 Thank you. It's been a very interesting summer school for me so far, so thank you very much everyone for putting this together and the opportunity to speak. This is all joint work with Tom Bachman that you heard from a few days ago, and we'll start with the application to accounting linear subspaces. Let's give ourselves notation for k a field and f1 through fj homogeneous polynomials that are going to cut out what's called a complete intersection on projective space. So there's the homogeneous coordinates of projective space and we can look at this variety which is the common zero locus of f1 through fj. And if this is dimension n minus j, that's what it means for x to be a complete intersection. So we're going to be counting linear subspaces, let's say explicitly what a linear subspace is. So on our plane in x is a copy of p to the r cut out inside p to the n by linear equations. We're going to let these linear equations have coefficients in some field extension, side e, and then we get it will be on our plane in x if it sits inside x viewed as viewed as a scheme over e. So there's our linear subspaces of dimension r. So the space of all linear subspaces is a cross monion. So this is a cross monion parameterizing p, r's, n, p, n. Equivalently it's the space of r plus one dimensional subspaces of an n plus one dimensional vector space. And we've got some different choices for notation for this, but let's go with gr, r, n. And here if we have a subspace w, we can take its projectivization and we get a pr, we can even make it a different color. We have the subspace w sort of a point in this cross monion goes to its projectivization. And there we've got some notation for cross monion. So the cross monion. Give some excuse me. There's a question for you in the chat. Like you want this. Great. Thank you. So Hain says it's just a closed sub scheme. It's not. So, oh, and this one too. I guess in principle we could have x be a linear subspace and then maybe it would be equal. So xe is some interesting variety cut out by equations of different degrees. And inside them they might have points, lines, two-dimensional planes, three-dimensional planes. And I'm just saying that explicitly here that we're allowed to have our linear equations have coefficients in e in some extension. Does that make great? Thank you. So we've got a tautological bundle and it's a vector bundle so that over the point corresponding to the subspace w, your fiber is w itself. And we're allowed to do vector space operations on bundles. And the one that comes up for looking at these complete intersections, let's look at sin d of the dual of s. This is the vector bundle on the gross monion, which at a point w we have the degree d polynomials on the vector space w. So let's give ourselves notation for the degrees of the fi. So let di equal degree fi and fi determines a section of sin di s dual w. Let's call this section sigma i by sigma i at the point w is fi restricted to w. Fi is a fi is a polynomial on the entire vector space. So it restricts to a polynomial on each of the w and this gives an element of sin di of s w. So from the choice of our equations we get a section sigma as the sum sigma sigma one plus sigma d sigma j section of let's call the vector bundle v for convenience of the direct sum of the sin di s dual. Now we can express the space of all r planes in x as the zeros of this section. So let's let f r x, this is the space of r planes in x. It's the subscheme of the gross monion determined by sigma equal zero. And it's a result of Debarman-Navell that for generic fi this is smooth of the expected dimension for generic fi. So why is the vanishing of sigma the space of r planes in x? It's because we have a plane p of w is in x. If and only if the restriction of fi to w equals zero for all these for all these i's. So we know what our dimension is. We've got the space cut out as the zeros of a section of a vector bundle. So the the dimension of f r we'll call this delta is the dimension of the gross monion minus the rank of the vector bundle. And the rank the vector bundle is the number of polynomials of degree di in in our variables and it's all to be explicit about it is it's di plus r choose choose choose r. So let's do examples when we get a finite count. We've got it's finite for lines on a cubic surface lines on a degree two n minus one hyper surface of dimension n three planes on a degree three d hyper surface of dimension two plus one fourth d plus three choose three. To give an example where it's more than a hyper surface we could take lines on a complete ci for complete intersection of two degree n minus two polynomials follies in p n. We're going to say n odd because we're going to want all these examples to also be orientable in a second. In any event there are there are tons of infinite families of these for not just lines and three planes but but but all the way up for two planes too for that matter. So in any of these cases there's some finite number of r planes and a question is what is that finite number? So when delta equals zero how many r planes nx? And the answer at least for k equals c over the complex numbers. When we want to solve equations we have this nice algebraically closed field. We have a nice topology on it and and this is a classical topological Euler number. We can see that it's the Euler number of the vector bundle we called v this was the sum of the symmetric powers and one property of Euler numbers is that they can be computed with respect to any section as a sum over the zeros of the section and this is valid because the zeros of sigma are isolated and so we're taking the zeros of the section gr for gross monion and w is a zero means this is equal to zero and it's of some some local degree. We can view a section of a vector bundle as a function locally function from c to the dimension to c to the dimension. c has a nice orientation fixed on it and then we can take the topological degree of a map of small balls. By the debar manavelle result the zeros of sigma are also simple because the zero locus was smooth and everything is orientation preserving so what this gives us is a sum over one of these same zeros of sigma and as we commented above the zeros of sigma are exactly the r planes so this is the number of r planes in x. In particular this first equality here shows that the the answer doesn't depend on your polynomials it only depends on their degree their degree only depends on the di as opposed to the fi. So one very pretty example is that there are 27 lines on a cubic surface a generic cubic surface generic. Got off the internet a photo from the Göttingen University that's credited to Ozaozig and there's a clutch cubic surface and there are 27 27 lines on it. To go up some dimensions which means we won't look at it but there are also for example 321 489 three planes on a generic cubic seven dimensional hypersurface and for for any specific ones you want you could you could get this answer using cosmology of brass monions the splitting principle and useful tools. So let's let's change k from from c to r so when when k is equal to r we want to use an Euler number again and for this we need v to be orientable there's again a way to express this just in terms of the the constants we have around so the this is equivalent to this the sum of di over r plus one di plus r choose r plus and plus one being congruent to zero mod two and all the previous examples so examples delta equals zero orientable all of those above and we can again compute the the Euler number it's again can be computed with respect to a section and again since the zeros are isolated we have that this is a sum over the points of the grass monion of these zeros and then there's this local term local degree of a function so if we look at sigma in some small neighborhood u of a zero of w it's going to look like a function from from r to the dimension to the rank which is which is the same and then in coordinates you'd have let's say sigma upper i be the i coordinate when you view it as a function and we can take the the partials with respect to the coordinates xj on on rn let's let this be denoted by the Jacobian of sigma the determinant of of this matrix let's make this a little clearer and then the local degree is the sine plus one when it's positive minus one when it's a negative of this Jacobian and so what this gives is some plus ones and some minus ones this is a weighted count of the r planes and it's also a lower bound so uh while the the number of r planes can vary um from uh say uh a generic seven-dimensional cubic hyper surface to to another seven-dimensional generic uh uh cubic hyper surface there is some side count of of these um three planes uh which doesn't and um Finesha and Karlamov write a paper about the whole family of of three uh plane counting problems and uh in particular so so they do this on the degree d plus one hyper surface in general and there is 189 three planes on a seven-dimensional cubic hyper surface um weighted count um over r uh so um with uh the a1 Euler class introduced in in Marc Labien's talks we can obtain a count uh over any field um let's sorry to interrupt before you continue maybe there's a there's a question for you in the q and a um that's right so um this so uh the the q and a is pointing out that this needs to be an integer um and uh it is uh many times thanks um let's uh talk about a1 Euler um classes and numbers so let's let v over y be a vector bundle uh with a section on a smooth scheme y a section uh s uh and um uh barge and morrell uh defined an an Euler class uh in the oriented chow of y twisted by uh the the duality determinant um uh and this Euler class was there was a there's a one in oriented chow and um you can push forward the one by the section and pull back exactly as um as marc outlined um uh last week uh and uh let's let's list a whole bunch of Euler classes um uh uh so uh although we haven't introduced um oriented chow it's a a sum of uh of cycles with with coefficients um and and let's uh let's not worry about that uh too much at the moment and this was further developed by uh fizzle and ashock fizzle and marc levine um uh and um another description of the Euler class is it's a principal obstruction to having a non-vanishing section um and uh in a1 algebraic topology over a field morrell develops an Euler class um encoding that that principal obstruction um in joint work with jesse cas we made an Euler class designed for a number of problems that was the sum of the the degrees over the zeros of s of of local degrees of s this is an Euler number um uh in a beautiful paper on fundamental classes and uh degrees jin and kan uh give uh constructions of of Euler classes and in particular uh marc uh presented um uh using this uh last week and one of the things this does is we get Euler classes valued in cohomology theories uh that are represented by s l oriented um uh spectra um another thing that uh marc uh talked to us about was a pairing on the kazool complex um uh and uh in joint work with between levine and raxa they show that um this is the Euler class for the the tangent bundle um my coffins actually uh after a talk i gave uh eyeball the the pairing on the kazool and and and asserted it was the same and i've heard uh marc uh say a nice story about um a similar um uh responsive of of ser i'll just put everybody's named down if that's okay um the the last list in our in our point of view on Euler class list that that we have here um is to take the opportunity uh to talk about my women in topology group looking at the hawkshield homology um uh and putting the pairing um that the the folks who think of coherent duality in terms of hawkshield homology um give us uh and showing this is this uh a one Euler class and this is work of kandace bethea uh nini arcila maya morgan opi um uh inis akarevich so um one of the things done in the in the paper that this talk is about is to check um qualities um for for some of these uh so um uh the the these local degrees and and the push forward and for appropriate e and the kazool um uh for for general vector bundles um uh as a warning um there's a very interesting um paper of asha kandacelle about comparing these two and they are equal up to a unit but in a in a context where there are actually quite a lot of units um uh uh so um it's it's an interesting it's an interesting class it's an interesting number and there are a lot of different points of view on it um and uh for moving forward uh with with the the story um let's let's summarize this by saying when y over k is smooth the dimension of y is the rank of the bundle and v is a generalization of orientable called relatively orientable then we have an Euler number call it e of v uh living in the growth and defeat group of of k um i'm going to spell out what the growth and defeat group of k is very explicitly because it's um concrete and our accounts record concrete um arithmetic information valued in this group um uh so uh let's let a um be a ring and g w of a is going to be the group of um formal uh differences of um isomorphism classes of non-degenerate symmetric bilinear forms and this group of formal differences called a group completion and for a field this has a very nice presentation so presentation for a equals k a field then uh any um bilinear form uh can be diagonal symmetric bilinear form can be diagonalized so the generators are uh one-dimensional forms for a and k star over k star squared as in uh mark's talk is on the one-dimensional vector space k k cross k to k and we take as a bilinear form we take x y to to ax y and it just has a few um relations we have the a b equals a b and a plus b equal a plus b when that's not one plus a b um a plus b and um this implies that there are a bunch of ways to write a special form called the hyperbolic form that um a plus minus a um is this uh hyperbolic form um h uh one one plus minus one um uh i'll give a whole bunch of um explicit examples uh so uh the growth and date fit group of c um is that there's there up to squares uh all all elements are the same in in c star uh so if we just take the underlying dimension of the vector space um the growth and date fit group of c is z by the rank and over r um when we diagonalize we have some number of ones and minus ones the difference between those is the signature and um that induces an isomorphism up to a parity condition um the uh for for fields like um a finite field with q elements or c t um we have that um the first two invariance of quadratic forms coming from the the miller conjecture uh invariance give uh give the whole growth and date growth and date fit group the first two are the rank and the discriminant that's a determinant of uh of uh of when you write the bilinear form uh as a as a matrix um so this goes to z to k star over um k star squared which let's say q is odd i give z cross z mod two and then we have invariance coming from the the miller conjecture um which was a great achievement of um of a one homotopy theory the the title of this talk was about integrality results and that means that uh we want to use where the vector bundle is defined um if it's defined over over z or some sort of ring of integers we want to um uh compute or say things about the computation of what the Euler Euler number is so let's also say that the growth and date bit group of the integers although there are many very interesting highly non-trivial um uh non-degenerate uh symmetric bilinear forms over z once you group complete you you only get z cross z it's a number of ones and number of minus ones for instance for example sitting inside g w of q um or d g w of r maybe maybe more straightforwardly um uh we have a miller exact sequence giving us the growth and date bit group of z one over n and it'll sit inside the growth and date bit group of q and then there's some boundary maps to the bit group is the growth and date bit group uh divided by the the hyperbolic form um for q not dividing n um uh the um the last thing we'll need to state to state results um are the explicitly giving transfers um uh we can uh assume the only thing we'll need to write down um theorems is transfers for an extension of fields k l that we can assume to be finite separable because the department of l result will say that our zero look is smooth and so corresponds to separable extensions and then we will have a transfer map from uh l over k from the growth and date bit group of l to the growth and date bit group of k um uh and it'll take a bilinear form beta v cross v to l and it'll take its class to the composition of um beta with the the sum of the galois conjugates the the trace from uh uh from from galois theory um uh so um let's let's talk more about the um uh about growth and date bit um but i also i want to pause for any questions so far um if you if you think of others uh please don't hesitate uh to interrupt so the the gw of fields pieces together to make an unramified uh sheaf by a procedure in morrell's a one algebraic topology so it gives um unramified sheaf my gw say on smooth schemes over k um and uh the uh sections over x they're inside the the sections on um k of x on the on the field the functions of x and you can extend over uh uh any um close subset of co-dimension at least two it's given by intersection of kernels of certain boundary maps for co-dimension one points um and it has an alternative description it's the sheafification in the niznevich or as a risky topology of um uh sending x um to uh gw of x where now this means the symmetric non-degenerate uh bilinear forms on on vector bundles um uh on our on our ax and this is a great sheaf um uh uh but from the perspective of um of descent sheafifying isomorphism classes is an odd thing to do when we if we if we want to glue together objects we not only want to know their isomorphic on overlaps we might want to keep track of uh of the data of that isomorphism have it satisfy um conditions so uh one thing that that that comes up is that you can have a sheaf of spaces version make it uh curly uh curly gw um so let bilinear of x be the space of vector bundles uh with a symmetric non-degenerate uh bilinear form and then we can sheafify um the group completion of this so let's let one half be invertible let's let these be schemes um uh over uh z one half and have this be uh um a sheaf valued in spaces and we'll let it be the sheafification x goes to uh group completion of um the uh this uh these bilinear forms um and there's a map uh to the sheaf of spaces associated um to uh her mission k theory we've got a homotopy invariant uh version on the right um and uh our our sheaf version of uh growth and deke um uh the sheaves uh on the left um we can take uh or their classes um in the in in her mission k theory um or related theories and get good functoriality properties um associated with them uh uh to to give us um integrality integrality results let's get back to um counting our planes and less questions yes um so uh it's necessary to invert two when when considering uh duality so uh bilinear forms there are a lot of different that there when when two is not invertible bilinear forms and quadratic forms are different turns out there are a lot of different variants um uh uh for um uh for four notions of um uh of a duality uh on on vector bundles when when two is two is not invertible um uh so um it uh it makes a her mission k theory defined with with current machinery and there are a lot of interesting things to say about inverting two uh so so thank you for the the question returning to to counting our planes on complete intersections let's list some things we know um uh in in joint work uh with jesse cas we computed the account of lines on a on a cubic surface which is the the computation of the Euler number for sim three of s dual on the gross monion and it is 15 1 plus 12 uh times minus 1 and uh it's this is also a song over the lines of uh an index involving the field of definition of the line and some some information about how the tangent um how the tangent plane spins along the line um uh this is for k a field and um mark levine developed a theory of uh bit valued characteristic classes and uh computes that uh uh the Euler number of uh sim of uh 2d minus 1 of s dual counting lines um on uh um um pd plus 1 is uh 2d minus 1 factorial factorial tell you what that means plus a multiple of h and the multiple of h is is designed uh to make so ec for a Euler number of complex points minus um 2d minus 1 factorial factorial over 2 times the cyberbolic form um so here uh 2d minus 1 uh factorial factorial is 2d minus 1 times 2d minus 3 uh times 1 and uh it's okay as a field um uh with with characteristic uh primed it to and times 2d minus 1 um and uh ec is the normal topological Euler number of the the complex points of c um uh Steven McKean uh um gave a uh enrichment of bazoo's theorem along with geometric interpretations of local indices and that gives the calculation of um vector bundles sort of counting points on p n that's just p n um the multiple of h and uh Sabrina Powley also along with uh uh interesting geometric interpretations um uh re does the cubic surface and does the quintic threefold with um a theory of dynamic um intersections uh you can hear her talk about this at motive and motives and um and whatnot um coming soon uh so we have uh enriched counts of airplanes um let's uh formalize this notation ec so let's let um uh it's on july 29th uh thank you um so uh let's let ec be this Euler number of the complex points and er um be the Euler number of the real points and um uh a theorem I'd like to to talk about from this paper is um the the the enriched count is the sum of ones and minus ones that you would get from the from the real and and complex point counts so let's let um r be a ring with uh one half um uh in r let and you can also do just a field of characteristic two um we'll consider the vector bundle we started with um the sum of uh the di symmetric power of the dual tautological or um um uh the gross monion and let's let it be relatively um oriented with dimension of the uh the rank of the is the dimension of the gross monion um and uh so this has uh we can we can rewrite this in terms of parity conditions and and sum conditions so i.e. we want the sum of di over r plus one uh di plus r choose r plus n plus one congruent to zero um mod two and the dimension equaling the rank uh dimension of gross monion r plus one times n minus r is the the rank di plus r choose r um uh and so uh then the these vector bundles that we were interested in um the the Euler number is ec plus er over two times one plus ec minus er over two uh uh times times negative one um and for example we can get an enriched count of three planes on a generic seven dimensional cubic hyper surface so there are 160 thousand eight hundred and thirty nine one plus 160 thousand six hundred and fifty minus one three planes in a seven dimensional um cubic um hyper surface and uh as uh to put this back in terms of um accounting r planes if we take the sum over the r planes we'll call the r plane p in our complete intersection x of the this trace we know it's for a separable field extension by de barre manavelle of the Jacobian we express um our function sigma in terms of local coordinates and take some derivatives then this is uh the the same the same number up here um and while uh the number on the right hand side uh is uh some ones and some minus ones these these are not in general sums of ones and minus ones they just have to sum to them uh uh and as suggested by the title uh this is proven uh with an integrality result the count of the lines on x reduced to a bundle on a grass manian that was defined over z it's the section that is defined over the interesting equations of x and that's why the um the left hand side here has all sorts of interesting summands and um the the right hand side picks up the fact that the grass manian is defined over z um so uh uh theorem two this is all joint with tom bachman um uh so let's let um v over y be a relatively uh oriented vector bundle with v and y defined over um z join one over d factorial and d is greater than or equal to to to two um uh with y smooth and proper sm for smooth and proper over the same base um then the uh Euler number has to be in the subring uh uh generated by minus one and um the two three all the way up to d inside g w of k um and uh in the case of d equals two there this leaves two possibilities so then either um e v is the the quantity i think maybe i'll still be able to paste it this this quantity here um or uh it's what we get by um putting in a single two um but but keeping keeping the the signature the same um uh so um uh we can distinguish between these um possibilities uh by uh taking uh an an Euler number over some single prime where where two two is not a square and in order to to prove um the result about the symmetric powers um we use a characteristic class uh argument to get rid of this uh this second this second possibility um uh um the the characteristic class argument owes uh a lot to Mark Levine's fit value characteristic classes um uh this would be a good time to to pause for for questions are there any um so theorem two um is proven by using an an Euler class for uh good SL oriented theory um for example uh ko um and uh the uh the d greater than or equal to two restriction comes from uh it comes from wanting to use ko uh and there is um currently uh a collaboration on uh ko without inverting two um with uh Baptiste uh Calmez Emmanuel Dotto Yonatan Harpas um Fabian had the straight uh Marcus Land Christian Moy Dennis Nardan Thomas Nicolos Wolfgang Stumm and um Marcus Spitzveck I um I understand also has work around this let's let's put that let's let's put Marcus Spitzveck um here too um uh so my understanding is that um that uh it is expected that um they will show that uh if you take k theory let's call that uh kgl and taking a vector space to its dual gives a c2 action and you can take a homotopy fixed points um of uh c2 and it's expected that the um uh if you make a substitute for Hermitian k theory by taking the homotopy fixed points of kgl and completing it too that it can be shown that um uh this is the growth of the group of z uh completed it too um so this exists over at at z so it exists over z um it's uh SL oriented and there's a um you can read about it in a paper of Bachman and Mike Hopkins um and while it doesn't respect base change it does have some maps and so this is sufficient the um to show the same integrality statement for d equals one to show um the integrality result theorem two for d equals one um uh because the the Euler number with respect if we let this this gadget be um k o prime then the Euler number with respect to k o prime it's um it's here and there is enough functoriality so that we'll have to map to the if we pull back to z um joining one half um to k o prime it'll it'll map to the corresponding number um moreover there there's a map from k o to k o prime um giving that the uh Euler number when k o exists over one half maps here too um and the um uh since these two elements map to um uh the same element in there um they determine an element of this of the fiber product of g w of z one half g w of z completed at two over g w of z one half completed at two and this is g w of z um so uh the the we get this stronger um integrality result um uh with with with that work uh in the spirit of summer school i'd like to end with some open um problems uh the uh the zeroth open problem is can be expressed somewhat facetiously like this i mean there are a lot of um uh great results uh um in uh isenbud harris's uh 32 64 and all that or fulton's uh intersection theory um that uh give uh interesting enumerators results and um uh there are beautiful enumerator results having deep connections to uh uh many areas of of mathematics so the zeroth problem is can a one homotopy theory enrich enrich them uh so take a problem from isenbud harris 32 64 and all that fulton intersection theory let's give more of a preamble to this question so um there are beautiful results in a one enumerative in enumerative geometry can a one homotopy theory enrich them and enumerative geometry uh can a one homotopy theory enrich them i don't think i wrote down um to uh taken um uh somewhat haphazardly from uh isenbud harris and i don't have time to write them down but let me just just read to so that we're on the same page about um the uh what what question zero is is trying to suggest so uh let v1 through v2n be general tangent vector fields in pn and how many points of pn is there a cotangent vector annihilated by all of them and uh another one is given four curves c1 c2 c3 c4 and p3 up degrees d1 d2 d3 d4 how many lines meet general translates of all four i'll add these to the notes and and put them in um uh to be less um hand wavy about this um in the corollary above we didn't have the analog of um the uh of interesting descriptions in terms of the the complete intersection itself so geometric interpretation in terms of x um they're the very concrete uh examples of this uh um for this jacobian which is sort of living on the on the um on the moduli space so beyond uh cubic surface quintic three fold the zoo um to connect up with marco rubolo's talk the um the the hawkshield homology of that matrix factorization algebra that that showed up it has a pairing on it and that pairing is the a1 milner number of the singularity and element of the the growth indeed vid group but there's also inside this um k nought of varieties this cut and paste relation there's this motivic um milner fiber and a compactly supported a1 oiler characteristic that mark levine told us about so uh these this this should be all all equal um and uh the list should continue um the you you lose uh certain tools and you gain others for example and then yes he's um beautiful splitting principle uh the multiplicativity of oiler numbers in exact sequences is no longer nearly as useful because you really can't divide but the kazool uh interpretation of oiler numbers gives you some control in exact sequences but in general it's hard so um are there better are there are there better tools uh there there is really structure here and an arithmetic information um that that lies away from c and what is it um i'll stop there okay thank you kirsten for a wonderful talk so um let's hear some questions and maybe i can ask by um by asking you if there's some connection between your work and oriented schubert calculus so matias vent um uh absolutely um so uh matias vent has information on the oriented chow of um uh gras manians in in terms of uh some generalizations of of schubert calculus and that gives uh other interesting ways to um to compute um these oiler numbers it doesn't directly um use it but um that there is uh interesting things to be said about that great i have a question um from my face and so this last this last question the a1 milner number equals the um oiler characteristic of the notific milner fiber you said the left hand side comes from a pairing on hawkshaw's homology can you say a little bit more about that yeah um so um the i think maybe the um a better way to start with what the a1 milner number is is if um if you have a point p of f equals zero the a1 milner number um if this p is a singularity this um is can be defined to be this this local degree at p of the gradient of f but then this is also the hawkshaw homology of this matrix factorizations um uh uh for for f um and that's by explicitly marco told us that this was the jacobian ring and uh this this local degree um can be computed as the jacobian ring and this can be computed as the code jacobian ring and the pairing on hawkshaw homology is uh is the same so we get we get this equality here so this one wouldn't be part of the problem and then this one um uh i think is open what what natural guess what pairing is there on hawkshaw's homology ah great so um uh litman and uh uh um i'm blanking um so uh there's there's even a six functor formalism uh in in hawkshaw homology and it it's a way of expressing things about coherent duality in terms of of hawkshaw homology and um i am embarrassingly uh blanking on on the important names involved and in constructing that pairing um uh but it's also uh in in this context we wrote it down in the the women in topology group um to uh yeah great so and you're too young to be blanking on these things thanks i appreciate that okay and there's a nice question in the q and a just if you can see that yes so um uh the so for uh over a finite field the three planes that um give you a square um so over a finite field we had um we have we have this um this growth indeed big group and um so in particular we're going to have a parity condition coming from uh three planes associated to um square or non-square local um local contributions and uh if you if you sum all of those non-square over odd degree field extensions with the square ones over even degree field extensions you'll have to get an even number of those and over things like cubic surface this corresponds to the difference between um hyperbolic and elliptic lines um and if you come up with uh a sort of intrinsic to the complete intersection definition of what it means for this discriminant to be square or non-square or more generally what that Jacobian means then you get a concrete um uh totally independent of any um of any a1 homotopy theory restriction on on that complete intersection but even in terms of the Jacobian we get this this even parity condition so um um by motivic by a1 Milner number um uh we meant the left hand side and then by the motivic Milner fiber um uh I meant the um the construction uh that gives you this element of of canada varieties um with this motivic integration um uh so it's a yes uh um uh the construction um the construction is a little involved um uh the way I've seen it doesn't directly use I use nearby cycle sponges but I wouldn't be surprised if if that's um uh my not knowing how to how to show the two are are closely related okay great then I have another question because in this homotopy limit problem you're completing a two is that uh so so you don't don't expect that we need to complete with respect to the huff map eta um the uh as far as I know but I think Tom would be the the better person to um to answer this question okay then and then the q and a there's another question yes yeah this yes this is the um thank you so Cas um uh Jesse Leo Cas um and I did work on um on a1 Milner numbers thank you Stephen great any other questions I don't think so so thanks again Kirsten for a wonderful talk and see you all um again tomorrow we start 1 p.m. paris time