Jet schemes and jet bundles are two methods of representing tangent vectors at a point. Jet schemes are a k-algebra homomorphism from S(V*) to K[t] modulo t^2, while jet bundles are objects such as KT modulo t squared or t to the cubed or t to the n. These are used to represent the tangent bundle and higher order derivatives of a scheme and are important in the study of algebraic geometry and statistical learning theory. Jet schemes can be used to compute various asymptotic quantities related to the learning process, and their canonical nature is of particular interest.

Algebraic geometry and statistical learning theory have strong connections, particularly in the resolution of singularities to calculate asymptotic quantities related to the learning process. Jet schemes may also be connected to algebraic statistics, although this has not been explored. The resolution of a singularity is bi-rational, proper and isomorphic outside of the vanishing set, and the choice of resolution does not affect the birational invariant, the RLCT. To gain more information than just the RLCT, one needs to find the irreducible components of the resolution that are essential.

Nash proposed jet schemes as a way to identify the essential exceptional devices of a resolution. These schemes approximate an arc by taking the limit of finite objects, which is equivalent to Taylor series expansions. Jet schemes involve taking quotients of polynomial rings and the completion of the algebra corresponding to the variety. They provide a way to study the Arc scheme instead of the resolution and can be used to compute various things.

Jet schemes are a way of defining the tangent bundle and higher order derivatives of a scheme. They are represented algebraically as a morphism of K-algebras from a ring to KT modulo t to the N plus one. This is done by taking a function from Z to a function space, currying it, and taking the spectrum of the resulting ring. Jet schemes are defined by a functor which sends a scheme to a set of morphisms of schemes, and is represented by a scheme. An m-jet to a scheme X can be classified by a K-algebra homomorphism from the quotient of a polynomial ring to K^(n+1), determined by what it does to x, y, and z.

Parameterizing a point in K3M involves assigning scalars to symbols X I, Y I and Z I, subject to the constraint that all functions of X I, Y I and Z O (F0 through FMS) must vanish. Computing the image of an equation is done by substituting each variable with a collection of terms, expanding and collecting terms. Symmetric algebra is used to think of points as K algebra morphisms and Taylor series expansions are used to talk about tangent factors. The Hessian appears in the constraint F2, while F1 imposes the constraint that the vector has to be at a point in the variety where F0 vanishes. Additionally, the data of X needs to be orthogonal to the gradient.

Symmetric algebra is an associative and unital K algebra formed by taking the direct sum of a vector space V with itself. It is not commutative but can be made so by modding out by a two-sided ideal. If V is finite dimensional, it is isomorphic to the polynomial ring in N variables. K algebra homomorphisms from the symmetric algebra to a commutative algebra a are in bijection with K linear maps from V to a. If a is a field and V is a K linear space, then K algebra homomorphisms are also in bijection with K linear maps.

A bijection between geometry and algebra is introduced, where a point in a vector space is mapped to a K algebra homomorphism. This homomorphism evaluates polynomial functions from the vector space to K at the point. Replacing K with a larger algebra, KT modular t-squared, allows for two linear transformations to be gained for free. The Taylor series expansion of an element in the polynomial ring can be done algebraically, with the coefficients determined by evaluating a linear functional at the point. This is related to the K algebra homomorphism.

Jet schemes and jet bundles are two different notions used to represent tangent vectors at a point. Jet schemes are a k-algebra homomorphism from S(V*) to K[t] modulo t^2, mapping a pair (p,q) to a PSI(p,q) which evaluates a polynomial f at p and takes the dot product of the gradient of f at p with q. Jet bundles are objects such as KT modulo t squared or t to the cubed or t to the n. They are both used to achieve similar things, though in different ways. Jet schemes may be important in SLT due to their canonical nature.

Algebraic geometry and statistical learning theory have many connections, one of which is the role of log canonical threshold resolution of singularities. Jet schemes may also be connected to algebraic statistics, but this has not yet been explored. Usually, a resolution of singularities is performed on a statistical model to compute asymptotic quantities related to the learning process. This resolution is bi-rational, proper and isomorphic outside of the vanishing set. Additionally, there are many different resolutions which can be used to calculate the rlct, an invariant which is independent of the choice of resolution.

In order to compute the resolution of a singularity, one needs to make certain choices. However, these choices do not affect the birational invariant, the RLCT. In order to gain more information than just the RLCT, one needs to find the irreducible components of the resolution that are essential, meaning that no matter which resolution is chosen, the same components will be seen. It is not clear how to formulate this correctly, but it is necessary to gain more information about the singularity.

Nash proposed jet schemes as a solution to the lack of canonicity of the irreducible components of the exceptional divisor of a general resolution. His theorem states that studying the Arc scheme instead of the resolution provides essential information common to all resolutions. The Arc scheme is approximated by a tower of finite truncations, called jet schemes. These jet schemes provide a way to identify the essential exceptional devices of a resolution.

An arc is a segment of a smooth path on a manifold. Gradient descent and stochastic gradient descent near a point can be thought of as an arc, where the point is the most singular point. Algebraic varieties and schemes are mathematical objects which include the set of zeros of a polynomial and the polynomial itself. A zerojet is a point, a one jet is a tangent vector and a two jet is a combination of a point and a tangent vector. This is an interesting research direction to explore further.

Jet schemes are a way of generalizing the behavior of some object near a point, up to second order. This is done by taking the limit of finite objects, and is equivalent to Taylor series expansions of functions. It involves taking quotients of polynomial rings by t squared and t cubed, and the completion of the algebra corresponding to the variety. Jet schemes involve algebra and can be used to compute various things.

Schemes over a field k can contain objects outside the subcategory of algebraic varieties. The idea of exponentials is introduced, whereby given two schemes X and Y, a scheme can be formed whose points are suitable maps from Y to X. Taking the function space from a platonic point to any scheme X yields X itself, and taking the function space from a platonic tangent vector yields the tangent bundle, whose points are pairs consisting of points on X and tangent vectors at those points.

Jet schemes are a way of defining a tangent bundle and higher order derivatives of a scheme. Algebraically, they can be represented as a morphism of K-algebras from a ring to KT modulo t to the N plus one. This is done by taking a function from Z to a function space, currying it, and taking the spectrum of the resulting ring. Finally, taking the home space of the spectrum yields the jet scheme.

Jet schemes are a concept in algebraic geometry which can be defined by a functor that sends a scheme to a set of morphisms of schemes. This functor can be represented by a scheme, and is proved to exist by a theorem. An m-jet to a scheme X can be classified by a K-algebra homomorphism from the quotient of a polynomial ring to K^(n+1). This homomorphism is determined by what it does to x, y, and z.

A point in K to the 3M is parameterized by assigning scalars to symbols X I, Y I and Z I. This is the data of three objects, subject to a constraint that F0 through FMS (functions of X I, Y I and Z O) all vanish. This is the equation of the empath jet scheme, which is used to take a point and factor it through the quotient. The constraint is necessary to satisfy in order to get the algebra amorphism.

Computing the image of an equation is done by substituting each variable with a collection of terms and then expanding and collecting terms to read off more equations. For example, a two hidden node one hidden layer bias free 10h neural network equation can be parameterized by points of an algebraic variety, the Spectrum, by replacing each variable with a collection of terms and then expanding and collecting terms to read off more equations. This results in the original equation plus some polynomial in the AIS, BIS, Ci's, and Di's, which is the second jet scheme.

Symmetric algebra is introduced as a way to think of points as K algebra morphisms. Taylor series expansions are used to talk about tangent factors. A function T is parametrizing an arc near a0b0c0 and for the blue tangent vector to be a tangent vector to X and not just a tangent vector to that point in K3, the blue line needs to be orthogonal to the gradient of F. The Hessian appears in the constraint F2, while F1 imposes the constraint that the vector has to be at a point in the variety where F0 vanishes. The additional data of X needs to be orthogonal to the gradient.

Symmetric algebra is an associative and unital K algebra which is formed by taking the direct sum of the tensor V with itself. It is not commutative but can be modded out by the two-sided ideal generated by pairs such as a tensor B and B tensor a to form a commutative K algebra. If V is finite dimensional and has a basis E1 through E n, the symmetric algebra is isomorphic to the polynomial ring in N variables, with Ei mapping to Ei and Ei tensor Ej mapping to the product eiej. This is another way of justifying why Jets can be classified by the polynomial ring.

K algebra homomorphisms from the symmetric algebra of a vector space V to a commutative algebra a are in bijection with K linear maps from V to a. Any K algebra homomorphism from V can be lifted to the symmetric algebra, as any element in the symmetric algebra can be obtained by multiplying elements in V. As an example, if a is a field and V is a K linear space, then K algebra homomorphisms from the symmetric algebra of V to a are in bijection with K linear maps from V to a.

A bijection between geometry and algebra is introduced, where a point in a vector space V is mapped to a K algebra homomorphism. This homomorphism, Phi P, evaluates polynomial functions from V to K at the point P. Replacing K with a more interesting algebra, KT modular t-squared, allows for something to be gained for free.

A K algebra homomorphism from V Star to a larger algebra can be seen as two points in V plus V. A K algebra homomorphism from V Star to K direct sum KT is just two linear transformations, one to K and one to KT. To explain this without choosing coordinates, a K algebra homomorphism takes a function f and gives two scalars, one for the first spot and one for the second spot. Choosing coordinates can provide a more familiar way of thinking about it.

A K-algebra amorphism is determined by what it does to the generators. The Taylor series expansion of an element in the polynomial ring or Sim V Star can be done algebraically. The coefficients of the expansion are determined by evaluating a linear functional at the point P, which is related to the PSI PQ from the amorphism.

PSI PQ is an algebraomorphism which evaluates a generator xi to pi plus qi t. Applying PSI PQ to a general polynomial f gives a Taylor series expansion with coefficients corresponding to the multi-index I. These coefficients survive when none of the ijs are non-zero, or when exactly one spot has a one in it and the rest are zero.

Jet schemes are a way of representing tangent vectors at a point in a direction. A one jet is a k-algebra homomorphism from S(V*) to K[t] modulo t^2, which maps a pair (p,q) to a PSI(p,q) which evaluates f at p and takes the dot product of the gradient of f at p with q. Higher order terms can be calculated in a similar way to represent higher order jets.

Jet bundles and jet schemes are two different notions with similar sounding names. Jet bundles allow for objects such as KT modulo t squared or t to the cubed or t to the n, while jet schemes allow for objects such as KUV modulo U squared comma V squared. Jet bundles and jet schemes are both used to achieve similar things, though in different ways. Frankel Franklin Benson has a section which treats jet schemes at a high level. Jet schemes may play an important role in SLT due to their canonical nature, as opposed to the resolution of singularities.

Jet bundles and jet schemes are two different approaches to studying spaces. Jet bundles are used to set up a language to talk about differential equations as maps, while jet schemes are used to study singularities independently. Computing the equations that define jet schemes can tell us theoretically about the structure of singularities, but the numbers themselves don't tell us much.

Arc schemes and resolutions are two different objects with different properties, but they both produce normal crossing points. The number computed from them is independent of the choices made. The geometry of the arc schemes is more tractable than the resolution, and it might be possible to relate the components of the arc scheme to the essential exceptional divisors of the resolution. The K's and haters of the normal crossing points can be completely different, but the quotients are equal.

welcome everyone so obviously we expect many connections between algebraic geometry and statistical learning theory we've discussed some of them the role of log canonical threshold resolution of singularities but there are others which are either completely undeveloped or latent and today I want to talk about one of those which as far as I know hasn't been raised in connection with what another's work maybe there's some connection between jet schemes and algebraic statistics which is a broader uh area and which watanabe's work could be Set uh the application of commutative algebra and algebraic geometry to statistics the broader area of algebra algebraic statistics is uh Watanabe has a sort of different set of concerns to to some of the rest of that field so jet schemes may have come up in that connection a brief search didn't turn up anything uh but for reasons I may not get time to explain today but may come back too I do think it probably plays an important role in some of the questions that we worry about so just to set the scene a usual setup is that we're giving a statistical model PQ verify I guess if I okay old Divergence k suppose we've succeeded in finding some equivalent polynomial h then as We Know to compute various asymptotic quantities to do with the learning process of this statistical model we perform a resolution of singularities or what in the algebra algebra geometric literature is usually called an embedded resolution foreign parameters inside w so w 0 is this the vanishing set of H equivalently The Vanishing set okay and the embedded resolution looks like this or an embedded resolution say it's quite important commutative diagram like this G is the resolution uh it's an embedded resolution because the pre-image of w0 is in normal Crossing form G is bi-rational and proper it's an isomorphism outside of w0 so the complement of U zero and U and W 0 and W are isomorphic and that's what we're looking for now as we've discussed many times G may be hard to find there's a second problem which we haven't spent a lot of time talking about if you've read through the gray book you know that it's not that the embedded resolution well it's not that an embedded resolution is unique there are many resolutions embedded resolutions which are in some essential way different now you can compute the rlct using any of them the rlct is an invariant it's independent of the choice of resolution so you just find one hence why you have these ad hoc procedures in the papers

and the books to find these blow-ups in order to compute the resolution you you do whatever ad hoc set of choices you want you'll get a resolution that depends on your choices but you don't care because the rlct is independent of those choices but we have started discussing uh trying to infer more information from the resolution than just these numbers right the K's and the H's and the rlct and the status of that is left a little unclear if uh the structure of U and u0 depends on the choice of resolution okay so there may be information foreign G or U which is not intrinsic to w0 it's just some artifact of our choices it isn't actually telling us anything about the singularities or the learning machine all right well what's an example of a situation where we care about more than just this birational invariant the rlct well for example in the abstraction seminar we've started trying to investigate some conjectural relationship or dictionary which uh links abstractions or concepts or perhaps some generalization of sufficient statistics or or principal components uh whatever you want to call them to essential well to exceptional devices of the resolution we haven't formally said what exceptional devices mean in these seminars I think but just to just to draw a picture u0 is a normal Crossing form so it will look something like this in higher Dimensions you can um pull it apart so that locally it looks like Vanishing of of coordinate functions and the irreducible component so the one two three four lines that I've drawn there are the irreducible components of the exceptional divisor of the resolution so the exceptional divisor is just the pre-image of w0 under the resolution map eating each other well that can't be right if what I just warned you about is true if the structure of u g and u0 depends on the choices you make in the resolution that can't really be anything important right it can't be to do with the statistical problem uh it's just to do with the way you chose the resolution so we can't literally mean take any resolution of singularities and take the irreducible components of the exceptional exceptional device so we don't believe that any irreducible component important uh so what we want is somehow the irreducible components that are essential that is the irreducible components such that no matter which resolution you pick somehow you'll see that component in the other resolution so it's the Monastery uh it's not clear exactly how to formulate this correctly but hopefully at a

conceptual level it makes sense that you want the irreducible components that show up no matter which choices you make the blowing up process for example okay so I'm not going to Define what an exec essential irreducible component is this is kind of setting up the motivation for the theory of jet schemes so Nash was thinking exactly about this problem that we just discussed back in the 60s namely the lack of canonicity of the irreducible components of the exceptional divisor of a general resolution and in this connection he introduced jet schemes so I'll just I'll State a theorem of Nash uh Without Really explaining all the parts in it but I hope I hope the content still it's clear okay so what did Nash do well uh excuse me it's a little awkward because I'm going to State aversion which is not about embedded resolutions so let me back up a bit for a variety X with the resolution of singularities G okay so this is a bit of a different situation to the previous board right but at some level the content transfers over so for a variety x with the resolution of singularities G there's some other construction you can make called The Arc scheme and an injective map from the essential um exceptional devices of the resolution to the irreducible components of the pre-image under Pi of the singular Locus of x and the Nash problem is when is this a bijection it isn't always a bijection it's a bijection for toric varieties and in some other cases I don't really want to get into that although maybe it's interesting later for the present purposes let's just pretend it's a bijection so what does this theorem say it says that if you want to just get the essential components uh the essential exceptional devices from a resolution of singularities what you ought to do is study instead of the resolution this Arc scheme The Arc scheme is not a resolution of singularities it's some other object uh but it contains somehow just the essential essential information that's common to all the resolutions at least something okay so that's the idea behind the jet scheme uh well there's no jet schemes here I guess uh as it stands so x infinity is an inverse limit of X ends the X ends are called the is the nth jet scheme right so the arc scheme I misspoke before the arc scheme knows the exception the essential exceptional devices of any resolution of singularities uh the arc scheme is is kind of approximated by some Tower of finite truncations which are these jet schemes okay um so the question I want to

pose and I think this is a an interesting research Direction I mentioned it in the open problems session okay so loosely speaking uh what the well you know what an arc is right so an arc on a circle is some segment of the circle an arc is yeah in general if you're on a manifold and an arc through a point sort of means a an infinitesimal piece of a smooth path right so that's the kind of object that an arc is so what kind of arcs do we expect to encounter in a learning machine well imagine that you're doing gradient descent or stochastic gradient descent near w0 well the idea is that you start somewhere and you're going to end up near this most singular point the idea is that in some sense I don't know how to make precise but which is clearly true uh SGD near here which you could think of as gradient descent with respect to a perturbed KL Divergence right so nearby in function space you take some kn you do great you look at the vector field Associated to that so degenerating KN to K is a little bit like taking a jet scheme in a family now that isn't quite going to work but you can by thinking about those kinds of things get some way to convincing yourself that probably the jet scheme of w0 contains a bit of information about what SGD near w0cs and its Dynamics so that's an interesting direction to explore further and that's that's what I have in mind for this question I don't I don't know how to make that precise yet okay so that's the high level motivation I'll pause here for questions and then get into some more precise material okay um but what's the etymology of jet schemes right and Arc schemes I was in the art part now but I also don't know what a scheme is yeah good question uh so for scheme you should just read algebraic variety okay so for example The Vanishing of of H inside um R to the D so that object's an algebraic variety as you know and uh an algebraic variety is a scheme so a scheme isn't just a topological space it's a topological space together with some collection of functions on it essentially to form a scheme from the polynomial H you're remembering the set of zeros of H but essentially also the function H itself so it's kind of slightly complicated mathematical object which subsumes the notion of an algebraic variety Jets yeah Jets I'm going to explain so you should think of the the following dictionary a zerojet is just a point a one jet is a tangent vector the two jet is well you don't have any other word for that uh it's what you get when you remember

the behavior of some object near a point up to second order rather than just first order so jets are the one way of generalizing tangent cool other questions can't understand you sorry okay you can always type sorry okay so that was very high level I'm still not going to get to precise details because it's a bit tricky to obviously introduce stuff to do with schemes when I don't want to spend several months introducing the theory of schemes properly so I'll try and yeah Ben's question in the chat is what's the reverse our inverse limit uh yeah I don't want to answer that question yeah it just means a way of a way of taking some finite objects and and uh well taking their limit so there's a there's a notion of limit of algebraic varieties so just as you can take a limit of um real numbers or functions you can take a limit of algebraic varieties uh it's a relatively sophisticated notion though so for the moment just imagine if I say zerojet1jet2jet dot dot dot where the dot dot dot ends is the arc space it's the limit of all these finite things sorry I can't be more helpful than that would it be appropriate to think about Taylor expansions yeah yeah that's exactly right so the discussion about as you'll see a discussion about jet schemes is really just packaging for Taylor series expansions of functions that's essentially what we're going to be discussing uh so that's like it boils down to something that isn't that sophisticated ask a question yeah for this for this kids came here if say x if x is variety is a yeah if you look at x x is a variety then if you look at this this Arc space then is that they you take the completion of the of the the algebra corresponds to that variety uh yeah that's almost right yeah so you're going to the the One jets and two Jets come from taking quotients of polynomial rings by t squared and T cubed respectively and arcs basically come from taking the power Series ring itself which is the inverse limit of all those quotients that's right uh but where you exactly put the completion and so on uh I mean it's it's a little bit yeah it's not completely straightforward but essentially it's just the completion of the level of the algebra yeah all right so uh let me pick up that last point about zerojet one jet two Jet and so on and sort of motivates how we're going to Define jet schemes basically the word scheme will just disappear and turn into algebra at various points that at any time that we actually compute anything so I want you to think about the

category of schemes of okay so just think about algebraic varieties of a k for you for us is is r so in the category of schemes over k well there are special objects which uh are not in fact varieties so they they live in this they're outside the subcategory of the algebraic varieties at least in the well ah they're not irreducible it depends whether or not you believe varieties are irreducible so okay it's they're not like the objects that we've usually been considering I suppose I'm calling them point with a subscript and I want you to think about zero as being like an actual point like a platonic point and one is a platonic tangent Vector I'm actually going to Define these in a second this is something uh that is like more information than a tangent vector so if you know what spec is that's what this thing is it's the spectrum of this ring over k now I don't expect that to mean much to you I'll explain I'll make the connection between this algebra and jets a very explicit hopefully by the end of today but for the moment just in case you know what the spectrum is I don't want to pretend like this is mysterious okay now I want you to imagine that the category of schemes has exponentials that is that given schemes X and Y that you could form a scheme sort of a so an algebraic variety kind of object whose points are suitable maps from y to X in fact this doesn't exist in the category of schemes if you know about topos Theory such exponentials exist and there's a risky topos that's outside the scope of these these discussions it doesn't matter but it's important to have the idea that this is what you're doing even though it takes more sophisticated techniques to actually formally do it but in some sense you can do this you can take two schemes and you can take the sort of function space from one to the other okay well what happens when you take the function space from the platonic point to any scheme X we just get X right the scheme of maps from a point to a space is just the space itself if you take x to the power of the platonic tangent Vector what do you think you get thank you what is the space that parameterizes all the ways to stick a tangent Vector onto X it has another name the risk of embarrassing myself is it affect the bundle yeah it's the tangent bottle that's right the tangent bundle is another scheme uh whose points are peers consisting of points on X and tangent vectors at those points okay now we can keep going right we don't have another way of thinking

about that that's a way of sticking a little platonic order 2 jet into X but xn the nth jet scheme is by definition x to the power of the platonic nth jet okay so that means that x0 is just x x one the one jet scheme of X is just the tangent bundle and then so on okay well that isn't really a definition because I told you that actually this exponentiation operation doesn't really make sense in schemes you need some fancy thing to to do it this way but okay this is the correct way of thinking about it and then we can find another way of saying the same thing which I'm about to explain to you and then uh turn detailed enough that we'll actually compute a jet scheme of a surface in maybe 10 minutes all right um okay so let me move on to the next board and try and turn this into a definition that's a little more concrete hopefully you get the idea of what it's supposed to do foreign consider the following calculation I'm going to put calculation in scale quotes for the reasons I just explained that this actually doesn't take place inside the category of schemes because the exponential doesn't exist there but that won't stop us all right so suppose you take some schemes in then the home space and the category of schemes over k from Zed to this enter jet scheme well uh if you take a function if you take functions from Z into a function space then there's an adjunction which says that that is this it's just currying and if Zed is that fun and we take this platonic in jet thing to be as I said the spectrum of this ring then that's a product is over k with a spec of this thing comma X but that's a tensor of a k with this that's just a t modular t to the N plus one and finally if you take homes out of spec all right so okay so the upshot here is that [Music] a closed point of this object here so that means taking it to be spec k is well if I take a to be K in that final line it's a schemes like that and if x itself is f on this is the same as a morphism of algebras uh the other way from R to KT modulo t to the N plus one okay so I apologize for invoking schemes and talking about all of that it's actually not uh we haven't covered that kind of material in the seminar series and I'm not assuming you know it but it's part of the justification for why uh the notion of an injet so an injet algebraically speaking is the same thing as amorphism of K algebras from some ring to this object KT modulo t to the N plus one that's one way of motivating it I'm now going to switch gears entirely and

motivate it from the point of view of a Taylor series expansion so this is the the kind of high-powered derivation of why this is the right kind of object um but uh we'll we'll see it from a few different points of view okay so well the calculations were in scare quotes right so this is actually isn't a definition of the jet scheme but what you can do is you can prove that this functor is representable so theorem so this is actually how you prove the jet scheme exists so theorem functor which sends a scheme Z two the right hand side of that first line there set of morphisms of schemes from this product to X is representable I.E uh let's just say x finite type exists in general but there is a scheme xn such that which does the job and makes that calculation an actual calculation not just a scare quote calculation all right so this is how algebraic geometry works at some level you kind of fly by the seat of your pants and invent an object then you construct a functor which if it had to represent rent if it was representable would be represented by your object then you prove it's representable and and there you are there you have it okay so this is where the jet scheme comes from and how it's proved to exist uh okay now I want to put a line under that and switch gears as I said and talk about Taylor series expansions well maybe I'll do an example first are there any questions okay I want to do the example so you see uh in practice how this is kind of how you would actually find x sub n okay so say x equals spec this another way of saying that is that X is the vanishing set of f inside K to the 3. just viewed as a scheme rather than as a just a set of points so this is this object VF is what x is so what is a two jet or just an m jet an m jet to X is okay so you're Imagining the surface uh sitting inside K cubed what is an mjit well we know a zero jet is just a point of X One jet is a tangent vector uh two jet is something okay well by the prescription that I've just introduced um which uh by definition an m jet to X is a k algebra homomorphism let me call it gamma star from The Ring to KT modular t to the N plus one all right well how do we classify all such K algebra homomorphisms well remember that a k algebra homomorphism out of a quotient of a polynomial ring is so we have the quotient k x y z goes to k x y z modulo f and we've got our gamma star okay algebra homomorphism out of the polynomial ring is completely determined by what it does to x y and z so the daughter in gamma star is the

image of X Y and Z and then there's some constraint so the data of gamma star is what it does to X and I call that x t and that's something in KT modulo t to the N plus one right which is which is K plus KT plus dot dot KT to the m so the coefficients are some scalars so let me call them x0 X1 through x sub m similarly with with y and Z okay now what's the constraint well when you feed X Y and Z into gamma star uh okay so you get these things here but it has to factor through the quotient by if which is to say that when you feed in F to gamma star you have to get zero so the constraint is that when you substitute x t for X which is what it means to evaluate gamma star on F and Y T for y and z t for Z you have to get zero modulo t to the N plus one okay but imagine substituting x t y t and z t into f f some polynomial in x y and z and so you have some brackets and you expand all the brackets and what you've got is a polynomial in t right right so you've got a constant term F0 you've got a linear term and you've got all the way up to the nth term and if you think about it in terms of the expression it's a little a little imprecise here because x i y i and z i are really just scaleless right uh Mr KT modular t to the N plus one so the x i y I's and Z i's are just scalars but if you think about the expression on the page when you expand out all those brackets the F I's or fjs will be in the way you wrote them down on the page some functions of the X I's the Y I's and the Z O's and then uh I suppose you have higher terms here um but you're about this so this equation this constraint tells you that F 0 F1 through FM vanish all right okay so this these are actually the equations of the the ampth jet scheme so the empath jet scheme of x is the space which has these X eyes y I's and Z I's as coordinates and oops F0 through FMS as equations why well that space exactly parameterizes gamma Stars right to take a point so what is what is the point of this space well it's to take a point in a point in K to the 3M that is assignments of scalars to these symbols x i y i and z i that is the data of these three objects subject to a constraint the constraint is on the right hand board that these equations all vanish but that is what we agreed is exactly the constraint for this I mean giving me some data like this just tells me uh specifying x t y t and z t just gives me a algebra amorphism like that to get it to factor through the quotient I need to know that it satisfies some constraint

and that constraint is when you unpack it exactly that these fi's vanish okay so uh an m jet is some data subject to some constraints and therefore we can parameterize it by points of an algebraic variety the Spectrum here so that's um that's that's it so uh Computing the imgit of something that's given in terms of equations is nothing more than making these substitutions expanding collecting terms and reading off some more equations okay um any questions about that part maybe I'll do an example that's come up in in this seminar just to make that more precise and so let's consider the equation whose Vanishing set is the set of true parameters for a uh two hidden node one hidden layer bias free 10h neural network is this equation here so how do I compute the jet schemes of this object well I replace each variable by a kind of collection that looks like this if I want to compute the um the second jet scheme then I replace a by this and similarly B buy this so on with c and d and then I just substitute so I just let me call this a of T I guess B of t so F of a t b t c t DT well that's going to be a0 plus A1 t plus A2 t squared b0 plus b one t plus b 2 t squared plus same thing for C same thing for d uh can you excuse me for just one second I'll be back um so you make that substitution for c t and DT and then you've got a square and you've got the other term okay so you can see what's going to happen when you expand that out right you'll get a 0 B 0 plus C 0 D 0 plus terms of higher order in t squared and the same in the other term and when you expand all that out you're going to get the original thing if in a0 or B 0 C 0 and d0 plus some stuff times t plus so on this equation here which is your original equation will be f 0. it's just uh it's F but in these new zero versions of the variables and then you've got some other polynomial here in the AIS and bis and Ci's and d i's and that's F1 and uh when you have an order two thing and then you have higher order terms but you don't care about those that's F2 and then okay so if we'd let X be the vanishing set of f the second jet scheme is going to be r in a0 A1 A2 and so on with a b c and d modular these three equations let's pick well it's the same it's the vanishing set of those three equations inside uh what's that R to the 12th right so in practice Computing these jet schemes is maybe a bit laborious but it's not not anything complicated okay look at the top okay um yeah let's pause here uh for questions and then I'll

um do some more foundational stuff so I want to I want to tell you what the symmetric algebra is I want to introduce how to think about points as K algebra morphisms and how to use Taylor series expansions to talk about tangent factors and um uh no there was no reason to go to second order it was just uh to go to first order would be kind of well in fact I didn't really do anything right like the since I didn't expand the brackets uh would actually do any work that derivation I just gave in the example would have been essentially the same if I'd done third order or amp order so there was no significance to second order is it bad to hold the um the Alp bot cot parametrizing so T is parametrizing uh and Arc near a0b0 c0 yeah that's right okay I I can't um figure out what the significance of having so so that's that's a function into uh yeah think about it think about the arc so if you fix a point a0 b0 c0 d0 uh then like imagine T to be a very small number you think about t as being Epsilon so uh as you vary T from minus Epsilon to Epsilon you're tracing out a little path that goes through a0 b0 c0 d0 so I can see what uh why we need the constraint of f capital F zero being zero that means that we are on the variety but um why do we want like high order derivative to vanish yeah hopefully that's what the discussion in a moment will will clarify but to to give the short answer uh if you look at on the left hand board here um the blue tangent Vector to X to it lies on X right that's a constraint and the constraint is that it's dot product with suppose X is the vanishing set of f as it is on this page right and the constraint for the blue line to be a tangent Vector to X and not just a tangent Vector to that point in K3 is that that blue line is orthogonal to the gradient of f right so you have some constraint which is this dot product is zero and this constraint is exact exactly one of the constraints that would be imposed by F1 in fact it's the Olay constraint okay so two Jets then you can ask well what are the constraints imposed by the equation F2 that's more interesting and that's got to do with second order information in F and and we'll see the Hessian appear uh in the in the constraint F2 but the first order it's simply saying that yes the vector has to be at a point that's actually in the variety where F0 vanishes and the X to the additional data which is telling you the tangent Vector itself needs to be orthogonal too the gradient right right okay thanks cool

okay let's move over [Music] so this is a discussion of a fairly Advanced topic jet schemes and at the same time some Elementary topics uh hopefully it's useful for those of you who haven't thought much about algebraic geometry I'm going to say a little bit about the symmetric algebra which is one way I think about the polynomial ring just because I'll be talking the way I'll discuss things makes heavy use of the Universal Property of the symmetric algebra so I feel like it doesn't hurt to say in two minutes or so uh what this is so let K be a field via Vector space then the symmetric algebra of V over k is well I take the following direct sum tensor so a tensor V with itself two times three times and so on and then I take the direct sum of all of those that's an algebra called the tensor algebra where I multiply tenses by just concatenating them so in this numerator here I have things like a tensor B and I multiply a tensor B with C by just forming the tensor a tensor B tends to C you have to check that's well defined and so on but it isn't makes an associative algebra the tensor algebra which is not commutative but inside that non-associative but unitol algebra you can sorry not associative it's not commutative I hope I said that right the first time it's a non-commutative but associative and unital K algebra and within that non-commutative Algebra I can mod out by the two-sided ideal generated by these pairs right a tends a B and B tensor a are different things in general in the tensor algebra but I can mod out by the two-sided ideal generated by all such pairs that quotient which a prior is just another associative algebra is in fact commutative algebra and that's called the symmetric algebra so that's a k algebra commutative K algebra Lemma has a basis so V is finite dimensional and has a basis E1 through e n then the symmetric algebra on V is isomorphic to the polynomial ring in N variables we might as well call the variables e 1 3 in the correspondence here is between in the obvious sort of way uh e i maps to e i and for example EI tensor EJ the equivalence class of that in that quotient is just the product eiej uh all right so I've already invoked the Universal Property of the polynomial ring which is another way of saying the Universal Property of the symmetric algebra so that misdated let me think for a moment if I actually needed yeah I might as well okay so Lemma so this is another way of circling back to justify why you should believe that something like in Jets uh classified by

K algebraomomorphisms into that funny quotient KT modular t to the in the N plus one so that VV a vector space a a commutative algebra okay algebra if Phi from V to a is just a second is K linear there exists unique K algebra morphism Capital Phi from the symmetric algebra of V to a making the diagram V so via obviously sits inside the symmetric algebra V right from V to here which just sends v a vector to the equivalence class in the quotient of V itself where you sort of stick V in that factor there okay so what this statement is saying is that any K algebra homomorphism out of V can be lifted to the symmetric algebra and that's that's pretty clear because if you can if you can map vectors in V to a and inside a you can multiply things then if I give you anything in the symmetric algebra well anything in the symmetric algebra is is some tensor and tensors can be obtained by multiplying so anything in the symmetric algebra you can just get by multiplying together stuff in v right so for example if I want to take a tensor B tensor C a b and c vectors in V well that's just a times B times C in the vector space V I can't necessarily multiply things but in the symmetric Algebra I can always multiply vectors in V to get other things in the symmetric algebra and um so for example if I take Phi of a tensor B tends to C or more precisely its equivalence class in the quotient then well that's that's a product of a with B with c Phi is a k algebra homomorphism so that has to send that to 5 a times 5B times 5c but when I evaluate Capital Phi on vectors in V I should just get five of them so this is five a times five b times 5c and you can turn this into an actual definition of capital Phi okay so this is just similar to knowing that if two linear Transformations agree on a basis they're equal right Phi preserves the structure of multiplication so it's determined to whatever it does to generators the vectors and Via generators and on the vectors in V Capital Phi agrees with lowercase Phi okay so the one way to say that is that uh K algebra homomorphisms out of the symmetric algebra are in bijection with K linear maps from V to a for any commutative algebra I'm going to pause for a sec to recharge my apple pencil all right uh so let's see I now want to maybe I'll write one more thing and then invite you to think about this a little so example let's take a to be the field as a k algebra and take instead of V V Star so V Star means the K linear joule then K algebra homomorphisms from SIM V

Star to K are what is that the double jaw that's right and the double jewel is homomorphic to be that's right I guess I didn't say V was finite dimensional but that was what I meant yeah so if I use the above bijection that's home k V Star V star k which is V double star which is canonically isomorphic to V the V is finite dimensional all right so this is a way of introducing the dictionary between sort of geometry and algebra because we see here how a point of V so I think about v as being Alpha and n space so we take a point p corresponds in this canonical way to a k algebra homomorphism which I'm going to call Phi p what is that homomorphism Phi P of f is what f o p well we have to I suppose we have to explain what it means to evaluate f f in the symmetric algebra on on a point uh as it stands the elements of the symmetric algebra just some equivalence classes of tenses but we can think about them as functions on V well the things in V Star are clearly functions on V by definition right the linear functions on V and If I multiply linear functions with other linear functions well I can just multiply them as functions right and actually that's compatible with the multiplication on the symmetric algebra so um maybe I'll say it like this Sim V Star is a sub ring of just functions on V I'm gonna run out of battery I'm gonna just say functions from V to K where you you make that that set of functions from V to k a ring by just pointwise multiplication in the usual way and you can identify Sim V Star with with a sub ring of that so the rule that we have for multiplication of these equivalence classes of tenses is compatible with just thinking about things in simply styrus functions so simvastar is the coordinate ring of functions of algebraic functions on V thinking of v as a of a variety these are the polynomial functions on V and Phi p is the K algebra on morphism that takes a polynomial function and evaluates it at the point p okay so the dictionary between points of a space p and v and K algebra homomorphisms is that P corresponds to evaluation at p no that's meant not to be very profound seeming but now the the cool trick is that if you think about a simple thing in a sophisticated way maybe there's a way of getting something for free so now I'm just going to replace a or rather K by a more interesting algebra namely KT modular t-squared it's got a ring of dual numbers but I need to let my pencil charge for a minute [Applause] [Applause] maybe you can think about it ahead of

time you can do this calculation as well as I can so using the first isomorphism let me call it I'll call it star um using star what happens if we take a to be the K algebra KT modulo t squared let's two dimensional Vector space equipped with a multiplication where t squared is zero let's do it on the next one foreign space that's got a basis which is 1 and T and the multiplication rule is the t squared is zero so let's compute what happens when I take K algebra homomorphisms from simvi Star to a well that's by the star that's V comma a and here I'm just taking K linear Maps I don't care that a has a multiplication right I'm just thinking K linear Maps sorry V Star right uh okay so that means I'm taking K linear maps from V Star to K direct sum KT now if I take K linear Maps from some Vector space to a direct sum of two things that's just kind of like a matrix with two rows right a linear transformation to K plus KT is just two linear Transformations one to K and one to KT and just take the two rows of the Matrix separately so that's that's just the same thing as giving a k linear transformation from V Star to K and these yeah I'm fighting the battery here um restart a KT this is just a one-dimensional vector space that has a basis T right so I'm just going to uh I'm just going to drop the T because it's really just a one-dimensional vector space but that's just two copies of V-Star b double star rather okay so that's V plus v what are we supposed to make of that so a k algebra homomorphism from SIM V Star to this larger algebra is really two points in v so what we're going to do in a moment is Trace through this isomorphism to see what a pair of points P comma Q in V plus v what cha algebra homomorphism they correspond to um but you might also be able to guess so let me take p and Q yeah and I think I have some notes I'll circulate I just want to get the notation consistent what I call PSI PQ yeah foreign okay so PSI PQ has to take in a function f and has to give you a vector in a so just to give you two scalars right because that's what a vector in a is uh one scalar which goes in the first spot and another scalar which goes in the second spot sort of the coefficient of the T foreign it's a little uh there is a way of explaining this without choosing coordinates but maybe just for the sake of time I'll now choose coordinates and explain um how to think about it in terms of those coordinates just because it's maybe a more familiar way of thinking about it

so let's do that so to understand what this corresponding K algebra amorphism is let's choose coordinates choose a basis V is finite dimensional in all this for V then I have a dual basis EI Stars and those are vectors in V Star so those X i's are a basis for V Stars so if you like simv star is the polynomial ring in X1 through xn it doesn't matter too much for us that you can think about it that way but okay apple pencil has has died okay so what I'm going to do is um well PSI PQ whatever it is so we're picking p and Q and then under this chain of bijections we believe in we get a PSI PQ but we don't know what it does uh but whatever it does it's determined by what it does to the generators as I said earlier right if you have a amorphism of K algebras out of a polynomial ring or the Sim V Star then it's determined by what it does to the excise so uh that's the data in this object right sub PSI PQ is determined by some numbers so let me take PSI PQ and evaluate it on x i that's some Vector in a so it looks like pi plus q i t for some scalars p i and q i right Define pi and q i to be those coefficients all right now coming back to Matt's earlier comment let's talk about the Taylor series expansion of an F so given f in Sim V Star or if you're liking the polynomial ring we can consider its Taylor series expansion we don't have to use limits and that kind of calculus to do the Taylor's there is expansion it works perfectly fine algebraically uh so the Taylor series expansion and N variables is the following that's an eye so I'm doing the Taylor series expansion at the point p you'll see why p is defined well uh uh yeah Maybe maybe now is the point to connect pis the pis to the p um yeah uh so let's do that just let me go back to this board here let's follow through how uh how we get what's the connection between the the P that we had here and the PSI okay um so we start with PSI PQ how do we get to home V Star a well we just restrict to V Star the copy of V Star inside simvie star the next line doesn't do anything the line after that just splits this into a pair which is PSI p q restricted to V Star and take the coefficient of one and then the second part is I take the coefficient of t and then I get the isomorphism from V double star to V and the way that works um is well if I take PQ then this here has to just be I take a functional and I evaluate it at P so this is just evaluation of a linear functional NP so if you track that through you'll see that uh you'll see that these here

well yeah I guess what I'm what I'm sketching and maybe I'll just leave it at a sketch is that these this isn't I guess as I said it earlier not a definition right it's an observation that if you evaluate PSI PQ the scale drama morphism corresponding to the PQ at the generator x i you get pi plus q i t where these are the this is the expansion of P the point in v it has an expansion in the basis and the coefficients are exactly these numbers that show up here so that's if you just follow through this chain of isomorphisms you'll observe that so the thing I really want to argue for is well that's fine we know what PSI PQ does to a generator it gives us this expression but what does it do to a general polynomial and that's what I'm Computing on the right hand board here okay so I take a Taylor series expansion of f and then I apply PSI PQ to it this is a k algebra homomorphism so it particularly it's linear uh let me just write this same expression I'm just going to simplify it this is just a number and then I've got PSI PQ of this stuff so what does PSI PQ do to all that well it's okay algebraomorphism red so I can this part here becomes I can take the PSI inside and it's PSI PQ of X1 minus P1 to the i1 dot dot PSI p q of x n minus p n to the i n but PSI PQ of X1 is just P1 plus q1 T and then I'm subtracting P1 so this is the sum of a this multi-index 1 over this exponential term and then this derivative and then I've got as I just said site pqx1 minus P1 is just q1 t all right so now we're I mean this this calculations just for tangent vectors that's just what we're about to discover but the same calculation tells you what higher jets are talking about right so this is something we'll revisit but at the core of the idea is already present here yeah because well what's this this is q1 to the i1 dot dot dot q n to the i n t to the sum of all the I's J I guess but the codomain of the PSI is in a and in a anything that has a power of T above one is zero so most of these sums I mean this is a finite sum because the polynomial only has finitely many non-zero derivatives most of these sums are zero so which ones actually survive well a multi-index I has a surviving contribution to this sum exactly when uh none of the ijs are non-zero so they're just all zero in which case I get t to the zero or exactly one spot has a one in it and the rest are zero okay so you can see hopefully what this the sum has to be so I get the case where the multi-index I is just all zero so there's no

derivative and I'm just evaluating F at p and then the rest of it is just doesn't do anything right all the eyes are zero so this is just a bunch of ones times t to the zero and then I get the all the contributions from where exactly one index in this multi-index I underline is one and the rest are zero and that is just the sum over which position is non-zero and then I'll get a derivative exactly in that position of f evaluated at p and then exactly one of the cues will have a one a month will be to the power one and the rest will be to the power zero and that will be QJ okay so this is f p plus the gradient of F at p taking the dot product with a Q okay so what we've discovered is oh yeah thank you to you yeah yeah that's right okay we didn't put in derivatives here anyway right so I introduced the Taylor series expansion in order to compute this PSI but even if I didn't know what a derivative was I would have just invented it by just thinking about this K algebra KT modular t squared and following this canonical bijection from a pair PQ I get this PSI PQ which both evaluates F at p and takes the derivatives of F at p and takes the dot product with Q so well that's the directional derivative right that gradient dot with Q is the directional derivative in the direction Q so the upshot is that this PSI PQ map is a tangent vector at the point p in the direction queue that's what this pair PQ means it's the tangent Vector based at p in the direction Q okay so that's what a one jet is to define a two jet you take a k algebra homomorphism so a two jet on V at a point p is a k algebra homomorphism from SIM V Star to KT modulo T cubed and then we can think about by doing a similar calculation what the higher order terms mean and what the information in them is but we'll do that at some point maybe not next week we'll see what we want to do next week but that's the next stage to talk about two Jets and higher Jets and how they fit together to form the the jet schemes but the the underlying calculations are more or less determined by by what we just did so I'll stop there and take questions yeah see you Matt I think I might have asked you this before then but do you have a recommended reference for jet skis yeah I probably don't have a better answer now than I did that time um yeah there's some references uh I don't know I don't know a textbook well I it appears in some textbooks so for example you can look at algebraic curves and vertex algebras by um bensphy and um

uh who's the other author on that book uh uh Frankel Franklin Benson that has a section which treats jet schemes uh it's at a quite high level there are some papers with better kind of um more Elementary introductions and I'll um I'll post some of those in the Discord cool so the Hope to come back to the relation to learning theory at the beginning uh I'm hopeful that so in the in the cases that are treated by information geometry so the regular case we we know how to think about higher order terms in the asymptotic expansion of the free energy in terms of G metric structure and it seems like in the singular case you should be able to do something similar but I mean in the regular case it's basically in the language of jet bundles already maybe that's so in the Bala subramanian paper it's not stated that way but you could state it that way uh and I I don't see any reason why you couldn't expect a similar kind of theory in terms of jet schemes in the singular case which generalizes that as far as I know that hasn't that hasn't been done but so the reason to maybe be interested in that is uh so far most of the discussion the relation of algebraic geometry to to SLT is is by the resolution of singularities but as I said at the beginning some of that information is already in the jet scheme and maybe it's actually there in a better way in the sense that the jet scheme is canonical and the resolution is not so um there's probably one would have to guess an important role for jet schemes in SLT which is so far an export any other questions what is the difference between as in other than the confectionery other than yeah they're really not the same thing they're different Notions of jet uh unfortunately so they have a very similar sounding name but the question is I read up these bullet ends before I read like what are the infinitesimal models that you're considering mapping into your space and for differential geometas it's a different kind of uh the the local objects they're considering are not the same um so it wouldn't be so here we're only considering things like KT modulo t squared or t to the cubed or t to the n uh the the differential geometry is jet bundle is more like you would also allow uh kuv modulo U squared comma V squared so it's it's kind of a achieve similar things but in some sense quite different so something like um not just Arc but you know even intestinal surfaces and volumes and so on that's right yeah exactly yeah so more information you would think so yeah that's right uh

so the theory of jet bundles is really about differential equations uh so it's sort of like got a similar aim you're probing your space in some infinitesimal way but the objects you're probing it with uh more General and so you get a a more general theory but it's kind of in some sense going in a different direction because you care about jetbun the jet bundles are defined for manifolds so there's no singularities to worry about here you're probing you don't care about jet schemes of things that aren't singular you'll just get back some boring thing you don't care so they have some similarity in their name and the kind of idea of probing your space with some infinitesimal gadgets and parameterizing all the ways you can map them in and making a space out of that but uh they're being used for somewhat different purposes in the case of jet bundles to basically set up a language to talk about differential equations as Maps and here to talk about arcs that are program the structure of singularities so I don't mean I'm not an expert in either Theory but they seem to be doing and I'm not sure there's a lot of useful overlap they may be right um I'm I'm going to ask a sort of aggressive sounding question so what's the um want to regain by um having this um uh so the purpose is to study sort of Singularity [Music] independently near w0 um so uh so what does what does the structure of the kind of the algebra here tells us about um something that we can um you know having having just a local chart and um have a local storage chart representation of our functions um what has that all right I guess the what is the structure of the algebra tells us yeah um sorry that's a very poorly worded question like um it doesn't kind of the general question between like what's the difference between sort of an algebraic geometry point of view versus just straight up our differential geometry point of view yeah I think I I think I understand the question um okay so at the level of well okay suppose we take uh one hidden layer 10h networks then we can write down the equations that Define these m jet schemes for all m maybe I mean there's some terrible polynomials they exist but okay maybe we don't want to compute them so what if we could what does that even tell us um well uh I'm not sure it does tell us much uh to be honest to just compute those uh in the same way that well you can go and compute the rlcts but what do those numbers tell you uh what we're interested in I think is uh what we can say theoretically more than

Computing things so um we know in some cases that as I was saying earlier the uh the irreducible components of the pre-image under the map from The Arc scheme to X correspond to the essential exceptional devices of the resolution now maybe that means we can say something about what those components mean right because maybe we find a way of relating this um these C's uh or the behavior of k n minus K near the set of true parameters maybe we find a way of saying something about that in terms of those components of the of the arc scheme um in a way that's maybe less tractable when we talk about the resolution I don't really know but uh yeah so the answer to your question is I don't know so I'm just kind of the resolution is not a canonical object right let's go over here do a blow up go over here do a blow up the arc scheme is not like that right the points of the jet schemes are really very simple objects that are clearly defined right they're they're truncations of arcs that pass through your variety so given that it's a canonical object it seems more tractable that maybe we can use it to say something um rather than just kind of saying your resolution exists out there and you know whatever it is we can we can do something with it and compute a number that's independent of the choices so that's the idea right that kind of makes sense I I actually don't have a good picture of how um resolution could be different like is it different in some kind of global manner but like locally around each point it's somehow similar as in oh no not at all there's no there's no obvious I mean that's a very difficult question uh so you can say something in some cases but in general they're just different right but they all produce a normal Crossing phone for um and the number you compute from them is independent of the choices but this the resolutions themselves have no obvious relation right so in in sort of um our SLT uh settings is like the K's are different the pages they are different but our CT is the same actually what we mean by the K's are different so in the case being the normal Crossing oh the little case yeah yeah little K's and little haters yeah yeah that's right they could be just completely different the geometry you could have completely different sets of essential coordinates and it's just like some miracle of the quotients are equal okay and and so what it means is that anytime we're talking about like we're tempted to assign some meaning to the resolution right like it's saying