 Great pleasure to introduce Kirsten Wickelgren. Let me just say very few words about her. She obtained her PhD in the year 2009 from Stanford University on the supervision of Gunnar Karlsson. And right after that, she obtained a prestigious American Institute of Mathematics Fellowship, which she spent at Harvard University for five years. She went to the Georgia Institute of Technology afterwards. And since 2019, she's been professor at Duke University. She's been working on the intersection or union of arithmetic and homotopy theory and has a long list of impressive papers and collaborations, and especially on what she's going to talk about today, which is an introduction to A1 homotopy theory using enumerative examples. Please go ahead. Thank you very much. And also, there are now notes for these talks. And I do realize they were posted a little bit late since tardiness is an almost universal human trait. You all probably know what I mean. It does mean that I was up pretty late last night. So if my energy starts to flag, it will be very much appreciated to have feedback and help and corrections. It's really a pleasure to get to be here and so the more I get to interact, the better. And thank you very much, Oliver. So we're going to use A1 homotopy theory to do some enumerative geometry and enumerative geometry. Let's say that what it does is it counts algebraic objects over C. And an example to say what algebraic geometric objects means, we can ask how many lines meet four lines in P3. So a nice picture of this is that inside P3 of C, there's R3 where we're all in our own separate rooms. And inside R3, we could take four lines in our rooms that don't intersect and actually just ask any friend, even who doesn't do math professionally, how many lines intersect all. So the answer, if you allow the lines to be complex, is two. And in your own little room, it might be two and it might be zero. The goal here is to introduce A1 homotopy theory and have fun. And but we could also make a concrete thread to guide us is that let's make a goal to record arithmetic information like whether those lines are over C or over R about geometric, those geometric objects that we're counting over some field K. And the number of these objects, these lines, is going to be fixed over the algebraic closure. But it may vary over the field K. And we'll use as a tool the A1 homotopy theory that we were discussing in Danny Krashen's taught lectures and Frederick Glees' lectures due to Morel and Belvatsky, as we've seen, but that's the homotopy. Let's write it down to homotopy. And let's start with some classical homotopy theory. We have a sphere in dimension M. So we could take real numbers whose, the sum of whose squares is 1. And another useful way to write this is it's equivalent to PnR over Pn minus 1, the R points of Pn over the R points of Pn minus 1. And there is a degree from the homotopy classes of maps from SN to SN to Z that roughly speaking counts the number of free images. And we were trying to count. So given F SN to SN P and SN, let's suppose that we don't crush the whole sphere or one dimensional part to this P. So let's suppose S dot T dot for such that that F minus 1 of P is a bunch of points Q1 through Q sum capital N. And the degree of F, we'd like it to be something like N capital N. It is. It's the weighted sum of a local degree of F at these inverse image points. And the local degree can be defined in terms of that global degree again. It is the local degree. So we'll let the be a small ball. We'll take a small ball in its inverse image to be a small ball just containing QI and no other inverse image points such that minus 1 of U intersects. So U is our small ball containing QI. And it doesn't have any of the other QI. It's just QI. So then we can map U to V. And we really want to map the boundary of these small balls. This looks like the over the boundary of V or SN. And this looks like U over the boundary of U, which is also SN. And we have an induced map. And the local degree is the degree of this induced map. And we have a formula from differential topology telling us that instead of sort of in some sense recursively defining the degree of F in terms of another degree, we could make the local degree with calculus or compute the local degree with calculus. So we have a formula from differential topology. So we can pick local coordinates near our points and express our function then as a tuple of polynomials. Or a tuple of smooth map if it's smooth. So let x1 to xn be oriented coordinates near QI. Let y1, yn be oriented coordinates near p. So then our function becomes a tuple of functions in the nxi of 1, m, rn to rn. And we can take this Jacobian determinant, we'll denote it by j, to be the determinant of the partial of fi with respect to xi. And then our local degree depends on this determinant. So the local degree at QI of f is plus 1 if the Jacobian is positive and minus 1 if the Jacobian is negative. And a great special case is if f is actually a complex analytic. So we've forgotten some extra complex structure. Then f is going to preserve orientation, putting us always in this case. So let's say that f is a polynomial map over c. And well, suppose it doesn't have a multiple root. So this Jacobian, it's not 0. And then because there's this multiplication by i that tells us which directions we're going, this has to be 1. And we do get that the degree is counting the points in the preimage. And this might be the number of solutions to something we were interested in. So it's the number of solutions to f1, f2, fn equals 0. And a great example is the fundamental theorem of algebra. So if we have a polynomial with complex coefficients of degree n in the usual sense, then its degree is also n in the topological sense. And we have n solutions if they're all simple. It's the number of solutions to our polynomial. And this is fantastic. The f1 were these folks here. So if f is encoding lots of equations at once. Now those f, they might have sort of a twisted target. So we can also count if the f actually corresponds to a section of a vector bundle, then we're not taking maps to some a1 or to some line. But that line is twisting over the space of our possible solutions. So remark, we can also count solutions to f equals 0. Or f is a section of a vector bundle, vector bundle. So here's our vector bundle v with its canonical map to its space x. And here's a section f for function. And then the sum of local degrees of this f is another invariant from homotopy theory. It's the Euler class. So the Euler class of v can be computed with any section. And it's a sum over the zeros of our section of a local degree of f. And in order to define this, we need to view f as a function. We can take a local trivialization of our vector bundle and some local coordinates. And we really must have an orientation so that we don't wind up changing the value of our Jacobian as we change our choices. So rank n, and we'll talk more about oriented vector bundle. So we can also count the solutions to such equations. Here's an example of doing that. So let's let our x be the gross monion. The indexing convention going on here is that it's parameterizing the p1s and p3, like in our initial question. So that's also dimension two subspaces, w and c4. Or equivalently, the p1s and p3s, like the lines, the projectivization of w. This is where we take out the origin and mod out by scalar multiplication. And w and lambda and c star. And this sits inside the projectivization of all of c4, which is p3, or the c points of p3. Over this gross monion, we have a tautological bundle. So here's a rank two vector bundle of the tautological bundle. And it's defined so that over the projectivization of w, over this element of the base, we put w. So the vector space over this line here is the rank two vector space, w. So let's consider that question we started with. Let L1, L2, L3, L4 be four lines in p3. How can you tell if another line L meets, say, L1? It's that in the intersection of the corresponding w's is not empty. So if you take the wedge product of the forms defining L1, it's 0 on L. And there's nice exercises that Sabrina Pauley has that's made that say a little more on this, but that means that the way we can characterize the lines meeting our four lines is as the zeros of a section of a vector bundle. So the lines meeting L1, L2, L3, L4, they're the zeros of f for f, a section depending on the LI of the following vector bundle. It's four copies, one for each line of wedge two of a stool over the gross money. So our section above a w, say, the first copy of this, it puts the restriction of the two forms defining L1 to w wedge together. And that means by the discussion that we were just invoking having that the number of lines is this order of their class or their number is E of the same bundle. So if you believe that the many points of view on the order of class from algebraic topology includes one where we can compute it with any section. And in particular, this means that this number is independent of our choice of lines. Of the choice of lines, as long as the associated section really did have isolated zeros, which happens when they're generally chosen or when they don't pairwise intersect. And for that, we needed that this was always one so that we were working over C. Then we can compute at least over C. But the answer is two, using homology of gross monions, the splitting principle for computing Euler classes from ways we'll get to later and tools of that sort. So we've been using a degree to count algebra geometric objects. And we'd like to do this over fields that are not R or C. So Lan and Morrell came up with a notion of degree for an arbitrary field k that reported more than what was happening over the algebraic closure. So an evocative place to start is for a rational map from P1 to P1. And above, we had this formula here. And we remembered the sign of the Jacobian at each point. And Lan and Morrell are going to allow us to remember more than just the sign of the Jacobian. And they'll do it by landing the degree in this group. So the degree of F will be valued in the growth index bit group of k, which we've seen from Danny Cashion's talks or the bit group, which recovers the growth index bit group. And this is defined to allow for formal differences of symmetric non-degenerate bilinear forms. So we'll group complete, meaning we're allowed to subtract even though we only started with an addition. And the addition that we're going to start with is perpendicular direct sum of bilinear forms. And we can also take the tensor product of bilinear forms of non-degenerate symmetric bilinear forms. And we've seen the quotient of this on Monday and Tuesday. So the bit group of k with the fundamental ideal and the Milner conjecture with the relationship of those quotients, say, tall cohomology and Milner k theory is also in the growth index bit group and also applies to give us tools to understand elements of the growth index bit group. And the bit group is what happens when we quotient the growth index bit group by this hyperbolic form 1 plus minus 1. So a useful invariant of bilinear form is the dimension of the vector space that's on. So this is the rank homomorphism. So the rank of a bilinear form, the vector space v over k cross v to k is the dimension of v over k. And we can recover the growth index bit group from the bit group. We have a quotient to the bit group. And we also have a rank. And given a rank and its quotient to the bit group, we get the element of gwk that we might want to pin down. So this is a pullback. We saw some useful generators for the bit group, which I'll put the notation for to get us all on the same page. So stably, meaning after you group complete any bilinear form symmetric non-degenerate can be diagonalized. In fact, in characteristic 2, you can diagonalize even without group completing. So that means our rank 1 forms are going to be generators. So here, this is the element of the growth index bit group associated to the bilinear form on k that sends xy to axy. And over a field, we have this nice presentation. So we take the tensor product of the two rank 1 forms corresponding to a and b. And we get the rank 1 form corresponding to ab. And an elementary diagonalization shows that this diagonal bilinear form is the same under a basis change as this one whenever u plus b is not 0. And these two together imply that for any non-zero u, u plus minus u is 1 plus minus 1, which is this hyperbolic element h. And as an exercise, we can show that 2 implies 3. And this form here is also the one whose brand matrix looks like this, which Danny Kruschen wrote. And you watch in characteristic 2. But again, the group completion is OK. Let's get some examples. So if this group is trivial, then all our generators are the same. And the rank gives you an isomorphism. For r, we have 1s and negative 1s for this group. And the difference between the number of 1s and the number of negative 1s is called the signature. We have a map, rank, comma, signature. And they have to have the same parity that this induces an isomorphism to z cross z. This is we diagonalize the form 1, 1, 1, minus 1, minus 1. And signature equals the number of 1s minus number of 1s. We saw in the Milner conjectures that the atolicoamology with Zeeman-Tucco coefficients gives us some useful invariance. And one of these is the discriminant. And so for fields where we can compute atolicoamology, like fq, we get the discriminant. And these are enough to determine the entire bilinear form. So this is an isomorphism to z cross fq star over fq star square. And we have another interesting term here in addition to our z. And we have another interesting z in addition to the z we started with. We can keep going and get more interesting extra terms. So if we had qp, we get another one. And more generally, for a complete, discreetly valued field, we can compute the bit group as two copies of that of the residue fields. We've got k complete, discreetly valued field, k residue field. So for example, we could take the p-addicts or the Laurent series in a formal variable. And we have that the residue field is this finite field. And we'll assume that the residue characteristic is not 2. And then we have that the bit group is two copies of that of the residue field. We're going to have transfers in a way lining up with Frederick DeGlese's talk. But I want to introduce notation for them and a formula now to be able to talk about them. So let's give ourselves a separable field extension. Let's finite extension of fields. We have a trace map or a transfer from the bilinear forms over z. And we need to get down to a bilinear form over k. And it will compose with a trace from Galois theory. So the sum over all the embeddings into an algebraic closure. So if we have a bilinear form over e, we can view the now as a vector space over k and apply the sum of the Galois conjugates to get down to k. And this is still non-degenerate by Galois theory and defines the wanted map. OK, so we have the growth index bit group, the bit group. And let's get back to Lan and Morrell's formula for the degree of a map from p1 to p1. So back to formula. So we'll have f from p1k to p1k. And we'll pick a rational point at the base. And we'll assume, again, that it's non-constant. So we just have finitely many pre-images q1 through qn. And for simplicity, we're going to suppose that the Jacobian, which is just the derivative, is not 0. Then the degree of f says Lan and Morrell is don't just take the sign of the Jacobian. Take the entire Jacobian and stick it in brackets. And we'll take this transfer. So this f prime is sitting in the residue field of qi. And we take the sum. So we're also counting the pre-images. And they have a weight that's picking up information about their field of definition and the function's behavior there. And then we have the very lovely fact that this does not depend on our choice of p. And there's some exercises in the notes to play around with this. Now, more generally than just a map from p1, Morrell is going to give us a degree map from the sphere, which, like in the beginning, we'll view as pn over pn minus 1, so the top cell of pn. And its value will also be in the growth index group of k. So this is going to mean here this will be unstable a1 homotopy classes of maps, which we didn't define, but it's just let's say the analog of homotopy classes of maps. Like we started with our degree function from the functions from sn to sn, but they were homotopy classes of pointed maps to z. This degree has the pleasing property that it interacts well both by taking real and complex points. So if we took the degree, let's say k over k is r, so we have pn over r, let's just say k is r here. And we have this degree, the gw of r. An element of gw of r was determined by its rank and its signature. And this gives the degrees of the maps on the real points and the complex points. So we could take c points and have pnc over pn minus 1c is s2n to s2n and the degree in algebraic topology. And we could take real points and get homotopy classes of maps from sn to sn and take the degree from algebraic topology. And the a1 degree simultaneously captures both. There are formulas to compute this that doesn't involve summing up a lot of terms. So Christof Kazemov and Thomas Brazilton, Stephen McKean, and Sabrina Pauley give formulas for the degree of lists of polynomials in terms of positions. OK, so we're going to say more about where this degree map comes from. And for that, we're going to also say more about pn over pn minus 1. This quotient doesn't exist in scheme, so it's already an achievement of a1 homotopy theory that we are allowed to do this and get something like a sphere. Now, let's talk about that. So the smash product is the quotient of the product by the, I should have said these are pointed spaces. So we're going to have a favorite point in y, and that's what that one is supposed to mean. And the favorite point in x, that's what that one is supposed to mean. And we have a smash product like so. And an example is that from classical typology that if you smash an n-sphere and an m-sphere, you get an n plus m-sphere. And in particular, if you smash s1n times, you get sm. And the suspension, which it turns out that there's a lot of phenomenon that sort of behaves similarly after sufficiently many suspensions, and this is the smash with s1. So p1 can be formed by gluing together two copies of a1. So p1 is a standard a1 and then a1 containing infinity. And together, their overlap is this gm, so it's a push-out. And the a1-homotopy theory comes from forcing a1 to be contractable. a1 is like a line. So as part of our introduction to a1-homotopy theory, we could make a lot out of the analogy between a1 and a point. That might have been a good thing to do, but let's do it now. So we're going to declare that a1 is homotopy equivalent to a point. c goes to 1 over z, c goes to z. And then it follows from the formalism of homotopy push-outs that when you take the homotopy push-out of a diagram to points like this, you get the suspension. So the homotopy push-out of this diagram is the s1 suspension of gm. And so we get that that's another way of writing p1. So we'll have several spheres. We're going to have s1. And half gm is spec a1 over z, or a1 minus the origin. And we'll then be able to talk about sp plus q alpha, or s1 to the p, gm to the q. And some other notation for this is sp plus q q. So another example that will be handy to have is that an minus the origin is s1 to the n minus 1 smash with gm to the n. So to see this, we can use induction. We're going to cover, say, a2 minus the origin by its ax, by the complement of its axes. So and in general, we can take a1 minus the origin, cross with an minus 1 minus the origin. This produces a cover, an minus the origin. And this is also a push-out. So we're gluing a cover together. And this an is contractable, and this a1 is. So we have the opportunity to replace this with, well, this one, say, with an minus 1 minus the origin. And then we're looking at a push-out along the projections. And it's a good exercise with push-outs to identify this as a suspension of x smash y. And in particular, we get that an minus the origin is the suspension of these two pieces here. So in induction, we know what these folks are, which shows the plan that we started with up here. So we also have the pn over pn minus 1. We could start moving this pn minus 1 in from infinity until it's the pn over pn minus the origin. And then we could get rid of the pn minus 1 at the end at infinity and get that quotient. And then this is contractable here. And it follows that this is the suspension. So this is s1 to the an or gm to the an p1 smash with itself and times. So that was our sort of summary of ways to think about some spheres. It turns out to be very useful to allow an algebraic topology desispending. So the stable homotopy theory in algebraic to classical algebraic topology permits us to desispend our spaces. And this gives us duality statements and allows us to represent co homology. And there are a lot of phenomena, which sort of repeat themselves in high enough dimensions or with enough suspensions that are then computed very effectively. And in a stable homotopy category. And that is also a useful thing to do in a one homotopy theory. We, we can desispend, we can unsmash with p1, which means we can also unsmash with with s1 and GM. And we get a stable homotopy theory that will, will, will denote it as shk, which is standard. Um, so morels degree homomorphism comes from a theorem of morale and morale hop and Hopkins morale. And it says that in the stable homotopy category, the maps from s not to s not. And remember we had some other ways to talk about these which which looked. A lot like spheres. This is this growth in the group of K. Moreover, we can in fact have a beautiful computation of the stable homotopy classes of maps to from s not to all of the GM smash ends. So whenever they're the same number of copies. Where did our spheres go. Whenever the same number of copies of s one. And we want to compute maps from a sphere to a sphere that have the same number of copies of s one. We lined up, we wind up in this, in this group. It's a group because stably, this will get a suspension in it, and you can add maps from from the suspension of a space, you can smash maps together. So this is actually a ring. And that's it. It is the combination of the bit ring and the Miller K theory ring we were looking at on Tuesday. So they combine to this ring here, which is Milner bit K theory so came Milner, which was the Milner K theory. And w of K, combine to this, but K theory. And let's define it. So it's a graded associated algebra, it's generators. So let's look at them as as maps from some spheres to itself. In a minute, but first let's get an abstract, just algebraic definition. So generators. We have you came over bit one K for you and K star and Ada corresponding to the hop from that. Subject to the relations. Yes. Okay, as a field, an arbitrary field. We have the Steinberg relation says you times one minus you as trivial. We have the relation that bracket AB is bracket a plus bracket B plus Ada bracket a bracket B. And Ada commutes with the hop map commutes with our generators, and the hop maps times an element that will correspond to our hyperbolic element from before is zero. And the hyperbolic element in terms of these, these generators is written Ada minus one plus two. So let's relate this to the growth indeed fit group. So remark, the growth indeed fit group of K is isomorphic to K zero Milner bit, and the bracket a element corresponds to the hop map from topology is like an S three to S two. So it's it's going down in the number of like in actually we're gonna see it in a second the number of GMs. We'll get to that. Because it's like an a two minus minus zero to you. And so the angle brackets a corresponds to one plus Ada a. And so our H. This was one plus minus one. This corresponds to one, then the one here is actually the additive identity so it goes away, and then plus another one, minus one. So, let's let's talk about how, how these correspond to maps between spheres. Here's some things about morale and morale Hopkins is proof. The map bracket a corresponds to the map from S zero so S zero is two points. And we're working over K so we, this is spec K, just went union spec K. And the bracket a corresponds to the map from S not to GM has to take the base point to the base point but the non base point it will take to the point a base point to a. And now, Ada is the map from a two minus the origin to P one that takes X comma why to the point in P one determined by by X and why. And on the C points, take C points, we get C two minus the origin which looks like a three sphere. And that's mapping to CP one, or the two sphere, which gives a half map, but on our points. It's a map from the circle, the circle, and it's multiplication on by two, in a particular. Over R, or if you're inside our data is not no point. And compared to classical algebraic apology. And the sheet is no potent. This is, this is different. This is a new thing. All right, let's introduce a little bit about the relations and and and and see a little bit more about about this, this degree. So, to do that. Let's talk about multiplication on GM, and we'll need a little bit of homotopy theory. The smash product was this quotient of the product by X times a point union point times why, and we could glue these points together, and the wedge is where you glued them together, blue base points and and the stability, the product is the sum of these folks. So if we suspend once meaning smashing with S one. There we go. Then, we can, we can make a map backwards like this, how do we do that, we have a map from X cross Y to X, which gives us maps like this. And the suspension of the projection of X. And since this is a suspension we can now add them. So what we'll do is we'll add this suspension of the projection onto X, and why, and we'll give us an expression for the product to be as some and morale shows morale and Hopkins show that the product on on GM is related to the hop map. So in our stable homotopy category. The multiplication map on GM that takes a rational point a and a rational point be to the rational point a be so gets called. It's all so motivation. And it can be expressed in these terms so stably we have product looks like that that some. And a map from a wedge product is just just maps from each of the wedge factors. And this is the identity on GM, the identity on GM and the hop map. So, this lemma allows us to justify the way we were looking at angle brackets a. So, we've got a map from P one to P one, sending Z to to AZ. If you remember the way we defined the degree was we took the inverse images of a point and looked at the derivative. So the degree of this map here is we can take the inverse image of zero, that is zero. Look the derivative that is a this has degree a by our by Landon Morales intuition that we stick the Jacobian inside brackets. And, but this is also, and we wanted to see that this corresponded to one plus a times a, so that we would have that GW sitting inside came on their bit of zero. So now, we can look at this map. This is the suspension of in GM, sending Z to AZ. And we can express Z to AZ is the product of a with Z. So, we take GM cross that K. And we get this map by the identity cross a GM, cross GM, and then multiply. Then we get to break this map up into the identity, and a constant map which is trivial, and Ada, and we get, we get, we get that. Similarly, this lemma. Here. This lemma also gives the relation that a B. So what would you get if we, where are we, if we, if we took the non base point to a times B over there is what we would get going this way. That's B. Ada lemma can be used to show the second relation. And there's a lot of great things to say about the relations. We'll, we'll end with that. And this will define a map from our generators and relations description of came on or that K theory of K to our ring of these homotopy groups of spheres, and then morel and Hopkins show that this is an isomorphism. It's a beautiful computation, or they, or they do. And in particular, we get the degree map that we've set about to count with particular. We can go from unstable homotopy classes to stable and then get a degree, even though in fact we are our isomorphism was really sort of defined the other way. So, I, this was very successful. So today about other homotopy groups of spheres they encode a lot of geometric information. It's not going to make it into this talk. But while while we're here, let's talk about some some big problems. So, let's do do the notation. So, this is the zero line of the stable homotopy groups of spheres. Another notation is these sheets of homotopy groups and we take global sections to get to get these these groups. And this is all the zero line. And then we have all the other lines. So the R line. So this is the sphere spectrum, just sort of all the suspensions that he wants suspensions of s not. And we know a lot of things about homotopy groups of spheres that imply very interesting geometric consequences classically. And we are beginning to know a lot of very interesting things in a one homotopy theory. But maybe fewer than you might expect. So what we've got, we've got an annals of math paper by Oliver Rondig's Marcus Spitzveck, and Paul Arnaud Auspair. I'm computing the one line in terms of her mission k theory and number k groups. And they know some things about the two line. So, what about over 2019 characteristic k, not equal to two. And we're, we're left with some, some questions. So what about over more general rings as you remarked this this was over a field. So, over more general rings. And then the analog of the growth indeed fit group over Z. So over more general rings, and then this is the group that will be using most just in these talks. What about that one. There's a result from this year by Tom Bachman and Paul Arnaud Auspair. So, once we localize. It's the growth indeed fit ring of the adjoin one half, which is the group completion of the symmetric non degenerate bilinear forms over the adjoin one half. And watch out over over non fields your presentation is not the way to define. And what about over more general data can domains and the and what do we know about that we can immediately feed that into problems that then then use this to get to get a new murder results. And then we had the one line and the zero line and the one line. What about the rest, maybe with a particular eye to the successful consequences and algebraic topology of knowing things about these groups there they're powerful. Now, once we have the stable groups, what about the unstable ones so it is not known. What kind of stable groups are equal to what kind of unstable groups so what is there a growing ball suspension theorem, which stable by star star correspond unstable groups. We're going to use such computations to count things, but there are other uses. I'm tempted to say something about vector bundles but I don't think I don't think I want to try to add live. Let's keep with the goal that we arbitrarily chose for ourselves and get back to counting things. And for this. Let's talk more about oiler classes. Barge Morel defined one right around 1999 and chow bit that Jean Faisal has a lot to say about, and Morel writes about one as a principal and obstruction in his book. And Ashok and Faisal compare them. And Mark Levine, as well as Jesse Cass, and myself use them for enumerative purposes. And dig lesion and con gave a beautiful functorial approach to please. And Tom Jushan in a deal con. Tom Bachman and I did a bunch of work with a growth indeed Sarah, which was also in work of connections with coherent duality in work of Levine racks that but actually my cockings was. I was calling one time when I gave a talk but they do a lot of other things like the much of the gasp and a theorem and they're there. They're many interesting points of view on on this class and start with one of them. So, let's give X smooth case scheme of dimension D. Let's take a rank, our vector bundle, and we need an orientation so that well from a certain point of view so that when we choose local coordinates we can't accidentally swap an X one to a minus X one and change the sign of our determinant. V is going to be oriented by the following data, a line bundle L and an isomorphism phi, where is a line bundle, the determinant of the is identified by our particular isomorphism phi with the square. And then we can talk about X being oriented when it's tangent bundle is and what we'll need to take an Euler number. Relative orientation so V to X is relatively oriented when the bundle where the Jacobian lives, the Jacobian of the zeros. It's a determinant of a bunch of derivatives that's a map from the determinant of the tangent bundle to the determinant of the where the determinant is the highest wedge power. So, for example, if you have X is PN, or the gross monion. No, Kirsten you didn't finish the sentence when home is what is oriented. Sorry. Thank you. So, the gross monion. It's tangent bundle is home from the tautological to the quotient. And if you remember our conventions, this is the gross monion parameterizing because the parody is what matters. PMS in PN so let's be particular about which which parody we're having this one kind of agrees with that. Then the determinant of the tangent bundles dual. This is the cotangent bundle is O of minus minus and minus one. So X is orientable. If and only if and as I and on P one is relatively orientable if and only if and is even. So things are positive over so there's a question in the chat about a square root of the determinant. If we take our intuition from from the reels. We, we want to know when the determinant of our coordinates doesn't doesn't change sign. So if we if we took a new basis, we want the determinant of the basis change matrix to still be positive if we're going to get the same sign of the of the Jacobian. So, the being positive is the same thing as being a square P one is relatively orientable. And is even. All right, so here comes, here comes our, our number in GW okay, and this this number is the analog of viewing a homology class in top dimension as something in Z, except now Z is replaced by by GW okay. So, let's suppose, again, X over K is a smooth, proper scheme of dimension D be to X with F is a rank D vector bundle is relatively oriented by this line bundle on this isomorphism. And let's give ourselves a section that has isolated zeros. In fact, so such that let's have the isolated zeros actually have multiplicity once we're imagining our Jacobian is not zero. So such that way to say that is at people zero consists of multiplicity one zeros. And equivalently, we have that this this map from the tangent space to V. So let's say for all x that is a zero, when when you have a zero of a section you get an induced map from the tangent space of x to the tangent space of f of exit V but since this is x zero, this using the zero section, this is just the tangent space of acts at the point, plus the fiber of the vector space. And so we get, we get a map from the tangent space to the, the fiber of the vector bundle. So this is what we have when we have a zero. And this map has non vanishing determinants of the map on tangent spaces as non vanishing determine. So then we're going to make our order number be the sum of all those Jacobian determinants over the zeros. So, the number of our vector bundle and let's remember its orientation, or forget it if it's clear. And with respect to a section, let's call it any of those sort of E as well. So let's pass this, this order number, the sum over the zeros, the degree of f used as a function. And so your two points of you want it so the degree X F can be computed choose local coordinates, and that's really any tall map from an open neighborhood to choose local coordinates. And we want it to be an isomorphism on residue fields at the point we're interested in, let's call this this navy coordinates on X, and local trivialization of the, which are compatible with the orientation to submit the definition of that with, there's the relative orientation, then you can express it as a function this. Now, X doesn't have to locally be some sort of a fine space, but nonetheless, you can jiggle it a little bit, and it won't be any longer. So then you can write F as a function from the dimension from a to the dimension of X so X is looking in these misnavish coordinates, it's like an open set of that to a to the D, and we can then make our determinant J F is the determinant of fi over X J. We had this assumption that our zeros were multiplicity one. So we get that this is not zero. This is in the residue field of X. Thanks, everybody. In the residue field of acts, and then the degree X F is the transfer of the Jacobian acts. And since it's, it's, it raised suspicions to say this although it's actually okay. There's here's an equivalent point of view. So or equivalently, we have this, this map, the induced map on tandem spaces from our section F that we were mentioning above that starts looking at like a map from the tandem space of X to the fiber of V. And then we can define J J of F of X to be the determinant of this map, which is then a map from the determinant of this vector space to the determinant of this vector space. We have all the same dimension now so the determinant that the dimension of X is the rank of V now so the determinant is the art or the deep right. The deep wedge power and our orientation precisely gave us information about this. So the orientation gives an isomorphism between this home space containing the Jacobian and the square of a line bundle. So if we pick any basis for this, our Jacobian just becomes a number. But if we switch basis, the corresponding number changes by a square, since this is the square here. So, this actually gives so here's J of F of X is sitting in here, but this is induces a well defined element of K of X over K of X is unit square. induces a well defined element that hey will also call J F of X of K of X is like trivialization of L of X over K of X is units squared by choosing a trivialization of L of X and this agrees with the above. Okay, so we have, we have four minutes. I had planned to talk about what happens if the zeros are not simple. But that's, that's a longer discussion and it's going to fit here. So, question happens. Well, we'll set it will last the question this time I'll answer it next time. If the zeros. Multiplicity one, we were missing this in differential topology as well. So CF, we can said that in differential topology, the degree was one, if the Jacobian was positive and minus one, the Jacobian was negative. So then, what happens when the Jacobian is zero. But, you know, something happens. So, we can use this to answer a further question that that sort of came up at the beginning of the talk. So answer will be next time. And the fact that it's possible to answer, and there's a good way to interpolate between the quadratic form that's giving the Jacobian giving the local degree. As, as you as your zeros come together, and gives us some leverage on this question. So why is the Euler number, the F independent of the section. F. In our four lines example, that's analogous to saying that the answer to works for any of those four lines that you chose. So the answer one coming from this is that sections with isolated zeros can often be connected by families of such sections of sections whose zeros are isolated but they might be higher multiplicity. So, our sections will be parameterized by an a one, and it'll lead to an element of the growth in the big group of the functions on a one or K of X, but this by a one invariance of the growth in the big group, or by harder serum will give us a well defined Euler number. Or as a pain in the neck, and I wouldn't know. So, alternatively, we can introduce the Euler class, inspired by DeGlesion cons, and and Mark Levine's and John fizzles perspective. So, another way to get rid of this dependence on the section is to have an Euler class and a co homology theory number is push forward of an Euler class. There are values and interesting formology theories. Like one Frederick de Gles ended with in this talk. And then we'll have for free and much more useful techniques for computation to control this expression. We will set that up. And get this control. Those those those wonderful powerful tools that I'm over. So, so let's end here. Thank you very much. Yeah, thanks a lot, Kirsten for a wonderful lecture. Let's thank her. Either by clapping real or an icon. Is there any questions. I had a question. I might have just over read into like an offended remark you made but when you introduce our notion of transfer here you said it was related to the stuff that DeGles talked about and this is all very new stuff to me so can you say more about how those two things are related to what happened in the last lecture. Yes. So, and the stable homotovic groups of spheres have to have a whole bunch of transfers, and which transfers you have is an interesting question. We saw some transfers that you that you have to get. The transfers here it's here they're coming from the looking at this, the sphere is going to map to Hermitian K theory and Hermitian K theory has as it's as it's K zero of these filenier forms on on vector bundles and has a nice point of view. So I'm going to give you to Mara, Yackerson, and Mark Boy, why an Elton, Elmanto and says, no, no, and Khan that that gives you a way to look at Hermitian K theory is this kind of uniform, you sort of universal thing with transfers when you have coherent duality sort of push forwards. What is done by this trace formula is sort of looked we looked at that vector space over E, and then we get the vector space just over K, it's kind of like pushing forward. Just just looking at it like that and composing with this trace sort of coherent duality thing. And there's a very lovely characterization of a permission K theory as this thing that has transfers when you work on recognition principles for for for cosmology theories and some. Well, that's one answer it's probably a little convoluted in this. He could, he could say a lot about about that answer maybe there's. I'll stick to that. Okay, thank you. Sorry, sorry, I want to advertise one. So that's for these infinite p one loop spaces and connected to that big open problem of we don't have a friend fall suspension theorem is, do you want to characterize for finite p one loop space is another very exciting sort of possibility in general field. Any further questions related to this question of all suspension, do we even know if, you know, smashing something a sufficiently large number of times ever stabilizes. So, is there a finite number but that finite number might grow arbitrarily with the connectivity, such that you stabilize all over. Can you can you take that one. So the problem is that we don't know a finite generation of, of the groups involved there. But for otherwise you, you, you would get, you would get stabilization of these coordinates. There are concrete computations done also by, by Kirsten and co authors, which indicate that stabilization occurs and behaves rather similar to, to what happens in topology if you combine it. What happens at sort of complex and, and real points, but yeah, it's, it's, it's a very interesting open, open problem. Any other further questions. Yes. Do you manage to recover these very classical results like 27 lines on a smooth hypersurface v3. I'm so glad you asked that question. We will Friday. Absolutely. Okay, so I think this is a good teaser for Kirsten's next lecture on Friday morning afternoon evening night wherever you are. Let's thank her again but don't forget there's a problem and discussion session with experts in the field, starting in about eight minutes. Thanks again.