Okay, so I got about two-thirds of the way through what was supposed to be my first talk, so I think I'm going to try to finish up the first talk and then give you my favorite part of the second talk, and so, and finish it in the hour. So just to remind you where we were, we were about to define a project of an OSOL transformation, which is a specific type of an OSOL representation into PSLNR, and the definition starts with the existence of limit maps. So these limit maps are row-equivariant continuous maps from the boundary of the group into RPN-1 and into the gross monion of N-1 planes, and the transversality is just that if you're two distinct points, the line associated to X never lies in the plane associated to Y, and there's one immediate consequence you get of this is that row of gamma fixes psi row of gamma plus, right? So right a lot where you found one eigenline, and in fact you found a second eigenline, row of gamma also fixes psi row of gamma minus, where gamma plus and gamma minus, this is the attracting fixed point of gamma on its boundary. So right away we know we have two eigenlines, and if we also assume that our representation is irreducible, then we know that this is an attracting fixed point and this is a repelling fixed point for the action on the projective plane, and that's another way if you want to, from the point of unilinear algebra, that says that this matrix, row of gamma, is a proximal matrix that has a unique eigenline of maximal modulus and a unique eigenline of minimal modulus. So right away, just from the existence of the limit maps, we have some dynamical information about our representation, and in fact it's going to turn out that if your representation is irreducible, this is enough to get everything we need, but let's talk about sort of the more general situation.  So U gamma is the unit tangent bundle, and then phi t, whoops, was the GD sick flow, and often it's convenient to work up with U gamma tilde, and we can parameterize that as the pairs of points in the boundary, well, you don't want those points to be equal, cross r.  So phi t tilde just takes a point x, y, s, and it's the silliest possible flow. It goes to x, y, t plus s. So that's quite a nice flow. With the level universal cover, this is a very easily described flow. So now we're going to have to, I'm going to try to unpack some words, which I initially found very scary, and turns out are not very scary at all when I was first reading about Nassau representations.  But for me, this sounds disturbing, but strictly less than zero. So here, m was a closed, negatively curved, closed manifold, and the curvature is strictly less than zero. So in particular, this G sick flow is a Nassau, which means it has a direction parallel to the flow, a contraction direction, and an expanding direction. So what is the flat bundle associated to a representation? Well, you just take this cover of the G sick flow, U gamma tilde, and this is, to be even more explicit, remember this was just the tangent bundle of the universal cover of M. It was not some weird object.  So here you have an action of gamma on, what is my notation, U gamma tilde cross Rn, and then it just takes a point, so if you have gamma, it takes a point Z and a vector T to the point gamma of Z, and the vector rho gamma of V. That's all it is. And then we quotient out by that action, and we get what's called the flat bundle associated to the representation. And then it turns out these transverse limit maps just give you a splitting of that flat bundle. And what does that mean?  Well, what's the line bundle? Well, let's work again up in this cover. And the line bundle over the point X, Y, T, well, what's the line associated to X, Y, T? Well, it takes psi rho of X. That's a line, so that gives you a line bundle. And then the hyperplane associated to Y to X, Y, T is just to be theta rho of Y. So you have a natural, since psi rho of X and theta rho of Y give you a splitting of Rn, this gives you a splitting of your Rn bundle into a line bundle and a hyperplane bundle. And this is all being done equivalently. So we can work up at this cover level and just quotient out later. As long as everything we did was equivariant, we just quotient out. Yeah. Why? Yeah, these are not smooth bundles. They're just bundles. The natural class of smoothness here is helder. So if you're finished, helder is like analytic, but for the most of the rest of us, helder is like continuous. So these are not that well behaved from a regularity viewpoint, but they are helder. And then they say, OK, take the flow, and then they say take the flow parallel to the flat connection on the flat bundle. And that seems even scarier, but in fact it's something very simple.  And then what is the lifted flow to the flat bundle? Well, at the level of this cover, you just act by the way the flow does. So you've got a decomposition of your lift of your flat bundle into U gamma tilde across Rn. And in the U gamma tilde factor, you just flow. You take X, Y, S to X, Y, S plus T. And what do you do in the other factor? Nothing. You just leave the vector fixed.  Right? It's not some really bizarre flow, but up at the level of your cover, you're just flowing along in the base, and then you're doing nothing in the fiber. Now of course, then when you project, you look at the action of gamma on this, it's getting twisted by the action of row of gamma when you'd go downstairs. But up in the universal cover, it's the silliest possible flow and the easiest possible flow to understand. So this is the flow parallel to the flat connection. And that by construction, this flow preserves the splitting. Because as you flow along in time, your X vector doesn't change. So therefore, your line doesn't change. If your vector started in your line, your vector doesn't change. So your vector stays in your line. And the same thing with your hyperplane. So this gives you, in fact, a flow defined on psi and a flow defined on theta. So the actual definition of being projective out in Ossoff is you have a flow. You have a bundle psi. You have a bundle theta. You can define a new bundle psi tensor theta star, which is hum theta psi. And you say, well, this flow should be contracting on this complicated bundle. And so now if you do a little bit of tensor manipulation, you see right away, and on the next slide, I'm going to give you a really concrete way of saying this without all that tensor analysis. But for the moment, it's just sort of abstract nonsense. And in particular, it says the flow is contracting on psi. Well, let's think about what does that mean for the flow to be contracting on psi? What that means is that you take some norm on e rho. So what does that mean? So that means you get, this lifts to a norm on e rho tilde, which is just u gamma tilde cross rn. And that's just a continuously varying family of norms on rn. That's all this. For each point, I have a norm on rn. And it varies continuously. And it's equivariant with respect to the action. So if you act by rho of gamma, then the norm here changes by the action of rho of gamma. So what does it mean for the flow to be contracting for this eigenline? So let's look at what happens to psi rho of gamma plus. So let's take some vector v in psi rho of gamma plus. And let t gamma be the period of the flow line associated to gamma. So that means that if you look at what is the flow associated to gamma, it's you sit over gamma plus gamma minus cross r. And if you recover, that line is preserved flow line in u gamma associated, well, flow line. Let's use better language than flow line. A better word is closed orbit. Oh, yeah. That was keeping me awake. That was helping me keeping you awake. Sorry. I'll work on it. I'll try to talk in a dull monotone. A nice, peaceful voice so we can all slumber. Anyway, so what does that mean? Let's think about this really concretely. That says that now if you flow phi t gamma of v, that just gets taken to v.  And if you flow along there, this just goes to phi t gamma of z comma v. And phi t gamma of z is equivalent to z. So this is equal to gamma of z comma v. So what is contracting mean? Well, that means that if you look at the norm of v measured at the point v hat t gamma of z, that that norm is less than the norm of v measured at the point z. But we said that this point is exactly gamma of z. So what does this tell you? Well, if you look at it, this tells you that phi t gamma of v at phi t of z is exactly equal to 1 over lambda 1 of rho gamma of v at z, right? Because this norm is supposed to be rho-equivariant. And so what does this rho do? Rho takes v. That says that the v at z has the same norm as rho gamma of v at gamma of z. That's what the rho-equivariant is. And so that tells you rho gamma of v is exactly lambda 1 rho gamma of v. So that tells us that, let me write some of this down, since I got myself v at z is supposed to be equal to rho of gamma of v at gamma of z. And this is exactly equal to lambda 1 rho gamma of v measured phi t hat of phi t of z, OK? So that says when you rearrange that, you can pull the lambda 1 out and you get this thing. So that says that this is equal to 1 over lambda 1 rho gamma times v at z. So that tells you, when you look at it, that tells you lambda 1 is bigger than 1, right? So, well, that doesn't, OK, well, that didn't get us very close. So lambda 1 is bigger than 1. We felt like we knew that already. But even more, now if you have a contracting flow on a compact space, it's not only contracting, it's uniformly contracting. So if you have a flow which is contracting on a compact space, at any point if you flow for some amount of time, you contract it by a factor of 1 half. But that varies continuously, and you're on a compact space. So there's some time t naught that if you start at any point and you flow for time t naught, you contract by 1 half. And so now if you contract, so now if you flow for time 18 times 2 naught, 18 times t naught, you contract by a factor of 1 over 2 to the 18th, right? So this is uniform contraction, follows from compactness. U gamma compact implies contraction is uniform, i.e. that the norm of phi hat t of v at phi t of z is less than c times e to the negative At times the norm of v at z. So it's being contracted by some exponential factor A. And so what that tells you is that, in fact, if we put this all together, I should have probably put this all in the slides. I just decided at the last minute this would be a good thing to do. So if we put this all together, that tells us that the norm of v at phi t gamma of z is less than or equal to c e to the negative At gamma times the norm of v at z. So if we play around, this implies lambda 1 rho gamma is bigger than 1 over c times e times A to the t gamma. But what about t gamma? So you've got a compact space and you're flowing on it. So what is the relationship between the period of the flow and the word length? Well, they've got to be related to one another. If you take the universal cover of the flow and you take the group, then those two are quasi-symmetric. And so you know that t gamma is comparable to gamma, and this is the reduced word length.  One is saying it's the minimal length of a word conjugate to gamma. Another way to say it is it's the translation length of gamma on its caligraph. So let's take that. This is translation length of gamma on its caligraph. So up to choosing a constant, I'm allowed to replace t gamma with reduced word length. And since, again, we don't care about constants. We're coarse geometers today. So this implies that lambda 1 of rho gamma, which is this top value, is bigger than 1 over some constant e to the negative k times the reduced word length. And so that implies that the log of lambda 1 of rho gamma is bigger than some constant times the word length minus c. And this should sound familiar to us. Let's think about back to Fuxian groups. What is the log of the top eigenvalue? Well, it's half the translation length on h2. So this is saying that if you have a Fuxian group, the translation length grows roughly as fast as the word length. And so we're seeing that same fact now transported up into this abstract setting of Projectivinossov groups. They have this same property, that translation length where we're interpreting translation length as log of the top eigenvalue is roughly word length. And this has a fancy name. It's called well-displacing. So a representation into PSL and R is well-displacing if log of another and a fancy name for lambda 1 is the spectral radius, which is another thing that used to confuse me at first. I had to teach myself that spectral radius meant top eigenvalue. So the log of the top eigenvalue is at least k times the word length minus c. And here's the more concrete formalization of this bundle ham theta psi is contracting. You can just say that if you put any norm on your flat bundle and you take any point in the g-sick flow, any vector in the line bundle at that point, and any vector w in the hyperplane bundle at that point, and you flow for time t naught, then you look at the ratio of the length of v over the length of w with respect to this norm, it's contracted by a factor of 1 half. So you can see that this right away, if we didn't know before, the lambda 1 is the top eigenvalue, this tells you that the vectors in psi are growing faster than any of the other vectors. So this tells you that definitely the lambda 1 is the eigenvalue of maximum modulus. And so in fact, we could have given this definition, but I wanted to at least put this sort of fancy definition up there, but this is totally equivalent to that. So once you have something which is well displacing, well, this also turns out to be in this setting equivalent to your orbit map being a quasi-isometric embedding. Well, why is that? Well, the log of the top eigenvalue is coarsely the translation distance on the symmetric space. So this tells you that the translation distance on the symmetric space is roughly comfortable to word length. And so if you think about it, your Cayley graph is getting embedded in your space.  And each axis is quasi-isometrically embedded because the translation distance in the axis is roughly the translation distance in the symmetric space. So we recovered the fact that our orbit map is a quasi isometric embedding from this sort of abstract definition. Remember, that was something we definitely wanted to be true. We wanted to look at representations which were quasi-isometric embeddings. But we realized that wasn't quite enough. So this is a strengthening of that condition.  And you've got to be faithful because your orbit can't pile up at any one point. So as a summary, we've got this definition of a projective and also of representation. It guarantees we're discreet and faithful. It guarantees we're well-displacing. And the associated orbit map is a quasi-isometric embedding. And the image of every element is bi-proximal. So you should think of the image of every element being bi-proximal as being, so if you're used to rank one Leigh groups, if you have a representation with this convex co-compact, that means the image of every element is a hyperbolic transformation. It can't be elliptic or parabolic. So this is sort of a generalization of that fact. It's that every element has to be bi-proximal. And one thing which I don't have on here, I should, but is I certainly don't want to prove is one advantage of this definition is this definition is in terms of dynamics. So this says that we have hyperbolic dynamics. And one of the really crucial things about hyperbolic dynamics is they're stable. If you wiggle a hyperbolic dynamical system, it remains a hyperbolic dynamical system in a very broad setting. So in fact, this setup is set up exactly so that you can now sort of mimic the sort of technology you see that goes back to like Hirsch, Pugh, and Schubes book and prove that. And in fact, another formulation of this is that there's a section of the flat bundle whose image is a hyperbolic with respect to the flow, whose image is a hyperbolic set. Well, yeah, anyway, maybe I have to be a little more careful about that, but there's something like that is true.  So that tells you that if you wiggle a projective Anosov transformation, it remains projective Anosov. And there are now many definitions available of Anosov representations. And one of the real advantages of this is this one is that it's set up to apply to very standard technology to give you stability. And I think that's one of the things that LaBri was thinking about when he made this definition. OK, so let's see what this means in a couple of our examples. Well, let's think about these Benoit representations. Remember, I told you that if you take a lattice, a co-compact subgroup of SON1, and you wiggle it within PSL N plus 1, it remains so the original lattice preserves some round disk. And the new one, Benoit-Approved, it preserves some convex domain. So if you take this wiggling rho t from gamma into PSL N plus 1R, you started with rho naught from gamma into SON1. So I'm going to draw in the picture where N is 2 here. Then you get this omega t. And this is C1, strictly convex rho t preserves, rho t of gamma preserves omega of t. So this is kind of amazing that it stays C1, remember. It does not stay at all C1 if we do the same thing in the rank 1 setting. But so now what is our limit map? Well, here you have an action on this space. This guy admits a complete metric called the Hilbert metric. And so you can just like when you take the hyperbolic metric on H2, naturally the boundary of H2 is its boundary circle. When you take the Hilbert metric on this guy, naturally its boundary circle is the boundary of this domain. So you get an identification, psi rho goes from boundary gamma to the boundary of this domain. And one thing you should be careful with is this identification is not C1. It's only held here, even though this boundary is C1. So it's like if you have two Fuxian groups. And if you take and you conjugate one to the other, that limit map is only held here. In fact, it can't be differentiable at any point. So this is something you often see in this higher-typing theory setting. The image of your limit map may very well be a nice smooth object, but your limit map itself is not smooth. And that can be very important. Now what's theta of rho? So here's psi rho of x. What should theta rho of x be? Well, we have a C1 sub-manifold. There's a hyperplane staring us in the face. The tangent line, that's theta rho of x. Theta rho of x is just tangent plane to omega t. So these two transverse limit maps are staring us right in the face. And what is the transversality? It exactly reflects the fact that I told you this was strictly convex. So if you unravel what strict convexity means, it right away implies transversality. So you can see in this work of Benoit, which is done about the same time as Lavery's work, a lot of the ideas of a Nassau representation is just very naturally occurring. OK. Well, what about these Hitchin representations? What Lavery proves is that, in fact, they're a stronger kind of a Nassau. They're a Nassau with respect to a minimal parabolic. So if rho is Hitchin, Lavery shows you get maybe psi hat rho, which goes from the boundary of gamma into the space of flags on Rn. So a flag is a line, a 2 plane, a 3 plane, a 4 plane, all the way up to an N minus 1 plane. And then, so once you have this, you, of course, have limit maps into RPN minus 1. You just take the first factor, and then the last factor is a map into the gross monion of N minus 1 planes. So the project of a Nassau property is just sort of like a restriction of the existence of this limit map. And another sort of setting which we haven't talked about, but we've talked about, various people have talked about ping pong. And going back to teats, you can play ping pong in SLNR. And if you start with a finite collection of bi-proximal elements, which are sort of in general position, which means that the attracting line of one should never lie in the repelling hyperplane of the other. Then if you pass the high enough powers, the thing they generate is projective Nassau. And if you can play ping pong on the RPN minus 1, and that ping pong construction produces a canter set. And that canter set is just the image of the limit map. So all that stuff is playing to you. If you want, and if you want to get the map into the gross money of N minus 1 planes, you just work on the dual projective space, which you can identify with. The projective space of our N dual is naturally the gross monium. So you can sort of use this duality and win. So there's these various examples. And so now if you start with a general semi-simple Lee group and a pair of opposite parabolic subgroups. So in SLNR, that might mean P plus is the stabilizer of a partial flag, and then P minus the stabilizer of a dual flag. So in our case, P plus was the stabilizer of a line, and P minus is the stabilizer of a hyperplane. And then your limit maps go into the associated flag spaces, G mod P plus and G mod P minus. G, SLNR minus the stabilizer of a line is exactly RPN minus 1, et cetera, et cetera. And then you can do this whole formulas. And I think I'm going to skip over this. But you can talk about the Anasov section of the flow. And this is a little slightly different formulation, but it's really morally the same thing. So you can do a bunch more examples. So for instance, if you can redo the rank one theory. And if the rank one theory, the only parabolic subgroups, there's just one of them up to conjugacy. And that's just the fixed point of a point in the boundary. So in PSL2C, this is just the set of upper triangular matrices. This is another thing that really bothered me coming from rank one. The parabolic subgroup does not consist of parabolic elements. Still troubling to me. It contains all elements which fix infinity. So the set of the upper triangular matrices, you can think of this as all isometries of H3 preserving the point in infinity. And then it turns out to be an Anasov is exactly equivalent to being convex co-compact. So there, we've done nothing new. We just have some fancy formalism for thinking about the objects we had before. But it tells you that this is, again, more confirmation. This is the right way to generalize convex co-compact representations. If you're generalizing convex co-compact representations and you restrict to PSL2C, you better get convex co-compact things, or you probably don't have the right generalization. And the hitching representations are Anasov with respect to the set of upper triangular matrices, which is the minimal parabolic subgroup of PSLNR. Another example, another thing which I would guess is an example, though I don't know for sure, is we saw in Pepe's second talk that he constructed these mere Schottky groups. And these limit set was a cantor set of projective lines in CP3. Well, what is that? That cantor set should be, so that means they should be Anasov with respect to a stabilizer of a 2 plane in C4. And then your limit map would naturally go into the space of projective lines in CP3, which is G mod P plus in that setting. So maybe that's an example, too. Just is sort of striking to me, so that probably should be an example. But then there's also this theorem of Guichard and Beinhard, which I've leaned on heavily in my own work, which says that in the end, you can always just think about projective Anasov transformations. I mean, you start with any old Lee group, and you start with any old parabolic subgroup, then you get some representation called the Plukar embedding of your group into PSLNR. And a representation from gamma into G is Anasov if and only if eta composed row is projective Anasov. So in some sense, if you're scared of your current Lee group, just embed your Lee group in SLNR in the right way, and you can work there. And so this is actually a trick which they make extensive use of in their work. It's quite powerful. And you get all these very general things. With this general definition, the representation is discrete faithful. They're quasi-isometrics. The images of each element is proximal. Whatever that means with respect to your parabolic subgroup, they're stable. And the action of outer automorphism group is probably just continuous. Oh, I forgot to say it. When we proved well-displacing, if you think about Francois' proof of Frick's theorem, the mapping class group acts properly as continuously, untightening their space, he exactly used the fact that your representations are well-displacing. If you sort of strip it down, that was the whole proof. And then a little group theoretic fact about that. The group theoretic fact was that you can have a finite collection of curves whose length determine your representation. So you put those two facts together. You have a proof of properties continuity. And well, that proof generalizes to show that the outer automorphism group of any hyperbolic group acts properly just continuously on any space of an ossof representations. So serve another nice confirmation that what we know from the rank one world continues onward. You said it needed to flip it, didn't you? Yeah, sure. Even in, oh, yeah, so well-displacing says translation length grows like word length. So if you have a parabolic element that right away fails, translation length 0, the translation length of the square is 0, the translation list of the third power is 0. So the translation length is growing like 0. And the word length is growing linearly. So they're not growing at the same rate. Or? What is longer than the other? Yeah, well-displacing implies discrete faithful. And properly is continuous, yeah. Well, if you think about it, suppose you have a sequence of things, suppose you're indiscreet. That means that there's some orbit accumulating at the origin, orbit of the origin, accumulating of the origin. So then you have an infinite collection of elements which are moving the origin smaller and smaller amounts. So you have an infinite sequence of elements whose translation lengths are converging to 0. Well, the word length of those elements is going to infinity and their translation length is going to 0. That's bad. But so it's even stronger. So in particular, it implies every group element is hyperbolic in the rank one setting. But it's stronger than that. So if you think about, well-displacing is equivalent to convex co-compact, is what it is, in the rank one setting. What? Yes. So there's a paper by Guichard, D'Alzant, Moses, Moses, and Labrini doing that. It's an awful lot of firepower for that result. They do have a very general discussion of the relationship between quasi-isometric and well-displacing. It's in Bob Zimmer's birthday thing. OK. Or if you take these five or subgroup of a three-manifold five ring over the circle, this is an example. Every element is hyperbolic, but yet it's not well-displacing because it's not a quasi-isometric embedding. OK. And now there are a number of different definitions. One thing I should say is we leaned on the fact that Gamma was the fundamental group of a negatively curved manifold to get our G-sig flow. But Gromov has defined a G-sig flow for any hyperbolic group. And it's a little tricky to work with. And then you can make this whole thing go through for arbitrary hyperbolic groups. But there's some definitions now which avoid that by two teams, one, Garry Toe, Gishar, Kassel, and Veinhardt, another one, Kapovich, Lieb, and Portee. And they both have developed definitions which avoid the use of the G-sig flow and even definitions which avoid the use of the limit map. So you can sort of look at things. So these are two. So you can, well, Fannie and Francois should feel free to yell at me if I've really oversimplified. But to really oversimplify, the work of theirs involves a study of the Carton projection. And they get a criterion in terms of the norm of the growth of the Carton projection, which I would think of as like some sort of hyped up version of the weld displacing condition. It's a fast strengthening of the weld displacing condition. That fair, Fannie? OK, good. And whereas the Kapovich-Lieb-Portee team, they work in terms of the action directly on the symmetric space. And so they have a very, and they look at the action on the symmetric space and the action on its teats boundary, et cetera. So they end up with some similar results but with really different and complementary techniques. So let me now do my favorite part of the second talk in 15 minutes, which is the second talk was supposed to be about the existence of a metric on higher titanar spaces. And in particular, there's a metric on the Hitchin component whose restriction to the Fuxian locus is the Vey-Petersen metric. So it's a generalization of the Vey-Petersen metric into the setting of higher titanar spaces. This is joint work with Martin Bridgeman, Francois Lavery, and Andre Sanberino. And how this work proceeds is we take a higher titanar space, we take a family of representations, and we convert them into a family of flows. And these are so the Hitchin component is converted to a family of flows. And each of these flows is a reparameterization of the geosic flow on a surface. So now we've replaced family representations by a family of flows, or a family of reparameterizations of affixed flows. But the study of a family of anossof flows is exactly the thermodynamic formalism. And so now, once we've made this transition from a representation to a flow, we have this huge toolkit called the thermodynamic formalism, and it allows us to do things like to define a metric. So we can talk, this metric in some sense measures how close the difference between two representations is going to be the difference between these two reparameterizations, the difference between these flows. So I'm not going to tell you all about the thermodynamic formalism, but let me tell you how you take a representation and turn it into a flow. And this is a very beautiful idea, which goes back to Andre Sanberino's thesis. And when I learned this, I was just learning all this stuff for the first time, I thought it was one of these ideas that had been around 100 years, because it was so natural.  So what's the idea of this flow? Super simple. So I'm going to define, by the way, I'm going to put a, I'm going to post the slides in my second talk if you want to look at and see the whole thing. But let me just show you this highlight. So you define a real bundle over boundary gamma, boundary gamma cross diagonal. This ought to remind you of that U gamma tilde of the T, of the unit-tanched bundle of the unit-tanched bundle universal cover. It had this structure. It was boundary gamma cross boundary gamma minus diagonal cross R. So it should maybe not be so surprising what I'm going to produce. It's going to be a repramtrization of the G-sick flow. And now pi inverse of a point x, y is going to be exactly, well, I want to get a line. Well, but my limit map gives me a line. So I'm going to choose a vector v. And I'm going to choose it in psi rho of x. So this is asymmetric, but I could write it more symmetrically. But the symmetric way of writing it is simpler. v and psi rho of x. And I don't want to choose 0. And then I'm going to identify v with negative v. So it's a choice of vector in the line bundle up to sign. Or you can think of it as a choice of norm on the line, whatever you want. And OK, so what is the flow on this space? Psi t tilde takes the point x, y, v. Well, I don't really know what to do with x and y. So I just leave them fixed. Well, what do you do with if you have a point in r minus 0, what do you do? We really can only take v to e to the tv or e to the minus tv. Those are almost the only choices for a sensible flow. So here's our flow. Pretty sure it's e to the tv. I always have to think twice before whether it's e to the minus tv. But those are the only reasonable choices. So we think of this as a flow space where we just flow in the line, preserving the line.  We looked at the flow that takes x, y, t to x, y, t plus s. It's sort of the analog of that.  And what does it do if you take some element gamma of x, y, v? Well, gamma acts on the boundary of gamma. And then you take v to rho gamma v. So again, it's acting by the group on the first factor and it's acting by the representation on the second factor that should feel familiar. And the fact that this makes sense, this is well-defined exactly because psi rho was rho-equivariant. So the equivariant's the limit map coming on here. And so now what I want to do is I want to quotient out. But the problem is I don't know right away that my action is nice. So it turns out that what I first do is show that I've produced a parameterization of the g-sig flow. And once I show that, then I know the action is proper and co-compact. So a little lemma you can show, and it's not hard. I'm not going to do it for you. We don't have time. But there exists a gamma invariant, or gamma equivariant, would be a better way of saying it. And it's going to be helder. Again, that's just the natural setting. If you're used to hyperbolic dynamics, everything is helder. That's the best smoothness for everything. Helder orbit equivalence. So what might be interesting is the question is the helder exponent feels like it behaves a little bit like the Hausdorff dimensional limit set. But there's some results like that. But that's g tilde from, yeah, it's in its infancy. It's in its infancy. So there's a helder, orbit equivalence means it's a homeomorphism which preserves orbits. It doesn't preserve the flow. So it preserves the flow lines, but it doesn't preserve the flow. So you can write this down.  You're going to take y to y. And then you've got to choose how to map t to v. So you just have to write down a formula for it. It's not that hard. So once you've got that, that implies that gamma's action f sub rho is proper and co-compact. So that implies that you can define, you get an orbit equivalence between u gamma and u rho, which is f rho mod out by gamma. So this is an orbit equivalence. And then it's a general theory that if you have two flows, so this flow is a NOS off, so if you have two, so if this flow is a NOS off, then also this flow will be a NOS off. And then if you have two NOS off flows, which are orbit equivalent, one, it turns out as a repramptrization of the other. So now you know that, in fact, this is a repramptrization. Then you can replace this by not, you can use g to construct a function f, which is to say that u rho is the repramptrization of u gamma by f. And so, and also, another nice thing here is if you start off with the Gromov-Gesig flow, and then you don't know that's a NOS off. You don't have this theory of Megalocurve manifolds. You don't know you have a NOS off flow, even in the metric sense. But it turns out you can take this construction and use the NOS off property of the limit maps to prove that this flow, f sub rho, is metric NOS off. So this tells you something about the dynamics that tells you the Gromov-Gesig flow of a group which admits a NOS off representation is a NOS off, which is something we didn't know before, in which we don't know whether that's true for general hyperbolic groups. There's something special about these hyperbolic groups, which are occurring here. No, not natural. No, it's just helter. It's just helter, yeah. You can sort of write that. You do an averaging procedure. So the point is, if you end up with something which is reprammrization by a holder function, it's differentiable in the direction of the flow. But this map, we don't start out knowing it's differentiable in the direction of flow. So there's a little bit of abstract dynamical systems, which is allowing us to make an upgrade in regularity using the NOS offness. It uses the NOS offness in a crucial way.   So now what we get, we get a map. So for instance, of the Hitchin space, h n of s goes into the space of helter functions on u gamma, right? So you start with representation, and you end up with a positive helter function defined on the flow space. And then there's also something called the pressure function, which is associated to the flow. And there's a little fact that p is analytic, and p of negative h rho equals 0 if and only if h is the topological entropy. What? Yeah, right. This whole, so for people from the rank one world, Ruel showed that the house dwarf dimension of the limits set of a quasi-fixing group varies analytically. So this whole project is a big jump up of that. So h is the topological entropy. So this is an analytic function. This tells you that once you can show that f rho varies analytically, which again is a nasty exercise in the Hirsch-Peschube technology, then you can show the topological entropy of this flow varies analytically. But what is the topological entropy of a flow? It's exactly the exponential growth rate of the number of orbits of length t. So you take the log of the number of orbits of length at most t and divide by t and pass to the limit. And the fact that the limit exists at all, normally you just have to take limb soup. The fact the limit exists at all follows from being an Ossoff. So the topological entropy, h is equal to number of limit t goes to infinity. You divide by t. On the top, you take the log and you take the number of orbits of u rho of length less than t. So you can, equivalently, that's going to be roughly the length of the number of closed gd6 in the quotient symmetric space of length less than t. And then there's some very famous work of Sullivan, which says this topological entropy in the rank 1 case is just the Hausdorff dimension of the limit set. So one thing that comes out of this is that if you take any analytically varying family of convex co-compact representations into any rank 1 lead group, the Hausdorff dimension of the limit set varies analytically. So this was previously only known for surface groups in PSL2C and free groups in PSL2C and free products of surface groups and free groups PSL2C. And Semuel Tapie, I believe, proved it with C1 in a fairly general setting. I'm not sure how general. But this is right. So right away, before we even getting to define the metric, just at this place, we've got analyticity of everything, analyticity sort of everywhere. Yeah, and this is just coming on the thermodynamic floor. This is just starting now. Now we start hitting the thermodynamic formalism hard. It turns out that the space of pressure zero functions admits a semi-norm, which is a positive semi-norm or a non-negative semi-norm. And we just pull that back by this mapping. So we pull back from this. We take from here, we go in. This gives us a mapping of the space of pressure zero functions. We have a form here. We pull that form here back onto the Hitching component. And we then have to show that it's positive definite. So that's nasty. When you want to show positive definite, it's kind of nasty. It involves a lot of trace identities in the end. And this is all a generalization. Also a generalization of Thurston's idea that the Vapeterson metric should be the Hessian of the length of a random GD six. So that's another starting point for this whole thing. But I better stop.


What's good with y'all it's your boy Ross back at it again with another video, so I got a chance to check out AEW full gear finally, I know there was people messaging me on Instagram DM DM and me asking me if I've seen it yet. I wasn't able to check it out last night We was busy recording the podcast stay tuned it out on the intercutch page should be dropping sometime this week But I did take some time off today after I got offered to check it out and the one thing I've been hearing on social media is that John Moxley and Kenny Omega match that unsanctioned match was ridiculous and I have to agree I will say about in my opinion of the overall show The first half of the show was okay I think for me the best part of the first half of the show was the young bucks going against Santana and Ortiz man. That wasn't it. That was an entertaining match. I enjoyed that match a lot. I also enjoyed the Pock versus hangman page match that was pretty cool. You know I'm saying I enjoyed those matches and Then everything else I was I was in and out for it. It wasn't that it was bad wrestling It's just I I really wasn't as entertained as I was for for those two matches But then once the Cody match happened with Jericho, I knew that was going to be a Marquis match and it was entertaining. I love it I had a feeling Jericho was gonna retain because he is at this present moment The best thing one of the best things going on for AEW and just one of the best heels Just in wrestling general like in wrestling as a whole not just you know AEW WWE just in the wrestling world. He's one of the better heels right now So it made sense to have him hold the belt for a while He's he's really bringing some prestige to the belt and he's just building up that heel heat But he's also charismatic at the same time. He's also funny. So I think they went with the right decision to having him retain Now what I kind of felt like what's gonna happen and they have been teasing it was MJF Turning on Cody and I think this was the perfect time for him to turn on him cost him the match He basically threw in the towel Caustic cost Cody the matching from Cody stipulation He won't be going for the AEW championship ever again So we'll see how long they last but hopefully he stays true to his word on that because I feel like He doesn't have to have the strap for people to be invested in his matches And I think this next few him versus MJF is going to be entertaining as hell because they both were great on the mic MJF is better as a heel and he's a heel that knows how to speak He knows how to heal it up and the fact that I don't know if this was a plant at the show When MJF is rock walking up the ramp at the gave Cody a severe nut shot after he's walking up the ramp Supposedly a fan through a beer at him. I think that may have been a plant I'm not sure. I think they may have been a plant if it wasn't a plant. That's good That's it's not good that the guy did that if it wasn't a plant But that's good to get nuclear heel heat real heel heel heat MJF just smirks, but I think that was a plant either way is a good visual People are gonna love to hate the guy can't wait to see this Cody Rose MJF feud So that's what I take out of that match match was entertaining and enjoyed it for what it was But the real bread winner the match that just had people Divided on social media when they were talking about it. There's none other than this unsanctioned match between Jericho and Kenny Omega You see the thumbnail. I really wish I could have did like a live reaction on my channel But one I knew it was gonna get demonetized because there was there was a lot of blood and violence So I knew it was gonna get demonetized and to AEW is probably gonna take down a clip I'm pretty sure that we're gonna take it down So I was like, you know what? Let me just give my thoughts and opinions on it just just on a just Keep it that way for this particular video because I already knew there was no way I was gonna be able to post this on the channel, but I will say This is one of the most Craziest unsanctioned matches. I've seen in a while.   They earned that paycheck.  There's no way I could see myself going through those type of spots like that I just couldn't see myself Falling in a bit. It was pretty much like a bit of bomb wire and they both fall through it Gave me the ECW one night stand vibes. Oh my god. Yo, that's Yo, I don't even know what to say that match was intense The way Kenny Omega sliced, you know the webbing between your fingers He's he had some glass and he was slicing John Marksley's inner webbing between his fingers.  Oh my god He was scraping it like he stomped on the bat while it's on his back and stomped on it Pushed it in even more. I'm like, yo, and then the fact that he got his revenge.  This is this was his MO I love hardcore matches. I can take pain I'll enjoy the pain like this is what he's known for so the fact that people wouldn't even make that comment They don't even know this is this is You can't fake this birth like they literally went out there put their bodies on line for our entertainment and I Have to say man. They ended off to show with a bang because I was just an intense a tense Battle man, I don't even know what to say. I'm pretty speechless now watching it after everyone else has been raving and Talking about on social media some people saying I couldn't watch it too much I had to turn away and some people are like, yo, this was the most insane match I've seen in a very long time and They went to war and it doesn't even matter like on the record books It doesn't even matter. He did all that just to beat Kenny Omega in a match that won't count to anybody's record Insane it keeps both rec a wrestler strong in the sense of in the kayfabe sense of the records But other than that John Moxley came out on top.  My god. They at the at this point He's tearing up the ring John Moxley cut up the ring you expose the wooden board of the ring and those I don't know even if you fall just with all the ring like set up like with the padded protection That still hurts like taking a suplex or taking a dive from the top rope That still hurts even when you hit the ring now you take away the padding you take away the covering and you're just left With the wood just the wood frame. Oh my yo My god Kenny Omega went for the ultimate ultimate dub He missed and when he missed bro, he missed hard face first into the wood.  You're hitting that if it's wood You're hitting that concrete. You're hitting that table.  That's legit Spots like that are legit. That's why wrestlers They have my utmost respect when it comes to doing sick and insane spots Those two matches alone. They are unsanctioned match in the aw championship match alone Definitely bumped up this show A couple points for me if I had to give the show a rating out of one of 10 one through 10 I'm gonna give it a cool Seven man. This was a seven show. It could have been better. I felt like the women's championship match could have been better um It had some down moments for me personally, but The ending of the show the ending half of the show was Just entertaining Wrestling at his best and this the last match was just if you want to see two people try to kill each other On a pay-per-view. This is a match you watch. This is a match. I could see myself watching as a kid But even then as a kid, I didn't see no crazy stuff like this like yo, this is on some next level Intense stuff, man. So I enjoyed it I want to know what y'all think did y'all enjoy the unsanctioned match What was the spot that made y'all cringe the most in that unsanctioned match for me Is them jumping off the stage into the barbed wire Letting like the barbed wire like bed. I guess you can call it a bit or a spider's web like Yo, that was so much barbed wire name They literally just laid in it because they had to get help to get out of it Like that was so sick. That was the spot that made me cringe the most Let me know what spot made y'all cringe the most and let me know if y'all were looking forward to uh AEW dynamite this wednesday if y'all are gonna check it out to see What's gonna happen what what fuge about to spark up on if y'all looking forward to mjf and cody rhodes fused Let's uh a few let's let's talk about it. Let's discussion in the comment section, man But aw full gear give it a seven out of ten very enjoyable WWE take notes That's how you do an extreme match. You don't do an extreme match Like an extreme rules match or anything goes match an unsanctioned match You only use kendo sticks, bro. That's not an unsanction. That's not an extreme rules match That's just a match with a couple kendo sticks And some chair shots to the back, bro. I get why they can't do it their t Their pg rated so I get why they can't do it But they need to get rid of the extreme rules pay review because nothing will ever come to As extreme as what aw just pulled off last night Intense enjoyed it. Let me know what y'all think and I appreciate y'all kicking in with me I'll see y'all on the next one. Peace


And now, for your strong Europe spotlight. More than 400 pieces of equipment from the European activity set were loaded onto a cargo ship in Bremerhaben, Germany. The equipment, including Abrams tanks, Bradley fighting vehicles, and Humvees, is set to be used in next month's exercise Triton Juncture in Zaragoza, Spain. The equipment will enable soldiers from Fort Stewart to support the exercise more efficiently and effectively. The Army right now is working to increase interoperability between its NATO allies. As a greater part of the mission, we're moving a lot of equipment around Europe right now. We're able to move equipment by land, sea, or air freely throughout Eastern and Western Europe. The European activity set is a combined arm set of equipment that is prepositioned in Europe, saving the US Army the money and time of shipping equipment with each unit rotation. For more information on the Army in Europe, visit er.army.mil.


Hi there and welcome, I am The Technician Projects and today's tutorial is going to show you how to add title screens and edit them in iMovie. So we're going to start our project today off in iMovie and we're going to select a title and we're going to click and drag it into our edit window. So if you double click on your title in your edit window, you'll be able to change the text on the edit window. So I'm just going to move the top line and type in the bottom part, so click and drag. Click and drag and I'm going to type in this is a title. Now if you click and drag over it and highlight it, you can change your text from the font settings. So I'm going to select a font and you can change your text size within that setting too. I'm just going to change that quickly and press the tick button when you're done. I'm just going to drag another title screen in now. Click and drag this end to one. As you can see this is non caps lock. This is a normal text one but this is a title screen. And again I'm going to hide like this and change the text to a regular text. Just this one here, the medium size one. Change the size and press tick to say you're done. If you enjoyed the video today, please leave a like rating and subscribe for more. Thank you for watching and see you next time.


I'm talking to the people. Black lives matter. Things go get much more gooder. Things go get much more better. We stand up and fight for all the black blood that splatters. Victory will be ours one day. Sing it with me. All people don't give up. We are really out here to let everyone know that we are here and we exist on this campus. We exist in the world. Just people who you can step on and be thrown away.  So what we're doing now is we're demanding justice.  We still get ignored by white professors in the classroom and we are still only 2.8% of our entire campus population. Racism lives here and a thousand other college campuses across the country. An act of terror on neighborhoods of color. On the father known to his community as a peacemaker. But to you a habitual offender. I never knew a rap sheet justified from these brutality. Tell me I'm crazy but freedom of speech still has a price. Defending your rights can cost you your life. You can say people of color, black students feel comfortable here because I definitely don't. When I walk the campus I feel like I'm being stirred up by every single white body I see. And not because of how I look, it's because of my skin color. That's pretty much it. We're still experiencing the same lack of recognition the lack of humility, the lack of treatment as human beings. We're here today to let everybody know that the game has changed and the script has flipped. Black power and resilience will triumph and we will succeed in a system that will set up for us to fail. Black lives matter from Berkeley to Mizzou, Ache. I'm bringing to people that struggle everywhere You won't ever win if you don't ever dare If you're feeling ripped off and you want to bear share The victory will be yours one day Sing it with me All people On the top I know we're going to


On December 2, 1805, Napoleon Bonaparte won one of his greatest victories in the Battle of Austerlitz. The Allied forces attacked the French line early in the morning, and for an hour, the army sparred for control of a small town. While the French line held the Allied forces at bay, Napoleon had 20,000 of his men march up a hill to attack the enemy's center. As they marched, the morning fog concealed their movement. But with perfect timing, with the men at the top of the hill, the sun broke through the fog and revealed to the French soldiers bearing down on the Allied troops. Caught by surprise and seeing the French infantry charging toward them, the Russians fled. The battle became famous, but the sun of Austerlitz became legendary. As the Confederate army at Pittsburgh landing lined up to strike the unsuspecting Union forces, General Albert Sidney Johnston described the scene quote. The two armies lay face to face. The federal host, like a wild boar in his lair, stirred, but not aroused by the unseen danger. It's foe, like a panther hidden in the jungle, tense and wait to spring. Long before dawn, the forest was alive with silent preparations for the contest, and day broke upon a scene so fair that it left its memory on thousands of hearts. The sky was clear overhead, the air fresh, and when the sun rose in full splendor, the advancing host passed the word from lip to lip, that it was the sun of Austerlitz. I'm Chris Calton, and this is the Mises Institute podcast, Historical Controversies. In the last episode, we looked at the preparations for the Battle of Shiloh, which was about to become the bloodiest battle the United States had ever seen up to that point. In today's episode, we will look at the first phase of the battle, beginning with the early morning Confederate ambush. After the battle, Beauregard would describe the opening ambush as an alpine avalanche, and the northern press would run stories about how the Confederates were able to sneak up on sleeping Federals and bayonet them in their tents. Neither description was accurate. The Confederates were running behind schedule, and their advance moved slowly. The troops were green, and the forest was dense with foliage, so it was impossible to maintain the lines. When the first shots were exchanged by skirmishers, you might think this would have alerted more of the Federals that the battle was opening, but most people thought little of the gunfire. General Sherman and Prentice, whose divisions were the first to face the Confederates, thought the skirmisher gunfire was hardly worth investigating. Colonel Everett Peabody, one of the brigade commanders in Prentice's division, thought that something more serious was taking place. In fact, Peabody was so on top of things that while the Confederates were still moving in a position at around 3 a.m., he was already up and trying to prepare for an attack that none of his superior officers believed was coming. Earlier in the night, Prentice had laughed at Peabody for saying he believed the Confederates were preparing to attack, so on his own, he sent out a patrol in the middle of the night. And when he sent the patrol out at 3 a.m., he shook hands with the men and said to the regimental officers, quote, I cannot say anything more to General Prentice, but he will soon see how near I was right. I must do this upon my own responsibility, but I will not live to receive censure or credit for doing so, end quote. So when the shots were exchanged in the morning and as the gunfire grew consistently heavier, General Prentice was angry with Peabody. He told him, quote, you have brought on an attack for which I am unprepared and I shall hold you responsible, end quote. Peabody responded, quote, General, you will soon see that I was not mistaken. So Peabody was the only union leader who believed that the Confederates were about to attack. And when he sent out his patrol to investigate the ambush, his commanding officer actually blamed him for bringing on the attack he was trying to warn about, if you can believe that. So the Confederates did still catch the Northerners by surprise, but not by as much of a surprise as they might have hoped for. And that was because of Colonel Peabody, who bought the Union Army time to form their lines before the Confederates could truly ambush them. By the time the Confederate line made it up to the Union camp, the Federals had infantry and artillery lined up and waiting on them. As the Confederates moved in, one soldier described the initiation of combat, quote, We trampled recklessly over the grass and young sprouts. Beams of sunlight stole a thwart our course. A dreadful roar of musketry broke out from a regiment adjoining ours. It was followed by another further off, and the sound had scarcely died away when regiment after regiment blazed away and made a continuous roll of sound, end quote. Another soldier gave this description of the Federals first coming within his line of sight, quote, I at last saw a row of little globes of pearly smoke streaked with crimson breaking out with spurred of quickness from a long line of bluey figures in front and simultaneously there broke upon our ears an appalling crash of sound. The series of fuselods following one another with startling suddenness which suggested a mountain upheav'd with huge rocks tumbling and thundering down a slope. In and again these loud quick explosions were repeated with increased violence until they rose to the highest pitch of fury. All the world seemed involved in one tremendous ruin, end quote. Because Peabody had been on the ball regarding the attack, the Confederates met with a fiercer resistance than they expected. As Southerners swarmed in around Peabody's line, his Union soldiers jumped up from where they had been taking cover and let off a devastating volley of close range fire. The Confederates were raw and this attack started a panic in some of them. As several Southerners started to run back screaming, retreat! Their panic spread like an infection to the other Southerners who broke line and started a veritable stampede of retreating soldiers. The Southern Generals were able to corral them back together and reform their lines. But because Peabody was the only officer who anticipated the Southern attack, his men had the unfortunate duty of holding off the Confederates while the rest of the Union army, which was taken by surprise as they were sleeping or eating breakfast, could jump in to join the battle. So had it not been for Peabody, the Southerners may very well have pounced on the Northerners while they were sleeping or eating. Instead Peabody's men bought the rest of the army time to grab their weapons. As one soldier, Corporal Leander Stillwell described the surprise at the sound of combat quote. Every man sprung to his feet as if struck with an electric shock and looked inquiringly into one another's faces. There was a low, sullen, continuous roar. There was no mistaking that sound. Suddenly, to the right, there was a long, wavy flash of bright light. Then another, and another. It was the sunlight shining on bayonets, and they were here at last. A long, brown line with muskets at a right shoulder shift, an excellent order, right through the woods they came. End quote. For soldiers like this, who were genuinely caught by surprise, they didn't know how to even join the fighting at first. The smoke was so thick that they couldn't make out any of the soldiers involved to know who to shoot at. As Stillwell's commanding officer started frantically yelling, Shoot, shoot, why don't you shoot? Stillwell replied, Why, Lieutenant, I can't see anything to shoot at. And his officer answered, Shoot, shoot anyhow. Stillwell was only 18 years old, and it was in this fighting that he saw his first comrade killed, and it haunted him. He later wrote, quote, I never shall forget how awfully I felt on seeing for the first time a man killed in battle. There was a man just on my right behind a tree of generous proportions, and I somewhat envied him. He was actively engaged in loading and firing, and was standing up to the work well when I last saw him alive. But all at once, he was lying on his back at the foot of his tree, with his leg doubled under him, motionless, and stone dead. I stared at his body, perfectly horrified. Only a few seconds ago, that man was alive and well, and now he was lying on the ground, done for, forever. The event came near completely upsetting me than anything else that occurred during the entire battle, but I got used to such incidents in the course of the day, end quote. The smoke was so thick that nobody could see who they were shooting at. On the Confederate side, Brigadier General Adley Gladden rode horseback toward the front of his line, so that he could try to get a better survey of the Union position.  A number of his men rushed toward him, and they immediately hauled him off the field where he could get his arm amputated. Gladden would die a few days later. He was hardly the only Confederate to suffer. The testimony of one Southerner paints a vivid picture of the morning combat quote. We could hear heavy missiles whizzing around and above us. Some of them too were distinctly visible. One great solid shot I shall never forget. As it came through the air, it was clearly seen. Captain Footsawd, as at Ricochet, and spurred his horse out of the way. Lieutenant T.J. Dupri was not so fortunate. This same shot grazed his thigh, cut off his saber hanging at his side, and passed through the flank of his noble stallion, which sank lifeless in his tracks. It also killed a second horse in the rear of Lieutenant Dupri. And finally, striking a third horse in the shoulder, felled him to the ground without disabling him, and not even breaking the skin. The ball was now spent. My own horse, Brimmer, and the excitement and joy of battle raised his tail high, and a cannon ball cut away about half of it, bone and all. And ever afterwards, he was known as Bob Teld Brimmer. Many solid shot, we saw striking the ground, bouncing like rubber balls, passing over our heads, making hideous music in their course. End quote. But the Union was taking the worst of it. One 16-year-old soldier recounted his experience. When the battle started, his regiment didn't even have ammunition. When they realized they were being attacked, they had to fall back to get their supply of ammo before joining the fight. When they finally did, the soldier described the fight quote. There was a crash of musketry, the roar of artillery, the yells, the smoke, the jar, the terrible energy.  Somebody calls out, everybody for himself. The Confederates were sweeping the ground with canister. The musket fire was awful, the striking of shot on the ground threw up little clouds of dust, and the falling of men all around impressed me with the desire to get out of there. The hair commenced to rise on the back of my neck. I felt sure that a cannonball was close behind me, giving me chase. I never ran fast before, and I never will again. It was a marvel that any of us came out alive. End quote. Peabody was trying to hold the Confederates off with everything he had. He'd already been shot four times, but he stayed on his horse, encouraging his men in trying to maintain the line. He sent his aide to check on a regimen at another end of his line. When the aide returned, he found General Peabody, the only Union commander who did anything to prepare for the ambush, lying across a log where he'd fallen off his horse, killed by a fifth bullet that hit him directly in the head. The Confederates continued to move forward, now charging for Prentice's battery. General Prentice had ridden up by this time and given orders to turn the cannons 90 degrees to hold off the attack on the Union right. The battery commander, Andrew Hickenlooper, later complained that it was a ridiculous move to attempt, quote, in woods filled with dense undergrowth, horses rearing and plunging and dropping in their tracks, which the enemy immediately took advantage of by a direct charge on our now exposed and defenseless left flank, end quote. Hickenlooper later wrote that he had just given the order to carry out Prentice's order, when there comes a crashing volley that sweeps our front as with a scythe, a roar that is deafening and the earth trembles with the shock, end quote. The result was a massacre. Union artillerists went down and the rest fled. Out of the 59 horses in the battery, every single one was either killed or wounded, including the horse that Hickenlooper himself had been riding. The Confederates captured four cannons from their charge. Peabody's troops did a good job of slowing the Confederates down while the rest of the Union army got their act together, but they couldn't hold off the massive line of charging soldiers. The Union line was collapsing. One Confederate later wrote, quote, my blood run cold and I felt my frame tremble to see the dead and hear the groans of the wounded. But in a few minutes, they ran like wild turkeys and left their tents and baggage and oppositions, end quote. When the Confederates overtook the Union camp where the men had been preparing breakfast though, their advance lost momentum. Instead of chasing retreating Yankees, the Southern soldiers, famished, stopped to help themselves to the abandoned food that's still sat over the fires. They also helped themselves to whatever they could find in the Yankee tents. One soldier admitted, quote, I had the momentary impression that with the capture of the first camp, the battle was well nigh over, end quote. Another soldier thought the plundering was an embarrassment later writing, quote, it was disgraceful to our army to see men by hundreds plundering while their brethren and arms were being struck down every minute, end quote. But the officers did what they could to keep their men pressing forward. Albert Sidney Johnston himself showed up on the scene and urged the soldiers onward. Officers were generally more disciplined when it came to plunder than the regular soldiers but Johnston was appalled when he found one of his lower officers, a lieutenant, taking part in the plunder of Union tents. The lieutenant came out of the tent with an arm load of booty only to find Johnston staring disapprovingly down at him from his horse. None of that, sir, Johnston scolded. We are not here for plunder. I don't know how the young officer reacted but his despondency must have been shown on his face after being scolded by such a high ranking general because Johnston quickly took pity on him. Softening his hone, he took a small tin cup from the lieutenant's pile of booty, smiled, and said, quote, let this be my share of the spoils today, end quote. Johnston kept the tin cup with him through the rest of the battle, sometimes raising it over his head as if it were a sword or clanging it against a soldier's bayonet as he gave orders and encouragement. For civilians, the fallout from the battle started almost as soon as the battle itself began. Nine-year-old Elsie Duncan, whose father owned some of the land that the battle was being fought over, went outside to her family's garden on the morning of the sixth. It was a beautiful day. The rain hadn't come yet. She couldn't see the battle but she could hear it, quote. When I heard the cannons roar and the guns popping and the horses screaming, it seemed as if everything was lost. It was terrible heart-rending. It was too much for my poor little heart to stand. When I went back into the house, I saw that my father had come home. He was bending over the bed and mother was tearing a cloth into strips. I went into the room and saw a rebel soldier boy all shot up. I gave one look and saw it was one of our neighbor boys. I couldn't bear to look at him. When they were through with him, his folks came and took him home. The fighting began at our gate just past the house. As the battle raged, it got further away, leaving dead men and horses behind, end quote. Before the battle, Elsie Duncan attended school at the Shiloh Church. The church was a small log building where the locals gathered for church and the younger children had school. Elsie remembered the last school event they held at the Shiloh Church before the battle. The adults watched as the kids put on a concert with the little girls all dressed in red, white, and blue outfits. Duncan participated in the event, quote. I spoke a bit about Abe Lincoln's goose that laid the golden egg. That was the South. And when he squeezed too hard, it ran away. Then another little girl came out waving the Confederate flag and the girls sang Dixie. Then everything was in an uproar. Men were throwing their hats up, yelling Hurrah for Jefferson Davis and the Southern Confederacy. Many brave hearts that were there that night went forth to fight that never came back or hobbled home on crutches.  Now on April 6th, 1862, William Tecumseh Sherman and his division of Union troops were surrounding the old log church. While Prentice was trying to keep his hastily formed line together at the Union center, Sherman was having his own problems on the right. Like Prentice, Sherman had received and dismissed reports of a coming Confederate attack. Only a moment after Sherman said, my God, we are attacked. Sherman's orderly was knocked off his horse where Sherman watched him spurt blood from the bullet wound he had just suffered. The attack against Sherman's division was headed by Brigadier General Patrick Claiborne who would eventually earn the nickname the Stonewall of the West for his battlefield performance. As the Confederates moved on Sherman's division, the officers hastily formed a line to meet them. The Federals were assisted by the rough terrain that Claiborne had to lead his men through. They had to move uphill through the wooded foliage and muddy marsh of Pittsburgh landing. For a time, the Union line held off the Confederates well. One soldier wrote, quote, we mowed them down by hundreds and repelled their charge made up into our camp and drove them back over the run leaving many of their comrades behind, end quote. Claiborne led a number of assaults up the hill toward Sherman's camp, but they were all unsuccessful. As dozens of Confederates, including some regimental officers fell, the Confederates were disheartened. One later wrote, quote, the mortification of a repulse in our first regular engagement was extreme. Some wept, some cursed, and others lamented the death of some of our bravest officers and men and not a few drifted to the rear, end quote. But that isn't to say that the Union line was having an easy time. At one point, one of the regimental commanders, Colonel Jesse Appler, panicked. Even as his men were holding off the Confederate advance, he started yelling, retreat boys and save yourself. He took off himself and several of his soldiers followed him, though others stayed to fight, encouraged to stand their ground by other officers. Appler himself would not be found again until after the battle was over. But Clayburn's attack was hardly anything more than a large-scale human sacrifice of his men. They fell by the hundreds and gained no ground. Those who survived fell back. Some of them weeping, others cursing, and all of them traumatized by their first battlefield experience. One Union soldier couldn't believe that the rebels were throwing themselves at the enemy the way they were continuing to charge even as they were cut to pieces. He wrote, quote, if they were willing to do that to themselves, what would they have done to us? End quote. But Clayburn's luck changed when Braxton Bragg showed up with nearly his entire army corps, two divisions compared to Clayburn's single brigade. The only one of Bragg's brigade that wasn't with him was the one led by Brigadier General Adley Gladden, who was being mortally wounded on the other side of the battlefield. So after Clayburn's troops had been cut to ribbons, Bragg's fresh troops were ready to overwhelm Sherman's line. Now Sherman's line was collapsing. One of Sherman's brigade leaders, Colonel Jesse Hildebrand, was 61 years old at the time of the battle. According to one soldier present, Hildebrand, quote, had displayed the most reckless gallantry, end quote. But when his regiments were cut down, he believed they were done, so he rode off and made himself available as a staff officer for General McLernand. What he didn't realize was that out of his 2,500 man brigade, 400 of them were still on the field trying to fight. They were now on the battlefield without a commander. Even though the Confederates now had new battle flags to distinguish themselves from US troops, the uniforms still created troop confusion, much like we've seen in previous battles. At one point, Lieutenant Ephraim Dawes, who left a great account of this battle, went up to a soldier and asked if the man knew where his regiment was. The soldier replied, I don't know, I was captured this morning and just escaped. Dawes replied, come with us. No, the soldier answered, I'm going with this regiment. He pointed to a group in blue uniforms, marching neatly to the beat of their drummer boy. Dawes, recounting the tale later, said quote, it did not seem possible that a Union regiment in such condition could be coming from the battle lined, end quote. Do the soldier, he said, they are rebels. The soldier wasn't convinced, they're not, he said. But as the words left his mouth, the wind blew so hard that the regiment's flag could be better seen. It was the Louisiana state flag. These troops, it turned out, were the Orleans guards, a French Creole regiment that still wore their traditional blue militia uniforms. So Dawes and the small group of men with him, including the soldier that wanted to join the Orleans guard, ran off and found one of Sherman's staff officers. Where's our brigade, Dawes asked the officer. I don't know where anybody is, the officer replied. And as he spoke, he noticed that underneath the officer's brown coat was a gray Confederate uniform. The Confederate officer, facing the same confusion as Dawes, didn't realize he was speaking to a group of Union soldiers but before either side could react. A shot of canister came flying in and shot the officer's horse out from under him, giving Dawes the chance to run off. He only later realized that he was speaking to the Confederate general Thomas Hindman. But Sherman's line was breaking at this point as the newly arrived Confederates swarmed a field like bees, according to one Northerner's description. Sherman ordered his men to fall back.  As one officer described it, quote, they were pouring in by the thousands through the camp, flanking us entirely, end quote. I mentioned earlier that Sherman's division had one detached brigade holding down the left end of the Union line. This was David Stewart's small brigade. It consisted of less than 1,000 men. Some estimates place it at no more than 600 and Stewart's battery had been reassigned to W. H. L. Wallace's division prior to the battle. Prentice's division separated Stewart from the rest of Sherman's division. So this was the meager force that David Stewart had to hold off a Confederate attack of roughly 3,600 rebels. The idea of holding off the overwhelming onslaught of Confederate soldiers was absurd but Stewart and his brigade held their ground surprisingly well before they were finally driven back. Stewart had the respect of his men and as they started to panic and flee, he was able to turn them back toward the enemy, a power that many brigade commanders lacked when dealing with panic-stricken soldiers fighting their first battle. But the standoff, as impressive as it must have been, was all the bloodier for it. Stewart's brigade held their ground for two hours against the Southerners. The experience was horrifying when one private, Robert Oliver, was taking cover behind a log to reload his gun when one of his regiment's officers came up beside him. As Oliver tells the story, the officer quote, "'Nelt on one knee with the point of his sword on the ground saying, "'Oliver, as soon as you get your gun loaded, "'take Ainsbury,' a wounded soldier in the regiment, "'to the rear.' Then he was hit with a canister shot in the head. He hung to the hilt of his sword until his hand came to the ground, bending the sword double. And when he let go, it bounded six feet into the air. That was the last command he ever gave." End quote. But Oliver did as he was told. He found the wounded soldier and helped him up, handing him off to another soldier headed to the rear. As Oliver was returning to the battle, though, he heard a familiar voice call out, "'Robert, for God's sake, don't leave me.'" And Oliver's words, quote, "'I looked back and saw James D. Godwin of my regiment.  "'I said, never. "'Put your arm around me, "'and I will do the best I can for you.'" As Oliver helped carry Godwin to the rear, he could fill more bullets hitting his friend. When he finally dropped him off with a surgeon, quote, upon cutting the shirt off to my whore, there were seven bullet holes in that boy, not yet 17 years old, end quote. Finally, Stuart had enough and he ordered his men to withdraw. But this meant crossing another ravine, forcing the men to expose themselves to enemy fire. As one soldier explained, quote, "'Almost instantly, the ground we had left "'was occupied by swarms of exultant and yelling rebels, "'who now, without danger to themselves, "'poured a shower of bullets down a pond "'and among the fugitives,' end quote." The man who gave this testimony nearly lost his life in the battle. After taking a bullet to the leg, another soldier, Parker Bagley, helped him limp away from the danger zone. With his arms long across his friend's shoulder, quote, a burning sensation passed along my back and we both fell together. The bullet hit crosswise under the shoulder and passed on, killing poor Bagley. Lying beneath him, I could feel his hot blood run down my side and hear his dying groan, end quote. Another Union survivor later said, quote, "'Only the excitement of battle "'could sustain a man in the midst of such carnage. "'As man after man was shot down and mutilated, "'a feeling of perfect horror came over me at times "'and I berated the powers that placed us in such a position "'and left us alone to our fate. "'Can it be wondered at when 43 out of 64 "'of my own company were killed or wounded "'in that short time,' end quote." One of the confederates on the scene later said, quote, "'It was like shooting into a flock of sheep.'" The episode could have been pulled straight out of a horror movie for the Union men involved as they suffered enormous casualties. Prior to the battle, Grant had removed Stuart's battery and reassigned it to another division. So with no battery and a brigade of less than 1,000 men against more than 3,000 Confederates, Stuart managed to hold the Union left until mid-afternoon. When the battle started, Ulysses S. Grant was nine miles away, sitting comfortably at the Cherry Mansion, where he was enjoying the hospitality of William Cherry, a Union supporter, and his wife Anne, who favored the Confederacy. 31 years after the battle, Miss Cherry gave an account of the opening of the battle. She was responding to a letter asking about Grant's condition at the time. She wrote back, quote, "'On the date mentioned, I believe "'General Grant was thoroughly sober. "'He was at my breakfast table "'when he heard the report from a cannon. "'Holding, untasted, a cup of coffee, "'he paused in the conversation "'to listen a moment at the report of another cannon. "'He hastily arose, saying to his staff officers, "'Gentlemen, the ball is in motion. "'Let's be off.' "'His flagship,' as he called his special steamboat, "'was lying at the wharf, and in 15 minutes, "'he, staff officers, orderlies, clerks, "'and horses had embarked," end quote. Grant was in poor condition to hurry to the battle. He had recently fallen off his horse during a rainstorm, spraining his ankle badly enough that he was forced to use crutches while it healed. As he hobbled off the steamboat, finally having arrived on the battlefield hours after the fighting began, he was horrified by what he saw. Wounded men were carried past him on stretchers while wagons of ammunition were racing in the other direction to resupply the soldiers in combat. "'We are whipped, we have been cut to pieces,' one soldier yelled as Grant passed him. Grant, now on horseback, where the crutches couldn't slow him down, raced ahead with his staff while shells landed around them. He found General Sherman near Shiloh Church. It was roughly 10 in the morning, right as the union line was breaking. Sherman looked grim, his uniform stained with dirt and blood. This was the first battlefield meeting between the two generals. One of Grant's aides turned to him. "'General, this thing looks pretty squally, don't it?' Grant replied, "'Well, not so very bad. We've got to fight against time now. Wallace must be here very soon.' He was referring to Lou Wallace, who was on his way to Pittsburgh landing at that moment, bringing reinforcements with him. Grant's recollection of this moment speaks to how much respect Sherman earned at the meeting. Grant wrote, "'This point was the key to our position and was held by Sherman. His division was wholly raw, no part of it ever having been in an engagement. But I thought this deficiency was more than made up by the superiority of the commander.'" End quote. Grant asked Sherman what he needed and Sherman replied that his men needed more ammo. The ammunition wagons were already on their way, Grant said, and he told Sherman that he was making an admirable stand against the enemy. He later wrote when speaking about the day's fighting, quote, "'During the whole of Sunday, I was continuously engaged in passing from one part of the field to another, giving directions to division commanders. And thus moving along the line, however, I never deemed it important to stay along with Sherman. Although his troops were then under fire for this first time, their commander, by his constant presence with them, inspired a confidence in officers and men that enabled them to render services on the bloody battlefield worthy of the best of veterans." End quote. This is undoubtedly an important moment in the friendship that Grant and Sherman would form during the course of the war. Sherman had won Grant's respect. The union line was breaking, but Grant was now in the field and reinforcements from both Lou Wallace and Don Carlos Buell were expected to arrive at any moment. But the fighting had only just begun and it would become even more intense as the day wore on. We will continue this story in the next episode. Historical Controversies is a production of the Ludwig von Mises Institute. If you would like to support the show, please subscribe on iTunes, Google Play, or Stitcher and leave a positive review. You can also support the show financially by donating at Mises.org slash SupportHC. If you would like to explore the rest of our content, please visit Mises.org. That's M-I-S-E-S dot O-R-G.