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2/4 Motivic periods and the cosmic Galois group

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2/4 Motivic periods and the cosmic Galois group
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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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my and this I found in the cost of so today's lecture is about and the periods and will be entirely mathematical did so classically on the way to think about periods are coefficients in and underarm isomorphism expanded that means we have to To Commodity Fair theories the score over to call Betty or Duran and the audio which goes from the category of smooth schemes With other varieties of 2 tonight to the category of finite dimensional vector spaces of acute suspend what is off and so Betty Uncle Monty there is also known as singular cohomology to learn about in algebraic topology courses in the I Betty cohomology of X will be defined to be either singular cohomology groups all of the complex .period of acts of Congress .period of axle is 80 and a smooth manifold so we can take its cohomology with coefficients and Q this is significant margin this constructed out of senior aims and then we have algebraic Toronto apology which was defined by cooking and so the general definitions is that the right the Oncology Group is the find to be the article noted the fall of X with coefficients the chief called Cregger differential forms on sex and and in all the cases will look at today some ex-Soviet fine In which case this simplifies enormously and the drunk monetary G B is the the naive things so it's it's uh uh closed forms modulo exact forms except instead of working with smoothie Infiniti forms you look without a break for and that the miracle is that it actually gives the correct answer the call commodity the following complex and the city is the usual differential of different platforms and so on on would adopt excuse here denotes but the complex of global at regular other rate differential forms which are defined of acute "quotation mark even exotic phenomena I'm so we have to different call Montazeri's plans they can be compared by integration so this is the comparison Mt I'm so it's can't be India which goes from the artist around cohomology groups of X canceled with because and periods of complex numbers general this is nice morphism and what is the man of the match is integration so we take a differential form Omega I'm sorry the to every commodity class Sigma we assign the integral of some of the similar so how do you think you have to make sense of this was Sigma here is an eminent and you will all convicted commodity and because we have CUC officials this is Hong each of the 2 x Q and this is just the usual singular apology although space center acknowledges that China's integration period is a subtype of number obtained by integrating an algebraic differential forms over some the perpetrator integration this is an example of the cold calls the left its motive Kerr said sex here will be 1 projective line of work you -minus 0 infinity so it's an it said the punch agreements fear into points and such complex .period is just the star C minus surgeon this and so on modern excuse is just the regular functions on acts of acts and the regular functions on this said just putting on mules and acts and 1 of the 6 as the anatomical 1 executed I just 1 form so the same thing times DX and so what is this complex it's on 0 Oh backs the yeah an informal the 1 next then services on cue it's 1 of the X Q X 1 X TAX so what is found a Chenault it's not a things in the corner of this mouth and so polynomials 0 differential and just the constants and what is H 1 it's a Coke on offsets the the elements and the 1 forms which cannot be written as of something and as we all know I'm the only elements In this ring that is not an exact differential is but IDX over tax so each 1 Duran security
us accuse DX the accident on in high school that X has a primitive just call the logarithm and that is not an algebraic functions but it's enough of homology so the punctured a cop explained homotopy equivalents 2 circles and so that means that um the Betty the homology it is again last offer to connected and it's 1 of the 6 it is not for the class of gunmen Mort where at German his son path winding around the origin 1st infant in OK so what about what is the other period as anyone Of course as as predicted as expected the and the drama better spaces of both one-dimensional they need to be isomorphic of detention would the bit of this and I mentioned to the period of H 1 it is on obtained by integrating In this possible for them you integrate over gamma to the differential form the acts of sex and this is the government approaches the states this two-part to attack there is need found something more complicated so the more complicated Richard relative commodity the something modular something else if you like but this is not said so what is the on way but that he did you read so if you have a I'm period is is where you take an element in In end-around which is a Q vector space and then you take endowment which isn't in the duel of Betty securities based on this comparison Will will not enable you to patent together but only overseas so produce a complex number if you like whiter brighter basis of this vector space on the basis of that is placed on right this linear mapping make as a matrix in Athens now appeared element of the matrix that that expresses the linear so the market is a matrix and it has complex professions but the analyst said the idea of city I think than say I wouldn't say choose a basis of the canonical these change basis is rational because the Russian workers he whole exactly so here I could check I could take 3 times on and 16 times the stamina so long under multiplying that there appears a ring of fears of meteorite I times rational numbers change will see earn them all interesting examples so now that I think 2 spaces that contain an excess an army slightly dishonest and let's assume that the smoothen out the examples are you as it will not be smooth it will be simple normal crossing widening what is the story of the this year's In the end of the month but of appeared matrix and and an element in this matrix it noted that the vector space bounded by the elements of the matrix will form a ring surrender that situation to any such amendment can be written as the new combinations of intervals I'm so a relative is something that sits in 8th and no exact sequence so here this applies to both military and Toronto margins so I don't want to spend a long time finding this and not just say that the elements in relative homology all I know represented by chains anymore but um represented by chains In X C singular center changed his boundary His boundaries are contained in and send the example that the chain of integration down had no boundary and that we need things a boundary and that's what forces us to look at relative that's a commodity and I'm the drama it's it's so defined in a similar way 1 way to think about the relative run form is on a differential form 1 X whose restriction to Z vanishes for example it's a bit more complicated now that I will be enough for example screen in this entire courts so in again as a comparison isomorphism similar to and which goes from relatives are cohomology tents admits city 2 thirds from what he considered safe in much the same way and Nunavut very special case so suppose that the dimension of X is and so dimension that it's a isn't much then on any algebraic I'm gonna stick eggs are fun to any the other and form Omega and because an form then automatically it's restrictions that will vanish and furthermore it will automatically be closed I'm so involved that any form of top degree plans and you can check that the differential form Omega and defines a class In wrote commodity prime times and now we can take any signal a chain contains logical thing change Monday's boundary Delta Sigma was contained in the substrates come and so that's been defines a relative homology class Ali tournament in the dual function on relative go margin and the
period the period of this all this data the differential form on the site and the the chain it is the integral of Omega along said so that the the relative diverted business just and to take account of the fact that we need a change of
integration I which have nontrivial boundaries some examples on Friday the same as before X is the 1 my 0 infinity and that is just a pair of points 1 and Alford where alpha is a rational number and so they don't want Dan and from from this long-exact-sequence it's a it's a arrested for calculation to check that Durango Margie His two-dimensional now despite details there is the differential form we had before and the class of the DX viewed as a differential form just on facts not related to Z is of course exact but as a relative form it is it is not exact and so likewise what's um the Betty homology relative etymology again has known this two-dimensional so there the class of common knowledge and the Jeremiah 1 so if we have 0 yeah and infinity out here then the government is a part of underground 0 and gamma 1 with is the the straight path from 1 to offer so Andijan relative homology crosses the boundary of as trivial so suddenly contained in the points 1 offer the the boundary of Gamal 1 is clearly contained in the 2 points 1 and which isn't gamble 1 indeed defines a class and from bases it's another the period matrix To answers the prayers question we take this basis we could choose another 1 if you wanted to find we are on the computer all intervals so I we integrate DX of abdominal Indiana 1 and the acts of X overcome general 1 likewise and at this point to make the answer on cleanup action going 3 scale this To be and divide by often minus 1 get a nice answer on you and so means a goal of of DX arm along this path is integral is just alpha minus 1 but divided by offer minus-1 precisely to make this number 1 and the integrated DX around a close look at on by Stokes formula is just 0 because the new president boundary it then it's good DX over X and 1 Alpha is is the logarithm there are miners and other than the 1 which is the logarithm of Alpha and finally the Dierkes of text around Goma naught for you to do that is to Madrid parts of the the next example I'm going to try to interpret and Zito too as a period in this language is obligated to grant of mining and it's a very nice examples his administration a lot of time on the phenomenon are going to come up again and again In the context of financial of transparent sometimes examples it's crucial to understand that the government had something to do with the what is the price of Pfizer appeared as well you can now you can cook up an example on Wall algebra numbers appeared and you can multiply by 5 His for rain so if you are allies appear to multiply to apply that I get to play monster but so is an example and I may have to cut the resultant too far out of town OK this so we want to interpret the following numbers of period Z 2 2 it is the integral I'm so this sensual such a trigger this is it a terrific physicists for a trivial you are just a um AT expands 1 of 1 minus 1 as a geometric series and and just I'm integrate students groceries in the EU will you find some of the water and squared invite central goes away that alignments this is the prototype for on all the Intracoastal City and all the financial OK so what we want to do is to interpret this geometrically so the differential form on we want to do is want to view this as a in between I'm a Duran cohomology closer differential form and some relatives cycle so we want to take I am X as a 1st approximation to be fine to space I mean it's removed about the singularities of the differential form otherwise it will be the final months so we have to 2 equals naught and and the 1 he tells 1 floor and the domain of integration I it's got to be demonstration is going to
be is among the 1st of all the differential form Omega is Didi 1 Over 1 1 1 D T 2 of the T 2 which is indeed a regular global differential form on the space X and the demand integration signal Is is the domain is given by the citic points to you wanted to where this inequality holds so it's this region this region here and where boundary contained in where's boundaries contained in did the the 3 white lines which are which is 2 1 equals union the 1 seeks to union to to equal 1 intersected with facts this is where the problem for mind is that the program as following is a 1 we when we remove the red lines from the space we I removed 2 extra points which actually In the boundaries of the domain of integration and so the problem is that the Tsing Ma does not define a relative monitoring class and this will be the case in all the interesting examples he was fine this phenomenon for the good reasons why every time you have appeared it's arithmetically so interesting this could have a phenomenon happens and it happens in abundance and in physics and unemployment is at the point 0 0 and 1 1 are both in these are the closure Of the boundary of the demonstration but the . 0 0 and 1 1 are not in fact N the space X sent some sense when you remove the red lines you throw out the baby with the bath water and use your chain of integration because you've just deleted 2 of the points in the corner so what is the solution In the decision is to it was not conditions "quotation mark blow In physics and indeed the were the relevant words and the figures had helped sectors that does the same job the quantum field theory and so we want to blow up the . 0 0 and 1 1 he the 1 in the case of fireman diagrams that had sectors at 80 Klaus said so the next time I'll do compact tradition in great detail I'm in the case of pharmaceuticals and that should be very familiar to people who know the old physics literature and so 0 0 1 1 was I mean it means you replace each of these points with a copy of the projected life so much of the 2nd 1 this is new space I'm sorry a new space that call a now in which is added to replace the 2 points by wants this is called a blowup on any label signed relabel these In these devices once and for all 1 2 as B-1 the 2 the 3 musical exceptional devices that someone the the views of those who hit users conference blocked by drawing required to all the people who you handily won the Norris there is a complex plot yeah but of course I can only draw your picture and that but I don't know I would identify the opposite but it said the inverse image and I wouldn't think of as a change of variables on integral so the inverse image of Sigma now it is the shaded Pentagon along with boundaries contained in not 5 devices not not 3 so be it is the union of B-1 the 23 some 2 and when you pull the differential form of today we have to check cannot do 1 of the cases in a manner that it has no polls talking opposite can acquire poles along exception devices but it doesn't have checked up on this season the other 2 a minus all 1 not too and so what this means is that is integral so such as it is in our simple Mona crossing devices and so I mean and so all we get is that the 2 renowned writer and cozy 2 vice-chairmen variables as plight inverse of signal prior to the start start so this seems like total nit-picking but it's it's rambling because it now expresses this number as a period all of a sudden relatives homology ecology group said that the pop Madonna has singularities but so singularities in all so minor and then retain relative to so that means is that as before this differential form that means that Monday again the Duran column Of this sort of thing and fight conversing overtures this Pentagon is indeed now at a relative to what lost so the 2 2 is a prime between the 2 again so that they can the crucial examples from now on explained how to do the blowups using and express support notes because that next week of a lot of lopsided 1 list example to be quiet so in that abruptly the at the origin said in the neighborhood of the origin the change core notes and Perspex
equals the of to White was teachers so what that does is an in 1 so when opinion and that he wanted to climb we have these just the sort of in this sector near the bottom left-hand corner and it's gonna corresponds to the ex-wife line scan the correspond to this picture so I 1 this is the 1 in custody this is it these are the exception the visor and 51 with no Andy we see that there was a previously appointed corresponds to an entire line in the new Cornyn's ancestors check the the differential form compared what happens to the differential form so he started off with so someone in his corner answers given by White calls not on fragile form on Monday is due to 1 Over 1 month to 1 utility to hands if you do the change of variables and you get the ax the wife of 106 why it's simple exercise and we noticed that in the new corners there is indeed no singularity at work was not that has no called and so it seems primed very pedantic but it's crucial because all on the information quantitative information about the numbers he determines the period is contained in this infinitesimal data what's happening in the corner and so on it's not data this time the governments um it's a illegal and so we have to study this in great detail in physics questions I'm so perhaps I'll give argued that the general the the I'll give the general coordinates for corners on this on his blown up space because its recently signed a long time ago but its recently become all of them as has several different applications and parts of physics In so young males 3 for example and I think you may be of interest to the 2 physicists so that we do this example and explain how to write down quantum Hall space and the general case I may postponed to the end on 4 half time and so on that's right down the coordinates on a so we can write down you won 3 the candidates to blow up simultaneously by choosing toward that's in a nice way I'm I'm afraid this on picture has a has a cemetery acting Simcha group acting on it and the chosen in such a way that they're completely symmetric With respect to the cemetery to die usual cemetery broader and idle troupe of auditors and history have on these corners will do the job of blowing up the origin and which is what we did before these 2 communities to corner but that the other ones also brought up the corner 1 1 for you would you like you use the yes we later he worried he wouldn't you not subsidies thank you very much this the because I would have think he would worker but so we get on so the space so these equations satisfied please corner satisfy certain equations this have done it right we get equations like this and that 5 equations of giant by Citicorp meeting indices incidentally these his across a variety in this particular case because Mongeon I'm and said a to be the variety defined by the solutions the point of this Rocha solutions to these equations the perspectives that you origin wanted these equations to MySpace cycle and the 5 Delta it's a partial compact for creation of the modular space of grievances with 5 more points and so on it there are devices DOJ defined by the vanishing of the special corner your articles not and a former Pentagon these corners to chosen in such a way that they vanished exactly then one-to-one correspondence with this the components of the boundary and they vanished precisely on each and so on and so the the commentary looking at this age 2 an 0 5 Delta relative to the by the In the union of the Joe so this generalizes a straightforward way I and the name was used so you can play the same game an express all the multiple the devotees said Zita 10 1 and 2 and try the find last time system seriousness into grows over sympathies at the end is the some of the arguments also known as the weight I can be written as its residents the 2 men where the ups and downs of the sequence 1 0 so is 0 2 the power something is annotation to mean a sequences of so many zeroes and for by 1 and another batch zeros 1 launched an online what so
if you do the same matriculation the only people
in the world I neglect the water young you RJ government stop being so you are generated equals UGA eyes and said the because the some of much positive values or can be written as periods in this and strict ,comma logical sense the the upshot Is that arm and Z visa which was examples appear to study last time little bit so are periods of I'm certain spaces so so this this is easy to end once and for for all periods of age and and 0 and 3 Delta some space which can be defined in a very easy way if may do it if I have time at the end relative to some special device inside I'm in other words they're pairing Dementieva differential form the drunken commodity across in here you have a gun Abedi relative cycle given 5 by blowing up the simplex in some way and by tying them together you get the most positive of the here In the same area businessman normal crossing so so that's fine when I define a drunken commodity you have some as just some simplicial complex and you are right you get if you take the hyper ,comma G of the substitutable triple complex so it's not a problem today is singular but is not a problem to define drunkenness yeah baby OK so you use hear also the shoulder of the yes so I mean taken show by general themes in mathematics that you here an artist them that if you write a general in schools the differential form that's algebraic over the rock then you can deceive you can you can now choose a nice compact fication like this they won't be canonical it will depend on some choices and the points of what's Muncy this ship later on the point is that in the physics case we can do it I cannot begin to express the the theater if you do it a different way than you might find you get the the commodity that you get maybe full of junk and this will mean that you expect to see gavel quantitative numbers accident occurred so you want to do this and that and the minimal cleanest way because the data both of these geometry is going to tell you about your period but OK so so what we do do next so the 1st stage was that the Boston we started off with a with a period which is just an inch below some some integral and we want to interpreted has coefficients In the pairings John morphism also on commodity some breakfast face some vector spaces which will which was Betty and Aramco margin but something so so far we've the commodities the vector space on and on the next stage is to explained that there is more structure going on here and forgot filtration on vector spaces and they carry "quotation mark army source structure and this is what will enable us to speak think about weights periods and and here a different types and then the next step up is to view these vector spaces not just as mix structures but representations of certain groups so and what I want to do now a started before the break is to explain to you what mix structures it's just something algebra and structure come on on the Commodity given by some filtration and then after that I want to explain something about Anakin categories and how we can and promotes this 2 representations of group and the point is that the the group this this group will be a version of it's got I mentioned in the 1st lecture and will move on the data this differential form in this cycle around and will enable us to define cattle conjugates all of the period so that's that's the plan this overhead he is under no it is not a ring of the during the state awaited the ring will call will come in here that action that you left can support hello yes yes that will come in and that the final step OK so so now argue us a summary on some of the watery apologized to mathematicians 15 minutes from the time that I think it's good to give all definitions so on resource structures and so this a pure round here structure always K it is a shame incident necessary that this we don't need the mix structures to define 2 we can go to the last step and I'm from here to here without the means structures but the reason I included is because it's very useful to speak about whites and and there's a lot of applications in physics the notion of weight and so I think it's important to emphasize in and secure structure of weight carries a fine on dimensional vector space the queue I'm always a decomposition all of Richard really plant
secure In 2 types the future of work here is equal to care I'm satisfied the PQ the cause of the Q fever so Grammy you can say that the city has a decreasing filtration F the hot filtration such that sp recently direct sun S Kerry -minus P plus 1 Bobby time is isomorphic into space so the example the brand mind here is differential forms with with P disease and cutesy boss and support .period and so the boss between these 2 definitions I'm then from the 2nd definitions the 1st he said city is that .period PC intersect Hughes of the city and in the other direction Boston 1st definition of the 2nd he said that the receiver equals .period songs are critical to pay we are asking themselves this is something differential forms of at least P the the disease Hong list appear more for components so example found Q minus and has the unique why is this the whole structure the following on structure Paul awaits 2 so to space Venus cases is just curious one-dimensional and knowledge filtration F Y I city it is the city if I it's critical to end so lesson for that and "quotation mark and it's 0 if I stripped begin you so on this means and secure means our it's the language space acute one-dimensional undermine sends annotations to take twist sanitation to tell you that you'll and he chuckled awaits him way to end so it's not just a vector spaces vector space with with a weight occurs in winter this notation that it originates from but it's forever standards and so this is the horse structure of tight end ,comma and so fact VC is isomorphic to the city and ,comma analyst this case you have Bloomberg Bowyer thank you so that the tide and the type of the the and so I speak you is a dimension all of the TQC and the hordes numbers of these dimensions to tight end and means that the only known by a single large number HPQ is 0 on less the equals CUC was that's what it means to him and just before the break so and of course the task of but if we take a smooth aren't objective variety over Q then the cohomology HK Q has a appealed structure .period care the government continued after the Anderson now a mixed force structure the the water this year the lesson is that in place this theory the canonical and functorial so morphisms of varieties induce morphisms of structures use the initials the not and natural he was the son of Houston yeah I was stupid that it's too trivial obstruction everything you wouldn't even know cause France and Hey mixed Hodge structure 1 is a finite dimensional vector space the of cure against accused the Q makes or structure I'm with 1 an increase infiltration W . the on video is called away filtration and a decrease in infiltration after all the on very soon can the horse frustrations I'm such that the associated with great pieces of the services W. would only came on as 1 of the and that quipped where's the filtering to halt filtration 1 is complex suffocation no defines a pure structure the way OK 2 and the King family unity to the meeting and that's in particular the sort of group centered in the wrote commentary singular political commentary a canonical and functorial makes obstruction the next key points is very important that
they will become clear in minute why it's important that the Qatar gimmicks structures and an abelian category so you have constant O'Connell I am and you can also take time cancer products tense approximates what structures and you so why that's important will become clearly chance the definition that importance and relevant for physics is at Annex Hodge structure is mixed tapes on this grazed the K is 0 I'm in the knowledge I and graded weight Carol V. it is a direct sun all take a course structures wait carry if carried equal 2 0 and all the examples we saw well this type so was it's because I'm so that the philosophy is that the periods of cohomology which are make state she diverse special and they are His I'm not positive values and imported and all of this type are usually what is the use all is a definition show 1 of incidents at 4 the contrary but is in no time 3 mounted the cohomology To some extent structure by adding these filtration and then I want to prove that a group action so for this I needed a reminder on Ontonagon categories another give full details could because it's very tedious and on to the idea of a tonight category is that it formalizes the and the the categories which occurs representations of groups so as you know if you have representations were group you can take .period songs In the cancer products representations in the duels and so the actions of attacking category formalized what's um the property satisfied by categories representations of groups and the game is the following that you think of group so far group or aid and after a major screw and you give me the category richer plantations and I will then reconstruct the group give you back your group that's that's that's exactly what's going on here yes yes and so is the setup cities you have to be patient with all these actions are not going to give the the full glory but I think in your mind Of the representations of such as a field of characteristic 0 and we start off with tea take a linear and category I convenience trade-off titanium means that it's the homomorphisms are character spaces on the composition of morphisms k by linear then came is what is called a neutral category and if it also has OK so I said you could take time to products of representations to the 1st thing in the tense product cancer TV the tete-a-tete which technically is inviting from and it has a unit so I think this is the true representation some extra data given by natural by some authors Of the 4 a tends to be is isomorphic to be tense a arms they tend to be Kansas City what's more freedom intensive being intensity want and so on and so forth MySpace details they satisfy a bunch of actions which you can imagine it also has a duality so this is a on contract font of the To the opposite category some debt-ridden about ending grossed and this satisfies all sorts of compatibilities with the previous data lines around the most important thing put here His was called a fiber functor so this is an exact faithful functor Omega From the category 2 vector spaces over the field so in the case of a representational group this is just the functor forget the group actions you have a vector space for the group acting on his forget the group action Nunavut to space the 1st time Esiason looks that city but it's Poland and so such that soviet intensive functor means that some Monday a tens of it is as small on very tense on 1 of the 1 it is isomorphic to the vector space of 1 dimension 1 and so on and so forth analyst elections incompatibilities between these actions but I don't want to write to you we have to he said it would be really really had anything to do with the release of liability sovereignty deficits practices in because and mutual tonight and that this is the Isa said the city decided on and if you're paranoid so you can check that it follows that ends His Arsenal offered carry follows from
actions to those missing thing that works for me considering that fails if you try to consider normal in neutral and in this case it's the definition of solid on it works fun .period criticism aimed for is that from that fact any category like this this is the category representations of so on what it means is that there is isn't and our final group schemes the article ecology Omega it's technically defined to be attends automorphisms farther from McGovern experiments on the map To try to representations to this is the category a final the finite dimensional representations of Chile on the map is just ended president again I'm such a sorry such that set as this market is an equivalent equivalents of categories so that the upshot is when I'm you have any category satisfying the axioms that would lead you to think that might possibly be category representations of group then indeed there is a group such that it is the category representations of that group you find if I dimensional here will hold the EU's lose his body of for me vectors finite dimensional for me his father nationalized yes thank you but this award is fight and that's what is this group so many describe its points application will hold the here on the issue of the common so you can extend this to for the match over at the presentations protecting the developments it was not I don't believe and so is is I want you know who you are Hey if your representations or limits the former dimensions and so is ports is 8 at a point in this group so far here will be signed a commuter to Unitarian I wrong then were hard to read if I did this could be just the biggest possible group it's got act on these vector spaces so it can be the biggest possible group which acts on all these vector spaces simultaneously and preserves all structures so I fluoride is given by and elements for M In G L on the other end subscript August means cancer for all objects so it's that cancer-free have we have to objects and an and morphism from end to end then the following diagram commutes this is for em and for an Irish ship and on his book the sofa for every for every morphism between 2 objects and an entity we get commits to diagrams and then there is the the dates of the town's apart so we need 72 objects such a but allows everywhere so this is a fly intensified Hamas's fight and tents and so the tense of product of John unaware this this this collection of matrices respect the principle this is and not as a helpful way to to think about on the tonic group for the target group is something slightly mysterious because it there and the hardy specifying element of the group but this in up part of the group where you have to specify and an automorphism for every vector space every possible objection category it's an infinite amount of data so it's a much better way to think about the talented group it is and its dual its functions so the better thing to do is to construct a functions on the group I thought something called matrix scuffle so functions on a good is that from the father brand and so we want to write down son so far as a brief run from attacking category so definition a matrix coefficients mom was sometimes called a friend or reject all also attacking Judge treaty is a triple consisting of an object and and read in Omega and and have in and duel and so let's so this is the way I think of this as a great thing about your mind you should think of medical Margie of something and the differential forms make integration this triple the fines on is this this is the date to which allows you to define an integral and hence a period and this as of cinnamon should be his reckless decided should keep this in mind so that ,comma go home under the note the character studies time
span but the symbols and the S model the Foreign relations which are going to be analogous to relations which you know to come of intervals the 1st relation is immunity announced that number 1 person on the TV too this is equivalent to wine 1 3 1 st the landed 2 and the 2 so to pursue this analogy with goes we know the tentacles of the linear and I'm in In the arguments in in the differential forms and also in terms of the plant integration the other so was the century have another seniority relations this has fallen to 1 day and then we need a hand another relations which corresponds to 1 changes of variables on and so too before every morphism and 2 and Prime Alford said and there is requirements and the equipment and crime the trial at time if I am the prime the calls on Monday of Alpha really a equals and after transposed from so we have on Monday and as prime never not on the under Transco's goes in the opposite direction from the duels OK said notes the program's class by square brackets and s and if I want to be pedantic up little Omega ,comma Omega and then in the superscript which reminds me that VAT is in an of isn't on dual event but will sometimes drop this other residents so that defines a river vector space of symbols and relations there's a product on the Net reduced and the earth multiplied by and primed his prime prime that's and tens and Prime the terms of the prime westerns font it is well defined and there's a Coke products and so it's not compete Mcgonagle here and intensity at and lands the formula which will be absolutely crucial warms next is simply a song you the so here the summer's over the eye basis In the end suddenly I duel as the dual basis and don't the choice of bases it was clear that this doesn't depend on the Tressa basis but of course the elements EDI Chechen NCI is the representatives all the identity In Hong Kong where and on the end which is church so we have some abstract algebra of symbols modular relations and from that redefines the guy at the top of the group In fact the definition of a tonic group this is beyond longer is we are fine range all of this I'm target group and in other words what this means is for all our commutative unitary care abroad take a suffered his 1st then the group 3rd that valued points of the group is given by the home office From emotions of rings from this ring To the ring off and the the group long GE is is encapsulated by this the product but he remarked the answer is that it is not the group the key remark is that it's the functions on a group which is a more fundamental object Moldova itself fraud espresso and element of the group I have to know what amount every matrix coefficient and all the point is that when we when we do this finding games and can look at intervals we only work with the finite on substrates of this place and we can do computations he worked with the group we have that some some In today to the goes into the defining and focus on our variant the men will be near the end of the technologies so different situations when we have to fight the frontiers 4 some of the enterprise the FAA dimensional vector spaces of the Kerry and then I can define he on own the prime the linear span of of symbols and the act exactly as before at this time V is in some of the
event for the 1st for the front and on the in the duo the image of I'm on the the 2nd floor of the from on we take modular exactly the same relations as before familiarity and equivalents as it is defined in and an identical way and now we don't get a the company to get Co action yeah Delta which is given by the same from now I think Oh all right again the on Monday this 1 going the print on the on the on the on side writes .period it was constantly so far only last summer after the tea there's little sister this you wanted to say so that after facts of this sort a year and a half years apologies on how many hot summers along so let me politically Foundation it was said that the army To give me more flexibility and the reason I need uneconomical primed to be different because 1 will be Duran written Duran functor be Betty functor and uh I cannot define appeared out of 2 differential forms in differential form another difference 4 sometimes can but in general I cannot define a number if you give me a differential form which is in Toronto and you give me I'm a cycle once economic integrate I get a number so anything about areas where it's crucial that we have 2 different farther from and so that will come in and in a moment so that the formula here and the enemy but Sigma tube to differ from the previous case is now songs on em so far I'm very pedantic on Monday the prime and V the override Church and cancer and the signal this year the servicing because here so too must be none of the primer Anderson as before and so hard to think about this this is a technical action 0 by definition to the functions on the group and so this is it this is equivalent 2 actions also g Omega on the other side of the problem so I think of the on the promise some sort of ring of periods times I'm purely formerly Regatta group which can act on it the media and now this lost on the 3rd periods so now I'm going to take to the many choices for tea the bare minimum the 1 needs that is not the district government on the bare minimum that I need will be the following categories all triples the data video and CD well the beach days a finite dimensional vector space of acute take the cake was cute and it has an increase in filtration quite filtration W 4 kabul infiltration we have another kid based "quotation mark the drama where also an increased infiltration "quotation mark awaits I decrease infiltration "quotation mark was filtration but the fun of Ecuador's time and see is a comparison isomorphism c is a nice amorphous and from on Kansas City the verdict such that it preserves the way filtration find such that the vectors of better at space equipped with weights filtration on the image of the horse filtration on this complex occasion it is a mix of structure the objects in the category they're triples 1 of them is a vector spaces and which isomorphic of detention with state and the morphisms are the most reasons of these objects which respect this data so this category also and this should be on category H scores category dates for polish coming through so rampages at American categories so the prefer this factored into the some times it has to further from dues Oh my God Betty Will Durant which drove from this category to 2 spaces here and directors pastry associate either the or Dr depending on which 5 perfunctory under the so long last week and defined None Abstract during the periods so P M H it's going to be defined to be plummeted around Betty In age so I like to write to the was a mnemonic and full the
pair on goods around diversity and I already all for the care of it around and so we have found a ring so we call this and I would call it may be the range of Page periods the hands we get a group so he Cheney rounds all Betty we had to groups in fact we can't take Spec of P Monday wonder way the Dotty closed around already we get to grips acting on this this this ring and frighten their guns at but preferred to think about Iran's so I should son this crap that and so does have 1 group and Audrey have as advertising 1st lecture to the elements of this ring all of this ring periods equivalence classes and the signal and where and is a triple consisting of 2 vector spaces and comparisons last morphism V is in and around and Sigma and Betty yield but I Hey thank you for being 3 of an abstract rain it has appeared homomorphism the complex numbers so what it does is it takes a triple mobsters or can you do that every time we have V we have some of the is in and around and then we have the data tells us that after tensing with see we can send out interview Betty the His Adams the tensest and Sigma is in is in the duel so we compare these 2 things together if Amendment in a vector space environment dual and at another time by this and that's a complex number so what gains is that we have a inverted commas Zagallo action was not much to do with got off now but it's it's this how won't think of it so we have this affront which came due around which acts on this ring all of experience 3 have some parts of numbers and I'm so there would be analogous actions if further to the Betty group often what else we have we have a weight filtration on so we can talk about the weight of the nature of a period of an element of this ring so the elements of weighed less than an Is there is no payment of the vector spaces spans why bicyclist classes where V the differential form if you like is of at most so we have not staggering with lots of properties and call some enough invite you can write this marvelous to be subjective you can write any complex number as a period around a mix of structure is not very interesting so the market is this ring is too big and because not all the objects In a come from geometry it's an opportunity to start offering but we will only the point is the key .period is that we will only we should only consider examples of the former 4 elements projected come from geometry and hence periods in the sense which I described earlier the objects will be Cremins causes of age better yet Betty so I H of this form and with the compiler the canonical comparisons which is given by integration and so we get periods of the form so this Wisconsin geometry under the defines a period in which the value of the yuan has course the lesions in the nation will be among the cases where the economy during the nearly all your the I will come to that so many of them but he said while the words the word that I'm not saying it is is the motive and so the Windyville normally set up such a tragic thing appearances by defining some Connecticut mountains and we we don't have a good definition of but tonight contested motives but there is a key point at any reasonable definition mixed motives that means you money considering cohomology of algebraic varieties with some relations given by it 7 of standards the exact sequence as train cohomology so I suppose there said existed such some 80 and began talking cats give mixed motives of acute so any 1 of the things you like when you have such a category is really like to have a functor makes motives of Q To this category H so much it should have a better utilization of dramatization as 1 of the better minimum requirements and and and it should have this this should be an example of an object in an to so it if if you have time and is also it's expected to begin map we get out of a linear map from the victories so this would be a metallic periods in there in the true sense of the word we had a category
motives I would not into the slightly more naive a version this ring of age periods hundreds 1 of the things that you would like about the catch commodities is this the the destructive should be fully faithful and I would imply that this is injected but I would answer your question that there should be no new relations here that you can't actually listed to geometry and that the key point of this which on obvious the 1st time you think about this is that if if you give me a category mixed motives all these constructions that we're gonna play within this category H begin immediately lift them 2 2 mn and the boiler means is that we cannot have we can define the analogous during a periods in this category and it's going to have the ring structure and it's going to have a collection in exactly the same way but the point is that this diagrams where I work in this I mentioned 90 category H is going to be exactly the same answer so if you pay the same Tamaki machine applied to the Chukotka motives for your collection you'll get exactly the same formula as a one-hour down before so this doggone commutes and is if you have ,comma period here that comes from geometry and if you could lift it to this captain talking Connecticut motives than money computes the and the the true Motiva Gallo actions here it will get exactly the same answers if we calculated in this category which sought to happen is 1 that is if you give me and my civilian kept talking category motives everything we do here we can immediately left To this and on the level of correction motives but if it turns out that you'll catch your motives does not have an injected not from here to here then it's simply the wrong thing to do I'm more remains as it is this game which was he didn't have to live with the the answer examples the examples where noticed some extent motives Over and were arrested s units and a ring of integers In OK where Kazan on the field this essentially the case where were we have that's the correct definition of mixed motives this is not clear you can construct toy categories of motives they will gain will get absolutely nothing we're doing that and so on so far by this this our reasoning it's perfect to work here so in some sense a completely circumvent all these conjectures on motives I believe the way so I will only concern summit to that period will be on I will be the Clostridium monitoring period for me plebeian elements In this ring so I'm going in my head and I want to think it up here but we don't have such a category of you in down here all the forms equals on something like this age and this will cover all the examples of needs on Sunday differential form and somebody cycle he noted that the all many are so I had to run smooth and smooth scheme of Q this music was objective scheme of a Q and Z will be simple no more costly devise all the final virtues the old enough and so that the subsidy here is that if you give me 2 different 2 different ways of writing a given number eyes appear commodity we don't have a problem I have to show that they define the same period that's inevitable problem because if you write you can write different integral representations for given number and you don't don't necessary know-how to prove that their the same the point is in this category this cat is much more flexible as much easier to prove I relations here than it is of tie your hands you say were only allowed to use certain types of algebraic operations and it's very restrictive so for all those reasons we work in this category H China gives on it gives the right answer will give the correct the armed character in the 2nd year in a year I'm I yesterday so here we take that that is is a on a trip equivalence class we multiplied 2 together that the death of the formula for the product and that the period of this times a period of this the close period that Florida along last some examples so because the classes addition normally I think of these the following examples in the truth ahmadinejad motives but because in this case it injects into he and H I'm certainly allowed to identified an analyst with the image that's a tiny piece of notation not with drastic so the 1st example restore was due Martin so this was H 1 he won -minus 0 infinity when the differential form to acts of sex and lower recalled ,comma nor has a loop around 1 0 0 so this is what is this page 1 it's it's it's the tech motive Q minus 1 and wait 2 at a time 1 1 and so the period well and was given by the integral objects of I'm gonna notes and it's too
Thailand is limited version so it lifts the number borrowing I am so sorry for of them and on the right hand side and the story of election the NYT so the Gallo action "quotation mark we have found G Tyronn His son his group we don't know the group is but how how could act on this thing but this is a one-dimensional vector space so the only way I could act is by it acts as a linear map on one-dimensional but space is just not the case and is not think about moving as it could be so fun How is Amendments Act on this number in this material .period G L and it going to be some rational number times out many of the .period so I think I give this formula people do want hold was there the whole of the decision to go to people hold not secession of productive group by prettyimportant group yes as a productive quotient they receive Africa's G gives a map so we think alarmism amount from GDR to GM some offices there is computed the the action of the Galabru prompted another example gave was the growth and this is the lower yes so yeah exactly so found another way to say said is that so let's examine the around the definition of the target group is that we take the the we project for the which in this case the drowned all what I wrote bodies notation as this I mean really the triple that you this is just another 1 the 1 bonds 0 infinity which as a as a structures Q minus 1 that is effective basis just the vector space cute I'm so we get so Fragment G here we have this amount from Q To cheer so what it does is is on this differential form as you say G. also among the drama G of DX over what it does it scales the differential form 1 number another forward skills in school by Russians you that come because appeared is non-trivial you can also use weight arguments that it because if it was trivial this would be the trivial representation and by the talking equivalents we have only Omega maps at Anakin category To vacuum is an equivalence so is if this was the 0 the trigger representation would be the 0 object and therefore it would be 2 0 object as an export structure the company because it has appeared it's not true when the economy because it can't be that has has waged as way Fleming said this so of confidence pursuant to the next example is the 2 on limited lover them with the same off-ramp .period Q after reading 1 says the triple destroyed this H 1 and it was different form was dear to the next and then the post Gunawan can anyone if your mom was apart from 1 of so on and so this is this thing here is called the ,comma motive K and fight it sits in an exact sequence come on mixed what is structures it's a mix of structure and so on the periods of smoke and after With the definition integral of Dixon selects on gunmen when I was a little bit this is imitating version of the the give the GAO action so How does the opposite I'm going back what the the time group acts on the army drama realization of this thing in such a way that it preserves the image this substance and it maps to the quotient arms and gives the action on June minus 1 40 computers so this time it will act is here and and that is the truth away and here by on this character landed you said the overall and there's going to be Oh images but offers a prime as psychological reasons then is it going to be logjam of P Plus I think a call that newbie last time thank you Jesus so let's have a go at the facts on on Glauber prime number you but when you are in future sorry and UPE and
rockets nuclear chief yeah so so this delay so give examples last time but we didn't really understand where they came from now on the aside holders representation bro last time g drama Hewlett said 2 you to and G maps to 12 some figurative the wrong way round onto the sand this gives it a representation of this on Group on gently a white GOT because acts on the drama realization of this the GAO to we think all of us G L 2 is G L around all but which is equal to the Geo also H 1 Toronto P 1 minus 0 infinity relative to 1 and as explained earlier this is a recall this two-dimensional vector space spammer to differential forms that's a year two-dimensional representation and that means that the that the worst thing that can happen to a lover them when he hit with the the gavel group is that it's gonna pick up a period of this integral to the rational the Union of the year where is the title of the year is yes because as a function equated there's a relation here that either which is the function equation of the robber and of course the law and and equals law and a personal and the says he likes every prime number I get a new degree of freedom anew map from teachers rational policy said that the used live in region had had to renounce and and although you have to do with movies music and only Yukon Gold would only be lead fewer is that you exactly so you have to do a little bit of inevitable geometry it's not more difficult months exercise and you have to check he had a look at what's the definition of of this equivalents means that the show the variety of debuts is as a periods and all of this as a period of something else and you construct a morphism of make such structures such that the such that the the differential forms matchup in the integration demands much practice that means you have to write this in school and this sum of its schools and find an algebraic change of variables which send a warning to the the thing here is the fact he alluded to the logic of this however and of course not now yeah I think so I don't know there are gaps schools on the this is on the many ways to do this unilaterally deleted but that is the point of time in general is not the case if you if that is an example of word design and so all of confessed to my honesty and I want to define limited Montpellier to value just a psychological reason it's good to see the definition what a definition I should say sorry I define the space in the case was model placed 1st in the case ankles to visited 2 wow so so this is the metallic and this is because I gave these formerly last time for the Gallo action I'm so here and are great included 2 and and is the weight the sun the indices Omega is is the form you get the relevant form on this model a space that was obtained from the in-school representation I gave earlier for the multiple victories and exes is the is the unlock the high-dimensional what I called high into the signal In an example when I'm close to so this gives you Madam what was after the end of period and all the calls the 2 and 1 and and a and then he so his Aqaba yet so there's another definition in what was it values so the usual definition involves a different geometry involves limited fundamental group but he 1 mostly points so we have to different definitions of limited mostly divided and we we expect them to be the same batch is not known expect this to give the same but it period means function means element of this trend so this is viewed and and so this is an example where we have 2 different ways of compact fine the space during these blowups to make it all minds acted extremely close to each other so I have no doubt whatsoever that this is there will be you have to do something and politicians who hold the community will be stationed at on he said same that you know do you know what he is and the 2 sensors so I'm not even in no the relations which are not known to be metallic In in either cents I can't think example but the other not everything is not limited where do you think that you have to you know that hold on that would bring world you have the army so that's the thing said so on I'm the organization of the so that's a censure question about the independence of the isn't it can reach reverse further than our relations it is here that the relationship news come from aeration factory check that such relations cannot occur and on and that's quite easy to do the
common practice in the situation because we have and wait we have all the structure have wakeful traditions we have this group action so snobbery difficult to to to show independence there is an example that illustrate example sometimes eaten 5 special case of this because of then in his case and the take is the way it is in fact a gratings we can speak of the wait this is a white fighters Nemzet V the ends of the weight that's on the section weighs 10 in the horses wait so it's impossible to something of type and 5 5 and then you take the number 1 In this range which is a weight 0 and so because this the way 5 the weight 0 we deduce automatically that and then independent of powerful easy defied speed to was 0 than alpha-beta equals 0 why is it true fractional numbers it is still an unsolved problem it is not known whether the 2 5 is a rational so in this setting its transcendent questions very easy because have all this structure some practice if you want to prove that 2 numbers are linearly independent in this range on you could recordings the title of Miss now but in some sense no missiles and is not the good question because I mean because you need to understand the relations in this room before you can even attempts to understand the relations as actual numbers because if there is a relation in this ring the pilot period you get a relation of natural numbers so before we can even think about transcendence questions need to know what you're supposed to prove which which numbers are independent and which numbers are not independent so you 1st have to understand independence in this ranked 1st and so I think that it's it's necessary to understand this before 1 can 2 transcendent questions this was a Tenneco bid down next week will be entirely Graf polynomials and physical physics can