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Thank you. Yeah, so I'd like to thank the organizers. We enjoyed the conference and really great. So I guess what I want to tell you about today, I guess we saw in the lectures that you can associate these different invariance to rings such as the F signature and a text. You know, when your ring is strongly rational or strongly regular. And so there's this another invariant that you can associate to a ring, which is called the F peer threshold, and at least on your hyper surface or complete intersection. It's telling you when your characterizes the ring being F peer. And I'm just going to find that for you. So this invariant was originally introduced by Professor Takagi and Watson Abbey. And there's an equivalent formulations called F threshold, which is later introduced by Professor Mercher Takagi and Watson Abbey. So I'm just going to call that for you. This is definition and theory if you want. So let I be an ideal contained in the homogeneous maximal ideal of a polynomial ring. And so you can do this more generally, if you'd like, but I'm just going to look at this case right here. And so, if for each team, natural number wanted to find this invariant new T of I, and that's going to be the largest or, which is natural number such that I to the or is not in the Frobenius power of the maximum ideal to the teeth, teeth Frobenius power. Let me recall you what that is. It's not just you take the generators and you raise them all to the TV teeth power. Okay. And so then the threshold of I is just equal to this limit. And it goes to infinity of new new T of I over P the team. And so the theorems that this exists. And so what's interesting about this is it has a relationships to invariance in characteristics here. Oh, thank you. It has a relationship to so called log canonical threshold. And so I'm just going to say that briefly. Let me put it right here. So, if for example, you took a I should say here Kate is characteristic P. So if you define your ideal of the integers, you can base change to different prime fields or the complex numbers. So, if you took your ideal and finding over the complex numbers after base changing, then it's just the same as the limit as P goes to infinity of that threshold of IP. So this is where you took your ideal to find over the integers, and you base change to Z my P, or did reduction my P. And so what this is saying is, you can get a handle on these get handle a lot canonical threshold. And so that's, now I'm going to tell you, I guess changed gears a little bit and this full screen, change gears and tell you about linear program. And then I'll tell you about how this linear program relates to a thresholds. So, linear program, you basically have some linear set of constraints, and you have some linear function that you either want to say maximize over the over the solutions, that satisfy the constraints. So here's an example if you take these constraints, you get some polytope in our two or. And then if you want to maximize it, you can let your x one x two. I mean you could say maximize or x one x two are integers or rationales and you get different. You can get different answers depending on if you let your x one x to be integers or rationales. So for here in this one, if you require the x one x to our rationales, then you actually get zero. So if you say the x one x, sorry, x one x to you solve it over the integers you maximize you require the x one x to our have to be integers, then zero is the maximum of this, but if you say, let x one x to can take rational values, then the sum of these is the maximum. It's any linear program that you can put in this form which I'll just say is primal. There's a so called dual linear program and the dual you basically the idea is roughly you're in changing the constraints with the. I guess the variables showing up. And so now I just want to briefly sketch for you. I'm going to tell you something how this goes in my monomial ideals and motivate motivate motivate the story for your monomial ideals, such that I can then tell you what happens in binomial ideals. And that's Professor Mondo was telling us on last lecture, we can attend a graph, say it's finance simple we've labeled the vertices one through and we're going to associate a monomial ideal to it's called edge ideal. We just do that by whenever there's an edge, we just look at this product x one x to so on. How much you devise a linear program to compute that threshold monomial ideal. So let's just take that. So the idea is here we have a six generators. So let's say we think of a plus B plus C plus D plus E plus F. We're thinking of these as some integers, and we want to basically represent like what's a generic monomial after we do some expansion where these are allowed to vary. And so, if you do that, if you write out what the product is, you've x one x two to the a times. Let me call this product capital X two x three to the D. So on, and in times this last one x five x six to the f. And so this condition over here is telling you that when you expand this out the degree of x one appearing an F as a most P the team. So you have that the degree of respect x one of F, which is just a is less than or equal to P and T. And if you're, if you're supposing that this is not in for being his power. And then so when you look at the other degrees, degree x two. And then a plus B plus C, that should be less than less than. And so on. And so this would give you. You could say, maximize a plus B plus B, or C, whatever, maximize.  These less than that thing. Sorry, let's see. And so one of the other ones. And so this tells you that this new team of I have this particular ideal is net is less is bounded above by the maximum this program.  So, if you modify your program, you might buy by P it's you can show that you get this as an upper bound for the threshold. And now this, you have to replace by you. And so, actually, you get a quality and you can show that. And so, more generally, if you take me know me, sorry.  Then to if you have in determinants corresponding to the number of edges, maximize some of the in determinants is this part, and then you subject this relation right here, you could rep that just this part right here. So that linear program actually when you solve it over the rationals is computing the threshold, and but even even more is true which is kind of amazing, which gives a connection. I guess which is what wants to generalize and I put that right here. So, and let me call this. So, the connection is linear program on one side. Over here you have some comment towards you understand, and then some one understand the connection between algebra. And so you can also solve this over the rationals, or as I mentioned at beginning, you can take some dual of this one. And the algebra connection is here you get manomial grade. Here, you get maximal matching. Here, you get either F threshold, log conical threshold, they're equal for minimal ideals. Here, what's called fractional maximal matching. And then the dual, you actually just get the height of the ideal. And it's a size of a minimal vertex cover. And then it turns out that this right here, and it's at least a partial generalization to binomial ideals, and that's what I want to tell you about next. So binomial ideal, same setup you just start with the graph, a simple label of vertices, but this time, each time there's an edge ij, you associate the binomial x i y j minus y j x i. And then you take that average of the graph, and you take that take that ideal, and you want to say like, how's algebra and common torques relate. And so, I'm going to skip this for now, basically, there's some nice classic graph called block graph, but somewhat resembling. It's important to watch. What I want to tell you about is that there's this concept called maximal packing inside a graph. And it's maybe what you suspect it is it's, how do you put the largest number of paths inside my graph. So they're vertex disjoint. And so, hopefully it's clear from the picture packs just what you think it is. And it turns out. Okay, so maybe you're kind of motivate you want to be like, how do I compute the f threshold, or a law conical threshold binomial ideal. And so, I'm going to say there's some linear program that is doing that and where's this linear program coming from. So, for each edge, we associate a variable. And so, what we did here is similar to what we did here. It's capturing that there's like degree relations. And the reason there's a two is that in your binomial ideal, you have both x v and y v showing up.  And then, where's this condition come from. This condition comes from other summers all of Lance Miller on rock sing and not tell Barbara, which says that if you take the two by two minors of a two by n matrix of indeterminance. If you take the f threshold of that. It's actually equal to n minus one independent of characteristic. And so that means you need to enforce this condition on every sub every induced sub graph of your graph. If you want to have a chance of bounding that threshold. And so you can show that the maximum value subject to these conditions gives you a bound for the threshold. And actually, it captures more common toro data. So if you solve this over the integers, you recover path packing this right here says, at each vertex, you have at most each vertex. There's a most like two paths that meet. And in this condition right here is saying that there can be no cycles rules out cycles. And so then, you can basically do a similar thing as you did over there, you can write down what's the house linear program late, the community value drawing the comment works. And the answer is, so if you solve that linear program of integers. So if you do binomial grade, combinatorial it's maximal path packing. If you solve the rationals. It's a log canonical threshold, but conjecture at least I can show that for some special cases, which you might call rational path packing. So if you look at the dual, actually you actually recover the height of the binomial ideal and combinatorial it's corresponds to like covering your graph by some certain sub graphs, where you've assigned some weight and you're trying to minimize this way. Just want to check how much, how am I doing on time. Oh, we're almost done. Finish. Okay, I'll just say so. Thank you. Conjecture is this path packing number is equal when you solve the rational is log canonical threshold and can actually show that conjecture. This is a block graph and actually show even a bit more that the F threshold agrees with the log canonical threshold, which is the gold pack and it's half rational. So, yeah, thank you. Thank you. Thank you.


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