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Feynman amplitudes and limits of heights
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Title  Feynman amplitudes and limits of heights 
Title of Series  A conference in honor of Arthur Ogus on the occasion of his 70th birthday 
Part Number  5 
Number of Parts  12 
Author 
Bloch, Spencer

Contributors 
Abbes, A. (Organizer)

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20244 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Feynman amplitudes constitute a beautiful little island of algebraic geometry surrounded by a sea of physics. Ancient AG's marooned on the island cannot help but feel skeptical about the seaworthness of the transport physics offers from the island to the shores of reality. With the advent of string theory, physicists understand another approach, realizing Feynman amplitudes as suitable limits when the string tension goes to zero. This talk will give an algebrageometric interpretation of the idea. The Feynman amplitude becomes an integral over the space of nilpotent orbits at a point on the boundary of the moduli space of marked curves. The integrand is a limit of heights of cycles supported on the markings. This is joint work with José Burgos Gil, Omid Amini, and Javier Fresan. 
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and on and on and on and on and on on and on and on and on and on about the source is it's a great pleasure to talk this celebration of barges progresses 70th birthday and I'm honored to be asked to talk about what I want to talk about firstly the chalked up this job and analysts acknowledged and get used to it that the enclosing wall above sofa in the world so this is so that was a longterm credit where credit is due joint work with holes skill and injury in phrase and then only in the coming and also it grows out of conversations I had with the physicists in particular that appear on hold and his son who was then student here broken In and around the world 1 the source of inspiration sometimes for me the most significant is a series of years really of conversations with Professor cattle are and he has undertaken graciously to explain to me the programs that he and his collaborators I have undertaken over the it's a massive program will restore some years to understand that to generations of Hodge structure and it's it's a very subtle and difficult business the and everything I would said it is in some sense known .period to these guys or power to these guys are attempting is the sort of on the bringing these guys my Marge don't know these guys and so we we want a sort of bringing together the various schools and in particular the massive and subtle program of cotton collaborators on yields many many many variants associated to generations of structures but only some of it and so on so some remember them Muller reduced those now I think I I think even Professor counselors he probably is associated testified that I think he would even admit that that some of them are really artifacts of external structure you put on how structures and others are completely fascinating related to regulators in relation to his so anyway that we proceed what are our amplitudes soon returned list of things once a year now so on the point is that while yeah let me say something about how a mathematician and tax visits to begin Minnesota have to set level here on this is a story I like to tell you when my grandson was for if he was interested in trains and so Christmas came and I bought him a trains and the trains Caymans massive box and it was immediately clear that I made a big mistake with the train set was much too complicated for the kids and to subtle and he could make no sense of all complicated pieces and I thought Oh dear Christmas is ruined and he will be miserable but not all because in fact what happened was the will of the train set was too intricate and complicated Sunderland and everything the box was fantastic with with a wonderful pictures of drains during all these exciting things and so the whole day was passed into fancy playing with the box and I think there's a lesson there for mathematics you can't it's not possible physics is too hard that for anyone but the dedicated professional physicist to to really deal with it on the other hand
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physics involves the structures which are completely fascinating mathematically and got sits without spirit that I wanted proceed so OK so what is what is that I want to talk about the quantum field theory and quantum field theory very typically kind begins with the with really a metaphor and this metaphor is what they call the path intervals and it's a it's a big game for the national that nobody really knows how to attack an so there's been developed at 1 school of 1 way to attack it is a socalled perturb readers away and this is based on the expansion also expansion which is again inspired by finite the case and it's an expansion over from where the index set is a certain collection of graphs so gamma for address for a collection of grass and so for each graft there is a coefficient alpha gamma and then there is the the variable which is raised to the power which is the 1st homology Nunavut rank of the 1st again so that most of the basic shape and we're interested in Alpha Gamma businesses local and so would achieve our is in front with a write down some handed over their hideout number intervals for Alpha Gamma and my enlarged none of them converge but we won't allow like that because I won't make any assertion I mean there are ways of regularizing we normalizing liberals but that's not our project which is what understand the emotions and In particular ones and the Intergraph so let me begin by writing down 4 different ways there 2 to understand this Alpha Gamma Civic you straight in their mind that I'm writing down things that don't make any sense to say that don't converge OK so In the 1st Way the change decision stick with my notes capital OK so the 1st Way would write capital a as an integral of 1 thing here on we an integer capital D which will be the dimension of space time so are to the duties space time and we will give the the Minkowski so our words X 1 squared minus the sum from 2 at school so then but what we In the 1st expression for the amplitude associated to the graphic account is an integral over all I also write you just general general notation if I have a graph gamma all right energy for the homology the rank of the the number of the move number of him so then it is the 1st Circuit expression is hard to The Times Gee that's the domain of integration and then would take the following saying we take the product over all the edges Indiana so Bloomberg the gamma misadventures the certain pathogen cajoling peaceably and the PCB is quantities so the bottom
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line is we we get a rational integral producing depending upon the various but each of these things has agreed to so at infinity and you you can see that there is the possibility for divergent and all kinds of complicated things can happen but at least as as any underground that makes perfect sense that will let me take a minute to say there's a slightly different way if we want to write down the homology Of the graph we know we have a little exact sequence wasn't real coefficients then we can take a year direct some following changes are there said and then here we can take the direct sound of all D Over here Over the edges and then here I am we can let me write vertically we can take the boundary man To the direct sound over the vertices the government is the edges in the endowment and universities and again we have b but we have to total constrain durable 0 because we know that the boundary this is just a topological is suggests Little statements and we had boundary the 2 endpoints of Of the segment and we know that the resulting element here the the coefficients some 2 0 side of 0 here I and so on here we have various for the we have various projectors which founders denoted by the duel so overtaken hegemony I can reject this direct to that particular that particular factor and here I can take my the Minkowski my Minkowski Mondragon that his me all Jerry so I get them for each hedge I get them a function on this after Spain's and I can then restrict those functions To the fiber Over a given Yeltsin outcomes and importance of additional structure I give myself inside here at that point in the the vector spaces which I call .period and this is the collection of external political external moment and so I basically can rewrite this interval as an integral over the inverse image of a given external momentum so this depends in other words on the choice of external momentum I should like to depends on external momentum all of us this the Chinese flags and then again the of these convened for a PDE now is this this fall on the other it was still too soon began campaigns can OK so that's the 1st expression for the amplitude but there's some others question then the 2nd 1 but 2 a gambler there is a factor here I think it's In minus1 factorial where all right and from the number of edges them so and be 5th notations and they can also write seeing playing important role is it is a the simplex so it's the same of all so we will be contained in the the I'll take a projective space dimension and minus 1 which I'll think about it because having a homogeneous coordinates labeled by the edges so the edges and so the corresponding projective spaces mentioned and minus 1 and saying I will simply be the locusts it's a material control duties nonnegative 1st 1 of the least has been answered I it was rejected all came his way In
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the case of the robust money and you you thank you what the stars doesn't come but if you want to come and it would come in here yes that's around anyway the the the 2nd expression then becomes an integral part of rapport Eugene crossed signals so product chain and here we have a deal tax and then we have all made us all right all may go for the standard integration form of integration on productive space it's not it's not really a form of protective spaces is sort of privilege 18 doesn't work so it's a asylum over plans minors ,comma have to be and then I'll have the TB 1 1 the 2 leaders the valve which is the standard form so I put this Omega here and then to 2 make the homogeneous work I take downstairs the song I take the universal the quandary sold those PCs were contracts so I take the SAT the index labels or multiplied by the college's corners .period and raise it to the appropriate power which is just now the passage between these various minerals is done by sort of standard standard tricks and this standard tricks depending upon the graph are probably completely illegal because they involve exchanging waters of integration in divergent situation so so you have to be very careful about about that but just
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again to just see the shape of the women work the 3rd guy involves a some extra data which will need to work with them and so it has the following changes a them again you'll be certain costs which are written down here but I do not guarantee that I got it right In the last factorial and now we just have an integral overseeing and here come to little polynomials the socalled 1st and 2nd semantic 12 was so the 1st announced I call sign it gives rise to a power which if it had got it right is an minus G plus 1 times did over to but again times this performance integration formal major but divided by the socalled 2nd cemented on so I call it the gamma and that raised power in mines the pitching deal to and that's the right so here so it is the 1st semester at the time it is the 2nd time I am and I have to tell you what those things are but but let that be postponed that for a minute I noticed that too little and 3 really are live another regattas comfortable with these because the rational rational forms and we're integrating oversaw chains and the answer makes any sense it should be a period of the kind of thing that no 1 is used to dealing the 4th expression is something that a physicist was comfortable with arms it's a sort of a tollway I would swell right down you you'll see I mean it's long and over again as a concentration of counting the messages for squared all right truly the whole thing to the Jewish power and now I have an interim and now I take signal twiddle so I should say I told you would signal was in the tunnel is sort of the final version so it's just a product over all are going introduce a 0 the index .period edges again it's the thing it's the call overseas the statement with the call was saying most of the are fine and role and this is not eligible generated unit track so here we come the exponential all of these same fellows now there are no no exponents is the 2nd Samantha the divided by the end of the 1st but Germany exponential and that I just take as my former integration minister D to the overall edges and I have to divide by the 1st semantics Saigon To the deal over 2 so I the most interesting
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cases Wendy's India over to use 1st OK so this is sort of a toy path integral itself you see because what what is this Sigma Sigma twiddle I have my graph was news program cannot an association which will it is the space or I can think of as the space of metrics right it's just assigning a nonnegative number to each edge With the possibility of degenerating 2 0 so this then becomes an integral over the space of matches now 1 of the typical versions of past integral that occurred In quantum field theory gave integration with the domain of integration is the space of paths but not on a graph but rather on interesting reminding bountiful so here is kind of a toy version of such end such a sort of thing but OK and so we want to algebraic geometry the answer of course we want to forget that and work with was was the 1 of the have seen the 1 2 4 3 but actually not In fact it the other way geometry we wanted to involve losses so I let me go on I have to tell you what what these polymers or is it it's kind of what is it To say short on time so let me say it quickly if I have out there in terms of configuration so configurations so I have some vector space age survive dimensional vector space which I the governors or is is given as being inside the it really is a matter is everything is algebra so it doesn't matter just take the field can end right but inside the sound the vector space with a given and when I do that then for each edge I can Bridgette doll on 2 the the corresponding the French Quarter and I write the dual also from the composition here so do will then become just a linear functional on stage and so evil squared becomes a rank 1 quadratic 4 but the former and so it makes sense to look at and think of it as your life I can't think of the dual squared as a matter of honor canonically from age to to fuel and I can look at the song I t he Times the square and I didn't cheat a little bit of is the better choice of me if I look at the determinant of this expression is not quite welldefined because sort of I have to fix a basis of but changing the basis doesn't change the output in these these variables and I really care about this thing as a as a polynomial in these variables so I will call this thing at sign "quotation mark College this is the 1st semantic and welldefined up to scale of history arm and of course the particular situation we're interested in is where age and the age and calls for each 1 class of inside well with with Kate coefficients analysis and I had the ideas for this that yields side were a "quotation mark again which is a homogeneous polynomial in OK now the 2nd semester is slightly trickier but not much if I look at this because of age or do you can't generalize H contained in care the labels vector space and where the right W for the for the quotient and a section called call here so there and if I know I have stage than of Ortega provides for any for all W and W I can look our at the the vector space not the vector space age but the vector space page lies turnaround the W the line online standby by but the guys and I can look at this the polynomial adjust constructed and the institute subscriptions from Willard H. SOW people's village close to the line spanned by the W and with the right to vote fall all the and people W it will be by definition of page summed up OK so it's a polynomial then which is also not a degree G7 member in the grass situation ages of dimension Julie so but added on 1 line so it actually has green G plus 1 so the so the 2nd semantic fight Of course what would also flies which depends on the TV To me but it also depends on I should have called it something else that in In the case of registered it depends on the external momentum for the precaution w here in our situation is the external aspects of external momentum and so this is homogeneous Of the 3 G plus 1 key and on contrary to there the instrument now there is a tricky .period amusement another tricky the .period here but for us we want things to be In an Audi we wanna be in space time what I described here is a kind of a linear I mean is the key here is not in time anymore so what we have to do it is we have to couples so I we fine 2 of the With the Minkowski metric and basically I'm on identical to in the telomere just say a word about how that works if I have a of matrix the words abroad the other is if I have so far
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redundant if I have natures that looks like this and here I have a domino transposing hearing of building in here and ask so this this is James and this is due course One and similar here it is G and here it people's land so the W's event role column vectors and is Justice scalar just 1 by 1 then I can yelled so this will be symmetric sorry so and Slattery then there is a classical formula for the determined so like all the whole matrix of socalled clean there hasn't been determined the game is over something like those in depending on parity of the day I computationally there is there is not a minus sign here on global you transport take the agile and agile merchants in town and I take the last times determined "quotation mark or if I underwriters differently than riders is determined to be divided by determining what and is minors W. transpose and here I put him in 1st because I know that adjoint matrix divided by the determination of the people that he will start the other once I left out of the status of the the W transforms and W the trend said yeah plus something that I'm now notice you see what I can do what I want to do is I want a couple of W true the this spacetime and so I have to then reinterpret this thank you wherever I see so there's going to be this Christoff was other quadratic in the interest of W so where I see a quadratic expression of I replace it by a the Minkowski quadratic form on most of the appears so from this point he was kind of easy to see how to coupled couple the W part d with wooden casket Majid and in that way using that technology you know him you seconds allows to me complained to I work with speed In well into the fall .period or do the index for the vertices comes In this quarter excellent thing is quadratic and B and ordinary people swarmed into so these are the 2 configuration polynomials which and played a central role in the whole day OK so now but the situation is that this I started out the year with a sort of generating series indexed body Graf and is generating series comes from this sort of metaphorical object which is the path integral but that whole process is extremely unconvincing to mathematicians it literally makes no sense at all and the solo that's where my life abstract and talked about a senior physics so that's that's the embassy and the question is how to get across that seat without indulging in some of fantasies that difficult for mathematician to understand and I don't know the answer but let me that there is a separate surprising that game that can be played as I want explained to explain OK so this is the basic system but now I want move to geometry arm and the idea is going to be there are graphs Gamal for a salary gamma but we interpret as being the doorway 2 a they are approved but stable rational can I gamble careful are we to assume maybe that vertices are all at least have 73 edges services more constraining amounted to get stable but as of right now I'm not a big deal of it beyond that so let me remind you this is a man of familiar game gambling line you have to have play the game and for each of the Council for each vertex In we associate or a monster elected P1 and that's what they think and if that we have an edge and the boundary of the edges let's the W then we go God P 1 gene and if you want something have until now we have to be careful I'm not claiming that there is a unique but there 1 July here if if but given Vertex has 4 1 more images attached to it then we will have 4 or more attachments To the corresponding P 1 and so there will be much so I don't claim that this is unique but I just do it somehow and all this going against us so this process yields a curve the stable rational curve which I seemed so this news at sea room from analysts without game those statements only at foot of the yet the event and if it doesn't have men but this is not enough a real difficult issue we can deal with stable security so we can then look at the result of the look the Brussels defamation all this Of this season on and as I say c mon itself can have taken them underlying so the picture that we get some idea circuit a shadow effect so the picture looks something like this we have and I was thinking the analytical resources drawing and because I wanted the topology Zionism and algebra John attended a formal things but 1 with analytically so we have a family there of Burroughs Over space spacecraft and thus contains all of a sudden variety close abroad which are called to the Internet fullback since so a thing like this so but the bears
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it is open this is essentially ceded to the trees G England's 3 and eventually I'm going to add more parameters so was somewhere more parameters but for the moment you can use any of its 3 G more parameters without any need to deal with March curves so this is going to be family friendly and he did you know about the is always the genius of the ground which is the same as the genius of the curve why should I should have said that in fact has seen a lot so the dimensions the Ancient Order of seeing all the old Seymour his duty and that's also a mention of each 1 of them on so what on Sunday the answer this is the oversold affirmation and for the team there will be devices today will be an intersection of devices D these days His wife for now is which is the same sort of know what is in order look the Indian look I am capital and I'm going to tell you that in a while this capital yeah that's going to account for what I have to put punches the word not not punches would have to look at March curves so this is will be extra parameters for market but for the moment we will not move doing that to understand the geometry so the point is the team is itself an intersection of devices so these are devices corresponding so we can think of us as is the versatile deformation so if we fix a crossing point and remembered as 1 crossing point for each edge if we fixer .period crossing point and we look at deformations of the curve where that crossing point stays crossing speak other crossing points can open up downstairs that defines a divisive so did parameterize deformations deformations for nations the risk the they didn't give it a name but I should give it a name of with with seed I fixed on the text of the curve other crossing points can open up with this Wednesday's verdict on then TV is the intersection of overall this devices this is advisers because I'm just putting 1 constraint on what information look at being and all this is unobstructed and by the classical because we're dealing with OK so a mail when the drop you know we have to composers so the androgynous to picture I and my artistry is not very good at all times no worse than most of them the study program thank you and then I draw the same thing again should have I the so imagine too served curious shaped objects kissing each other in 3 places OK so this is seen on and what is seen not was the graph should have 3 of the 2 vertices and 3 edges so that was easy to draw down is just says controversies 3 but I when I when I look at the vessel deformation I just it becomes that it becomes the usual gymnasts pitcher and so what have I done I've squeezed so vanishing scientists that 1 and there the Jones these are vanishing cycles and so are drawn here is the seen not and let's call this see With the Arsenal and for some base .period so I'll take a base .period in here which is which is isn't away from so as not is not more and not in any of the many of the devices that it corresponds to occur with no singularities again and then some by the generous when so that's a classical well understood the church and some remarks 1st of all we have a specialization and recording speech from the homogeneity of the general fellow also policy is not just smooth curve it would ration competitions to the mileage Hong the senior Her and it's consortium that's the that's the 1st of all the existence of such a map is classical it amounts to the statement that in the fibers here if I take my senior fellow I can always find a tubular neighborhood which is a defamation retract and then I can imagine my small guy living in that neighborhood so I can map the homology of the smooth guy to the homology of the neighborhood and then by the deformation the maps to the knowledge of the special fiber so that's a classical game and the fact that its symmetric you could just see him because you're looking at clothes pads in here and they will have to to close power to pads here which which stand at the vanished inside with the eye just go around the engine cycle and just amounts to saying the vanishing cycles are connected easy exceeded the seductions broken you know the fight the 2nd fractions of the 1st practices projection the 2nd
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factor flyover right there 3 the kernel of inspections emissions so that's the space spanned by the the vanishing cycles notice the managers cycles are not linearly independent because when when we have to can to irreducible components we get a relation between the engines whose firm manages and not linearly independent here the 3 of them but I've been a user space spent by the halls of and the assertion years it was the assertion that is the is a natural isotropic source don't it's 1 additional on serious there you have to think a little bit and you have to give bridges over Israel's isotropic recalls if I get close enough to hear then and the vanishing cycles are gonna be forcibly separate from each other because they live in little neighborhoods of the points in the pointers so the vanishing cycles are clearly separate from each other which means that the pairing between 2 of them will always be 0 and the fact is maximized dropping just simply becomes read because this surjective and the says dimension G and this has dimension to GE than a has dimensions G cancel so the reason is just the mention of his G 1 1 or more of his University of Tennessee what we are a form which is part of just and this must be intersection of the also there is an alternative form 1 page 1 Of because it is physically just intersection is with with orientation Soltau who lets her still have enough time I think I yell so I now want to talk about the declined Lachance transformation so through this accord clashes at the end of the current lunches there is is like that but I rights and for the vanishing cycle associated to me In the remember that these guys index the the the points here and the bad points a singular points the regional as I have a circle that is contracting to it that gives me the vanishing cycle which I call the In the end the luscious but says that if I look at the effect on the homology I get by winding around so far wines around DDE the divisive debate associated to that particular eh then what happens is a general 1 cycle goes to the and there's an issue orientation when received plus being the intersection of being with the the times the designation for of the legality of the Sinai is minus looks as wants you to know that it also visit by a lot of yeah I mean we have to figure out which way we're winding here so that the doesn't it is not important in all his arm so that This is a classical familiar the fact in salt ,comma if I call this transformation was recalled this healthy Of all the linear form then we know that and and to dispel the minus In the event company which is also long half of the year there aren't just sense or any company this is just the prime intersection b times so this is all again and familiar the start of the and the noble Norman In my mind is more than the AMA someone into logic coming out of the bottle here and see if I can do it Of course there were then the debris then the noble toward serve the following month it's just that the collection of all and the morphisms in call of the form some of the edges trade unions so some real in negative real the constant times this year the and so it can call the collection of and it's easy to see that in fact it is noble it's not perhaps quite obvious but you have to think about what each of the Andes or other international donors is in fact squares but it's a question is certainly no part it died is and I certainly not only individual example division "quotation mark using a physically it began as this issue that because the all physically they don't know if I apply the cause of this is so the product of differences in social actually squares working as a kind of thing is difficult to think when you on the Lobato which reached out this is it this is exactly the point of an unregistered and his 2nd point I was going to say these are new potent has an morphisms of page blond the school system but there is another way you can think about these things which is true as follows you can write H 1 of the dead the that is the same as page warned all the same worker and adds isomorphic to stage 1 of the smooth curves Margalo this which remember was the matzo isotropic some space spanned by the managers cycles and then he any 1 of these entities kills all of a sudden it gives a map from this quotient In other words the the image is which is isomorphic had to H 1 of C. S module a duel but I don't find these
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things maybe spring to mind or
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this is then isomorphic to page 1 bogey on the way so for each there we get a bomb on the Net From this dimensional like the space to install that is we get a quadratic form and the denies that would allow a lot but it's a little exercise so "quotation mark proposition it is that this man feet the using equal to but I think I screwed up by not giving in the name of is equal to anything which was so remember I had H for gamma from inside their and so on hedges of gamma and eventually reached edge I could project to are in this case the a functional Michael duel and then he was associated was essentially the associated the quad formed into and the proposition is that this did this the cement and when there is a symmetric matrix of quadratic form on is the same as this 1 that comes from John it's not hard but thanks have altered so that with that in
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mind we want to relate the 1st and 2nd cements on on Wheels which came out of just abstract discussion of the draft program linear algebra associated to the 2 the limits of heights on this family of curves not all heights we need to all I got were time so far I'm going to go fight over the chairman of the chairman can can start his watch and so I need to talk about heights so let me list say enough to to make statement so I consider it now but let me draw the a picture of me because it wanted to hear so here this mostly as here it the bad fiber and here is this he here see what single occurred and so then here is a smooth curve this is .period more so what I want to do so again of futuresrelated nominal no I think it's the normal instances disconnected I'm so above so I went and In parameters and when I had parameters that will enable me to enrich the picture by adding some sections so I will and some sections "quotation mark so these sections sections subsections oracle sections would say new you know thank and then by control very carefully How the sections need Henderson I think so what what data do I have to pause this section meets this compiled well this component remember it as member the components of the index by the disease so this is the the component so revered vertex on them Monday so what I do 2 things I I want to get I want to deduce from this collection of sections which I think it was a family of 0 cycles are in fact I think of it to families and your calls I want to the you but external momenta at infinity now external momenta or linear combinations of the vertices with coefficients in already OK so what I have to do in the 1st instance survived a couple the sections To the vector space so I couple the sections 2 for being awarded at nearly as many sense but what I'm interested in is the height so I write down recently can that was whether it was gone 1 of the parade down the Hyatt which is a cell R. V it's called you New once or I would say knew I knew all eyes the sessions so that call this and I'll take another 1 would say only prime on a prominent J and holders of the prime and I look at the height which is any prime and because I want these things to yield of external momenta I have a constraint that this summer all the archives should be 0 and similar here the somebody ought to Prime's equals 0 which is perfect because the talk about a high I need to talk about 0 cycles of degrees you so I don't have time to explain this idea of coupling to a vector space but it's not hard once you know how to define Heights you use easy to see how the couple to vector space and then the here while were young and so it at the end of yes and then it it is the number 1 and this is it this is it this is the height of if I have to 0 cycles of degrees 0 which are disjointed I have always defined essentially you can computed by taking differential forms of polls on eBay and integrating over chains here so this is the classical or I'm sorry that my time that way but the ceremony was going to be there at the bottom if we look back the forth preserved that the the exponential Paul Tsongas or foreign all it was the limit but as a certain parameter alpha not much of a newly have time to explain as Alvin not goes to 0 which essentially amounts to saying if we look at the structure Of the base answers so base as we can think of is a product of copies of GM's alike punctured disk and reimagined the parameters as all always going to 0 the notice that this is what is often lot things and then there's exponential here is a limit of exponential of all time high a prize where In a prime are both
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0 cycles corresponding to the given external momentum he so this is a function of external momentum so ANA prime are 0 cycles corresponding to a given the In the sense that I explain that this is isn't the across the right vertices with the right and the bonds of the spacetime at this point so the bottom line is that the the the term that occurs In the amplitude there's a limit of Hi now so in you office so alpha 0 always measuring I don't really have time to go into detail but alpha 0 is measuring how we are approaching In view of the detainees so remember we have passed and we have to so I'm thinking of it as Smitans TD has being something like a product of contradictions an alpha 0 measuring how fast we are approaching the punches it's just as adviser normal crossings and we're approaching simultaneously although all the parameters and having speech and say this is on the way Arkansas you can get the details but I think the stock which we disclose the Thursday of the passage the ship loaded New Brighton loaded with cases of this flu to those of the other good not assault not exactly they are we just say a word the phone this will get rid of the Member Everything is a function Of of the external momentum which are in the home all are don't V 0 OK on the the 1 of these the components "quotation mark so for each V freestyle In I give myself section OK so I have a section was called New something although and this section needs please the body it's about but that it just needs to to because it it's it's guy "quotation mark the yes because he has a good point again siren time so that there is a title .period about the heights which we need to assume then the this 0 cycles in question disjointed so what I would like to write this bank but that doesn't make any sense so what I do is I take an and prime which it also meets there with the same with the same thing so that's that that's we knew that there I think the question is should high handinhand with of the other Larijani and my friends in between cooling effect on the fashion Finland division it is roughly speaking they they are the same they are the same height is the Green's function on there are in fact I like to think of it of the heights because the the philosophy here what the what the mathematician wants to inject are 5 structures and the heights are typical examples all realvalued functions associated to variations Hodge structure so there should be a general game when we have to generating variation on structure we should be able to get interesting ample as In the girls all over the Nobel award associated to this this variations of ads that said it would you right and maybe that's why I say that the new guys move but you maybe didn't think of that from the point of view of our actions it's the only way he was going to visit yes but you don't want yeah this is actually a professor credited were presides over this and I can but you don't want to let the various Edu's goal at different speeds because there and in that way can be if you if you choose that the different speeds badly you can get the limit can you get what you want to that he has the system for his new position not all as I should have called back Prime yeah I should called the Alpha Prime is His measuring the the approached the podium this season as he won the match the 777 profanity )right parenthesis has I'm still working with the box you it instead the engine and want to see the region known review is moving from among the analysts who where is that was not on his arrival at 1 end of the year in the U.S. journalist in the direction the head of the provisions of the bill the soul of the children were you we know what needs to be taken closely at the very end of the and year and Jane and political grounds in the city although the components analyst managers by the name of the Father of season you go to the not yet it is not yet but there also president and so openly conjecture of the price the based on the the securities funds which have been in who can fall 2 of the variety of things that's been very interested in the work that was all the injuries and the variety of cultural and political leaders in the world and World Book of the Month considerably so .period cases in recent years in private and in the world in London the land is another of those killed in the the forward it was the story saying the story was a significant over here to ask the cost of this world from my mind for a review our eyes notice is just 1 1 remark just about what you said that this is absolutely the most generous we're going 2 the worst place why wine should physics be preoccupied with the world's worst .period this is there any physics to be found by just degenerating the Union of an armed assault on there's lots of possibility for interplay between the national all this really who was injured have where can we need we need the notice we need the hand emotional situation because we need ultimately are integral is going to be over the full nobleman not just over 1 1 we don't the question is do you question yourself with the world must and yes yes I'm sorry I should have said this is all Marsalis but it's a nice it's a nice question what happens if witnesses and I have a full consideration of premier who rationalizes for exchange of fire in the grass and usually surfaces with his hands ages 5 through these disasters the work of the socialist system many Muslims yeah I know how I could hear people manager allegedly correctly is that not by me in any case no not going my what was the the proposals was made on where can he said a lot of things the let me everybody was just say that the chain of integration is replaced by irreducible components of the real grass mine end of story for me and I can go from there start
00:00
Point (geometry)
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Mass
Student's ttest
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Frequency
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Manysorted logic
Lecture/Conference
HodgeStruktur
Cuboid
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Energy level
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Regulator gene
Generating set of a group
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Set (mathematics)
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05:08
Axiom of choice
Homologie
Greatest element
INTEGRAL
Multiplication sign
Decision theory
1 (number)
Food energy
Mathematics
Positional notation
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Ranking
Flag
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Infinity
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Projektiver Raum
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Hausdorff dimension
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Coefficient
Local ring
15:51
Polar coordinate system
Polynomial
Trail
INTEGRAL
Multiplication sign
Kontraktion <Mathematik>
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Mathematics
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Produktraum
Divergence
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Expression
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Integraltafel
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Chain
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21:46
Axiom of choice
Algebraic curve
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INTEGRAL
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Ranking
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2 (number)
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39:30
Homologie
Group action
Correspondence (mathematics)
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Orientation (vector space)
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Mereology
Fraction (mathematics)
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Circle
Fiber (mathematics)
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Product (category theory)
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Moment (mathematics)
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Delay differential equation
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Order (biology)
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Morphismus
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57:15
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Functional (mathematics)
58:55
Greatest element
Group action
INTEGRAL
Multiplication sign
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Parameter (computer programming)
HodgeStruktur
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Cuboid
Fiber (mathematics)
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Infinity
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Chain
Linearization
Normal (geometry)
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Number
Prime ideal
Frequency
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Family
Matching (graph theory)