I want to thank, obviously, the organizers for the chance to lecture here, although I wish the original plan, when this was set up a couple of years ago, it was going to come with my family, and it was going to be this two weeks, and France, and all that. And obviously, this is slightly less than what I was hoping for. So let me begin by giving some motivation about what I want to talk about, which has to do. So for motivation, let's just start off with the case of a smooth curve of genus G. And then I can look at its Hilbert scheme of points, of endpoints, which, since C is smooth, is just the same as taking the symmetric power of the curve. And so there's a very nice formula going back to, well, probably in this version it's older, but on the level of, say, homology, goes back to an old paper of McDonald, which says that if I take the Euler characteristics of these Hilbert schemes, and I sum over n, well, this just has a very nice formula where I just take 1 minus q to the 2g minus 2. I can then kind of make more complicated in kind of a dumb way by writing it as 1 minus q to the 2g over 1 minus q square. And now the numerator of the right-hand side itself has kind of a geometric meaning. If I take the Jacobian of my curve and I take its homology, well, we know what that is. It's just the exterior algebra of h1 of my curve. And so if I take the Poincare polynomial here, the Jacobian, where I'll just introduce a minus sign, that's exactly the numerator of this expression. And so now what I have is I have some kind of slightly silly identity where the left-hand side I'm kind of summing up overall n, these Euler characteristics of the Hilbert schemes. On the right-hand side, I just have a single space, just the Jacobian, but now I'm doing something a little bit more refined. I'm taking the actual homology instead of just the Euler characteristics. And the q kind of has a different meaning depending on which side I'm on. On the left-hand side, the q is just indexing which module I space I'm working with. On the right-hand side, the q is this kind of homological variable. It's helping me keep track of my homological degree of the Jacobian. And so in some sense, one of the things I would want to try to explain in these lectures is kind of a non-sox due to many people. So I'll do the attributions later on when I actually get to it. But the iteration that I'll be talking about has kind of joined with the Toto, you can know with Toto, which basically proposes a way of extending this to kind of much more singular curves. So for instance, pervs in the kind of setting of Kalabi out threefolds. And one of the things that I kind of want to get to hopefully is that, OK, so generally it's a conjecture. And you can believe it or not. But in the cases where we can prove it, already kind of gives you examples where something like this holds for extremely singular curves. And the technique of proof is kind of nice because in some sense what the technique of proof really does is it reduces it to the case of the smooth case, where it originally just McDonald's formula. So there'll be some kind of chain of logic where kind of the final step will just be applying the original identity. So let me give kind of, I'll start talking about this properly in maybe the third lecture. So let me just say now kind of what I'm hoping to kind of cover in these five lectures, which is in the kind of first two lectures. I want to give some kind of overview about Donaldson Thomas theory, some version of which you saw in Richard's lectures last week. But I'm going to focus really again on the setting of Kalabi out threefolds. And in particular, the kind of perspective that you get when you get to in the Kalabi out three setting is that instead of doing things intersection theoretically there's kind of an alternate approach where you work in this kind of constructible world. So constructible functions and then that'll be today and then eventually tomorrow constructible sheath. And then kind of in the remaining lectures, I want to then kind of talk about this kind of picture that I just sketched out, which is this notion of an approach to thinking about what are called Gokomar, Gokomar Vasa invariance, which you should just think of the analog of this as the analog of the right-hand side of this McDonald's formula, some analog of what to put in the numerator in general, where we'll be using the kind of technology developed in the first couple of lectures to pursue that. And then in kind of the last lecture, I want to kind of talk about some related conjectures to this story, in particular a conjecture of Tota in the last couple of years, which is related. These kinds of topics are related to kind of defining right-hand side of this McDonald's formula. Are there any questions before I get started properly? So again, so if there are things that show up in the chat, I'm not really probably going to be able to see them so easily, I think, so hopefully Andre can read them aloud in his dulcet tones. So today, I want to start talking really in some generality about numerical Donaldson-Thomas theory. And so again, the setting for today will be focusing on kind of the geometric setting, and specifically in the case where we start off with some kind of Kalabiyao threefold. It does not necessarily have to be a projective. So just some, you know, smooth algebraic threefold with nowhere vanishing algebraic threefold. And the kind of basic moduli space that you associate here is going to be some moduli space, m of x, which is going to be some moduli of stable coherent sheaves, moduli space of stable coherent sheaves, or more generally, complexes of sheaves, on x with a fixed discrete invariance during classes indexed by some vector v in the cosmology. And so the kind of classic version of this story is where you just take, maybe you fix a polarization, and then the moduli space you look at is just, you know, let's say, you know, geesecker stable sheaves on x to avoid kind of stacky issues or issues with the obstructions that you often will kind of trivial, you know, fix an isomorphism of the determinant of ease. So here I'm giving some kind of examples. For curve counting purposes, it's usually better to work with some kind of variation on these spaces. So, you know, the old version that we kind of, the subject kind of started off with was to work with the Hilbert scheme of curves, or rather, Hilbert scheme of one-dimensional sub-schemes on x. So here maybe I'm gonna fix an element of H2, and then I also fix an Euler characteristic. And so this would parameterize, you know, sub-schemes, one-dimensional sub-schemes, where the support, the kind of one-dimensional piece is last beta, and then the Euler characteristic of the structure sheaves is n. And then one example that, you know, we'll see if I have time to talk about maybe in some, is the special case when beta is zero, and then you're just considering the Hilbert scheme of points on x. Now, in these examples, the way you kind of put it into this framework of modular space of stable sheaves is that instead of thing about the sub-scheme and then the kind of surjection from the structure sheaf, you just remember the ideal sheaf. So you think of this, you look at the ideal sheaf of your sub-scheme, which is a rank one sheaf on x with the trivialization of the determinants, and this turns out to be equivalent to looking at the Hilbert scheme. So the way you put it in this framework of modular space of sheaves is by forgetting about your sub-scheme and instead remembering just the ideal sheaf that cuts it out. The version that's cleaner, and maybe actually Rahul already has spoken about this in his lectures, or if not, he will soon, I'm sure, is a variation. Again, for curve-counting purposes of the Hilbert scheme construction, where you work with what are called stable pairs. And so this theory was developed by Pondre Ponde and Thomas. And so here the modular space, again, you kind of fix a curve class and you fix an integer. And then the data here in this modular space is, there's two pieces of data here. First is a sheaf e, which is one-dimensional support and is pure. Pure meaning has no kind of zero-dimensional sub-sheaves. And then the second piece of data is just a section of the sheaf. And then the stability condition is just the statement that the co-kernel of this section is zero-dimensional. So what does an object of this space look like?  So again, the support of the sheaf, just the cycle theoretic support is gonna be in class beta. And then the Euler curve is Euler x, is n. So the way to think about what this space looks like to first approximation is, let's say here is x. And then imagine the support of E is, let's say, smooth curves inside of x. And so you could ask, what are all the stable pairs with fixed support like this? Well, the simplest is I just take the surjection, O x to O c. So that's an example of a stable pair. But what I could also do is I could also take, if I give you some line bundle on this curve, I could take E to be its push forward, which is now a coherent sheaf on x with one-dimensional pure support. And then if I just any non-zero section of L will then produce a stable pair on x. So given non-zero section here, it defines for me a section of E and the co-cernel of the corresponding section of E on E will exactly just be the zero locus of the section on the curve. So the way you can think about this is that you have some line bundle on this curve and now I have some section. And so the same curve will contribute many different stable pairs. Just, you take any line bundle with a section or equivalently any collection of points on my curve, there will be a corresponding stable pair on x. So if I, the contribution of C to all of these stable pairs fit ends up looking a lot like just, taking the different symmetric powers of my curve. And so in particular, this will be kind of, the left-hand side of this kind of McDonald inequality in general. And so again, how do we wanna put this in this kind of setting of DT theory? So again, the setting of DT theory that I wrote it is that we're gonna wanna consider a modular space of sheaves or complexes of sheaves on x. And so here I have this kind of two-term complex. I just think of the corresponding object in the drive category. So I take this two-term complex and I think of this as an element in the drive category of coherent sheaves on x. And this is how I'm gonna think about this modular space. It's a modular space of a certain kinds of two-term complexes. If you, again, so one thing that's kind of instructive to do is, if x is projected then this space of pairs is also projected. And so kind of a fun thing to try to do is just to understand what certain limits look like in this space. So for instance, maybe exercise. This is a local question. So it doesn't really matter what the ambient space, what the ambient three-fold is, but imagine I have two lines that are kind of colliding, two skew lines that are kind of about to collide in some limit. And then we know from Hartron or something that the limit in the kind of Hilbert scheme of one-dimensional subschines is you get two intersecting now coplanar lines and a little fat point there. But this kind of limit isn't allowed in the stable pair space because the support of the sheaf because the sheaf isn't pure. And so instead what you can try to work out is what the limit of this kind of thing in the stable pairs moduli spaces where the support of the sheaf is still going to be these two lines but now there's going to be what used to be a kind of a fat point now gets replaced with some kind of non-zero co-kernel. So these are kind of, the examples of the kinds of moduli spaces, geometric moduli spaces that one can look at. But actually everything I'll be talking about today it doesn't really have to be a geometric setting. So they're kind of the kind of main example of non-geometric examples to look at or come from looking at representations of quivers with potential. So everything I'll be saying today and tomorrow kind of makes sense in that setting. In particular, there's a lot of the material in Marcus Reineke's lectures I think will be relevant. So these are the examples I want to look at. And so as always, we're interested in some kind of virtual structure in these moduli spaces. And so in the context of Richard's talks last week the kind of initial piece of data that you want to understand is something about the deformation theory. So you have a deformation and obstruction theory for understanding these moduli problems which because I'm just working with moduli spaces of sheaves or maybe complexes of sheaves they're given by X groups. So the deformation space, the tangent space is given by the self X one of whatever your sheave for complex is. If I fix the determinant then we usually do some kind of traceless thing. And then the obstruction space is then given by next two. And so again, my understanding is Richard kind of talked in more detail about how these show up for moduli of sheaves. And so already the first nice thing that happens in the Calabi out three case. So this is of course, so far everything is completely general which is you get to apply serduality. So if I take the dual of the obstruction space I get X one with this twist by the canonical bundle of X but because I'm in the Calabi out three setting this is just trivial. So I get exactly the deformation space. And so in particular it means that the virtual dimension of my moduli space was just the difference in dimensions is zero. And so if X is proper or at least if my moduli space is proper that's really all I need. I have this virtual class which is a zero cycle and under the properness hypothesis I can take its degree which will give me a number. I'll call this kind of the virtual number associated to my moduli space and it's just some integer. And the procedure for doing this again this is something that Richard sketched out is the way you produce this virtual class from all this data is basically by using some techniques from intersection three. So that's kind of the world where these constructions live most naturally. Okay, so what do I want to explain first then is again in the Calabi out setting what another way of thinking about what these numbers are. So this is this notion of what we now call the Baron functions. The setting here is that let's say I have some, I have some moduli space M and I've equipped it with this kind of perfect obstruction theory meaning I have some two term complex that calculates the deformations and the obstructions to my moduli problem. We say that E is symmetric I have a quasi isomorphism between E and it's shifted dual. So you should think of E as being kind of supported in degrees negative one and zero. And then E dual is going to be shifted supporting degrees zero and one and then I shift it back. So it's again, supporting degrees negative one and zero. Well, I have some isomorphism like this with the symmetry condition.  The easiest way to get a kind of data like this is that for instance, if I have what they have F as a vector bundle with a symmetric bilinear form alpha then you can produce an example of a complex with this kind of symmetry just by taking a map from F dual to F and then you have using this bilinear form you can produce a map like this which exactly has this kind of symmetry. And so the baby example of a moduli space with a two term perfect obstruction theory with this kind of symmetry is where your obstruction theory is kind of dumb. So let's say M is smooth and then your obstruction theory basically consists of the map from the tangent bundle of M the cotangent bundle of M, which is just the zero map. So this is like, I have an obstruction space but it those obstructions are all unrealized because the space itself in fact happens to be smooth. And in this case, if you calculate what the virtual class is so it should be zero dimension. It's a zero dimensional virtual class is the definition of obstruction in dimension. It's just going to be, it'll just given me that the Euler class of the cotangent bundle, which is my obstruction bundle. And if I take its degree, I just get up to assign the topological Euler characteristic of M. And so the first observation everything here I should say is I'll be saying is due to hi Darren except for the name. He didn't of course name it after himself but at some point it caught on. So in the Calabi out three settings all the examples that I said before again, just the same seriality calculation I did before tells you something a little stronger. It tells you that the obstruction theories. And so let me give the kind of key local example of one of these symmetric obstruction theories which we'll kind of use again next time which is an imagine I have some ambient smooth space V which is just a fine space and I have some function on it. And then the kind of space that I'm looking at my actual modular space is just the zero locus of all the partial derivative just the critical locus of this function F. So because it's some, it's a space cut out by a bunch of equations and in particular has a nice two term of obstruction theory and then you can just write down what the obstruction theory is in this case and it's basically determined by taking the Hessian matrix or F you take this kind of symmetric matrix given by taking the partial, the second partial derivatives. And though this defines exactly that kind of a symmetric obstruction theory and this will be kind of the main example for us. So this kind of baby case is the case where the function was zero and it's kind of dumb but in general it's more interesting. So this is the framework for us. We have a modular space we have this kind of two term obstruction theory and it has this symmetry property. So definition, a function, if I give you any kind of complex scheme, a function from the complex points to the integers is constructible if the set of points where the function has some value. So new inverse of A, this is a constructible set. I have some variety M, there's gonna be some open set where it has some value zero and then maybe there's some locally closed set where it has value one and then maybe some other stratum where it has value negative one. And then given one of these constructible functions I can kind of use some version of integrating it. It's a discrete version of integrating it where what I'm gonna do is I'm gonna sum, I'm gonna look at all the kind of strata where the function has some value and I'm just going to add up those strata weighted by the, add up the Euler characteristics of the strata, weighted by the value of the function there. For instance, if I just had the constant function one I would just be getting the top logical Euler characteristic of M but of course in general I'll get something else. And then in particular I can look at, for instance just a billion group of constructible functions Z value of constructible functions in M. And one way of thinking about this is if I just look at characteristic functions this is a basis indexed by irreducible subrides. Do you only assume for actually many non-empty fibers? Yeah, yes, that's right. I'm sorry, M is gonna be just a finite type thing. So that's right, only finally many. And so the key theorem that kind of kicks off, for me at least this is the whole direction of the subject is that if I give you M with a symmetric obstruction theory in particular any moduli space of, she's or whatever on a clobby at three fold, there exists associated to M and E there exists a constructible function on M such that if M is proper in a virtual number of my moduli space, meaning in the sense of taking the degree of a virtual class is the same as what you get by integrating this constructible function. And so what it means is that, this intersection theoretic quantity, it means that you can kind of study it using ideas from kind of constructible geometry or micro local geometry. So let me just say a couple of remarks about this. I'm gonna say something about why this is true in a second, but let me just kind of say what's kind of so interesting about this. It allows you to do a couple of things that you couldn't really make sense of intersection theoretically. So for instance, this virtual cycle, it's really a cycle class. And so if I give you some subset of M, it doesn't really make sense to talk about, what is the contribution of Z to the virtual class? Cause you can't really localize. There's not a clean way of localizing this kind of zero cycle class to all the different, along some stratification of M. On the other hand, if I give you a constructible function, it's very easy to do it because I can just restrict my constructible function to Z and I can integrate it there. The right-hand side makes sense even if M is not proper. So usually the left-hand side unless you're in some kind of equilibrium setting, like in Richard's lecture, if I have a non-compact modular space, it doesn't make sense to take the degree of a zero cycle class on it, but you can always just integrate this constructible function. On the other hand, the left-hand side, of course, because it's defined intersection theoretically is deformation at least in the proper situation, I mean, which is the only time it makes sense. And something like the right-hand side, if I just take for instance, the actual topological Euler characteristic, that is as M varies in flat family, that's certainly not going to be a deformation invariant. And so there's something going on with this kind of specific choice of constructible function that's kind of correcting for the failure of the Euler characteristic deformation. This is really not at all obvious from how it's defined. And so this ends up being an extremely useful theorem. Let me just say, maybe I won't break this down. One way that gets used a lot is when you study how these invariants change under change of stability and wall crossing and so on, which is that when you kind of cross some kind of a wall and your stability condition changes, your moduli space usually changes maybe by some kind of flip or flop or something like that. And so understanding how the zero cycle trans, the cycle class transforms might be kind of delicate, but this kind of weighted Euler characteristic, if there's some open part where the two moduli spaces are just the same, then you can just throw it out because the contribution to this kind of integral is going to be the same. And you can just focus on the kind of the actual strata where the stability is changing. So this ends up being kind of an extremely powerful tool for those kinds of analyses. So let me sketch the proof of this result. And it goes into how this kind of virtual class is defined. Which again, I believe Richard covered in his first couple of lectures. Well, the idea that if you have M embedded in some kind of smooth space, let's say, then the way you get this virtual cycle is that you have M sitting inside V and then there's some kind of vector bundle over V.  This is kind of a conical cycle, not a cycle class, an honest to God cycle inside of this vector bundle. And then when I intersect it with the zero section, I get exactly this virtual class. So this is true just in general. But what Kai showed in his paper is that if you now add the condition that the obstruction theory is symmetric, again, like in the Kalabiyaev situation, you can actually refine this picture so that this vector bundle F is actually the total space of the cotangent bundle of V. And this conical cycle inside of the cotangent bundle is not just a cone, it's a Lagrangian cone. This cotangent bundle has a natural symplectic form. And when I say this cone is Lagrangian, I mean that every, the smooth locus of every irritable component of this cycle is Lagrangian in the usual sense. The symplectic form restricts to zero and it has middle dimension. So why is that so special? Well, so there's a natural, there's a isomorphism between on the one hand instructible functions on V and this free, this free building group of conical Lagrangian cycles, which is known as the characteristic cycle map. It takes a constructible function here and sends it to what's called the construct of the characteristic cycle of this function. And this is defined in some sense via some kind of a Morse theory type construction. The fact that there is an isomorphism like this shouldn't be kind of super surprising. Each of these spaces has a basis that's indexed by these irreducible sub varieties. So I already talked about how irreducible sub varieties just by taking the characteristic function defines a basis here. Similarly, if I give you an irreducible sub variety I can take the smooth part and I could take its conormal bundle which is the Lagrangian side of here and I can take its closure. And so that gives me a natural basis here but that identification is not what's used to find this isomorphism. It's a little bit more subtle than that. But what's great about this construction is that on each side, there's an evaluation map to the integers. On the left-hand side, when I take the constructible function I can just integrate it. And on the right-hand side, if I give you a conical cycle I can take the degree of its intersection with the zero section. And the way this characteristic cycle construction goes is that this diagram can use. This is what's called the index formula. I don't know, due to many people. So maybe that's in. So this is just a very general statement about constructible functions on V and Lagrangian cycles of the cotangent bundle that you can kind of set up an isomorphism which makes this diagram commute. I'm not gonna, I won't actually, if I had more time I would actually, I had a, I'm sketching a proof of this kind of index formula. There are a lot of proofs. The one I like the most is in a paper of Schmitt and Volonen. Or basically they just reduced to the case of understanding kind of past of the real analytic world. And then you reduced to the case of understanding like a tiny ball. And so this is great. You see the right-hand side is exactly what we want to define the virtual class, to the degree of the virtual number. The left-hand side is the kind of thing that Barron's theorem is about. So to produce this kind of Barron function in the statement of this theorem, I'm gonna take the conical Lagrangian cycle that's associated my obstruction theory and just move it over to the left. So this is now what we now call the Barron function is just it's whatever constructible function maps to this obstruction cone under this characteristic cycle map. And so that's exactly how you kind of go from the intersection theory to roll over to this kind of constructible world. So then this begs the question, what do we know about this Barron function? How do we, this is kind of a somewhat abstract statement. How do we kind of compute it in any examples? And in general, it's quite hard. So I would say if I give you a kind of a random moduli problem and I give you some random point in the moduli space, it's not so easy to kind of compute this thing. But some cases, we have some kind of statement. So for instance, the easiest case is when M is smooth and it's smooth, you can again kind of put the stupid, you know, this is the baby example where I just have the zero function, the zero section defines for me a symmetric construction theory and then the Barron function in this case is just constant, negative one to the dimension. And then Barron's theorem says exactly what I wrote before. If I integrate the Barron function, I'm getting negative one to the dimension times the topological Euler characteristic of M, which is the degree of this kind of virtual cycle that we associated it before. What about this local example I did up here? So here I kind of wrote down this key local example where I take the critical locus of a function. So M is the critical locus of a function S and N variables. And so in this case, if I give you some point on M, the value of the Barron function, again, it's something pretty nice. This is related to what's called the Milner number, maybe the reduced Milner number of my function at P. So it's some kind of notion and singularity theory, which let me just state what it is. So given a function and some point in the critical locus, I can take the Milner fiber, which is just I take a tiny, I take a ball of a closed ball of some tiny radius around my point P, and then I intersect it with the fiber. So let's assume that F of P is zero. Make my life easier. And then I just intersect it with a nearby fiber of my function. So here epsilon is much less than delta, much less than one. And so then what the Barron function is in this case is I'm just taking, again, up to a sign, taking one minus the Euler characteristic of this Milner fiber. So this isn't super explicit, but it's again, something familiar from singularity theory. So there's a question in the chat. Do you assume that the singularity is isolated and what if it's not? Oh yeah, I am not assuming the singularity is isolated. You can still, this definition makes sense in general and it makes it a little harder to think about, but this definition is still not. So for the whole thing, you wrote X is in V, do you mean X is in AM? Oh, sorry, yeah, V was AM, that's right, yeah. So let me give a more complicated example where you get to kind of see this. So let's say I wanna, so I'm gonna take the following kind of Kalabi out three-fold, I'm gonna take a three-three hyper surface inside of P2 cross P2, which if I kind of pre, this is a Kalabi out three-fold, complete intersection, I project onto P2 and I get an elliptically fibered Kalabi out three-fold. I'm going to pick it, pick the defining equations such that, this is some elliptic. So that one of the singular fibers of this vibration is sitting inside of P2, just given by X squared times Y. So one of the fibers is this reducible, non-reduced cubic inside of P2. So this fiber looks like maybe two C1 plus C2. And so I'm gonna look at the following, modular space of sheaths, I'm gonna look at sheaths, which are one-dimensional sheaths where the support is C1. So that kind of, I take the non-reduced component and I just take the underlying reduced curve, which is just a P1. And I basically, I'm just gonna set my discrete invariance to be whatever the train character of the structure sheath of this curve is. So the support of the sheath is C1 and the Euler characteristic is one. And so I can look at the corresponding modular space of sheaths on X and set theoretically, it's just a point, this is the only object in it. But it turns out, and this is the calculation, this is from Richard 30 years ago or something, is that this modular space scheme theoretically is non-reduced.  So this is the same as looking at the critical locus of the function U cubed plus V cubed, because these up to three, these are the partial derivatives. And so, okay, so then, so this is a pretty explicit function, you can work out what this, what this Milner number calculation gives you. So if you, well, it's just a point. So the value of the baron function at this unique point ends up being negative one squared, one minus negative three. So this negative three is exactly this Euler characteristic of the Milner fiber in this case, which is value four. On the other hand, this thing is zero dimensional. So the virtual dimension equals the axle dimension. So the virtual class in this case, if I just calculated it, this is just the length of the zero dimensional scheme, which is four. All right, so this is the main theorem. So what I'd like to kind of do in the, I guess I'll start this now and then I'll continue this tomorrow, is I wanna kind of sketch how this, give some indication about how this theorem gets used. This ends up being a really useful theorem for understanding, for calculating these numbers. And so, okay, so what I'll maybe do is I'll just do one example now, and then I'll just, I'll write it now and I'll kind of finish it tomorrow. So let me state this theorem. And so this is gonna be a theorem about these, you know, the stable pair invariance. This was this space of stable pairs. And then we can define a generating function where I fix beta and then define what I'll call the PT series. We get virtual number of these spaces, summed over two. Which again is kind of, should be reminiscent of the kind of generating function that I started this talk with, where I took the summation of the Euler characteristics of all the Hildred schemes for a six curve. So this is gonna be the kind of general version for an arbitrary Calabi-I-3 function. And so the theorem is that, so again, the X here was my Calabi-I-3 function, is that this generating function is the Laurent expansion, Q of the rational function, symmetric with respect to Q goes to Q inverse. In other words, it's basically built out of things that look like, you know, Q to the one minus R, one plus Q to the two R minus two. You can express this generating function in terms of the specific rational function. And so, okay, so what I wanna do is I wanted to kind of sketch the proof of this in the special case when data is invisible to prove it's due to Ponder, Ponder and Thomas. And what's kind of nice about this, so I won't do it now, I'll do that tomorrow. But let me just say why this is a nice result, which is that, you know, this rationality is something that we expect for any threefold. But right now we can only prove it, because this is expected.  But in terms of a really general, this is a really general statement. If we, the only case where we can prove it, really in some kind of generality without knowing something really specific about the geometry of X, is in this Kalabiow setting. This is not only known, but for general, in general, we don't know. What's special about the Kalabiow case is precisely we get to use this kind of constructual approach to prove this kind of theorem. So I'll maybe say, like, you know, a few lines about this argument tomorrow. Let me, thank you. In the examples you wrote down in the Baron function, is it worked out by going through this sketch you outlined, or are the functions found differently? You mean the examples of what the Baron function is? Yeah. Yeah, so right, so in the examples I wrote down, you can calculate, this number number is something that you can calculate. We have techniques for calculating it. And so that's basically how you can do it. So like, for instance, it turns out that finding the Milner fiber, the Milner number for like, you can, that is something you can more or less do by hand. And then you just work it out and you get this negative three popping out. And in general, there is kind of a procedure for calculating it. If you have like a function and you have like a lot of time, then there is like an algorithm for calculating what the Milner number is. But for, you know, usually the geometries we're interested in get larger and larger dimensional. And so getting your hands on it isn't really feasible in fact. If you wanna do like, so an example of, here's an example of something that I, you know, that we like is to, you know, if you do like the Hilbert scheme of points on C3, the function that you would wanna find them, you know, the Milner numbers for is in some sense pretty explicit, but involved, you know, three end-by-end matrices. And so if you wanted to actually kind of, you know, do that for any given point in the Hilbert scheme, this is actually quite difficult. More questions? Yes, we have one. So if I take an opinion scheme or a fed point, is it true that the bearing function is bounded above by the length of this fed point? Sorry, say that again? If I take a fed point, so a spec of an arachnian ring. So I have only one close point like in the example by Richard. So is it true that the length of this fed point is an upper bound for the bearing function? Yeah, I mean, again, up to a, I mean, so I think, so I mean, if the, you know, if it's coming from one of these, maybe it's just equal to it even in general. I, it's definitely equal to it. If you know it's coming from a critical locus, that's just what the theorem says. But if it's not coming from a critical locus, there's still a definition of the bearing function and then, you know, maybe it's still equal to it in that case. But in the cases that show up naturally, it'll always just equal that length. Right, certainly it can be smaller. I was wondering if it can be bigger as well, probably not. No, I think that's right. Thank you. Any other questions? No, the last thing. Wish again. Thank you.


Look folks, I'll be perfectly honest with you. I have been a little bit more cynical than usual lately Particularly because of this year and look, I know what you're thinking. I feel the same way, but I mean it just it feels like you know humanity Isn't up to the task of dealing with some of the biggest issues we face like climate change And when you consider the fact that we see every single day It seems like a new video of a Karen throwing the fit because he or she won't wear a mask or you see People still following Donald Trump even after he's bungled this pandemic and you see people on our side You know in the Democratic Party primary voting against their own self-interest for an imbecile like Joe Biden And like you can't help but just feel like nothing is ever going to change or get better Having said that though I do want to share with you a Different perspective for this video and then we'll get back to being doomer because there's a couple of polls that are kind of Helping to make me see that you know, maybe we're not as bad off as As I was thinking at least from the standpoint of you know American citizens right our public officials are trash I will maintain that they are garbage our institutions are failing us our politicians aren't doing a good job But I will say, you know, Americans know What the right policies are for the most part? So this poll from Hill and Harris X found that 82% of Americans actually support a National mask mandate and only 18% Opposite so this is actually really encouraging to hear because I I was under the impression that the numbers wouldn't be that high like if I had to guess I'd say a slim majority of Americans support a National mask mandate because if you look at like the Facebook comments underneath a particular article or YouTube comments Which you should never do Like it seems like Much more people are anti-mask than they are in reality But really what we're seeing is a vocal minority, right? It's presenting this type of skewed image of what people actually believe I mean just the other day I covered a video of Judge Jeanine Pirro being anti-mask after she was previously pro-mask So, you know, we see all these viral videos and we see Republicans respond You know in these groups saying I'm not gonna wear a mask and this is tyranny but overall I think that people are Persuading everyone else to do the right thing like the right side in this Debate which it shouldn't even be a debate and I hesitate to call it that but I mean it's America So we politicize everything, but I think that the right side ultimately is Winning out and not just with regard to a national mask mandate, which is absolutely the right policy But with the perception of how our government Trump in particular is handling COVID-19 a pull from NPR and Ipsos found that when it comes to the United States is handling of COVID-19 and Comparison with the rest of the world two-thirds of Americans think that we're actually handling it worse than other countries with 41% saying they believe we're handling it much worse now additionally a survey monkey and NBC news poll asked people Who they trust when it comes to COVID-19 and a majority of Americans at 51% do in fact trust Dr. Fauci in spite of Trump's attempts to discredit him with 29% distrusting him now when it comes to Donald Trump These results are almost mirrored with less than a third of voters at 31% trusting Trump compared to 59% who don't trust him So make no mistake about it That's still too large of a number like 31% of Americans trusting Donald Trump is way too high But understand that a majority of Americans are Correct here. They're you know, rightfully skeptical of Donald Trump and they're trusting Dr. Fauci more They support a national mask mandate. This is a good sign, right? Common sense isn't entirely gone in America. It's still there. You sometimes just kind of have to look for it, right and Encourage people to Practice common sense more frequently. Um, so yeah, this is this is encouraging but there's another poll that I want to look at here because During the Democratic Party primary support for Medicare for all was actually decreasing a little bit as you know we worked to increase public support for Medicare for all between 2016 and 2019 Individuals like Pete Buttigieg and Joe Biden who were constantly fear mongering over Medicare for all Actually managed to be successful at driving down support for Medicare for all and while support for Medicare for all was decreasing Support for a public option was rising. So, you know as someone who is a proponent of Medicare for all a purist That's my number one litmus test I was absolutely distraught by those results, right? But now during a pandemic we are being proven right and even as people casted their votes for Joe Biden during the Democratic Party primaries They still supported Medicare for all according to exit polls, which tells us that they weren't necessarily voting for him because they believed in him and The policies that he's espousing. They just wanted someone who they believed was the most electable better suited to take on Donald Trump And they mistakenly believed the mainstream media now I say mistakenly loosely because at the time with the information that we had it didn't seem like Joe Biden could actually beat Donald Trump But COVID-19 really did change everything, but I mean people didn't necessarily Determined who they were going to support based on policies They just voted based on electability because that's what MSNBC and CNN Constantly talked about but now during a pandemic I'm happy to report unsurprisingly now that more people than ever in America need healthcare Medicare for all is very much still popular So as Jerry Martin of Newsweek reports Democrats overwhelmingly threw their support behind Medicare for all according to data released Tuesday from a Hill Harris Expo Increasing support for the national health insurance program may be linked to the coronavirus pandemic many businesses have closed permanently Leaving their former employees with no employment based coverage Medicare for all which would create a nationwide single-payer health insurance plan for all Americans was the cornerstone of Vermont Senator Bernie Sanders suspended presidential campaign presumptive Democratic presidential nominee Joe Biden has not embraced the Medicare for all platform Although many of Biden's potential voters support the program out of 948 registered voters who participated in the poll 87% of Democrats said they were in favor of Medicare for all Independent voters also lean towards the idea with 69% supporting the concept less than half of the Republicans surveyed Agreed with the idea with only 43% supporting providing health insurance to all Americans in total Medicare for all was supported by 67% of registered voters and opposed by 33% Voters between the ages of 35 to 49 showed the greatest support for Medicare for all with 80% of that demographic saying they were in favor of the program surveyed voters between 18 to 34 also thought highly of the idea with 78% of them giving Medicare for all their support the Hill Harris expo carried a sampling margin of error of plus or minus 3.18 percentage points so I think that this is encouraging to see because what it tells us is that Americans aren't too far gone like we're not beyond redemption like we can still have nice things because most Americans Have bought into the idea that they deserve nice things. They support Medicare for all. They are you know Competent in determining who to support and trust during a pandemic. They're reasonable with regard to you know trusting Fauci over Donald Trump and that's encouraging to see the task that we have ahead of us. However is Getting them to not vote against their own self-interest So if we get them to agree with us on the policy that's step one But I think the hardest step is getting them to actually Vote for politicians who agree with them because politicians are out of step with what voters want So how do we combat the mass of propaganda that they are constantly being bombarded with? I think that Bernie Sanders would have won the Democratic primary comfortably had mainstream media not been completely biased against him But because for so long they you know talked about electability this electability that and Bernie's just not electable and Bernie didn't really actually challenge that narrative sufficiently People believed it so they voted against their own self-interest because you know Even if they may not support Joe Biden, they would rather go with someone who they don't agree with on everything Then risk losing to Donald Trump. That's what we saw So, you know the goal here is to not just get them to agree with us on the policy But to actually get them to do what makes that policy a reality and that really is I think The struggle with you know leftism in America. That's what we have to do. That's the task ahead of us It's not impossible. I mean people agree more with us than they do Republicans or Democrats, but yet they keep winning. So what do we do? how do we combat that and There's a lot of approaches that we have to take but ultimately that's the goal this tells us You know what we need to do convince those people who already agree with us most Americans that they have to stop voting against their Own self-interest. There's no right or wrong answer to that. It's tough Propaganda is going to be louder than you know, what we're saying. It's gonna be more persuasive than indie media But nonetheless, we at least know what we have to do. We at least know where to look in terms of like Shifting, you know the country It's just a matter of getting that mass of Americans who agree with us Mobilized and activate them politically and get them to vote for politicians that agree with them. So, yeah I thought that these polls that all came out this week were pretty encouraging and When we don't really have that much good news lately, I'm gonna take anything I can find and this is certainly good news


we're going to be looking at chapter 15 verses 33 through 36. So let me read to you and as I normally do I'll give you some reminders of what we've seen, a correction that I last week I I was using John's notes and I gave you the wrong information so actually I I spoke out of I spoke wrongly and you'll see why in just a moment but anyway let's begin at verse 33 and read to verse 36 we'll get into our study. Mark chapter 15 verses 33 through 36. Mark writes now when the six hour had come there was darkness over the whole land until the ninth hour and at the ninth hour Jesus cried out with a loud voice saying Eloi Eloi Lama Sabahtani which is translated my God my God why have you forsaken me some of those who stood by when they heard that said look he's calling for Elijah then someone ran and filled a sponge full of sour wine put it on a read and offered it to him to drink saying let him alone let us see if Elijah will come to take him down. Now Jesus has been crucified he's been on the cross at this point for three hours. Mark 15 25 tells us he was crucified at 9 a.m. it is now noon now last week I mistakenly stated that it was a Roman manner of keeping time I should have said it was the Jewish or the Hebrew manner of keeping time. F. C. Gilbert wrote in Messiah in his sanctuary the light part of the day began at 6 o'clock in the morning the third hour of the day would be 9 a.m. the third hour of the day would be three hours past 6 a.m. which was the beginning of the Jewish day making it 9 a.m. so last week I said off the hand and I should have I said Roman when in fact it's really the Jewish time and so with that correction Jesus has been crucified between two robbers as I've mentioned to you repeatedly these were violent men they were capable of murder that again fulfilled a messianic prophecy Isaiah 53 12 said therefore I will divide him a portion with the great he shall divide the spoil with the strong because he poured out his soul unto death he was numbered with the transgressors he bore the sin of many and made intercession for the transgressors and so Luke had told us that in chapter 23 verse 34 that he had prayed father forgive them for they know not what they do that would have been in fulfillment Isaiah 53 12 at this point Jesus and the robbers have been on crosses for some time it's given them opportunity these men who were dying alongside of him it has given them opportunity to take notice of him now ordinarily the man next to them wouldn't draw any interest they were too busy hurting themselves they they would have been engulfed in their own torment and their own pain so at first they wouldn't necessarily have heard what Jesus was saying but eventually one of them did and we noticed this last time very little is known of these men outside of the fact that they are violent robbers are described that way as mentioned they were more than likely part of Barabbas's followers now sometime after being crucified Jesus had prayed and one of the robbers undoubtedly heard him pray it gripped his heart you see Jesus had a ministry that that reached and impacted many people Matthew tells us in chapter 4 verses 23 through 25 that Jesus went about all Galilee teaching in their synagogues preaching the gospel of the kingdom and healing all kinds of sickness and all kinds of disease among the people then his fame went throughout all Syria and they brought to him all sick people who were afflicted with various diseases and torments and those who were demon possessed epileptics and paralytics and he healed them great multitudes followed him from Galilee from Decapolis Jerusalem Judea and beyond the Jordan and so his fame had gone throughout Syria and various other places and so his name by this time was well known you see the verses I just quoted spoke of the beginning of his ministry it would have spread even further over three years these thieves would have been part of a group that were rebelling against Rome that would have been that would mean that they would have been aware of the things that Jesus had done because his reputation and his fame had preceded him the undoubtedly had heard of Jesus Christ the Bible tells us though at first the they began to revile him the word revile means to mock or to taunt the chief priests the scribes and the elders also taunted him Matthew 27 42 and 43 they said he saved others himself he cannot save if he's the king of Israel let him come let him now come down from the cross we will believe him he trusted in God let him deliver him now if he'll have him for he said I am the son of God and then Matthew went on to tell us in verse 44 even the robbers who were crucified with him reviled him with the same thing now as we've been watching this and going through the scriptures we've seen that Jesus didn't stop ministering even on his way to his death he was on the way to Calvary and Simon the siren was next to him and as he was walking there were women who were along the side of the road and they began to weep for him remember how he had said to them daughters of Jerusalem weep not for me weep for yourselves and your children so he is still warning them and calling them even as he was walking on his way to Calvary now on that way Simon of siren came to faith in him in him and so did his wife and so did his sons now he's on the cross and as he was on that cross he prayed he prayed for forgiveness for those who were killing him and when he prayed and said father forgive them they know not what they do that is what is called the first of the seven last saints of Christ on the cross now we've seen these saints already beginning to be fulfilled I'm going to point some things out as we go through the study it's been said that the first word was a word of forgiveness when he said father forgive them the second word that he gave has been referred to as a word of salvation because as he had prayed father forgive them the next thing we hear is in Luke 23 43 when the when the thief said remember me when you enter your kingdom Jesus said to him I say unto you today you shall be with me in paradise so chronologically the third saying of Jesus on the cross is found in the Gospel of John and I want to read to you and develop this with you for a moment that's the third word it's called a word of relationship in John 19 25 through 27 it reads therefore the soldiers did these things but standing by the cross of Jesus were his mother and his mother's sister Mary the wife of Kloppas and Mary Magdalene when Jesus when Jesus then saw his mother and the disciple whom he loved standing nearby he said to his mother woman behold your son then he said to the disciple behold your mother and from that hour the disciple took her into his own household now there are four women at the cross there's Jesus's mother Mary there's her sister Salome who's the mother of James and John there's Mary the wife of Kloppas who is the mother of James the younger and Josie's and Mary Magdalene but there's another one who has spoken of who was there at the cross and that's the Apostle John now it says here when he's speaking about that that there's this other disciple there this other disciple being John is one who's referred to as the one whom Jesus loved five times in his gospel he referred to himself in that way I think it is safe to say that the love that he had for Christ was something that would draw him it drew him to the high priest house John 18 verses 15 through 18 tells us about that the Bible tells us that he and Peter went to the house of the high priest and John entered but Peter was outside the Bible tells us that John knew the high priest and he got permission for Peter to enter that took courage on the part of both of them but more so for John now John gives us firsthand knowledge of what happened in John 18 verse 15 he said Simon Peter followed Jesus and so did another disciple that other disciple is himself mark 14 verse 54 says Peter followed him at a distance and and I pointed out that the word distance there speaks of a long distance so there were a distance away they were following as as as Jesus was being taken and all but John had gone in and Peter stayed outside he stayed outside of the gate until he gained the entrance to John so it was love that drew these men to the high priest's house and I want to develop this with you for a moment because I think it's very practical Peter's love for Jesus drew him to the courtyard but that wasn't enough under pressure we know that when he finally entered in he denied Christ three times so his love for Christ wasn't enough for him to remain courageous and strong on the other hand John not only went to the courtyard but he also went to Calvary you see in the garden when Jesus was taken all of his men for sook him and fled according to mark 1450 and that would have included John but though he initially fled John had made his way to Cairophus's house it was the love Jesus had for him that drew him it was Jesus's love for him that drew him and gave him the strength to go all the way to Calvary one of the things that I think that we make a mistake and I've said this before but it bears repetition at this moment perhaps some of you haven't heard me say this kind of thing I've been around for a while I've been a Christian for for 5 7th of my life and I can tell you that there are times in my life when I've told had I've done this many times when I've said God I just really love you I really love you and I found myself to be like the Apostle Peter I have found myself to be telling him how much I love him and I haven't been basking in the love he has for me and the love I have for him will take me so far it took the Apostle Peter to the entrance of the courtyard it took him into the courtyard but the love that Peter had for Jesus wasn't enough the love he had for Jesus led him to deny him ultimately because he didn't have the courage strength or faith to understand it wasn't the amount of love Peter had for Christ and listen carefully I want to remind you it wasn't the love that he had for Christ it was the amount of love Christ had for him and that is very important for us to understand the Bible doesn't point me to to tell God how much I love him I worship him and I do that but the Bible teaches me to be compelled by his love for me the Apostle Paul said in 2nd Corinthians chapter 5 verse 14 and 15 he said Christ love compiles us because we're convinced that one died for all therefore all died and he died for all that those who live should no longer live for themselves but for him who died for them and was raised again Paul said I'm willing to go anywhere where the name of Christ is yet to be named to preach this and it's not because of my love for him it's because his love compiles me you can go to any class you want to you can get a degree in college on you know theological degree you can go to every Bible study that church has to offer you can get up early in the morning and pray you can stay up late at night and pray you can read you can go through the Bible but if you don't understand that it's not your love for him but it's his love for you you're going to be frustrated ultimately the one thing that compelled me and still does has been the love of Christ for me not the amount of love I have for him because my love for him is faulty it's incomplete but his love for us for me is not it's complete that's what Jesus is showing on the cross that his love for you is demonstrably above anything you could ever I could ever imagine Christianity isn't based on how much I love him Christianity has been based on how much he loved us it's based on the love of God for God so loved the world that he gave it's that kind of love and so the apostle Peter had a great love he said I'll even die for you though these people all the rest may may forsake you I never will when in fact his love could only take him so far but John learned something and it's what drove him and drew him to the cross of Christ with with Jesus his mother and these other women it was the love that Christ had for him it's that kind of love that draws us it's not my love for him alone it's learning to rest in the love he has for me that gives me peace and and gives me hope and and takes me through the tougher times of my life now as this is taking place I want to develop it a little bit further Mary was watching as her son was slowly dying and as a mother there's no doubt but her heart was broken as she's seeing this as she's watching her baby now she's there standing she hears the mocking of the priests she hears the crowd she hears the mocking from the pastors by she hears the things that are being said to Jesus by the thieves she sees the indifference of the soldiers as they're gambling gambling for his robe I would ask any parent mother and father to think about that for just a moment how you would have felt the pain that this devoted and virtuous woman endured would have been incredible it's very possible that as she was there watching her son that memories of this child began to crop up and his dedication at the temple she had been given a prophecy it's recorded in Luke 20 rather 2 verses 34 and 35 it said Simeon blessed them and and said to Mary his mother behold this child is destined for the fallen rising of many in Israel and for a sign which will be spoken against yes a sword will pierce through your own soul also that the thoughts of many hearts may be revealed when Jesus was eight eight days old he's taken to the temple being dedicated as a firstborn who opened the womb according to the law demanded his prophet Simeon came and spoke to her in that way the thoughts of many hearts will be revealed and that happened when the people's rejected Christ for a while Jesus was very popular there were multitudes following him the Bible speaks of these multitudes and yet at the end he was totally rejected there he is on a cross and so their thoughts are being revealed in the rejection of him the brutality that he endured the mocking revealed the hearts of these people and so the sign that was spoken of openly revealed his complete rejection but also a sword pierced her now from the very beginning Mary would have endured many things because of this child the angel had spoken to her and said that she was going to conceive a child by the Holy Spirit because she had said how is it that I a virgin am to have a child and he said that holy thing within you will be of the Holy Spirit and so the Lord had through her done a miracle for her to have a baby as a virgin and then Joseph the one she was betrothed to he had to be spoken to by the angel because he was minded to put her away privately why because he was a just and a righteous man but the angel had to say don't be afraid to take her as your wife because that which is within her is begotten of the Holy Spirit so a man and his betrothed wife have a baby they're to have a baby she's pregnant and she from the beginning in the small village of Nazareth would have endured a lot of taunts a lot of gossip a lot of backbiting because it was a small village there are those who say that the village of Nazareth during the time of Christ could have numbered as little as 60 people others say could have been as high as 200 but Nazareth isn't a big city it's a small village and so the people knew of this woman this woman is pregnant this woman had a baby she and Joseph had not consummated their marriage vows prior to her conception and therefore she was unfaithful and that's what she lived with from the beginning she said let it be done to me and according to thy word and she was willing to do it even though the pain of such a thing is going to continue and that's bad in and of itself but inwardly she always comforted herself at the truth and that would be something that one day she would be vindicated of as she watched him grow up as any mother watching a child grow up this child would have brought her great joy during that day when the Jewish mother would would have their baby the mama would nurse the baby and she nursed Jesus and she would speak to him she would pray over him she would sing to him and it was a tender thing for her as he started to grow a little bit older and he began to walk began to form words she would hear him speak she'd hear his little voice and many of us in our who are parents grandparents can remember in the back of our mind those little voices of our children when they were young and how we wish they'd shut up but we remember their voices you'd hear him outside do you think that Jesus played with the children I'm sure he did and the mama could look through a window and she could see her son playing they're enjoying himself and yet in the back for a mind she knows that this child is destined at the age of 12 they go to Jerusalem to celebrate another feast of Israel and they leave but they didn't notice that Christ hadn't come with them in their day's journey they have to come back they start searching for Jesus we all know the story they find him in the temple and he's speaking to the scholars asking them questions and answering theirs and he's already starting his ministry and she walks up and she says your father and I have searched frantically for you why'd you do this to us he says don't you know that I should be about my father's business and she she hears that and she ponders these things as he grows up into young manhood he's now 30 years old and it's time for him to minister and they're at a wedding in Cana of Galilee she walks up to her son and she says they have no wine and he says what do I have to do with that my time hasn't come mama I know my father's timing and you're not the one who tells me when he tells me you don't have that responsibility and yet there was something within Mary apparently that they made her aware because he went on and he began to multiply he turned that water into wine not so they could be drunk by the way but because it was a symbol of festivity and joy Jesus it's time for these people to have a joy don't move me into the direction my father needs to move me in but she's aware that prophecy must have stayed within her heart all those years his child this child is destined for the rising up in opposition and a sword will pierce your soul and so there they are and I can't help but wonder if perhaps that she's looking at her son and she's hearing she's hearing what's being said the taunts she hears the moans that are coming from him this is the one that when he smiled the sun would seem to shine even brighter when he left the joy her joy would would leap that's her baby and they're just standing there imagine that for a moment just watching and as he's there she's hearing him speak and then he looks at her and in John 1926 Jesus is not thinking of himself he's thinking of her he says woman behold your son now Jesus honored his mother making provision for her care you see he was the firstborn and that was his responsibility in the book of Exodus 20 verse 12 it says honor your father and your mother in that verse to honor speaks of giving respect it speaks of highly prizing caring for it's been said that honor is the internal attitude the outward evidence of honor is showing care and Jesus is there providing for her future care you see at that time according to John 7 verse 5 his brothers did not believe in him and because of that he would not entrust her into the care of these unbelieving brothers when my mother was about to go home to be with the Lord we would not have wanted to entrust her into the care of those who could not feed her soul who could not give her fellowship who could not pray with her or pray for her who couldn't provide those things for her when my mother was about to die my mom was ill all of her life but her in her last year she had fallen she had broken her back she was bedridden her last year when my father had passed on when my dad went to heaven my my mom went to live with my sister Rebecca and for 12 years my sister Rebecca took care of her and she did it out of love for my mom Rebecca told me over and over again she said this is not really a sacrifice for me David it's my honor to do this to care for mama we didn't want my mother in the care of those who didn't know Jesus my mom was someone who liked to talk about the Lord who liked to pray who liked to have fellowship that was my mom so she was entrusted into the care of someone who loved Jesus and Jesus entrusted his mom into the care of somebody who lived him also because when he says women behold your son he's not speaking of himself he's not saying look at me with pity or look with me with sorrow he's speaking of John because he went on to speak to John and he said behold your mother in John 19 27 it says to John the one whom he loved he said behold your mother and what he did is he entrusted her into the care of one who loved him and who would love her and in doing so he honored his mother even while he was dying and that is taking place and that brings us to this verse verse 33 it says now when the sixth hour had come there was darkness over the whole land until the ninth hour from noon until three there's darkness over the land on the night that Jesus was born the sky was filled with supernatural light he was the light of the world and it was fitting that his birth would be accompanied by light in Luke 2 8 9 it says there were in the same country shepherds living out in the fields keeping watch over their flock by night and behold an angel of the Lord stood before them and the glory of the Lord shone around them and they were greatly afraid so when he was born his birth was accompanied by light but on the day he died when the sun is at its brightest the sky became dark from noon it says until three in the afternoon darkness filled the land now why was it dark well in scripture darkness is often used as a symbol of judgment you see that both in the old and the new testaments in exodus chapter 10 verses 21 and 22 it speaks of the judgments of of Egypt and and one of the judgments one of the plagues was three days of darkness in the old testament book of Amos chapter 5 verse 18 it says woe to you who long for the day of the Lord why do you long for the day of the Lord that day will be darkness not light when Jesus was speaking of judgment in Matthew 25 30 he said throw that worthless servant outside into the darkness where they'll be weeping in a nation of teeth so the cross was a place of divine judgment the sins of the world were being poured out vicariously on the Son of God and supernatural darkness expressed God's response to sin in the Old Testament book of Habakkuk 1 verse 13 it says to God you are a purer eyes than to behold evil and cannot look on wickedness you cannot look on evil and wickedness with any pleasure or any permission one commentator wrote a veil hides from us the incidents of these three hours and all the details of what our Lord shrouded in the supernatural gloom under went for us men and for our salvation someone said the light returning after three hours may foreshadow the resurrection well as this is taking place verse 34 in the ninth hour Jesus cried out with a loud voice saying all the way all the way llamas bhaktani which is translated my God my God why have you forsaken me that's Aramaic Matthew records Eli Eli which was Hebrew now why did he cry that out you see this is the fourth saying of Jesus on the cross it is the saying of abandonment why would you cry out why have you forsaken me he cried out in anguish because of the separation that he's experiencing you see he's in anguish because of the separation he felt for the only time he ever felt any separation in eternity and taken upon himself all the sin of the world he felt the isolation that sin produces Isaiah 59 to says it like this your iniquities have separated you from your God your sins have hidden his face from you so that he will not hear and today there's an awful lot of of chatter going on that there's no such thing as sin we're all designed this way we all have been created to do the things that we do and there's no real talk about about the the sinful nature of man olive sin and fall short of the glory of God every one of us has been born totally and utterly in sin because that's our nature that's why we need Jesus Christ and sometimes people will say I feel that I'm distant from God well if they're not saved there's a good reason to say that sometimes people if they're not saved say well my my prayers seem to just go as far as my voice carries and then fall to the ground useless and it may be because your sin had made separation between you and your God what's the problem with man today is we don't understand the fact of sin we are unwilling as a nation and as a world to recognize our own sinfulness and that's the problem I remember my mom was speaking to a woman and sharing with the woman the gospel mama used to like to evangelize and she was sharing with this woman and she said would you like to receive Christ the Savior and the woman sees it I would and so my mom said repeat after me father forgive me a sinner and the woman looked at my mom and said I'm not a sinner and I told my wife Marie honey that wasn't the right thing to say no I'm just kidding but she did she said that I'm not a sinner yes you are we all are all have sin and fall short of the glory of God there's none righteous no not one we were completely and utterly born with the Adamic nature is prone to sin because it's sinful we don't become sinners when we sin we sin because we're we're sinners and you don't even have to realize that when you're a kid you don't realize there's a sin nature so if you see something you want you just take it I'm five years old I go into a supermarket we don't have a parakeet but I still remember my first knowledgeable sin because I stole some bird seed we didn't even have a bird but I remember taking it it was green it was a parakeet with a yellow you know it just was a good-looking bird and I wanted it I took it home I put it in my pocket my mom was doing the wash and she sticks her hand in my pocket to pull out whatever is in it and there's a bird seed can I still remember her marching me to the supermarket I had to take the can and hand it to one of the people who worked there and I had to tell him I stole this I didn't realize I was stealing I thought it was free here you go into a supermarket it says take one so you do right when you first began to lie you weren't very good if you lie enough you get good you can practice something till you become good it doesn't really matter what it is but you can that's what happens it's a natural thing for us to do that and Jesus on the cross was given to us opportunity to realize who we are and there he has taken upon himself the sin of the whole world he's becoming the sin offering all of the sins not only prior to him but the future ones too all included are falling down on one person as the lamb of God who takes away that sin and second Corinthians 521 he made him who knew no sin to be sin for us that we might become the righteousness of God in him he took our sin so that we might have his righteousness what a deal why don't we want that why don't we want a righteous life why don't we want the joy and the love and the peace the goodness and the kindness and the patience and all those things that come along with being with Christ well there he is on the cross experiencing our separation now the question has been asked what kind of separation did he experience while on the cross my God my God why has thou forsaken me it wasn't a separation of his nature it wasn't a separation of essence it wasn't a separation of substance it was a separation of fellowship someone said that Jesus had a taste of such broken communion the first and last he ever experienced in those desolate hours when darkness lay upon the earth and upon his soul Jesus was our forerunner in every kind of experience even to the feeling of God's frown of disapproval on sin that he might become our high priest understanding all our infirmities and being tempted in all points like as we apart from sin he felt the way a lost sinner feels without himself having sinned someone said he did not cry out when accused he did not cry out when crucified but he did cry out when he was separated so he's there and he's speaking Eloi Eloi Lamas Bhaktani my God my God why have you forsaken me verse 35 some of those who stood by when they heard that said look he's calling for Elijah and someone ran and filled a sponge full of sour wine put it on a read and offered it to him to drink saying let him alone let us see if Elijah will come to take him down now this was mockery it wasn't out of curiosity it wasn't out of religious fear what they're doing is they're twisting his words into a cry for Elijah and not for God my God my God is what he was saying but they they said always call him for Elijah you see there was an Old Testament prophecy that Elijah would introduce Messiah to Israel in the Old Testament book of Malachi 3 verse 1 behold I send my messenger he will prepare the way before me and the Lord whom you seek will suddenly come to his temple even the messenger of the covenant in whom you delight behold he is coming says the Lord of hosts and so verse 36 someone ran and filled a sponge full of sour wine put it on and read and gave him to drink now that brings us to the fifth saying of Jesus on the cross it was a response to what Jesus has said recorded by John in John 1928 it says after this Jesus knowing that all things were now accomplished that the Scripture might be fulfilled said I thirst now by saying this he fulfilled a prophecy given by King David in Psalm 22 in Psalm 22 verses 15 through 17 my strength is dried up like a potsard my tongue clings to my jaws you have brought me to the dust of death dogs have surrounded me the congregation of the wicked has enclosed me they pierced my hands and my feet I can count all my bones they look and stare at me now this Psalm prophesies Messiah and that Messiah is going to be crucified and it includes the picture of dehydration my strength is dried up my tongue clings to my jaws it also speaks of the piercing they pierced me it speaks of the mocking dogs have surrounded me the congregation of the wicked has enclosed me it's a picture of Christ on the cross and David had written this psalm is a messianic psalm and he had written this psalm a thousand years before Christ again it's a picture of crucifixion a thousand years before Christ the Romans had introduced crucifixion they had they had taken it from other peoples including the Assyrians but the Romans didn't start crucifying till 300 years before Christ this is the thousand years before Christ and it's picturing the way Messiah is going to die on the cross you can see his rib cage as he's pulled up and he's trying to breathe it's a picture of crucifixion well as this is taking place again in verse 36 someone ran and filled a sponge full of sour wine put it on a reed offered it to him to drink and so the sour wine that he was offered was going to give to him the ability with this dried out mouth to speak it was high in water and low in alcohol it was used to quench quench people's thurs again that fulfilled scripture Psalm 69 21 they also gave me gall for my food and for my thirst they gave me vinegar to drink here's going to moisten his mouth so he could speak his final words but the responses this is taking place is even more mocking let's see if Elijah will come let's see if you'll formally introduce us to this man who refers to himself as our Messiah let's see if Elijah will come they they didn't realize that John the Baptist preaching in the spirit of Elijah had already come it was John the Baptist who came to prepare the world for the Lamb of God it was John the Baptist who pointed Jesus out and said behold the Lamb of God who take away the sin of the world it was Eli it was John the Baptist preaching in the spirit of Elijah and they didn't hear it they didn't see it they didn't recognize that he had already come he had already prepared the way for him he already had called people to repentance he already had come but they didn't see him they didn't recognize him and so what did they do with John the Baptist they killed him they they cut his head off because he spoke concerning the moral lapse of that nation and spoke to Herod and told him and it's not lawful for you to have your brother's wife and for that he was beheaded they killed John the one who proclaimed prepare the way for the Lord make his path straight they killed John the one who prepared and then they killed the one he prepared the way for they killed Jesus they killed him and now they're killing their own Messiah if anything shows the depravity of the human soul even anything reveals our own nature to us it's the fact that mankind is willing to kill their own God mankind was willing to kill a man who only did good and by the way we're seeing that attitude even today in the society that we live in because anybody who speaks forth things of Christ with faith in him is looked at as being a lunatic today we're looked at as being off scouring but that's what we always have been we're looked at as being bumpkins and ignorant we're looking at those we're being looked at as those who are who are blocking progress you know what we are and what we ought to be is a voice crying out in the wilderness make make the path straight for Jesus that's what we are that's what we're called to do and even though the world does not want us even though the world really doesn't care for us that doesn't stop us from talking because we know our Savior we know our Redeemer lives and we know that one day we will be with him and until that moment we preach the cross Jesus crucified so that people might have a relationship with him that's what the church is doing and I'll close it with one last thought when I got saved so long ago in a movement I've mentioned so many times but some are not familiar with that term the Jesus movement that's what we were told I'm 20 years old and I'm being told this Jesus Christ he's the only remedy for what you're going through he's the only one who can set you free for me the alcohol the drugs the relationships everything else it was all hollow because once you drink of this you're going to drink of something else once you've done this you want something else everything progresses down not up so the first time I smoke a cigarette the next thing I'm smoking is pot the next thing I'm doing is dropping drugs the next thing I'm doing is getting drunk and alcoholic those things all follow and then you hear you can be set free your sin has made a separation between you and your God but God loved you so much he sent his son Christ to die in a cross so he could set you free and even on that cross he said father forgive them and he wants you to have a relationship with him and when I heard that I said why me because God so loved the world that includes you he loves you too so it's not the amount of love I've had for Jesus these 52 years it's the amount of love he's had for me that has kept me going and gives me the ability as a man to say turn from your wicked ways follow Jesus Christ because he loves you and he gave his life up for you even on that cross he was still crying out for us father forgive them they know not what they do may God help us to come to faith in him because he's the only way he's the only way that this nation will ever change it's by person by person it's not voting another sinner into office it's by receiving Jesus Christ who transforms sinners into righteous people who affect their nation for God's sake that's what God is calling us to today is a follow here and that's what Jesus came to do and father we


At 10, I see Denver going offensive line. Andre Dillard from Washington State is an excellent player, terrific in past protection. I could see Denver going with Andre Dillard. At number nine, Buffalo might go with Dexter Lawrence, the defensive lineman from Clemson. Lawrence is getting solid comments after the combine. There is enough momentum for him that I think he can overcome some of the things that people have alluded to that might be red flags like BED suspension. But I think that the Bills who need help on the defensive line could be looking towards Dexter Lawrence. At number eight, the Lions could go with Cleveland Farrell, defensive in from Clemson. Ziggy Anza hit free agency. The Lions are going to need some help up front. Farrell is terrific. And some of the traits that I think he has, I think fits what the Lions want, in particular with Matt Patricia. At number seven, I think Jacksonville could be looking to the offensive line. They need to sure some things up up there. But Montez sweat on the board right here. I think he's been too good in this draft process to pass up. So from Mississippi State on the edge to Jacksonville, Montez sweat. At number six, the New York Giants have got to address the quarterback position. Now, they might do that through a trade. We don't know. But if they don't, Dwayne Haskins is a great option right here, the quarterback from Ohio State. He's a developmental guy that at some point in his career, I think might have the top upside of anybody in this draft, at least at that position, the quarterback position. He's got a great arm. He's smart. He works hard. I think he could fit nicely in New York. At number five, Tampa Bay might be looking to the defensive line. Rashawn Gary from Michigan would be a great option there. I love how Gary can play all four positions on the defensive line. He's shown that ability throughout his college career. Tampa might look on the offensive line. But if they don't, I think Rashawn Gary might be the pick. At number four, with the Oakland Raiders, I like Josh Allen, the edge player from Kentucky. He's terrific. And with Bosa already off the board and getting most of the headlines, I think Allen's got monster upside. He's quiet, but very athletic and long. He's going to be a force on the opposing backfield for many years to come. At number three, the New York Giants looking on the defensive line and really looking at best available here, Quentin Williams from Alabama. I think he's viewed by many, including me as a surefire pick in this draft. Maybe one of the only ones along with Bosa and maybe Murray, but he can stuff the run. He has excellent pass rush ability from the interior. This dude's a monster. So with Murray going number one, Bosa's got to go number two to San Francisco. The defensive in from Ohio State is an absolute freak rush in the passer. And the 49ers are going to run to the podium to put his name in. Nick Bosa to the 49ers. With the number one pick in the NFL draft, the Arizona Cardinals select Kyler Murray, the quarterback from Oklahoma. There is not a quarterback that I can remember with Kyler's combination of passing acumen accompanied with his dynamic playmaking ability with his feet. It's just second to none. The guy's outstanding. There's not better tape that I've watched in a few years. Let's just put it to you this way. If this guy was six foot and not five, 10 and an eighth, everyone would be talking about him as the second coming of maybe one of the greatest prospects that we've ever seen. So good his tape is and how dominant it has been over the last year. He needs more time, needs to develop a little bit, but rest assured, if you take Kyler Murray and put him in your backfield, you're better next week.