RLCT is a technical tool used to calculate asymptotics of functions related to program synthesis and statistical inference of SLT. It involves using KL divergence to measure the difference between two probability distributions and to calculate the fiber ideal. The Morse Lemma is used to show that a Taylor expansion of a function can be written in terms of coordinates on the function, and that it is equal to the function that calculates the RLC of a fiber ideal. RLCTs are finite constraints applied to a transmission that match the KL divergence on a local level, providing an intuitive understanding of the models.

RLCT is a technical tool used to calculate asymptotics of functions related to program synthesis and statistical inference of SLT. It is defined as a pair of lambda and theta and is calculated through a partition and theta function. Lemma is used to control estimations and bounds on the RLCT. Two ideals, f1 and j, are compared in a ring of analytic functions using Koshy-Schwartz lemma to show that if c d is greater than zero, the LCD of f1 and j is comparable. The parameter space w is assumed to be compact, giving the functions boundedness.

Ideals are more flexible than functions when calculating the RLCT. This is because the RLCT of an ideal is the sum of the squares of its generators. A theorem states that if a function f and g can be factored with a point w such that f(w)=0 and the Hessian of g at u(w) is positive definite, then the RLCTs will be equal. The Morse Lemma can be used to show that a Taylor expansion of a function can be written in terms of coordinates on the function, and that it is Lipschitz equivalent to a sum of squares. This is equal to the function that calculates the RLC of a fiber ideal.

RLCT is a technique that relates local and global RLCTs by defining a local RLCT on a neighbourhood of a zero and the global one as the minimum of the local ones. It involves using KL divergence to measure the difference between two probability distributions, and the fiber ideal is generated by the difference in the probability that a model assigns to some element of the set minus the probability that the true distribution assigns to it. This makes it easier to calculate and provides an intuitive understanding of the models. The theorem applies when the KL divergence is regular around the point q in the probability simplex.

RLCTs are finite constraints applied to a transmission that match the KL divergence on a local level. They identify the fiber ideal and its generators, and are less than or equal to twice the order of the ideal at a point. Analytic functions on two distinct parameter spaces can be combined using projections, and the inversion of a junction theorem states that when a variety is embedded in a non-singular space, the log canonical threshold of the restricted variety is less than or equal to that of the original. Jet schemes can be used to prove this statement.

RLCT is a technical tool used to calculate the asymptotics of certain functions. It is linked to program synthesis and statistical inference of SLT by linking each model prior truth triple to an ideal in the ring of analytic functions. RLCT is defined as a pair of lambda and theta, and is calculated through the partition function and the theta function. The partition function is an integral over a compact and semi-analytic subset W, and the theta function is a meromorphic function on C. The smallest pole of the theta function is lambda with multiplicity theta.

Lemma is used to control estimations and bounds on the RLCT. It is proved by looking at the asymptotics of the partition functions and taking logarithms. Monotonicity is observed and the inequality is swapped around. Asymptotic expansions are used to recreate the log of the c n and o one. Naughty editing details are in the notes for further interest.

Two ideals, f1 and j, are compared in a ring of analytic functions. Koshy-Schwartz lemma is used to show that f1 squared is less than or equal to the squares of the gjs, where each of the fis are in the ideal j and the h i j are in the ring of angular functions. The result is used to show that if c d is greater than zero, the LCD of f1 and j is comparable. The parameter space w is assumed to be compact, giving the functions boundedness.

Ideals are more flexible than functions when it comes to calculating the rlct. The rlct of an ideal is defined as the sum of the squares of the generators. This differs from the definition in Lin, where a factor of two is added, but this works out since the rlct of an ideal is important for the following result. Inclusions, sums and products of ideals also have nice properties with respect to the rlct. This makes calculations of these ideals simpler and more flexible.

A theorem states that if a function f and g can be factored, with a point w such that f(w)=0 and the Hessian of g at u(w) being positive definite, then the RLCTs will be equal. An example of this theorem is when w parameterizes probability distributions over a discrete set, and u maps into the simplex over z. The theorem is proven by assuming u is compact and semi-analytic.

In a neighborhood of a point, the Hessian of a function can be changed to the identity matrix using the Morse Lemma. This Taylor expansion can be written in terms of coordinates on the function, and it can be shown that the function is Lipschitz equivalent to a sum of squares. This inequality holds on a set, and when composed with a function mapping to the origin, it is equal to the function that calculates the RLC of a fiber ideal.

RLCT is a technique used to prove a theorem, which states that the local and global RLCTs are related. It involves defining a local RLCT on a neighbourhood of a zero and the global one as the minimum of the local ones. An example is given of a model with a truth and a prior, where the parameter space is compact and the probability distribution is over a finite set. The map p given w is a map from the parameter space to the probability simplex, and k of w is the KL divergence between q and p given w.

KL divergence is used to measure the difference between two probability distributions. The theorem applies when the KL divergence is regular around the point q, where q is a point in the probability simplex. The fiber ideal is generated by the difference in the probability that a model assigns to some element of the set minus the probability that the true distribution assigns to it. This makes it easier to calculate, and also provides a more intuitive understanding of the models. The ideal and the function have the same effect on the parameter w, at least in the neighborhood of the true parameter.

RLCTs are finite constraints applied to a transmission that correspond to the Hilbert basis theorem. They match the KL divergence on a local level, but can't be applied to neural networks due to the complexity of the function being computed. RLCTs have appealing properties, such as being able to identify the fiber ideal and its generators.

KL divergence is a measure of how singular a function or ideal is around points where it vanishes. The order of an ideal is a coarser measure of this. The RLCT of an ideal is less than or equal to twice the order of the ideal at a point. The last result states that the RLCT subsumes the order, and that for sums and products of ideals, the RLCT is the sum and product of the RLCTs of the original ideals.

Analytic functions on two distinct parameter spaces can be combined using projections and the sum of the ideals is given by the sum of the lambdas. The inversion of a junction theorem states that when a variety is embedded in a non-singular space, the log canonical threshold of the restricted variety is less than or equal to that of the original. Jet schemes can be used to prove this statement.

looking at regularly parametrized models which is basically a technical tool to calculate the rlct of certain functions um oh quickly do i want to attach to the orb as a speaker yeah that's right okay cool [Music] um okay so uh basically a technical tool to uh calculate some rlcts but it does have a kind of appealing conceptual content which links their program synthesis or in the statistical inference of slt with geometry by linking each sort of uh model prior truth triple to a um to an ideal in the ring of analytic functions um anyway so before we do that we're going to have to define the rlct of an ideal and then we prove a theorem that links the two so a certain ideal to each to the function in question anyway so throughout we'll be working with some subset w of rn which will be compact and semi-analytic that just means that it's cut out by just the non-negative lockers of some collection of analytic functions and aw will be the ring of real analytic functions on w um so i'll just quickly remind everyone i believe this would have come up when we're talking about the asymptotics of the free energy but what i mean by the rlct so um definition if f and phi a are non negative analytic on w um then we write rls uh rlct on w of f with the prior phi will be a pair lambda theta such that um so there are a few equivalent definitions and the reference that i prefer is probably lynn's thesis but i assume it's also in the gray book um so either we can look at this so-called partition function uh zn which is the integral over w of e to the minus uh um and uh that will have the asymptotic expansion as follows uh oops minus logarithm of that n will be lambda log n minus the minus one um and so this is the sense in which we use it in slt because the partition function is exactly the form of the integral of the posterior over some subset w and so this shows us that the rlct lambda um controls the asymptotics for large n but we often don't care about the log log n terms but we might as well keep it because it doesn't really change any of our analysis for the moment um so in practice the way we actually calculate this is more often through the so-called theta function so for a complex parameters said theta of z is also the integral like that um uh so its poles are real uh it extends to a meromorphic function on c again these preserving uh lin or at least their references there and the smallest pole is lambda with multiplicity theta um and so this is proved by a resolution of singularities which is

probably a story for another day um yes and so when we are i will approve a couple of lemmas that talk about ordering these pairs lambda and theta and basically what that ordering comes down to is ordering these asymptotic expansions so what i mean by that is that lambda theta will be um strictly less than lambda prime theta prime if lambda strictly less than lambda prime um right does that ech is this all familiar to everyone is there any questions or clarifications i can make on that before i move on cool well i'll jump over okay so this lemma is kind of what uh controls uh a lot of these estimations we're making and the way we get bounds on the uh rlct uh it's a remark in the grade book where it's stated incorrectly and not proved and then everyone just seems to point to this remark and so anyway we'll actually prove it um so if f and g are really analytic and non-negative and there exists a c greater than zero so that right so to prove this we will look at the asymptotics of the um partition functions you can also look at the resolution of singularities if that tickles your fancy um so we can if we look at this integral of e to the minus and f of w then uh just by monotonicity you can observe that this is going to be larger than the corresponding integral but with c j in it minus c n g of w by f w um which of course is just said g of c n we usually assume n is an integer but it doesn't really matter in this case ah okay so why does that help us but if we take some logarithms and then add a negative then the inequality swaps itself around so we know that minus log said f and will be less than or equal to the minus log of z g n um so we can uh use our asymptotic expansions this tells us that sorry interrupt tom yes um could you either speak up a bit or maybe walk your character back towards the orb a little bit it's actually a little hard to hear no problem does that help yeah that's better yes okay so i want to stand next to the orb is kind of how this yeah so people who are attached to the orbs with the halos on their heads are listening from the position of the orb so you're standing next to it then your volume will be okay gotcha perfect um right okay so if we're just um so uh this is just recreating the asymptotics that we had before um g log of now c n um uh and then plus o one i'm not sure if i put that in the previous one anyway it's that's what i need and so i won't go through all the naughty editing details they're in the notes if you're interested but uh you

can kind of see that the constant part of that pops out as a constant and uh likewise with here you get an extra one term and so what that tells us is that if you make n big enough so that the log and dominates all the o1 terms then lambda f log n minus oops will be less than or equal to lambda g log n minus theta g minus one block n uh for n uh and so this exactly tells us that we have the required relationship between the rlct's okay so um this all over the place is that in the right place now yes cool okay so the fortunately for uh all of us the actual order of the inequalities doesn't usually matter um because the way we use this result is through the following corollary is that um if we have c d greater than zero so that c g w um then the lcd uh and so we'll refer to this as these functions being comparable that's lin's term uh it looks a lot like lipschitz equivalents so we sometimes use that term as well um okay and so that follows fairly directly from the previous result um so in our specific case the uh when we're trying to look at the rlct of an ideal the way we do that is by taking the sum of squares of the generators um and so that requires a choice of generators but it ends up being well defined via this corollary here so say we've got two ideals f1 and j inside the ring of analytic functions um if i is a subset of j then um e r l c t w of f one squared plus is lesson three two um right and so this given that we've got an inequality we're going to be using uh the lemma from earlier and so the way that looks is um well first for each eye we can write it in terms of the gj with functions for coefficients that's just what it means for each of the fi to be in the ideal j so that says that f i equals sum j plus y naught r h i j times g j um where each of these h i j are in this ring of angular functions um so then by koshy schwartz let's get the consonants in the right order um that tells us that f i squared which is just uh is less than or equal to squares of these hijs times the squares of the gjs so that's great because then the function we're interested in uh this sum of squares of the fis is going to be less than or equal to just all of these that's not exactly what i want um oh sorry i've got this entirely wrong um these squares should be on the inside that's how that works um right and so that makes this all work out how we want uh okay so we've assumed that the parameter space w is compact so this is bounded which gives us so the these are the functions that we're

interested in the rlct of and so that's just we have the sum of the fi squared less than or equal to c times the sum the g j squared um so that proves the corollary right and so again we're not super interested in this but what that tells us is that the following definition is well defined um so if i is some ideal we're now we're just fixing a choice of generators then uh we define the rlct on w of oops i with this prior to be just that sum of squares we wrote down earlier okay and so obviously if we choose um two different sets of generators then we have both inclusions between the ideals like in the previous corollary and so the rlct's must be equal so just as a little note of caution uh this differs from the definition in lin which is basically the reference for all of this stuff uh and so as a result uh well you think i'd get that right by now uh if we have a given function which has an rlcd of its own then the rlct of the ideal it generates might be different um well it will be different by a factor of two um and so you could see that by looking at the zeta functions because this yeah anyway the square in the generator is what gives that to you um this is kind of mild because we care about the rlct's of ideals because of the following result and in that case it turns out we want the factor of two and so um that's yeah and so this works out for us um before i jump into proving that theorem this uh i'll just say a couple words about why i think looking at ideals rather than functions is kind of appealing and so the main reason for me is that it's a bit more flexible so if you have a fixed function um you have to like at each point prove that uh if you're modifying the function you're calculating with that it doesn't change the ideal so you keep proving these lipsticks equivalents whereas with an ideal uh you will have the flexibility to choose a set of generators and because you're in these the ring of analytic functions you kind of have a good stock of units and so that makes choosing these generators quite easy which yeah can make calculations of these ideals significantly simpler the other point is that these ideals this notion that we've got here of these rlcts is has some nice properties with respect to words so we've already seen with respect to inclusions but if you have um sums and products of ideals also kind of work nicely and all of these i think become clearer looking at the ideals rather than at the functions so if there's time after which probably there should be

after i've proven the next theorem then i will sketch some of those properties but otherwise there's a link in the notes too just some little thing i've wrote written on that okay so without me continuing to speak i'll pop onto the next board and state the theorem um is there any preference for should i motivate the theorem or should i prove it first and then show you the application that we care about any advice on that term or others yeah i think it's probably just worth stating the theorem i think many of the people here can uh have some idea what it might be useful for cool okay um right so uh this again comes directly pretty much from the work of lin so say we've got the function we're interested in calculating the rlct off um and we can factor it like so um so through some other where this is also compact and semi-analytic ridiculous um and so on that we will assume that uh and so we'll be interested in a specific point um w hat in w such that um f of w hat equals zero so that means that it possibly contributes to the asymptotics we're interested in um and um then also assume that with u hat equal to u of w hat um the hessian which i'm going to write like this the hessian of g at u hat is positive definite um then with will define an ideal to be generated by each of the components of um and the rlct's will be equal so right and so we call this the fiber ideal uh because yeah it's just the fibers of the map u around or no the um vanishing locus is related to the fibers um the reason we talk about regularly parameterized models uh which is the title of the talk is that if um f and g are the kale divergences of two different um models uh associated to some truth on a parameter set and g is associated to a regular model then the kl divergence around the true parameter will be positive definite um the hessian which is the fischer information matrix uh will indeed be positive definite and so this is satisfied and so uh the example which i can sketch in more detail that came up in my thesis is that uh if you uh if our map w is in fact just parameterizes probability distributions over some discrete set then um u might be the the map into the simplex over z and in that case we've yeah we'll win this uh is satisfied but i'll do that actually write that down because that didn't make any sense right so before we do that i will prove the theorem uh over on the first of the boards okay so we will assume that you had um because that doesn't really change anything and so

since it's positive definite um we can find tea linea so uh h which will define to be g composed with the inverse of t um from v e so uh so all where oh and um so all this is saying is that uh we're just changing the coordinates so that the hessian of h um is just all one it's just the identity matrix and so this is just an instance of the morse lemma i think um and then this expression that i've written down is just the taylor expansion written in its own form and so here um just because it'll be useful for us later uh these are equal to um t of the x's where x is coordinates on u um that's confusing i'm sorry but i've already used the letter u for the function um okay so now we're going to shrink our oh sorry i didn't say this in the statement of the theorem but it's actually only on a neighborhood of w hat that's the equality holds i'll say that more precisely when it comes up ah right okay so now we're going to shrink to some set b some sort of b so that h tilde of b prime is uh bounded so might as well say minus a half and a half this is just by continuity um uh so then we can write that uh if we abbreviate this sum of squares it's just b squared that a half v squared is less than or equal to h of v respect to three over two b squared for v in b prime uh so this is already looking promising we've sort of got some constants but other than that we it seems like h is lipschitz equivalent to a sum of squares um and so now we just want to extend that to all of that to the original coordinates x so we can write v squared oops as v transpose c um yes and so that equals where x are our coordinates on v times t transpose t um i'm going through this in detail because i sort of had to think about it for a surprisingly long time um and so we have uh so t transpose t is symmetric and so um the eigenvalues are real and positive i think yes um all can be chosen to be uh and so let lambda and mu be smallest and largest eigenvalues of t transposed t um and so that will tell us that um lambda x squared be less than or equal to b to mu x squared um and so this uh implies we have amp over two um okay but um and if we want to look at the function f then we just compose with u and so x is just equal to u of um w so x i equals u i w and so x squared equals ui u1 w squared plus okay and so because we assumed that uh ui hat was just the origin this is exactly the function that calculates the rlct of our fiber ideal and so this is that this inequality holds on the prime um or u prime

and so that tells us i'll write it down precisely on the next board um so therefore on um we have that rlctv okay um just as a side note i forgot about this business with shrinking the neighborhoods because once you do the whole story with resolution singularities it doesn't really matter the um uh so yeah you can define for any given zero a local rlct which is just on some neighborhood of that zero and um the global one is just the minimum over the local ones and so uh yeah kind of this uh on some neighborhood of where f vanishes is all you really need um right and so that proves our theorem that the rlcts are related uh so now i'll give an example if that if there are no questions on that so far yeah sounds good it's a very useful trick i in some sense the proof is not very hard but uh yeah very useful to think about it as a composite this way yeah yeah and so um well i'll just give um where it came up in my thesis and so that's kind of why i care about it and so i suppose um so we've talked a fair bit about these um we have some model a truth and a prior and so we can assume the parameter space is compact and also that the um so q is the probability distribution that will be just over some finite set and this is just the standard simplex oversaid um very clearly my palm doesn't touch the whiteboard in the way that the rejection code likes um so this is just sorry about that no it's clearly i'm joking it's not my it's not your fault um and so that's just a formal sum where each xy and the sum of the x i equals one and so yes that's all i just mean by that is it's just a probability distribution over some finite set and so that means the map p of um said given w will be a map from w out a parameter space into this uh probability simplex um okay and so for example in my case if i were to be talking about program synthesis on turing machines said might be the collection of tape letters and if we're examining uh the output square of our turing machine at some time then p is just the smooth relaxation of the um the transition function of our turing machine that tells us given some distribution which will be w in w given some distribution over the input squares what is the distribution of the output square and so this extends to multiple squares or to the state or in whatever way you want to think about it right so it's going to erase the theorem of that board so then k of w will be the kl divergence confusing um oh between q and uh p of blank given w um so this is

uh for any fixed w well anyway let's just write it down so this in some sense is p ah so this factors like that dkl against some q um where q is just a point in that probability simplex and so in the notation of the theorem our u-hat will just be the true distribution q that's the only place where the kl divergence in the sort of traditional sense vanishes is if the two distributions are equal and so um yeah that's out the point in question and so all of the singularness potentially is wrapped up in the function p um yeah and so it turns out that the kl divergence of where it's just the map from the probability simplex to r is indeed regular around the point q and so the theorem uh applies and the fiber ideal in this case is i equal to is generated by um p of z given w minus q i've said for set insert so this is just um these are well we will assume some analytic function of the parameter w here and um uh and there will usually be some uh perturbation away from the boundary but that kind of doesn't enter into it exactly and so this has kind of a nice conceptual uh content right so this ideal is just generated by the difference in the probability that our model assigns to some element of the set we're interested in minus the difference between that and the probability that the true distribution applies to it um so at least to me this makes a lot more sort of like intuitive sense than the well k of w equals sum of i'll skip the predivation for the moment just assume it's all on the interior uh and so it turns out that this ideal and this function k have the same rlct on w at least again in the neighborhood of a true parameter um yeah and so the quite apart from that making the calculation significantly easier so in the example of turing machines this p is just a polynomial whereas this has logs and all the rest of it in it um it also feels like you're associating a more geometric notion to um these models and so if you have differing models say by adding or removing constraints they're related by simple operations on the ideals and so on and so forth um okay that's probably a good time to pause for any questions uh if no one has any objections i'm also more than happy to talk about some of the nice properties of the rlct of an ideal but that's kind of the main content of the talk i would say it doesn't seem like there's any questions right now um yeah i think you can go on with any of those things remind me what the what are the limits of uh using this regularly regular

parameterization and i remember there was something we were trying to do with it that we actually in the end couldn't do with this technique but for the vague question but i can't remember what it was that's fine and so what we what where we lost out is that we assume that said is finite um right and so that that corresponded to applying finitely many constraints to our transmission and so that's interesting because by the hilbert basis theorem even if we apply infinite many constraints this ideal is still finally generated um and so finally many of the constraints suffice but you don't in that case you only have one inequality on the rlct's um even though the ideal is kind of still well defined you can still associate an rlct to it but it doesn't necessarily match that of the kl divergence but also is probably more deep than the finite case that's spencer's talk well um is that what you were thinking of yeah that was one of the things i also remember being so the fiber ideal is defined point by point in i don't know what the notation is in delta z or or is it w um in w is what the theorem was getting at um but uh and so yeah the rlcd's match on some neighborhood at that point but uh yeah somehow the um because there's only one because the model g in the notation of the theorem is identifiable there's only kind of one fiber ideal at least in the regularly parametrized case yeah yes at some point we were considering whether this result of lens could make it easier to do the kind of thing that spence is doing where you try and find a global polynomial that's comparable um yeah sure it's not it seems like you sort of need to know i mean exponential models are regularly parameterized so in principle you can apply these techniques to neural networks i think yeah but then the question is what um can you identify the fiber ideal and it's generators maybe that's the issue yeah i think that is also complicated because this only is a simplification insofar that p is simple whereas in the neural network that would be the um the actual function that the neural network computes right and so that's not necessarily like much better than that function but with logs right yeah i think it's part of the issue yeah okay well there's about 10 minutes and so i mean obviously as usual no obligation to stick around and listen to me wrap it on but i might just write down a couple nice properties of these uh rlct's because i don't know it's kind of appealing um so yeah i think these identities are

well worth knowing and not at all obvious from the the kl divergence formulas this year as you pointed out yeah yeah and so um we'll just work as before with uh so i and j might be ideals in the ring of analytic functions with w as usual and we'll just fix some prior throughout and so we've seen that um if phi is a subset of j then uh will be less than just j uh associated to a given prior um and so that's just well we did it by actually writing the generators of i in terms of j but you can also just add some generators uh to extend the ideal i to j and you will kind of more obviously have the inequality between um well so say i equals uh will be a subset of j uh but that's just the same as if you just add in all the f um which is the lazy way of proving the same result um also if you have an integer which is positive then and the rlct of some ideal is then you can compute any power of that ideal uh we'll just be lambda over r theta and so this just works by uh comparing the generators of the powers with just um with that function uh so the sum of squares are the generators and they you know that are pretty straightforward there are notes and so that's kind of useful because um so define for an ideal um and point yeah so uh for p and w the order at p of your ideal equals k where k is the um supremum over 10. such that um uh such that the ideal i is contained in the maximal ideal power of that number n um and so for a function this is just your usual order of vanishing um then we can calculate the rlct of this by um at some in some neighborhood of the point p of m p um and we'll just yeah all right use a constant prior in this case is going to be um oh which was a terrible letter like that r so n is the dimension over two r and then one will be our multiplicity um and so um will be less than or equal to two times the order at p of i um one for all p in the vanishing locus um so this is just uh kind of a coarser invariant the order than the rlct but this proves that they're kind of related in an appealing way so one way to think about the rlct is measuring how singular so how flat your function or ideal is around the points where it vanishes and so the order just um is a course measure of that and so this proves that the rlct kind of subsumes this easier um uh this sort of coarser invariant um so the last results i will state on the other boards uh for sums and products of ideals which is also very useful um so let's say we have um w1 and w2 are semianalytic

and we have some ideals in their respective rings of analytic functions and set w to be w1 cartesian product w2 um consider just by composing with the projections um now so any analytic function on the wi is just an analog function on w and so settings um oops uh with some prior on each one then we have the formulae so we can take the sum of the ideals um uh and you just get oh this is um we'll just give you the sum of the lambdas and for the product so um and so you can demonstrate the first one by looking at the partition functions um and the second one by looking at the z to functions um sorry i've been using partition function all the way through laplace integral is usually the term that actually gets used but it's the same thing um so just before i let you all go we've assumed that these are kind of somehow separate ideals um they're on disjoint parameter spaces uh it would be very very nice if we could prove that um for any ideals we have this same uh thing but with an inequality and so that's not clear at the moment it's true for complex uh kind of log canonical thresholds um but yeah anyway unclear in the real case and probably deeper than what we've been getting at so far uh anyway that's all from me thank you very much for coming and let me know if you have any questions thanks tom thank you yeah thanks is that related to the junction of inversion yeah i think that's what they they call the uh um yeah inversion of a junction something like that all right yep that's not the way around yeah um close enough uh let me check if that's actually the same statement um but yes that's basically what i'm getting at i saw some notes somewhere about jet schemes was that you trying to yeah that's right i'm heading in the direction of trying to understand that that theorem it's very interesting yeah so yeah so the the inversion of a junction is the statement that if you have um so in the case of varieties if you're doing this if you have y embedding in x k non singular closed captioning not available though this is all in the complex case so we call it um and then the inversion of a junction is that l dt at p of is less than or equal to the log canonical threshold of the original pny um yeah so things get more singular if you restrict them to a sub-variety it's the thrust which lets you prove the statement i gave before about sums right yeah yeah just at a higher level so the connection to jet schemes is uh there are lots of things you can do to a singular variety or singular scheme to