we had extra contained PNG minus 1 and we also have the scorn things and you up this precise description is not accept Kenya's based page and in it we have strict transforms about the services and in the case when no no maximum and we only have nor the 2nd part of this strict transformed of the respective excesses 1 year
14:31
ahead adviser called the witches has devised 2 types those which just but facets of the came from "quotation mark happens downstairs and no new ones which come from blowing stuff up we had today because emoticons precisely that which needs to be blown up to get a minimal compared applications uniformly fall from grace the last time and the main result was that we got a service is strictly crossing and last time we got America the structure on on everything the graphic services on these devices found space me right now and so 1st of all for the the devices day yeah .period "quotation mark g /slash means you can track the edge in the ground P G especially and for the things going out with a support structure began his piano Cross P G want dominates the quotient graph for you contract for the connected components of Canada counterpoint to the text and these profits of this this 1 and remarkable structure that when you intersect with the boundary they likewise decomposed as a kind of and why and section began equals the why gas "quotation mark she the union has 2 components he began cry why additional and these identities were approved by proving some asymptotic factorization form the fall the craft was in the name of some of the offer variables go to 0 we saw that the size and type on your miraculously factored into products putting on the thing the role of the 1st agreed yesterday and which is the key to meeting at this and I want to I interpret this integral as a period of cohomology as it stands now so we need to think about the the integration the mine this is the betting I want to sir To make sense of this we need to define a good space the front space for the kinematics so what about the word the generalized including region I'm sure Magnus perhaps is a better than another name audiences which systems this year's the reason be set you which should be too some space of of kinematics space possible cues and always will be delivered a slightly and where we demand that the real parts of the squares announces a positive for all ages which are thick and which carry nonzero masses I think in the 1st lecture called the edges which had 0 masks the vanishing the and vanishing my Spanish and also imposed the real power the fastest growth momentum so the squares of the partial sentiment have opposed to reapply for all subset external vertices edges which carried and and this is contained in a space 1 In this space which but in Rotterdam last time should done so this is what I vaguely referred to last time as the space Internet amended so that this assumption all these books conditions will conditional on being in this region of enactment and now I'm restricted a strict space that "quotation mark about the about and this is the best was the best in the 1st lecture and this was the set of challenges don't carry and this certain extent alleges which don't carry on so maybe system I should have used this means all edges which carry massive defections which of the compliance we mustn't under it's strictly positive and if the momentum nonzero minister because we cannot we have some planes which carried 0 disarmament and business condition 1 the commenter Yeah yeah yeah should usually the firm and for old masters and movement in this space find important this is the the just from last time that the full backing the closes for the athletic topology of the pullback of the interior of the house "quotation mark simplex does not meet the strict transfer the graph of son opportunities at the red red line does not meet the politics of which integrated no and so it proved this is and this party to stratified but so in respect of stratification during the draw we have the big
22:22
and big opened settled in the middle and then we have the open I could mention 1 faces and so on and the big open structure let's call it Singh told the G interior this is contained in minus definition the block this is isomorphic too P & G minus 1 and minus the union the corner and this contains the open that opponents and so to people and and this isomorphism sends this open streptomycin Leon ,comma onto the specimen or so this is a point that was said was an important point last time the scenery trivialize crossed over but that's that the growth putting on you is not identically 0 get last time and this is what's important the 2nd this site polynomial is also not identically 0 In the case went to the house any nontrivial kinematics at all this is crucial in the definition of motor that this book now what has kinematics but never the case Q 0 of incorporated because we don't look at we reviewed the growth of the by donors and don't have a genetic scifi have notes on kinetics this is the son of an empty set you get 1 another Maria part of an abusive but yeah you have cannot we should have decided they should yeah yes the props identical but like in the In the examples like it is as an artist so I'm not going to the coefficients In a Gruppen ideas wanted to topple plus Was there 0 but it's not identities era so that implies that the growth going on strictly positive on the the locus where we offer our assistance post and assembly coefficients and inside you all all some things like squares environment and and he's going of madam that idea because it's an answer that is positive about the future American all "quotation mark you where users estimated him so that means that wide intersect certain due to the it is executed in city interior and that's empty so shown that DDI the graph of doesn't mean that the interior of the some of the 42 and now by induction resistant to check all the faces pursued by induction on the but the faces have a good measure of being of exactly the same paragraph the final 40 types of quotient melodic and so on and quotient graph this form that we should not and so I induction the deed the components of the graph of surface on the faces of again Crawford surfaces and revolution the the corresponding I'm funds I'm going to find out the girl's against the use of the word mountain it's really thinks or structure not the which will be the cohomology and minus 1 minus relative to the minus the intercept wide and we think of this as an object In some categories of a more precise about this village on we think of this as a tyrannical Monty group and betoken Mojica comparison more for them that wasn't it section 2 of the precise about that and the and so the point of view the them is that we can take the homology class of demonstrations this fund it and include a welldefined plan which is in fact universal the many depend on the construction the same for grass segment Mister said connotation will mean the commodity the relative homology class of this from 42 in the Betty homology and on the better naturalization of dual use of all the DDT is the devices these devices here and integration domain at the time of of the year you have to be so we can't we haven't done different formula the the boundary of 1st of all this says that a priority Singh detailed is contained MPEG right so it's a timehonored cycle PG relative to the but because it doesn't mean Wyatt told we can restrict too the
30:09
PGA miners with impunity and the boundaries segment of the gene is contained in the and in fact is contained in the minus the intersection G because of the fear and that's precisely that's that it defines a relative from underclass about and that was the whole issue but we have to go through With the 2 example to he was getting the the domain of integration to be welldefined commodity the remark is that amounts I'm angry uses the same definition where this graph mounted defined by blocking on crime the special case where there are no masses Monday I'm possibly some further restrictions that is not always so we have a demonstration of monarch 90 days a something to integrate we need the ,comma history had a full apology it's a matter blown up space as the blowdown too and productive space and we want to write an integral the Financial Group Q we want to put it back to to the space upstairs going integrate over the phone and Florida and we're going to integrate the pullback the differential form on which here and we're going to take you on them and of course in this space suspects generalized kinematics genocide the generalized UK his regions it was the also was conditional Tottenham has this depends on the fact that we're in the generalized you couldn't regions bacillus this ground and cook until diverge the whole problem renormalization and I play for difficult season with general fact that are denying integration but the 2 remarks 1st of all that the domain of integration upstairs is a compact 42 and secondly that the Real the thing we want to integrate strictly positive so indicating such opposed to functional compact the people who were also sees the I have been the following you know it is in the interest in deposit and the loss of human all I hold on all 4 the during new chairman of I'm on the more thank you and I know that my attention so but they may not seem like a small domain the audience is is absolutely convergence on some small demands quality be if and only if part of the state I'm the GQ and it has no called In the where can I OK renewed strength in the main event so To get some wasn't sufficient conditions for convergence readers need to compute the Order of the Polar and all along the device the so yeah but a definition In the superficiality of as the calls the number of troops times dimension to minds and therefore I am which is really easy to prove to the order of the poll all of this stuff GQ did on a boundary the argument is I 1 who has assess the job some grass I'm not and amendments mastermind spanning that the subgraphs connected to every external amount which was it's a momentum which is nontrivial and contained all the masses In case when is announced across Boston infrared divergent we get this formula can and approved it you know
38:07
factorization Kansas City last time mn means I was a definition I gave last time last momentum despite and so many that I am a bit from us is not as if there is a massive lines then the edges in the engine of growth and it means that and some connected components of gas meets a few nights or contains couldn't contains all external legs some of the hostages which can all I wanted to the story that I feel a connection to the success of the factorization this simple calculation is the use of factors in them which gave us time and then use corner block the fact that there a form last time where we computed the order vanishing of a graph polynomial along the devices using this factorization and broke on was simple changes of variables like Kewaunee was a form of I and Sony's plugging these variables the 1st reported on political power counting the as reached we get
40:10
conditions for a fundamental to convert from a mail satisfied so so let's call this condition has no polls the convergence conditions that has no poles along the devices did the and we can define the I put up a In the dry group associated amounted to which and you might want the the arms of the whole thing is JeanFrancois J. relative to the 1 of the inspection was so close and money when it doesn't all that we have a problem and that's the subject renormalization it happened when you pull back your into grant upstairs you find these 2 poles along the devices do and and not about the case we know how to do that you normalized it's a different topic of what I'm talking about but it's just an inclusion exclusion on this on this picture since you find you have people like a differential form defined has the polls along some these devices but because of this recursive structure the residue will be this symbol of the residue will be a product of into of firemen and 2 grounds of sovereign "quotation mark growth and then you need a recipe which comes from physics to find a different form which has the same residue among moment devices and subtracted the 2 inclusion exclusion singularities along with .period bond auction dimension and that is exactly the prescription for the external and you get the people said prescriptions and forced former comes for free out of the geometry and the because equation comes out of such as well again from this sort of factorization structure since the geometry which determines everything that's a sort of idea for infrared and analysts the rule of all of you all he said thought I'm just saying that I'm not considering the case about week we could put the some more work I just 1 to do is to incorporate the case of the renormalization so it was with the primary defines the corresponding motive for divergent processes which have just a single sky different from the we're looking at and look ahead and causes concern here exactly European friends it's very funny yeah Simon amplitude and OK so najib connected final Graf above here and in you on and let's assume that are integral convergence In this genetic sense and then we can define the meant to find an amplitude 2 so more precise about this in a minute subscript the GQ and announcement of a reign of 2 periods of some category the fact that the aged about which sells this morning this is something that I data that's the following the construction in the 2nd lecture of thinking about the middle of the season it is the environment of the firing Of the course I was more than with the around pretty from this category and the the game is a week ago action on I'm on the period of course please stand up .period map appeared almost was America's without saying that the period of 2 complicated and it is of course the opportunity started off with so it had better converge otherwise known as well recently need to restrict ourselves to the present and for the time being the 1st step but also for with the defining what is in the final opportunity is what is it the triple major Oscar certain the motive of Jews then trying to book that of the farm differential form Probst and the logic of Bloomberg Finance 42 and I this this this class is indeed a glass in the Durango Monday up and this is a class due on June of the Betty and I think that the company will define material .period now and the next official slightly confusing and provide the Confederated terminology now instead of looking at so we can change this but we have to look at this differential form we can look at different differential the convergence and redundant staff so we can actually get any Duran and could make it's not confusing None of other the Q and depend on cue so as subscript GE will be the canonical 1 the comes from the physics the actual skater
48:07
financial wouldn't looking at this will be a generalized Intergraph so we can ask us to look at is more general In the girls and recall this summer to make fine and not amplitude maybe this is of about terminology but I want to use the word opportunity to mean the original meant to go we started off with and period for a generalization might reconsider any a more general Intergraph and so G but dropped the case and will be on Monday so again from interfering in some ghastly quarterfinal and the will to fight by ideology on so what is this and let's look at some examples and example of such a generalized From an opportunity to find would be something like that In the case we have kinematics we could take any new numerator and we can erase we could take the denominators arbitrary power that would be an example of the final period is not necessarily find comforting it's a B I don't know you know we could put down we could put polynomials in the kinematics ineffectivebut elements of Clifford algebra if you consider engaged the last of course thank you understand that ABN said compete in as rational coefficients at any point on the island In addition to such that the whole thing is homogeneous took a meticulous homogeneous and the degree has to match up to such the entire differential form defines a former space industry homogeneous degrees 0 only that's the next edition thank you degrees 0 and I'm not old Uganda most so why simple Parkinson argument can rights some explicit inequalities that is going on the human vision conditions example of such things all theories In most cases there is when you write to mitigate Syrian you write it in a pragmatic representation you do China you get an inch kind of this fall and the new writers will contain In addition to to brief example contains all the data of case so crime has been working on this recently in an efficient way to produced the only ones it's an example of a final period is and is an integral part this convention yeah of the anniversary of the theories for the question the in 1 automatically conversion it was was about time and again you have to be normalized this is an offhand remark to say that when you consider opportunities in a gage theory you know the Wright Amendment in space is something more then when you write a if few Redmond ,comma to form it will naturally produced in 2 kinds of this type with higher powers of the golf polynomials trying to monitor and to be extremely complicated new writers involving traces of government to seasonal but the general shape is of this in particular the amplitudes which a convergent you get engaged there will always be new combinations of the what is the same so if you know the period amounted to know all possible opportunities for possible the deductible the use of the whole thing all factories sort I was going to say is later on whether maybe now and in some sense that the keyword professors would be the 1st monster into you work trying to reduce opportunities to a small class of Mostar aptitude out of which you can express everything and the methodical way to think about this it's just write down the basis for the drunken mortgages that's thing that 1 if you write down you can compute commodity that this is the end the down differential forms and calculate the periods when you're done you on all possible he competed for possible opportunities and whichever 30 look will pick out which particularly in combinations of so I wanted you wanted only computer finance just once once and for all and then never again In particular choice of variable will select which opportunity I always thought it would the hard for this is an example of what I call Fineman periods and commas not feeling artists and funny ,comma go so girl here he said connected find where but with few external momenta so in effect the number experiment that is what we observed some political coming in and coming out and on and interfere with any possible masters these possible masses the articles without and we can define a face kinematics some somewhat crudely to be a deal this is the possible moment we could have and his a possible massive and the field of fractions of this I find space recalled it's just a field in which join comments on the animosity because this is like a dumb thing to do because this is not extremely
56:03
well we were unused to them in the environs on your fearful of they would be better to look at just the doctor the masses of momentous along In other words you modest variables in the the current year because temperatures by him the opportunity depends on some subset of of the full data of masses moment so we can do better this is not it's extremely well there's no need to got detailed Of the course you so you think I'm so that unanimously called on we don't really just what this analysis and so HQ and will be a category objects some reporters have been studied technical it be something better something the way filtration Hong filtration and ParisNice amorphous and these will be on Sunday as a rescue opens up subset all space kinematics reflects the number of masses and some rivals the masses can occur with repeated calls to the number of possible compromises that and the the category consists of tripled over With where this is a local system performer dimension Q space on the contrary points of some of in the space the drama is not about to the former type With the effect connections and regular singularities infinity on rescue Q and as a whole bunch of actions that I don't want to hear this is I'm so I think of this as we had this before this comedy before we had the attitude had better homology we had a pretty homology of something so March OK so far otherwise Monty had cut is betting on a match and why the appeared minus 20 C right to the minus the .period I also at the door which is special but it turned out not so thinking wide use static it's a fixed and molecule and moved to this vector spaces and this is a fourdimensional cube but the moves so but the finds a lowkey constant chief "quotation mark local systems and on such a thing you have a moment on me action of fundamental group which encodes ocean monitoring no differential forms now again we can meet the differential forms have prompted an and we get a victim of undergoes a different the specter masses of different perspective and promised and that gives users the gas money connections and each fiber we can compare Bettendorf after detention with the and that's what I explained the 2nd and the package will have thing 1 would like about the possible structures we have some category in which contains all the data we might want to use some but it contains different this professors this is the the culprits equations in contains all information about differentiation of Spector masters and movement on this thing contains information about so it's like a technical sees comparisons so remember we that these often tense enhancing integration that tells us how to take a differential form and pair with an integration so this was about actions it's slightly long but I have spoken to them and we have for front of the door the simplest is to view this category Of this big field we 5 from 2 to vector the fact that the spaces of a specific field fractions which is the 5 the generic .period all elected by the door and secure the 219 category and over the field you I and if you don't like all the stuff that we can just take the case cured and to be 0 no monsters no momentum ministers Q and this category or is exactly the category called age and 2nd nature but it's a fun this to find cause so so we have a friend 2nd actually have the automorphisms is far from His fund groups given acts on the jury of the commodity on the differential forms like football Graf instead very few people I
1:03:12
don't know about you want to you want to say that the new wonders in the sense that that you are nervous about the I'm nervous about it and the reason is because the periods will will involve rational functions of maximum of every if you had wanted to have a talking cat Syracuse and the periods of weight 0 with were the government would be would be rational numbers we don't get Russian numbers up but clearly we you can write down an integral to gives a rational function and Q this is all this with the final period the announcement followed you and announced by this because the fans and yesterday that the kind you can construct the Tanzania the coefficients of the periods will have to be functions of rational functions of care yes you can write it is your right under the 1st the first one inch go weak and it will be at down and and it's good that you know and once something this is a period of a trivial representation and that means you have to have he was emotional and denied that it had will and in business that is crucial to the work of his bigger field on and if you want to if you could find a point that's where you could and specialized and take 6 maximum antigenic occupied only due to the station and In another could be ramifications everywhere that's what what what makes me nervous about this 1 is that you could these pharmaceuticals could have singularities a space of kinematics base the condensed when you look all farm diagrams to get ramifications as a dense subset of the space and that's what makes me nervous when you want to do with protection so that their results in the physics literature claims that you can continue on the physical she decided don't understand I don't want to and we have a target group acts on all these things and has assumed huge subgroup which is trivial military here and In a subgroup acting all right all but from what define GM yeah I have finally reached the final cosmic the long lost year because Q Is this huge group 1 day went incident which is the 1 that actually more fun and I said I find groups came this field so it will begin this group can act on all the time Q for all the In brought here that can happen at your craft the prime number 2 juror initially interested in doesn't converge but it has other periods but the enumerators of for which does convert consider all across cultures the convergence are you must go because because the OK so and some consequences in the form a break so the 1st market than what guy conjugal this is look at the principle of the the fund amounted to original 1 got a because of convergence plan attitude of the final period all the way to client so if you're interested in In the structure of the periods your forced to consider these more general types I'm sure and there is a formula for the corresponding Co action the use of money from the 2nd match so we have a correction not from the ring material periods 2 the final of calls it this the action of the government honoring of don't this is that enough the amount of sorry and so performances there take notice coefficient supplied to jam for the last 2 from the church and the sun is over where the all of the direct colleges which is something we can compete and at the outset of the dual basis so a complete computable former but you fall so that the the gonna he start off with
1:10:09
opportunity and force to consider a gallop on of an these guys that's forced to consider more general into ground but the start of this idea USA John a break right
1:10:32
yeah mentors have appearing now and not give the consequences of OK I'll just give a list of consequences and come back to them was questioned so what's the upshot of machine tionary it's that every but fine In particular from 2 I am to among now defines In fact it generates a representation finance dimension representations all of this huge group commissioned group seriously it's something like Have you got a group of Q with something that huge what Winston is that of these fundamental representations and that's something that we can compete they in In case using this kind of form so in particular at every opportunity now defined representation and has a dimension example we get a new variant of a final growth which is the dimension of the corresponding representations we also gain await filtration all farming on another remark is that the period of variables such of give small list of consequences but that there be a lot of applications of I can't even scratch the surface of it he said Vermont the period of the the actual Foreman amplitude because of the the geometry .period it's actually intermediate serve as a single valued function on region so as my that we we also gets 3 we get on the Gulf of equations the connection so we can differentiate the
1:13:22
specter masses and the meant to
1:13:24
comply with the terms Marjorie geometry and we have a Johnny and all these structures are compatible With this catalog With this catalog actions collection the the collection so we say something about the couple's equations K will have to do if you want to start off with an ground depends on maximum Memento and you differentiate on identical signs we different Omega then it it's the same as different shooting of 2 here among the doesn't appear here so you see immediately that there's an equation the date that the action of a number of factors the the left hand side of this this form of development things 1 can do in that direction we can also been rigorously speak of mix tapes mix tapes but the period so those are the but the final opportunity the mix structure is equivalent to 1 which is mix tapes and these Dundas have weight filtration have heavily grading so we can we can actually speak about the we've signed a degree to each of these things and they typically give .period over general films about variations of such things and from this weekend and defined symbols so I mention this because it's become a very big industry and in physics at the moment I so essential and this considerable confusion about the symbol but single it would seem to me would be what you get from this material proof we forget almost everything we forget that the main integration I just retained the differential form and his connections to and from constructs some simple format and people have been using the selloff but in some sense some was a bad thing to do because you lost a period of lost connection with numbers what is the way the domain of integration you're not allowed to integrate anymore and so it's much better just to work with them this has given us the application of theory I'm so for example now we we have some idea enormous management moment of electrons which is a sequence of numbers and now we can replace it with a sequence of representations numbers can be lifted to 2 fundamental representations of the group and what that means for physics I have no idea but I think you may have some impact on questions of reasonable responsibility on Wanderers on you adding with might be adding numbers of together and we hope that it's convergent all parolees hummable but these numbers are the numbers their representations so perhaps we should someone intelligently and take into account the types of representation yeah there is a loss for all of the sections of these are the things I gave very 1st lecture I gave what you missed but it I gave some examples that I gave examples like that I didn't like the look too I am the collection and the gavel is just 1 so the representation associated skewed locked some it's a twodimensional because they think the twodimensional representation lands How does the group that will give Energy the group the it sends it acts on the block to block but it not scares advice some national multiple some the rational numbers for this is accused this is case and so the the action of this group and factors through the image of the government in this case is an additive group and and expected 1 1 6 because it's a rational invariants under this so I gave examples like this in the 1st lecture and the In some something Johnson something and something that would be absurd so when we have and so I wrote down the motor cross .period to this I think in the 2nd and we computed that is a twodimensional we completed a basis for that commodity and just from the definitions we worked out that this was the way that the girl had to act and so you have numbers and you know how they transformed on the Net got to this 1 has weighed in the Duran Duran category so there is citizens that the Senate vote so we have known as we know how how they they transform I'm not positive values and now we have financial rules and we understand a minute about something about how the transform Murray put the 2 together we will get some very strong constraints about the possible moment and such ,comma too much question about around the reason I'm I looked at the Toronto action is because this is this better glass is given to us in the last thing I want to do the move around it's it's absolutely economical and it's really simple the differential form another hand is something that changes for every graphical this is complicated science sign so I think that it's already complicated the maximum number of enormous for the better the final and on all of this group the cosmic got group by definition every finite dimensional representation of his conduct our group is there year that's that that so 1 way to define another half of this group it is by the exactly the symbols and with the cold products platinum United down to the the existing it's the whole files were generated by symbols master this and the mayors equivalence relations in in his 1st couple of forget and then the Cobra is the same formula exactly as I as I wrote in In the 2nd the end of this year so the enough so the drama music drama means annotations that much of the drama around some of the season Wallstreet Duran and it is more you if you know this is concrete way to do it you look at every phonograph we just fix phonograph acknowledged in all of them and we have this equivalence classes of tripled and this is explicit formula for the Copa at the front hall fungible and spectrum is the quotient of this huge group which acts on this particular crime so that the the huge Cooper's office like a projected demand of all these all possible With the help of resentment of all possible forms of it that it is fully yeah yeah this because of the modification of the cosmic and the a the 2 of you who you are you will have to use who told you that we should take the call regarding sources it's just a projected demand of all all possible quotient that we'd ever be interested in it's a part of a group the focus of the consequences of this set up but in some sense were done so far is nothing because there is no evidence that that the definition of homers and actually make you do something useful for us mean affair so that's this was coming next uses for a special geometry "quotation mark face maps and defined relations between beatific periods for different grass the report we
1:23:26
Gaby container page and the began his 1st said and we have included in the office which includes enough assets and more generally faces but its 1st so the compliment all of the growth of services Fossett can be embedded into the big space as part of the boundary and likewise P can miners were gone cry PG Montana minus 21 down in bed and so I kind of more commentary on it's a comedy Betty and her this is a sort of standard factor let me and try to motivate it with a very simplistic example and so when the directive commentary as I mentioned in the 2nd lecture there a long exactly chronology what sequence when you have such basic contained accessing the circulation and so the goingson sequence gives you a map from something in the boundary into the relative calm and changes the degree to which the smart way you the boundary goes into we we get some information from the I for every facet sir it would be in the form of the 1 of these devices to or watereddown blown up 1 there is a canonical mine why and according to the 2 different situations where an it will the motive of the "quotation mark corrupt governments reverse contraction edge into the mountains for graft attends a part of the motives of the subgraphs question graph and montage in the case where the closed the gap what is it on the better so this that something terrible on differential forms it's quite hard to compete with this doesn't affect forms but on the better things overnight and that's 1 reason why I don't want to change the classes so did given not the other way and what it does to operator a takes such a defining in the big space and you just take the corresponding "quotation mark fast but you were on the way I did decide to try was lazy with notations I called signature was the homology crossed all of the 2 I wanted to simplify notations that in the event of a war everybody on the block to adjust damages the restriction on the map when used in the search phase and the number of people out of work homology classes does this the dramatically it's just restricting intersecting the taking up of this implies faced relations between the final and what they reasons about the definition of the majors coefficients amorphous and gives its relations what the the Murad amounts to the majors coefficient more to murder signature "quotation mark the history of America men signature and reunite the relations equal to the signature so what is what is this this is an equation that takes a generalized find period for big raft and says it's actually equal to sir Thomas Simons isn't a big office actually equal to a period of us more grass grow from which you can trust and likewise saying so .period vigorous actually products appears for "quotation mark conclusions so the relations between parents and the key to the whole story likewise we can we can either a trade all look at the faces of how could I mentioned the 2 fastest justice simplicity the remarks on some running behind so that the market words is that on a curiosity here is that if we start before the graph with general kinematics it's always the case that either the sum all the quotient has no kinematics all and that means V the amplitude of diagrams of no kinematics please respect the yawned uninterested in finding diagrams with lots of exotic particles when you started the period there will always dipping into the space period of grass with no cannot accept or and this is of citywide looking at for the full theory you know it's not very relevant theories and got master's meant that we don't not interested in that the lots of good answers to that criticism but this is probably the best because even if you are interested in graph many momentous many masses you will end up finding these periods In the master and case coming in we like that is now on it's something so that the the key points now is that every every farmer .period of small weight will actually come from a sample cautioned graph tonight about weights and main so this ring has an increasing weight filtration and it was defined by the wait filtration realization on the weight of the differential form In the 2nd actress obsession money that the states material period and ominous signal there is a way to than and Of those in weight and and around the of you according to the waiting so
1:33:29
what's with the with the help of a runaway and ,comma the general fact about weight and we had the motivation is dramas this relative it's a general we know that it has the dead weight minus 1 part this is 0 and the weights to anything thing is the place that means it's it awaits a contained in 0 2 and sometimes social tradition of course but we can write the way to contain between 0 and 2 I'm to think about this is that the baby example to the way we had this relative come on sequence and then he goes into a man's Islamabad is 1 you thank you and this maps to the miners wages and so on and so forth so that's the sequence and think of way so this thing has has wait anywhere between 0 and 2 and and I this thing but by summer a man has waited 17 0 and 2 and minus 2 because the chronological degrees and minus 1 and bathroom words but this has always in the interval endtoend since this is not I don't know how thanks the blown up in front of me and you got all these exceptional sir and this is sort of a simple case to motivate within the next answer since we know that there was a defect in the weights filtration is a strict filtration and means that the greatest weight is an exact functor and that means that if we had any good drank montage which is why less than or equal to and minus 1 means that is weight and minus 1 Jeter then I will go to 0 here because the weight of between 1 and 2 and it would London W and minus 1 of this which is 0 and therefore she has to come from the boundary has come from this thing so it actually lies In the image Of this which is W In the end constant somebody and minus 1 H and minus 1 he did the slogan is the differential forms of weight must necessarily come from about you know what the on activismoften these amorphous and cement construction was underway I that this whole of the degree of differential form the weights so so good around classes no come boundary what's the general said that the rights of all for the face occurred I mentioned the convention size and indexing tiny which which devices Dionne wanted to say so the face I could be disappointed it's the 1 261 3 I'm slightly behind all right all the necessary attention and each face defines a a motive in the same at the same way as we saw in this In the previous year so we have a corresponding growth motive for everything on the Net which uses all this this this geometric structure of the government the we signed facets of the program mentioned cause Angie minus came on this 1 and look at the way capo part of the drama Margie fast and we have is that all the face mask from the previous the following year and and my March to 3 and that is not subject means every differential form on the big phonograph which is of low weight must come from part of the boundary that await 0 period comes from the 0 rations the way stuff comes in onedimensional and so on and so forth to total control on the low weight of this and the proof is his geometric product structure we have books on a spectral sequence and and the theory of waste last week to the man Maine connected from and that's the way it's very focused on finding period which is the sort of guy will close to the misses every gallon ,comma she and to on you and your cause it appropriate as you and so every ,comma which is of weight this is something that we already know about it comes from the small grass it's witness and care then it is a linear combination where gamma all quotient motives grounds aggression is contracting edges and as such the number His most gave this 1 this is the previous plus the face you will where particular regard local year with the army and the suspension of the What Is the What is this term saying assignments that's not every doubt conjugal is the period of small and we've got 100 years old more weight to the theory of more of them want to think of this as a as a
1:43:14
statement of the kind client period closed closed under the action of this kind we're in sense this single so that's a very strong constraints on on attitudes and so I explain explained how to make this work the principle of small this is starting to talk about what I said in the 1st section on Monday at the point is that there are a few there are very few grow with about 1 inches and so remained firm so we should write them down compared to the main firm will give us a constraint on all opportunities to all the reporters a major constraint on possible amplitudes importance contend that the new judges like in the same way as well and sample the spirits and the spirit of the 1st so here some of the program and to and the competing opportunities numbers but we should be doing increased computing the motives Kincaid Mark J. Duran and very small grass and pathetically small also in a manner that's enough economic statements .period simple calculations and and as I said before the call monitor if you computers vector space that tells you something about all possible monster intervals wanted this up to a certain point you know thing all possible must and go up to that point and you know something about all possible periods up to that waits for any conflict so what happens if if we're thinking about peers have await filtration you say OK what this look at the of away 1 or 2 but like the logarithms thought that to looked to hands and you look at the office of 1 of 2 new computer and you get some numbers and a priority is you keep going up and up and rely on reporters to keep getting more and more periods of low weight and become spaces that control the main film tells you that can't happen that if you've not seen period add capers 1 edges then you'll never see her again so let's do it he will give some examples of how the new give myself an extra 10 minutes of Kaiserslautern start off with arms list of graphs of masses moment and do some really terrific summer and pathetic calculations but the point is to illustrate the the power of this so once crosses is just 2 children and 1 of them to inches so we want to do is write down grass to edges graphic to edge the 2 masterminds momentum coming in going up Q and instant Gruppen on here is the so which and so this the the type applying a winwin P 1 it's pretty simple this justifies appointed P 1 so the it just a 1 P 1 minus points to blow the boundaries of the fickleness of pictures just 2 . 0 infinite and solicitors you so the on generalized period you could ever get from this graph rational number it's kind of noninterest an immediate rational rational functions of cure maybe there's another study more interesting 1 brought to its profits as this 1 and now the growth going on the CSI 49 is a bit more interesting defines quadratic function of powerful enough to do so we can computer motive is also which is a form of motive is to 1 P 1 minus 3 points relative to the infinity and that we can we know that it is just but some of to come extensions and sediment appears on you know what I want the 2 of them some quantity x y and maturity other quantity to which you can compute just some functional Q and I think it's called the Chilean function for the under the motto examples in which a country so that understood all classified all possible until periods can come to that's 1 of his sons some some function Q Q squared and and my squares where I given by the intersections of the purpose surfaces with the various quarters so a matter functions some some horrible expressions square root of Qschool minus and scrapers and commitments for 1 and 2 system when it informed I didn't want to write it down in the case Curic was 1 of them is just I am 1 of em this is 1 of the 1 of the quantities you can get out of this and people accuse critical it's it's simpler and cheaper and 1 cosmonaut you get I think something like look you square as someone square of the general things "quotation mark some complicated I don't like to use this equation ya this equation equals 0 he still saw these 2 equations and projectors space you get some quadratic iodide in 106 of the thinking and I can't think it's called competence sure OK so the classified all possible farming .period army at 2 edges and country and that all of the former lover of some rational functions such as algebraic functions of the masses and Memento which I did write down and this is a special case what the amount of time squares because of news from now we will realize that doesn't wanted 3 edges and ,comma let's look at the Triangle Groff with masses everywhere 1 2 3 and now we will look at the mountain age to the 2 minor support a mail may quoted union have vanished relative to the generic dubbing of developers will be reverted to the Cornhuskers and in any case is clear to see that this is a mix tape and the way greater pieces of of type Q 0 minus 1 can manage to do that without any calculations and so we know our general facts about
1:53:06
variations of mixtapes structures that the periods of new combinations all dialog with them but it does it's it's come collection is given by reduce corruption president by this form this is where them plans will be made Is this the right hand side of the collection and this was slightly modified in this case we now we know that the right hand side the collection on the cheap it is necessary the former some 40 computers House the refinement all of the quotient graph to edge the 2 it's quotients this so you do is you contracted edge here and you get back this craft and we already computed its amplitude and that tells you what the right hand side Of the corresponding dialog time isn't for expression for the reason I mention this because that many some recent results in the physics literature repeated exactly this is computed and some small apertures like as logic functions for a complicated expressions dialog those crazy arguments and then they've written down the symbol it is not unrelated to this the computer could propped on the symbol and the neighbors observe from conjecture that it can be really expressed in terms of farming .period and the main fan says that we know this is always true that the right hand side always be expressible in terms of fund this explains a lot of a lot of phenomena in the physics literature to keep playing that game when I stopped them switch switched this is not masses and armament like in the 1st section nanoseconds and more exciting examples from us far as the 4th so I did that so the only way to think about how we can get period a period of waves of to to some of them by the is from H 1 and H 2 so you need to look at it 2 or 3 graph some of the most interesting cases 3 great so what we do on the back 9 we write down 3 edge Ashcroft with no Masters moment as this 1 and this 1 that's 1 thing and now because we were working in the mass this mountainous case were only in the 1st thing I form suffered 2 3 so we have this for example that in each case you want to write down the mountain Her and we can do this but in fact just by staring at it's obvious it's very easy to see that's it's make state it's just about 3 tackles .period together yes there was a time generation has right the conduct your personal problems because we need to consider when we look at the Gallup congregants of an opportunity it would involve the quotient Grafton which we contracted edges and if you look at start of with the Fulbright and theory we may be forced to contract this line and it will fall gives us a function so that the government will necessarily go outside and gross fixed degree but that doesn't have this is rejected they will make state and unratified personified means that when you reduce modulo and prime something funny doesn't happen and this is obvious because all the coefficients in these parts 1 nothing new nominee will disappear for example when you reduce modular 2 or 3 so without doing any thought in fact but the calculation of the there no nontrivial periods coming from 3 crossed there's nothing I mean in particular there is no to periods of up to wait to talk ever and for the full fare as chemical so special we certainly get no all the prime and in particular regional looked as a gang Konjic "quotation mark In but expansion the that is yeah what was but because of because appeared real as the demonstrations were being environs so the for being as fixed the the betterball also announced the real numbers ,comma so he so we had a corollary to this is that fair a statement is true to all orders and perspiration that completely reversed I don't think you make it even more special let me take you grow invite the full 30 which is primitive of divergent restricts this statement is very special classes across way at the motive is mixed and ratified it to me this is not Van and then the corresponding amplitude this is a guy that's a statement about information from and as affront to 430 has the good fortune of giving examples of expect to satisfy this condition for example we can occur there are some 9 and 10 examples I'm in for the full 30 which which lie in wait 12 14 15 which has been completed but on the Internet so just a few years ago would have taken hundreds of years of computer time to even come close to this sort of order of magnitude these guys arrested computers such examples and they gave on the songs an experiment attest this the sort of sums of up things like minus 1 to and forms the Carolina so is time for selection and sold on the sunny periods of this this promoters of this type and so you can as a friend 1st that you write down these these for the songs he
2:02:59
placed some of the material version
2:03:02
convert that's well defined a new test this you see where the gavel conjugates involve locked and a priority such such a period sets and a former dimensional vector spaces something knowing that it doesn't have a lot to a Congo immediately reduces downturn space about half the size that's exactly the sort of think that I was talking about this is obtained by completely trivial competition small last year Varitek's Tomkins and highly nontrivial constrained very hard port transmission so again next could go up we look awaits you want to understand all periods of weight must fall dialogs and things like that we look at how far grass and kind you know forth and from without doing any work whatsoever it's completely obvious that the geometry of these hottest never produces a 6 unity so a dialog provided security duties certainly never come as a period of such across I have to work at all to do that is completely clear the answer from another priority grass and fight the theories about such that motive it is mixed Q joint security unity on Friday nite and said this race most family grass by the maintenance now never as he can the 2 meters 6 as a country and convenience the friends ponderousness have found examples in for 430 wait and 8 33 late thirties and then evaluate and the secretary so they sit in the space of periods of my statement to the Securities announced the gigantic space at this weight but we have this constraint that we know that the capital constitutes can never be of this type immediately but pure we know that the aptitude is very difficult graft has 2 lines on a much smaller space possible and so on and so forth that's exactly right promised in the 1st lecture this is America so as I explained to you you compute something like this is 1st that numerical it's affair is undertaking proven to be exactly that the and then you list that number 2 the metallic version and you compute thing the the GOP action and you find that there is this the strong constraint it never has so we keep going for my expectations Is this the main and plus a small grass principles but eventually completed explained all the all the structure of the amplitudes we saw in the 1st should explain 1 destruction conjecture that I mentioned in the 1st match of origin and payable if you like explained all the symptoms consequences this congestion so the 1st thing to do and that I didn't initiate something different to the 1 used to check the and need to check that there no no need to tell and work your way up this explains why the amplitude of the structure conclusion is the amplitude and quantum field theory by sense Hall so we used to putting more of the act and this is the Swiss cheese more of competition this piece of Swiss cheese if you is a picture of all period and so the the the cheese is going to be 1st have calls and she is the aperture in coming from financial and as shown this this group huge groups this cosmic got G calls is in this sense going to movies peers around is you take it appeared like it did at 11 30 to something crazy period we have hard the trouble understanding the gavel group may send it into appeared in various weights we have already eliminated that means they can't occur to me there must be a hole up here all the things that map under the action of this group into a place where there was none .period cannot overcome so that's sort of intuitive picture of all the opportunities and with for me to be done to final :colon woman student questions to connect with time to understand gage theories we know that the amplitudes her son very special structures that cancelations and so forth are untrue this regional efforts to understand the connection with the normalized renormalization the whole list of questions but that I think you we use the more so so yes we have some amplitude I'm really have some differential form in some drama to space and the group music the group acts on it so it hasn't come together in the same place and if it's a small weight and we know that it comes from his face amount comes from the boundary all general films about most of the region and wait and in the face maps because of the special geometry the faces are themselves happened to be and given the Crawford of if elected
2:11:30
general periods if you were done under general period you have some complicated singularity underground and it's Carol constituted the arbitrary complicated because when you go into the boundary and blow up and take the limit you'll find more and more complicated anonymous appearing and all situations out of control the thing that constraining physics is that V and that that doesn't happen and when you go to them the only find properly number of small graphs and that's what constraints constructive question the comment in general terms about weights and the fact that the the geometry is very special it's exactly the question that we never go outside the graphite services and this explains why the 2 galleries is in this sense closed on send sense from from the whole thing we believe that the only yards away increases the risk is way around the world no not that 1 has to ask the right questions it is not possible To keep competing opportunist on our reporter when looking at said the 1st letter we look for the truth for all opportunities to treat all orders and especially and here's a structure that is true 2 orders I it's a negative constraints but maybe this is not really the point is just to give examples of very powerful predictive firms 1 can say about competent physicist a competing amplitude that is something the American try fitted in summer to space and the structure collapses the size of vector space down to something small as a practical application but I think it's the main point of the the main point is the existence of the oldest mathematics most of the time the 2 this is restricted to convergence yes so if you want to normalized opportunities of this during normalized and then you can do if you analyze space what happens is that you and you can pick up counted for this type of the loss of some new practices which are very very close to and you have to control the geometry of the guys and is very close to the original juncture so I don't think that the enormous opportunity shareholders captains will change much and experience shows that the normalized opportunities across the subject is actually simpler on than the ones which converge so of better the important task is to this with the fate nomination but the damage is a little more than a block the structure the question was asked what our possible periods and we don't know we have no no so I know days there these conjectures but in the case of 5 to 430 that they would be almost positive and that's so that's been a lot of work and it myself and collaborative China that's completely false and so on if that true physics is it is much more complicated than we expected and what what the terms of the left to say about and the structure nevertheless survived all these these negative results which of the 2 of them of the type from an opportunity so much more complicated than we thought they're very complicated but they still of extreme special classes of numbers functions and the possible possible continued using the old 1 None of this is the 1st such I I stayed census Cochin conjecture which stated that the periods of 5 to 4 himself closed very strong conjecture I'm skeptical about an experimental to the Czechs hundreds of examples that gives all possible constraints if you know the loss of numbers if you look at those which are not for the 2 values and you imposes corruption conjecture then in fact that can successfully predicts exactly the place of all the opportunity to give the rights that I was going to win yeah get new numbers coming in control so what we know that we get modular forms and certain of myself the prove a hoops we get grasses motive is a model monitor form and to get a new type of number that's not the this we expected to get arbitrary competence so as as the as the so the program of trying to describe explicitly the types of numbers coming in To conifer trees out think because we have a lot of examples where things every border historically every reporter there's been a conjectural also conjecture that died a horrible death but what we expected to be true that a completely false and this has gone on 1 of the last 2 decades I think what and not be too pessimistic about the nature of the 2 the was people so you have it you have girlfriend of native grass it's in some 18 dimensional space heating something it's Harper's office of degree 9 and a million times in it and the claim is that its core origin has some question which is the motive model 4 and you prove that by firing its accessibility and doing some operations on the graphics surface to decrease the damage and on and on and on and you get a K 3 surface and the case resurfaced security services grew up in the classification and it's not that it's it's it's modern the model and you know that this the motive symmetric square and that rules out the musical possibly be a period of time extension to the continent positive Compean is nice numbers we and so we have a lot of examples like this wild nature well With the commodity looks as if it's going to be arbitrary so that the program original program expand Qatar's original conjecture was given the expected all the Zapatista ends at the end of the summer group reacting on them there should be a compact enough at to the original trying to control all find opportunities is just not possible because these exotic counterexamples but the exists and that the structure still still works and is all the more surprising because he aptitudes and going completely orthogonal directions to multiple the 2 of what we would have liked so I think that the program of trying to write down opportunities this specifically year is Beyond what's reasonable for the foreseeable future but instead we have these ideas coming from other spectrometry we have new environs of opportunities things have representations and just numbers and we have weights we have different equations we can use all machinery I think few