3/4 Motivic periods and the cosmic Galois group


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3/4 Motivic periods and the cosmic Galois group
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Brown, Francis
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Institut des Hautes Études Scientifiques (IHÉS)
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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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man To have em on to Monza for the masses and 2 and 3 and 4 as era just ignore them the six-time momentum is considered to be 0 3 other arrived and we had momentum conservation Q 1 must conclude a 0 answer to such a growth in the 1st lecture explains how to associates Bancroft polynomials never 3 of them and there was the careful of polynomial which was the son Over all spanning trees In accordance spanning 1 trees and then the products Of the edge is not in that tree after and then the earned although this was a call the 1st semantic this is the 2nd semantic putting unknown is the sound of spine territories and you take the products over the same thing again but this time you multiplied by the total momentum coming into 1 of the trees to wanting to make to friends total the momentum entering 1 and then the thing I'm most interested in His polemical CSI which depends upon the masses and external mentor and it is defined to be I far GQ plus the sum of the squares of the masses times SIG Sauer not an offeror example we have society the costs and for GQ so here we have leave this right here squared equals Q once critical thinking skills this equals key squared off for 1 hour for 2 of the 3 the 1 after 2 years of the fault 1 4 3 of 4 case so and we were interested in the the financial integral GQ and as a function of the masses in mentor had some some gonna effect is understanding I essential over said the main born over 1st sometimes to the deal to an area of the NGA mine safety deal too on the Jane so MGB and the number registered the graph and THG is the number 2 fence cutting and the lawyers and so as explains the 1st such good degree subside use homogeneous degree equal to the number and the degree of fight and hence CSI In the offer variable some apprentice is Hg plus 1 as for important but as always Sigma Sigma wars the the main projected they above-mentioned engine minus 1 and it was the coordinates simplex signal it's the region with projected corn nuts and I'm such notation of are real and positive and all energy energy is the minus 1 the guy of the awful warned the medial supplies the and so the goal and his team but as I explained in The Last Lecture if you want to understand their size I'm a period of time cohomology and I will take some time talk today and it is the beginning of the end of next week so want to write OIG Iran has a period of family of areas all commodity so we want to think of it as a pairing between the dire I'm closer differential form and song cycle integration and so it was latest this is the science of the the capital CSI's like this and I hate writing and because it doesn't look like anything so on the board of voter rights online through a logistics like the random lines the rules of the Jedi but I know I'm on my greetings well would the appalled would with this is supposed to be the 1 they
could make him 1 of his sleeves the U.S. a school a couple of years and uh the year should be the year of the moment primitive from primitive if you will not general no no provision whatsoever no connection whatsoever and I Nevada 1 considered from until much later that the I'm so it did they wanted even because that's the case where most understand because as we all know we live in the 11 dimensions which is you have to get out of favor the pull the lame jokes and so obviously Wendy's is squared here you have to do something doesn't does not problem I think 1 can 1 can work with that but and I haven't thought about it I want not for you to many decreased 3 . yeah I know this so surprising everyday people eat some of the 3 dimensions will conform to the doctor expansion from 4 dimensions of the 3 but I am sure that everything works for great it's a different kind of fish so that the issue here is the same issue as I explained on the example of the two-tier last time if we want to interpret the domain of integration as the homology class so boundary which is contained in the place where all the off-ice vanished musical the corner about the planes and then the problem areas which is exactly the issue with visited to integral last time is that the boundary of wore them in the domain of integration itself nuances boundaries in his scrap um meat the singularities the into Grant the differential form and the sport poses problems to compare the to to example the last week so now wanted to explain this issue let me do you I'm an example this a circuit on this graph and for reasons that will become clear later on I'm going to label the edges 2 3 and 4 the answer the call for an unknown here it is after 2 of the 3 we suffered too after a fall of 3 after full time so cool X subscript society be the vanishing locus of this growth .period onion impedes this is the 0 progress where the crop and then washed his hands before so I the 5 Cornyn hybrid were off on adventures and so we have the following picture I'm profit surfaces the summit this year will I love because it would appear as a as a quotient of this graph on a contracted 1 answer later on it will use examples of purposely and the corn and other planes would like this and so on so this quirk of Walker said from a total of 3 and I'll fall this is Texas I and the domain of integration is the simplex here so I'm in this situation I'm assuming say that Ng equals a stimulative that I'm not considering the phone I'm excited but it's just that the toy example and exactly as in the last lecture them about points here which normal crossing and and for the 3 back .period show the battle but points out well the domain of integration means the graph surface intersection Sigma so G and they are exactly the 3 and the vertices all the the simplex so others ran last time the solution is to blow up these points to make the situation more consent and the victory got off to blow up ISS there and then the 3 about points because copies of P 1 the 2 I'm sorry about that and should run this times the white lines no longer meets growth of the service and the picture we get a new demand of integration which is no longer triangle but a hexagon some recall until the Treaty for defining general and these are exceptional devices Epsilon and to Epsilon 3 4 and so to
3 so what today and we need to understand that the the general domestic situation the 1st of all we want to determine the bad guy in general and then would bottom out and then the 3rd part will be to study the structure of the resulting space quicker so he said the of us all you know what I did in the 1st section of sitting in the office "quotation mark China parameters I answered our physical shrewd parameters anyone who believes in Korea we exactly on economical way uniform for all forms of so I I'm very slightly probable differs responded of course is the families of hype assesses depending on massive movement and is gonna work for the but fiber because that I want to say a lot about families but it that's poses no problem whatsoever I want to be very precise about how works brothers pretend the Q M fixed and generic and want to be very precise about what generic means that this 1st of all summer and a lot of properties of coffin unknowns the first one is called construction deletion so is it can be half an hour of just graph theory minimal certainly there is an internal edge Of all rather than define G minor which is the deletion fall of energy and body on the delete the edgy and the you retain its end .period so for example if we had a gruff like this and security then deleting the edge it gives you a new grass which now has 2 connected components the clearly this achievement so for most of the rest of the lecture values answer may be that this talk about this I'm going to consider subgraphs defined by sets of images so an subgraphs I'm so that means you take a subset of edges and you include all the endpoints of vertices so sometimes I may write this I may forget to write Gamez contained in edges but we think of it as a sub graf of G and send this case we define the cautioned contraction this year the this is because you always take the virtues of the and the voyages unknown more than most of us his exactly we take the minimal subgraphs contained the virtues of the grass army so what is the contraction what you did you delete all the edges Indiana and then you identify all the vertices lying in every connected each connected components so if you have a as a bunch of connected components is connected components gets contracted down to single verdicts it's squashed and no masses but they they're gonna come later as but they add that to thank you that's my next comment to the quotient but yeah I think it's the question 1st of all the deal 1st of all the momentum cushion G gamut will inherit the external kinematics and in general and it will inherit the master messages through a massive before they will there will remain massive exit depends on the situation all the more about that later so it is an example that Stage and contract the subgraphs 3 4 this is just this craft and so on duty contracted 1 um is this is the other example which is why I considered would you use the 1 of the messages on the way to the issue of this is a must you know you not as narrow the plane did that aired the corresponding edge does not appear in the quotient some Fernandez talking about the quotient and so that must know does not appear it is on edge of questions have gone but were later on explain exactly what happened to the mass in the mountainous quite subtle that'll come here because you know that the Treaty of about he hadn't thought about but that's probably a good explanation here 1 day so unfortunately there's another notion of contraction call for some stupid reason today with this contraction vision and instinct a man with a mind-set arenas such as defined such different and its so defined another national construction firms is G double /slash can answer enormous notation for I think it's helpful such G /slash gamma a Gamez forest so it has no illusions but it's the anti-graft 4 0 Graf whatever you prefer and if the cell growth the contract contained in at least 1 common in the construction division I identity is the following me take any internal intelligent growth and in the 1st draft for a nominal satisfies intermediate scrutiny and multiplied by the cross-pollination parameter and the cost and time is the contraction in this so brittle sons and the 2nd
semantic polynomial satisfies a similar identity so but he is going to be 2 you have the same external vertices so date they keep their external lakes there's no ambiguity about what this means and what can happen of course that when you delete announced the gruff becomes is no longer connected and in this case that this graph phenomena will vanish that's what you want that DDD equation the critters are correct precisely because of that fact but this is the only anything the women need from this is on the foreign identity but the 2nd so the 1st couple enamel restricted to the locusts at the close is just some adults nationally the definition of the restriction to Alpha equals not it is stodgy double session Q so this tells us something about the John tells us what happens to this graphite surface when intersect with according to plan and there's a beginning a recursive structure because you get the the corresponding hot the surfaces of coaching crossed the contracting edge so there will be important so the 9 states somewhere factorization films is absolutely crucial for everything that I'm of top crux of the story is hidden in To the identities and I'm going to write down now and everything relies on that so JB connected again that the phonograph army and Gamma sub graf defined by a subset of images where and several connected components again and so the number of rapes endowment it just the song the number of which couldn't on and the 1st results and so this is not new because it's been inoperable mind of the crime and arm and I'm sure these identities are in principle but known for a long time but not necessary in written form so we can rights the 1st summons according on ultraviolet don't ask me what it means In firms should this this formalism paper by by blocking on crime and following want in the trumpet with crime and an ever-expanding minute why these days in principle but not for a long time on so far UV damage so here we have to approximate factorization of the Grafton onions on Monday explain what without means what forests we have to to go through the use of force I didn't really not done socalled reason I'm glad he spotted that yeah and Bob got in either case commentary 1 is of degree strictly larger than age gamma they the degree of this part this is degree age cannot In both cases and the other has remained time has hot water Indians variables after the 1st is In subclass gamma so example the so in this example we have on this is the 1st growth for nominal following so maybe we can write the 1st Gulf for anomaly can check that it's can be written off a 3 Pacella for 4 times often want to suffer too plus after 3 of 4 whether subgraphs 3 fault the quotient grass 1 to the disarmament of playing a role for the 1st "quotation mark polynomial and it's also or to the factors into different ways and offer for 1st for 1 after 3 after 2 of 3 and here's subgraphs and the quotient is temple so on so here we have that the subgraphs here has variables of offer full and and deleting order in these variables is a degree 1 the remainder is quadratic and agreed to so if we think of office 3 enough for going to 0 she close to 0 the Board of Exxon squared and this goes 0 at Sony limits as these are some Crawford was go to 0 the remainder towns will disappear and just gets the product on the left so covered remarks and so in a very special cases and very special case where gamma sub divergent then these factorisations are sufficient to drive
all the main world must remain results at many of renormalization so you can prove it the EPA said there and you can drive accounts semantic equation I'm just phoning the existence of these factorisations In a very special cases raise special funds of all this was done in a paper with myself and and the couple years ago so and that's why I say physics knows about these factors editions and has done for a long time because its success equivalent to being able to normalized but the funny thing is that the renormalization doesn't use the full strength of these equations it only uses the case where the subgraphs graphs of divergent and the general case the general case of these identities the time is not used but it will be absolutely crucial in the construction because McDonald a 3rd remark is that's and you can ask since this this factorization property is but is very closely related to normalization you can ask what are the possible polynomials that satisfy this factorization identities and if you think about it it events you can check that they uniquely determine on the the gruff 1400 so that the factorization formally plus Lindbergh In fact uniquely determines the condom optimism paper with chronic but 1 thing we never got round to was to explore this idea and see how How rigid the final rules if you demand that they be Renong was within artifact from which elicited the on it is for me and I think it's just anyway putting on a little be homogeneous but it in there in those variables cross .period subgraphs it has hired it was you and the wealth of Suffolk side so if Western's what you can choose Aleksandr you want if you give yourself across the grass seed these graphs and you want this former to hold and he wants on this formula to hold and you specify what side of an edge is inside of the tadpole then those 2 actions will need to determine the 1st semantic point on that there is no choice physics gives us the unique possible choices for which these equations and so that's a thought experiment that we never around here is to try to try to on the there's an obvious going to place a hot how unique is of the final rules if you impose a set number of design conditions but isn't becomes an important point that was interview 1st but will be important next week and that there growth hormone in the course of growth is never 0 on day that can happen the company that's the time the 2nd semantic Coronado of the quotient graph 0 I'm sorry I kind of I did it was in the case here because here is a little bit sloppy here here this this discussion graph here really has to Q 1 and Q 2 but because jumpers due to 0 this is equivalent they spend in the 1st lecture tour graph no incoming movement and so this has circuits among opponents so this is an example where the quotient has vanishing 2nd demand for the head OK so don't understand on this phenomenon which is related to infrared divergences the definition and a sub Jammer I which is connected so they connected subgraphs is on momentum spanning this is a phrase I made up it's not standards terminology was unaware Madam spanning the for every Vertex V can the big raft which carries some momentum so says the total incoming Amanda this is the moment that the total entering further momentum entering the verdicts tremendous finding very verdicts which is carry some months and then verdicts in the grass His in the sub some profit that touches every on external aid that's that's the carry some momentum earned the General Growth gamma another 2 subgraphs not necessarily connected Is momentum stunning In if some connected component it has East 1 connected components and hence exactly 1 connected components all down Is momentum spanning so where momentum spying subgraphs is a sub graf that has exactly 1 component that touches all the external legs non-zero-sum and all other components do not touch any of the external links example so here in this case and inspiring Graf would be this grass because it meets the to from momentum kind vertices and here and here in the middle of a different color here here another momentous paragraph would be to 3 because it meets its care has attacked a component that means everything on and no 1 else was not meant inspiring and so this of 3 4 was not moment Spanish because it
doesn't CDs but including the so you lose the people playing the time any actual yeah I'm override the full list of all movement inspired you take the time and I need to find a condition later 5 I won't will double in other conditions to the masses and then I will I will give the complete list of of those across some of the use of the vote is the 1 with nontrivial momentum so in this case so this is not this verdict is not carry memento because and because the incoming Montana 0 Parliament and conservation mission resistance fighters involved in the 1st my next comment during deaths so far yeah with with a cover yet the copier that we have to assume generic generic momentum In considerable and if all cheese grater equal to 0 for example and you can easily construct examples for which this does not we don't consider yeah so we sort of generic kinematics 2 of the favorites actually it's very easy to write on condition is such that the proportional songs should be non-zero conditions for generic kinematics and Jan it is momentum spanning the finale the caution has 0 bashing 2nd Sonata .period but it's another sees to an infrared factorization formula and now are meant to believe that this is it this is nearing factors having never seen anyone which I don't think is living will dinner he met with him in the square where were the norm this fall is the sum of vectors and d dimensional space and this is the occasional would you want for all I care for any subset of external legs you take the total the some of the momentum needed you could handle it the duty of the 4 who is in his he was leaving absolute after the right says all I heatstroke thank you I was not planning to talk about this that you have the right of yeah is not I'm not to I'm I'm these amusing Outerbridge so so the masses in front of Out very variables and complex place front and have makes sense the job you make sense of a competence in the way 1 the square the Euclidean the kitchen all losing him he may have led to believe that the Fed has an unusual moment on each known of the summer I said it again that there some good have 0 momentum and the and we specify which reverses have 0 mountain bike at the bottom of the others after non-deliverable forward he said he had the was sorry sorry sorry this is what happens when I or my family my when I improvise tho I contains on the other is called the Ext what I know I had I think a defined set of of vertices which carried no kinematics infrastructural half-a-dozen attention I use with external orphanages the character yeah I don't wanna get into these users it's very simple but it's some just to get them such a tragedy was questioned and a former governor of others the Aso's all happened will have a family over some some some space and you Cajun region will be the region where accuser real some of this and in the neighborhood of that space that thing will buy beautifully and we can have well defined analytic functions and we we can extend over complex space and then we'll have some terrible discriminatory don't understand news earlier this is where it is yeah so they visited condition of yes if there really is this condition is driven in part by the ones I had at the end of but this an infrared factorization fan so the downward an edge subgraphs which is momentum spanning along with components down jam on of after the gamma 10 where gunmen Is the connected component which is momentous funding and I should add remarked that when you have a momentum spanning sub graf of course it inherits it inherits To the extent automatics it's all elects its own kinematics because it's precisely connected to every single external like the president and but if so then we have the following factorization formula 1 Figeac here the cause 5 Gemini Awards Q I makes sense because gunmen or inherited the external legs many multiplied by the 1st semantics and the other graphs and then we take the 1st semantic quotient graph and remainder Thomas and uncle the infrared remained at where on the same same-store as before the remainder time has 5 degree rent than than the leading pace so has degree bigger than age cannot plus 1 because this time this this part here has degree it's Canopus 1 so as as to get a degree in the end variables all the subgraphs gum so as I said before mentioned I don't think this is an extremely anywhere and time to my surprise I think it's a new results be implicit in some of our literature 1 not stated specifically that there is no reasonable size no she said is that there is a resident in a cemetery in this formula because so I always it's sort of um the fact is it is thought only involves for decision fight involves boasts on 1 side and from the other side and it by contrast to the contraction nation which involves fire on both sides and as big as exercise you can try to prove the
contraction deletion from the factorization and vise versa and sets coroner's exercise and when he do it using this you you realize recognize that to get the cemetery in the 2 fights here tells you that there must exist a symmetric formula with the file on the other side and so and they have to both exist 4 and this asymmetry that the jury that the fight involves the size and the flies and that means debt-ridden formula for 5 only 1 4 times it was the surprise winner focus but to this example the new not convert infrared but now that so that they connected subgraphs so let's take some grass on 2 afraid his quotient is want full and then the and Q squared prosecutors office 3 times I for 1 have full and the men's time also long after 2 of the 3 I missed another 1 we should take this approach given by this massive lines postmasters play no role in the discount discussion and he rejected on cue squared off 1 after 2 out of 3 himself and if degree reasons remain there happened to be 0 OK so there we have 2 different factorization so this causes begs a question whether what what 1 can do with I'm regularizing infrared singularities and done I have no idea but I've been tempted to think that the thing was the study we did with doubtful ultraviolet somebody's can carry through verbatim by using this formula in the last couple of 105 securities law for now In 5 minutes off the question because we haven't considered sci-fi doesn't occur on its own in the financial it occurs in this combination when you write it because in this combination of outside Chile here and the close fight Q last summer the and then squared off SIG so of course as you rightly say we now need to consider masses we want to understand factorization formerly subsides so that's what I'm going to now and another definition Witaker subgraphs the final substantive edges and it is called again invented this phrase taking the interests of mass momentum spanning and abbreviated despite an commencement spanning from its 1st of all on the edges of gamma contains all the massive edges but what and to understand what is meant and spending the base of the wall of the verses contain all momentum and the edges contains all the monster monster so obvious and meaning and up again if we have just generic kinematics there's a very weak condition on the disarmament and gamma and and its finale the CSI 29 of the quotient managers here you don't who legally not just enough to believe that the message will be 1 humble has said he has in the last year and that's a very that's of regular among and for that reason and I'm going to stay slightly a simplified version of the next Proposition to make sure have very cluttered nothing but it looks to me works fine 1 of those registered in the Nordic break turnover the proposition and so let's take cannot contain under suppose connected just for simplicity but it really makes a difference and then we have to factorization forming for so I it's the thing were most interested in it's so I just saw gene on Q and class remain time is the ultraviolet factorization times sigh G Q and sorry this is was for any grass but that's just the cannot not last momentum spending an infrared fertilization occurs land come is momentous stunning why are chemistry to endowment is announcement from Spanish and of go somewhere degree almost all citing German GDP in the offer the and subgraphs it is extremely good and scanner and the degree has the floor the All ultraviolet In the same
vein is taken cannot respond so that this this provision follows from the previous propositions you just have to take the definition come up here definition of CSI and you plug in the previous factorisations for information for me and you get this coming out and you notice that in through to answer question that in the 2nd formula successfully uses the the the 2 ultraviolet factorisations subside and for 5 the 2nd former uses the infrared factorization fight with the ultraviolet factorization the but gives so that the factorization applies both infrared and ultraviolet and sometimes his the views the where history last year used 1 of the the goal of the game no we have you know and it has been in the hearing projective space so it's meaningless so another red underfunded view is why said I don't understand what you feel Our means because we have we have a picture like this and we have some bad points and that and that's why notes and so the renormalization only cares about ultraviolet mathematical point of view the infrared is in there as well and the 3 works fine without any problem probable cases start explained that toppled the life of the loan all exactly so mathematically we don't see the difference so the these letters you artificial it's just for physics intuition and so will stop here and I'm not continue farming for case so discontinue use modified now I want to determine the the bad news sigh which the better need up so that didn't I and subsets stricken contained In the southern colleges and so here where we're working Internet which is again a precise but weak conditions that I dumped acute-care to write down so let's L I contained in projective Ng minus 1 space be the locusts where after the vanishes football team I it's a corner preparing the cornered nearness of space and we have come to have a services what's often called a profit the surface the effects seige equals the stodgy In pejorative and minus 1 space and we have a family all of oil services 1 which Article X and this is very the vanishing locus of the polynomial Q closer to family but we just didn't think it would further father and so it's fiber Over over some points in some appropriate space kinematics but I can't down it will be difficult and so the amount that subsets on subsets by contains that any subset contained instead images of GE has he remarked earlier this is equivalent to that at families of subgraphs edge subgraphs gamma defined by its adventures and so now let's viii for example of the 1st among the phenomenal the order vanishing all of society alone devise a L I and so we know exactly what the result from this factorization formally the factorization formula so it as None of this adviser of sorry hear for other conference non-native English speakers of factorisations an expert I think that is endangered letters so I take it upon myself to you observes in words otherwise there's no use for that is that English using the army so complete evaluation moments of graph falls outside well we applied the island the factorization formation of the 1st putting on your ISP's idea side Kemal must remain determined has grown but by definition of remained time the men the term vanishes the higher-order then so this is I um recall that this is all over the Greek age cheating and this thing is that remark I made earlier that non-zero that means that the valuation is just the valuation so gamma said 1 ,comma and what's your the vanishing of Saigon what it's when all the variables good Xerox is the degree so this is just the age can and likewise on the valuation of the 2nd was so 29 0 it is the page gamma therefore ,comma is not must spanning it goes up by 1 and exactly 1 when ,comma please mass movement in Spanish so that is to say this isn't an ultraviolet subdivisions cameras positive well and in some dimension
and this is sort of the infrared
ones and have 1 degree I'd go I'm gamma interchange on the other side a subset intercepted Gamez is I think of a subset as a surprise and so now I need a definition so right and committed about because I knew the words to define the bottom and I thought about it and the
the the adjectives I could think of were worse than the 1 that I'm gonna proposed so far and try this Saturn and see if you think it's ridiculous we can abandon thank you different words but motivation is there's a widening of the motor which is a speck of dust for a particle this is to notions of their disability and there is no word in English does not adjective relating to merge English but the letters M O T C stands for members of the circle by coincidence to this lends notions of interconnectedness and Of course the motives of Graf will be related to its Motyka subgraphs so I define amounting against this what does not exist but tried out amount except Downer contained in energy is defined by a subset of edges it is the 7th graf such that stroller strict edge sub graf also to gamma emitters of graft began this addresses of all strict subgraphs defined by strict subsets edges which is master spanning then the number of troops of the subgraphs strictly the number of bits of so on In the event the sense irreducible Tommy made smaller without changing something and it is also there isn't it 7 years of his life so if there is no full circle which is mass momentum was also there is no such Groff earlier than then and then so there is no such graph witches Mass is 1 thing that the world of England center because it seems to me that creation OK I'm so there's no grafted his mask momentum spanning many think this is usually the area then that the other that need to be eaten important .period so remark on it if the 2 cases said it is not quite entrancing notion so yeah so it must be understood in Canada sorry for those of you I forgot to keep people must understand abreast of Ghana it's a relative motion and it's a quality health the society around motion but if gamma is must men planning In Jean apologized to study go and then we think of it as a final subgraphs inherits all the masses Amendment of the Beecroft and then and then this condition is non-trivial but if cameras not is not Nascimento spanning we think it's time we give its no and no uh masses on Monday said it is considered it is considered there was no kinematics and in which case every subgraphs is mass mention spanning so and if insists that we forget the condition about mass mementos of free gift and I have no masses no momentum than this condition is equivalent to being 1 particle irreducible so on the organizational so in this case so Gamma Gamma not ,comma has no masses all momenta then this is on gambling motorcade Cliff 90th this one-pot cleared usable that means that it's not he doesn't have an edge that you can cut so an edge and growth is not 1 partly reduced book as Niger you can cut and and the corresponding subgraphs has the same number of it so this is something more subtle motions related to the existence of masses he was he was he did so this is a physics terminology it's recalled 1 particle irreducible so this is when you consider ultraviolet the evidence is that this is a very standard notion physics because you on this comes up all the time on that because we're considering infrared stuff as well that this this subtle this this condition has the definition has this extra conditions last month can point is that gonna amounted subgraphs in implies that from the previous Our formerly have evaluations it because evaluation along gamma calls the crop anemic side instantly be given 0 because scam restricted bigger than a slowdown and so on and that implies now that the core nets Hi proclaimed coordinates so space El down is contained in the graph of and so yeah about locusts so that's something we need to develop cities of that magic is a subclass of precisely the minimal set of subgraphs that we need to blow up in altogether could come bifurcation and because I was worried I wouldn't have enough time I should prepared some some pictures and some to hand out few led customer and stadium so now the nascent properties of of photographs where sigh of course whatever it what's on that and is on this board here and so you know as bottom thing has not yet these are the sets of Mount except graphs in the graph G so let's give names this 1 this 1 is not cement and spanning because it contains the massive edge that anyone Antonito external momenta this 1 is not and this is and this is and this is an so when the variable of 1 goes to 0 you can see In his pouring on awful 1 divides everyone on you so it vanishes along warning was not and that's an inference of belligerence president of divergence this 1 went after 3 and a half or full go 2 0 again you can check that the whole thing vanishes this is an ultraviolet subdivisions and to to go back to the the question idea these gross
here both infrared and ultraviolet don't know what to call them and we don't really see the difference in this genetic pictures of the sort of mixed In the both infrared and ultraviolet whatever is and you can check that that all they have as to condition said he you cotton edge here in this graph it's an the fog well you have to look at it as you come as alleges massive so so you can't be conquered the edge and its mounted in these grasses this edge the number of its goes down and it is also the name of the state .period might take means and are either or infrared problematic and problematic at all in any dimension of space time sufficiently large dimensions space and it's minimal without proper is minimally problematic so it's not just a problematic sub crosses the minimally minimally problematic subgraphs bid for some arbitrary high-dimensional where this is due to the possible during the genocide awardees divergent because divergent refers to a specific well from the refers to a specific elements this time you take 2 pages of graphs the property on cue precautions general ones containing down at 2 climbs down to "quotation mark injury then Gannett to quotient the quotient of emotive crosses mounted extensions gamma 1 containing dammit too and both Gamal 1 hands quotient down Montana 1 and this implies that the big raft damaging amounted unions if ,comma 1 endemic on take them down on the Union Government to cause also murdered and to mend contraction that he is an urgency and edge contraction is then it comes from us then either Clinton East 1 of gamma or gamma union as erotic I have a suggestion that U.S. and just wanted to call and portable I have regained my problem the variegated problem allies of announced a problem under so because the citations and I wish I'd thought of that when I can hear now I will and is now on parole officer so but lots of government manages 600 million EUR blowouts and space In the case of a finite set page as a projective space of dimension modest minus 1 and with coordinates exactly what we've been doing all along are phosphorescent and phenomenon eschews any subset baby b a subset of the palace set events that so impediment to pass it so the 1st editions to TTS is the set of subsets the but the property that's close to 100 on the unions which is the of all and J. R & B comprised of union is indeed but this anomaly defined the integer blowup and so quoted King b to blow up balls it space PS and we did blowout foreigners answers standard procedure along all the L. I where always in assets so be
sensitive about locally will hold most of bad will will blow up the things you want to go up and you do in the following ways the 1st of all 1st of all we blow up the the points so the and the substance of spaces which are an asset to be Vermont such that the dimension of that space is 0 Belfast all points and then after that we can blow up In in the new space in the new blown up space in the eyes of the law while I'm here yeah so the number of elements is is not evidence of as minus 1 yes if his services and alright as site of 1 of your own and then blow up districts transforms the sun the original Oh then we skip this step we might straight onto dimension 1 so brought this community said and then brought bolster transforms all of the alibi the such that the dimension of Allied equals 1 so I know this is the Achilles is that of the lines the P ones I need you got to keep going up In order the dimension to it and so on the and so when you when you blow up again so so when you when you blow up you can look at look at the the the the compliments In projected space of all about .period to substance in an open subset you have on some some devices like this as expects Taiji this amount that we look at its it's it's inverse image on not open spaces just this part here I think it's a risky so it's exactly this part here the the the the full inverse image also involves these red exceptional devices so that the the the the popular the map standard here is all the rage stuff strict transforms the rest of the does not involve the use the line such the bigots strict enough the total transformers is everything including exceptional also the value of the much of the is also home and that the 1st point is it's it is well defined picky .period and is because of this condition style advocates stage on the strict transforms the air although I won the fight too for dinner I 1 the there 2 things you want to blow up an and 2 because we've already blown up the intersection by the Commission staff there disjoint so the post each other and it does not depend on the order so long as as long as you grow up in the strict order dimension it doesn't matter in each step which would you choose so it's very well known facts and so paid B has a stratification and accelerated so and it's could I mention 1 strata so this serve the rest of nonfiction mentioned 1 structure are following and that is where are the DTI which the the UN so it's not the right picture Michael is a mixed up it said and I think that the stratification of the DTI which is the total an inverse image all the original called corner high-performance ROI so that's what does the size of this triangle we started off with an energy we also have new sides the facets which are the total image although the L. Paul I was going to everything everything's Brunner on the other so I'm not time yet these devices as well as grades having 1 and as a product structure I'm so this devise a is isomorphic 2 P a space at the same time which is a blowup of I cross P S minus where B R I B and who the
be is the set of subsets which were contained in line and b lower I is this sets J minus where J is in and G contains has not structure that every every hour the visor is a product of spaces of the same type 1 the following I I was going to talk operators but on amount of time for just over of of and and then the DTI that often gets neglected in this business is isomorphic to appoint cross please b subscript and that's a blowup of Pt month as as nice repressive structure and of course the union of these devices and all the duties is strictly crossing the case so that supplies for phonograph definition so is that this is why I mention this example last time about these modelers spaces to have this very nice operatic structure that I actually forgot to mention this this product structure which can be thought of as particles going in different directions and that was on a mission last time in any case it doesn't matter because time if retained find graphs In general we get the same structure because we can said James P. J to be the projected space blown up in Willimantic motives of graph and others this condition on a very close on the Union's when I was a property that was satisfied promoted so I see what I should have done here the restless to to draw circles and white so so far we've been discussing on the widespread that defines this block is going to the red line yet this is the white stuff so this is this is because the structure and in the White on white devices which have just played projected space and nothing to do with strong substance now OK so now on let's why G and Whiteside secure the contained in I was sorry now contained this the back 1st base for grass to be districts transforms so in this British the Red line now have chance occurrence all respectively paragraph the surface so G. McGrath for surface firstly remains upstairs and the trade the firm which is absolutely key 2 Everything the following the festival this space pages is well-defined and that was the property that in most except Russell close under unions it has this arm stratification but devices so what are they and they are of 2 types there other subgraphs which amounted With the least to edges and they are just the edge devices that's that's just the that is triangle in this case move arms so again by the so we had disregards the structuring of the general setting gamma professor Chris when gamut is itself a product of and the same structure .period Grafton cautioned graphs and now uses the cautions and extension property markets of graft and the edge
device it is .period Cross that was actually and that uses the contraction property of of grass and now we want to use the factorization formally so why the graph at the surface when it intersects with a young corner 100 planes wall of social upstairs then by the contraction deletion formally that's exactly the same situation before the contracting those contraction division then the factorization properties and said that when you into sectors profit to surface with 1 of these facets you get to irreducible components service nice for structure and likewise for for the other had to surface and when we intersect with on the blown-up facets we gets so 1 of the young officers rejects on give the other half profits of corresponding to the 1st semantic so this is something gamma is not an answer it's pure ultraviolet divergent subdivisions "quotation mark potentially utterance of the and in the case and is a man we did on this and so it can be a citizen and and considered the essential defines a sort of operates structure on on this service geometry the I'm in a generic situation I'm always in a generic situation so this is this is a fight this is a family and the blow up the the the the fibers and that it will be nice to behave on some bigger and others reservists is as none of the house US-appointed absolutely highly singular extraordinary singular and it's good that they should be singular because on otherwise such say on the user swipes Irish Whiteside very singular and we don't understand the singularity in general therefore they need to be singular because on if there was smooth we wouldn't do the hard numbers will be all wrong and we wouldn't get multiple the 2 values so and because it's most positive values we should expect this to be very simple and you can also in the city cannot the discriminants on again it's something fiendishly complicated and intuitive to study of Landau singularities in physics and that it's a tricky business this is not the end of the month so that this divisive 2 types there there are so each devices of here that the 6 devices there are the ones corresponding to each on edge to this edge to and so I should draw the quotient graph here subcontracts the edge to and inclusion here contrived the edge 3 I get this craft and here contracted for and then there are other devices which come from blowing out and they corresponded to the subgraphs so here we get the subgraphs was 3 4 right 3 4 cancer too and it will be a product of the corresponding spaces of the cost photographs of this 1 is 2 3 and for some labeled so that the Diaz is the set of all these views white devices that 6 of them here and they fall into 2 sets those that corresponded just 2 edges of contracting an edge in the graph and those which we actually grew up and they correspond to promote excelled that is a very quickly idea Of the proof of this is real simple this just look for example at the local coordinates and for single blow and help by convenience are used as the set warned not to buy and clearly and once 1 then the local recorded for a single blow up and in on some of fine shots are given pledges changing variables so as I explained last time we can blow up still happens locally by bypassing the new variables and what we do is we rights I'm sorry G let's say for example the costs ii so gene-modified plus or take the factorization formula and right in the new corners the ideas In the new coordinates society computer strict transformed and Table Limit but you don't have means to them and just scared of the exception devises to something more factor UN and then I limit you will get on you will get the although 2 0 because of higher degree In the end of the 1st rivals and it will become sigh so I more and so the 0 locus I'm so that this the intersection of the strict transformed with this this course .period Fossett will have this equation and the zeros of this equation is the union of 2
pieces in its exactly this time that's essential reason why this works it follows from the factorization and for physicists acquainted with your literature they should recognize that these types of ideas in a different language for use by perhaps so I think from Stanford he blew up everything this option work by .period sphere where he defines and a sort of minimal set of coordinates to To understand the the the singularities of these polynomials on I suspect is very similar to the story I'm telling here so remarks In the case the way I she has no masses or memento and primitive then this 1 final divergent then this is the same space pages the same space as defined by blocking on crime so they did this 1st in the case of and in this particular case this another case that and I did was 3rd ,comma from single single Scowcroft In order to understand the normalization in the context of a spectrometry and it's a difference slightly different from this 1 it's a different rectification in algebraic interpretation organization that presume 1 can re-normalised directly in instrument used by jumping out of OK so long as it is the example of land Of this crop of this situation is on we had this is the graph is the list of Vermont subgraphs answer tells us we should blow up and so in the developed 1st the points which the the sobriety is defined by the user blowout .period 1st of all points 1 fellow 1 2 3 L 1 2 4 on 0 1 3 4 in any order and then suddenly we should blow up the reliance L 3 4 and this of golf is normal so on his a picture of these this is a picture of him the corner have abandoned P 3 3 bad points already don't we blow them up that the status that disjointed does matter which would we do it and we can think about is simply taking a piece of cheese and cutting of corners so after blowing up on the pizza cheese will look like this that's the 1st step will be blown up these 3 right corners and then there remains to blow up this line which is the red line here and then the 2nd step we brought back to read read cuts this edge of cheese and rejects a a figure of this Of this show and you can check that such a picture of MySpace PG and it has some facets and one-to-one correspondence with either edges of the graph or motives of graft no no no no is not has triangles is so I'm so we saw what happened to the corner simplex we had segment in P & G minus 1 Was the accorded simplex and serves just as the locus where all the protective coordinates are real and positive that's detector on left that figure and redefine and Fineman fine and points of for want of a better man fine find and 22 on is Sigma tell the duty shares the inverse image of signal as contained in this blown off and the reports this brand of space and supply this the map from patient to patient and once was so Singh until there is is this part of and it said the product is a nice way to visualize over G. that's going on in there in the pharmaceuticals and hence the mountain so the facets of segment of the gene in one-to-one correspondence with edges in the original draft was cleared by the human you mean the news that the likelihood didn't think the that what you have to monitor rights hold on yeah Abbott and undertake monitored the voting yes or at the very end of the signal but the interior announced the on so that the differences on the edges of the graphs and the demotic subgraphs and of course has nice structure G intersected with the gunman is isomorphic to products of fun 22 this repressive structure the and so on so from this rejects
a whole Father Brown so I think that the correct concept is is all that and operatic but some sort of time so it's quicker to define photograph so let s bring generated the the vectors the freezer module generated by disjoint unions fine France Armed so it's it's great it's the normally arms when you do hope fathers across a greater respect number and that's possible but I prefer to grades while the number of for reasons that will become clear and this is an algebra weather multiplication and it is this destroying Union of grass and buttons side-by-side side I should not say that the woman castaways labeled it is labeled and on this we have a lot of all absolutely absolutely so we have a Coke products following town :colon ,comma on which to connected graph Associates in any combination can tens of began Motyka subgraphs energy 2 this is much more general than the ,comma further but because it contains the infrared so my understanding is that that they're the kind ,comma morphological sees the altar vital information on and maybe on In into the literature but it seems to me that the infra case hasn't been considered on this contains all the more so damaged charismatic yet this is a problem of course he also in there special case so special case With no masses no momentum this is part just your ultraviolet then we get this is called the the call father bro 2 Sunbird blocked and crime and in some sense the limit of the the compromise of in all 4 of the full From then on this we have a differential so the fact that this is still follows from the properties of cautions and extensions of time the mounting across a stable on the caution extensions we also have a the differential every edge we have a different show after that which sends G actually contracted and then we can define the To be the song for a maximum points once the and then the results the corollary Is is the theft is a different sort of aerial photographs I would actually need is just the you it uses numerous and ordering yeah I model or otherwise you have irritating you have to work over support someone who was playing in the league is not I just think it is John union of graph which have ordered edges and in the light Mitchell tells me how to differentiate products such as cell the low the lowest edging each component that had their different ways to get around this something and that's 1 way do so that means is that 1 Delta is ,comma sensitive which is which is the key factor falls the normalization group equation hold None of us not so yeah so this is and the knot absolutely not pro-competitive yes it's a function of some groups as delicate explained to us the I hope further progress gives you the threat go explanation for the 1st formal plans all the the the structure normalization and the army renormalization group equation so but that some absorbed holes Infratest as well and I don't know what that means and the differential on structures this equation and of course is differentials as contractions communities so so that the proof actually what's nice about this surgeon genetic picture is the proof of this is completely obvious if you have in your mind at this point because if we think of this as a on a simplicial complex there is a matter that there's a boundary maps so that the interior the interior of the interim stratum is indexed by the graph G and we want to take over the boundary of the sporty To the boundary appointed is the son of all the facets the facets it's being around so the facets of two-time either that the edges and I gives the differential although not all of them the blown and facets the except and then you know that he sees no need to close a helpful you would like to the U.S. so you are going will that is that the whole thing the the years near it's every 104 yeah and um so that the amount is it that the boundary operator D squared equals 0 so if you take your the boundary the boundary you get 0 announced to the statement the fact gotta Hopper differential critical Fogerty is just this fact and In the handover explains and how to see that we can look for example at the 2 skeletons Of this point out that it is here and this edge can be viewed by the
in the boundary of this said on abounded that and they give a gives same match and that's exactly because sensitive so what you have a tragic picture this summer the statement is completely trivial this approach look at the teams look at the Curragh mentioned to skeletons also US until June and so next time and we want to study the degree motives and and which should be a relative calm on G 8 minus why G for relative to the honesty and Section 108 so here why G as union of In generic cases will need both and both had the surfaces and so the 1st thing to do next time is to define a Betty homology class which will be the same until the final point and from that define into the period and so on then there will be this amendment and then speak about the Motiva period associated to find aptitude and main pheromone which says that the the conjugates the most the confidence of the same type of water from some and special sequence argument apply to the geometry of this Lucille he could use a little the Lunar New year and believe it could be used as I haven't had time to define the category H which is when you think of appeared as a number in the classical sense so here we would you need to do is you need a good notion of appeared function family of periods and you need attacking category of of not make such structures but variations in export ostriches so the Betty will be they the category of and fine dimensional so of local systems of finite dimensional Q spaces Cook increase infiltration the drama will be trouble algebraic vector bundles with Rangers singularities infinity crippling increase infiltration and a decrease in concentration sank first-rounder study the comparison begin voting held correspondence and you can set up their visitor Anakin category you could set up the notion of metallic periods and families exactly the same way 1 of the things that were going to do it later the on the so good as to what the the city specialize appoints and no you don't you can't take limits sought accounts I don't understand what happens if it from the momentum goes to 0 the community can jump so it I don't know how to do that In that's something that someone needs to look at a map to be very complicated but Over the generic .period yes the the superficially the formula for them into the collection will be the same the unit of topological 16 different relate to get something like that the Motiva over them the view is a function and then you look at them at a government appoints and this particular rational point it could have some since special functional equations that aren't true in general so it's it that no specialist is a tricky business but then the center with generic and what their ideas it will have a space we look at all diagrams of the set number of incoming mentor certain fixed at masses that's a space kinematics some Hugel consent unless we have Anakin category of variations in export structure and every graphic that kinematics defined Sematech .period honest Nikon category and so we have a cosmic got group acting on orders from and it works fine but I won't talk about that because it's the Museum of thanks to them my


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