 Uh, hi guys. So yeah, uh, I'm Andrew and I am just a senior at Amherst College undergraduate institution. Um, so yeah, and I'm here to tell you about enumeration of formative classes on discrete Morse functions. So the contents of the talk will be just kind of a background of where discrete Morse theory came from. It started off as well. It's still Morse theory, it's still a thing. It didn't start off as with Morse theory. But, um, so these are the origins and then we're going to talk about discrete Morse theory. So just some of the basics and how it came to be discrete Morse functions, what they are and how they differ slightly from Morse functions and, um, what the equivalent classes are precisely and the approach to the problem, which I'll introduce at one point, which is counting the equivalent classes. So smooth Morse theory was developed back in 1925 and it was popularized by John Milner. They studied smooth real valued non degenerate functions on manifolds. Um, so these Morse functions alone encoded valuable topological information, so it told you it's diphyomorphism type. Um, it told you it's, yeah, diphyomorphism type, homology, um, a lot of valuable stuff, buddy numbers and so on and so forth. Um, and one, for instance, of a nice, what is it, statement from smooth Morse theory that we're going to see an analog of it and discrete Morse theory is this right here. And what is it? So smooth Morse theory having, what is it, having a Morse function on a manifold, uh, gives you the ability to get a cell decomposition of that manifold, uh, quite easily. Um, so yeah, so just kind of remember this for now because it's going to pop up in a different form and discrete Morse theory. And this is precisely why discrete Morse theory came to be, uh, because we had all these really nice results in the smooth case, but we wanted, uh, to get similar, uh, things when we're talking about messier structures like CW complexes. So what is it? So discrete Morse theory came quite a bit afterwards. It came in 1998 and it was developed largely by Richard Gorman. So again, it was kind of mostly to bring all these nice results about smooth manifolds into talking about regular CW complex, uh, regular CW complexes. And all, uh, regular means is that what is it? Um, is that for each cell, uh, the complex, uh, each cell in the complex, the attaching map is a homeomorphism onto its image. So, but for simplicity and just to make the statements a bit cleaner, uh, we'll stop talking about general CW complexes and we'll just talk about some partial complexes, which are an instance of a regular CW complex. So let's see. So a discrete Morse function. Um, so recall that smooth Morse functions were just, uh, what is it? Morse functions in the smooth case were just smooth functions on a compact manifold that were non, had non degenerate critical points, meaning the, the Hessian matrix had a non zero determinant. Um, in this case, it's a bit uglier. Um, so for each K dimensional face, um, in the complex, uh, there is at most one, the higher dimensional face that contains it, uh, that has a smaller value at that point. So, uh, right here, the function is the, uh, associates to every cell in the complex, a real number. And the second condition, which is kind of just the backwards of the first one for every lower dimensional face that is a boundary of it. Um, the value there is larger, uh, and there can be at most one of these cells. So this is a very messy way to think about them. And it's a bit clunky working with, but soon kind of like how in the smooth case, so smooth Morse theory also allows you to talk about gradient like vector fields for, uh, on a manifold, a discrete Morse theory gives you a similar thing to talk about, uh, a discrete vector field on a simplicity complex. So we'll get there soon. And we won't really have to think about this clunky definition of assigning a real number on every cell. But for now, this is what the actual definition of a discrete Morse, uh, discrete Morse function is. Um, furthermore, kind of like how in smooth Morse theory, we care about critical points. So the, the, what is it? The minimums, maximum saddles, and so on. Um, discrete Morse theory has a similar analog. Uh, we say a cell is critical, uh, in the complex. If essentially around that complex, uh, the values of the values the function takes, um, corresponds with the dimension basically locally. So locally at a single cell, uh, the function corresponds with the, uh, with the dimension. Um, and here is, for instance, a really simple function to keep in mind, just knowing that like discrete Morse functions do exist on any complex, uh, just assigned to every single cell, its dimension at that point, in which case you would get that actually every single cell in your complex is critical. So to see what a discrete Morse function looks like, well, here's an instance. It's not necessarily pretty. It is just like great with numbers, but this is actually a Torus and this is a discrete Morse function on the Torus. And if you were to check on any one of these cells, you'd find that those conditions that were listed, uh, were indeed satisfied. Um, and if you look really closely for a really long time, and I won't put you through that, but you might find that for instance, like, uh, what is it? So if we're looking at vertices in this, um, that it seems like this vertex here is critical because, um, because it satisfied that condition. And it's also a sync of the underlying, uh, skeleton, the underlying graph. And we'll see that pop again soon when we're talking about gradients. So here's a discrete Morse function, but again, we won't have to think too much about these numbers for too long. So what is it? So just to, again, uh, show how, like, how, uh, Forman was able to bring all these analogs, all these nice theorems into the discrete Morse theory case. Um, what is it? So here's one thing that we saw earlier, uh, popping up again in this case in the case of CW complexes. Um, what is it? So if the, if these are the critical faces, uh, after we put a function on a complex, and these are the dimensions associated with them, then it's homotopic to a complex that could be decomposed, uh, with those respective cells. Um, and so here's now the equivalence relation we care about. So in a lot of Forman's work, he re, he keeps using this one equivalence relation and that's, what is it? He says that two cells in a complex are Forman equivalent, uh, sorry, not two cells, two complexes, sorry, my mistake, two discrete Morse functions on a complex are Forman equivalent. If for every pair of simplices, essentially the inequalities match up between the two simplices. So if you put them next to each other, if one is less than the other on one side, the same thing happens on the other side. Um, and this is used for cleaning his proofs, and he seems to care a lot about this equivalence. Well, then that leads us to the question. Well, um, if we fixed a complex or he fixed in this case of graph, just so things are a bit more straightforward, how many classes are there in this equivalence relation? Um, and it was fairly messy, but in the end, the answer was actually a bit interesting. So what is it? So now let's try and trip away from looking at these numbers and, um, we'll find a different way to think about these gradients as well. Uh, not gradient, sorry, we'll think find a different way to think about, um, these Forman classes that doesn't rely on the numbers so much. So notice that for any discrete Morse function app, so this is, this just comes out of the the definition of the function. Um, exactly one of these are true at all times, and that's because, um, the condition that was given for the function is actually said, well, there can only be, there can be only at most one of this and at most one of this. So kind of, phrasing that a different way, you get the statement here that says that, well, there is a unique one for this and unique one for this. And what this says in other terms is that non-critical faces with respect to the function can be partitioned into pairs, uh, where the, what is it, where one is a face of another one, and that matters a lot because that allows us to start talking about orientations on the simple show complex. So going back, um, so this is going to get something a lot easier. So going back to this giant tourists, uh, with the discrete Morse function on it, um, essentially says, uh, you can take an arrow and have it point towards, uh, the cell with a lesser number associated to it. So it's a really, really straightforward way of assigning orientation. And this is how we're going to think about Forman equivalence classes because it turns out that, uh, what is it, two functions, uh, sorry, this is a type right here, but two functions on a fixed graph or form an equivalent if and only if they induce the same gradient. So that means we don't need to think about numbers anymore. We just can have to think about these orientations on simple show complex. And when are they giving rise to the same gradient? So in the case of graphs, uh, we now started looking closely at gradients on graphs and wondering, um, what is it and wondering what is essentially the pattern here. So in the case of gradients on graph, there are something does, uh, there are some things that were immediately straightforward. A vertex is critical if and only if it's a sink of the gradient, uh, in edges critical, uh, because we have critical cells, uh, if and only if it is undirected under this gradient. And if the graph contains a cycle, there must be a critical edge on the cycle. And this last one follows just because essentially if we were going around a cycle, that would mean we'd have, uh, the value assigned here is less than the value assigned here, less than the value assigned here. So when you get back, um, that's a contradiction. So what is it? So yeah, so these are very straightforward. And then there was this extra thing we found that if you take an arbitrary directed graph, how do you know it's the gradient of something? Well, it turns out there's just these two simple conditions. There are no directed cycles and no two edges share a tail. So so far it isn't terribly interesting, but it does. What does it get a bit more exciting? Let's see. So if you have a, uh, graph and we wanted to basically figure out how these critical edges were arranged on the graph. So if you consider a set of independent cycles on the graph, it turns out you can prove that, uh, what is it? The psych, uh, sorry, that it turns out you can prove that the critical edges, there needs to be at least one critical edge on every single cycle. And no two cycles can share a critical edge, meaning if you take a set of independent cycles in the graph, um, you'll never be able to find a single cycle where essentially you don't have a critical edge because that's a problem with what we mentioned earlier, because it turns out if you have a gradient on overall graph, you can restrict to a subgraph and, um, and then you still have a gradient on that subgraph. So then we could just keep going down to the cycles. So in this case, uh, the one on the left is indeed, uh, what is it? So if these black edges had arrows on them, so they were a full-on gradient, and these red edges denoted the critical edges, in this case, this is just fine. This could possibly be a gradient of something, uh, where again, these would have, uh, directions. Um, but the red ones are the only critical edges where the one on the right, um, is not, would not represent a gradient because you might notice that this cycle right here does not have any critical edges on it. So in, we would just restrict down to that subgraph and therefore show it's not actually a, um, a gradient of a discrete Morse function. So we need one more thing in order to, um, in order to kind of get closer to this formula. Um, so we have this lemma that tells us how the critical edges are arranged on a graph, and now we want to know how the orientations work on this graph. So there's actually this, um, theorem, which will be very handy, uh, that was proved by Forman back in his original paper that essentially says, um, what is it? Essentially, it says the critical, the number of critical cells of a given dimension bound the Betty numbers and the other characters that can be expressed using the critical cells. So in which case, notice now if we have a tree, so what is it? So the previous, the pairing lemma essentially was telling us that if we put the least amount of critical edges on our graph, uh, and then we in a sense kind of deleted them or ignored them, we'd be left with a tree that we can leave orientations on. So now consider this theorem when we're talking about, um, just a tree graph, in which case the statement on the right would be saying that since the order characteristic of a tree is one, and we now go over to the statement over here, it would say that we have exactly one critical vertex. And if we have one critical vertex, well, if we go back a bit further, that means we have one sink in our graph. So in other words, if we kind of just use some notation to note the set of equivalence classes, we just, um, noticed this, that the number of equivalence classes and number of gradients where you take out the least amount of critical edges, that is B1 many, the number of independent cycles, or of course they satisfy that pairing lemma condition, then we're only looking at what is it? It turns out we're only looking at the rooted spanning trees of the graph. So that's, um, what is it? And that's what the pairing lemma and the weak Morse inequalities are saying together. It's saying essentially take out as many critical edges as you need in order to have no cycles left over, and the next one is saying pick a root. And yeah, and then the order of this, just by Kirchhoff's theorem, is the product of the non-zero eigenvalues of Laplacian matrix. Um, but we can say more than this. Uh, what is it? So it turns out, um, what is it? Given the previous assertion, it's not difficult to see, like, you can just keep doing this. So once you have spanning trees, well, you can keep taking out edges, and now you aren't worried where, uh, where your critical edges have to be, like where the ones that you're taking out are gonna be from. So in which case you're just essentially disconnecting that spanning tree. So what you end up actually counting is, uh, the rooted forest inside your original graph. Um, and now there's, there's actually this theorem from an algebraic graph theory booked by Biggs that says, um, so these are edge subgraphs that we're now counting, and that subgraphs are actually the coefficients of the Laplacian matrix, um, of the graph. So it turns out actually these gradients are precisely the coefficients of the graph Laplacian. Um, yeah. And that's, that's the main result. Uh, so some, I guess, final comments are, what is it? Is there an analog to this lemma for higher order, uh, higher dimensional simplicial complexes, and could the gradients be counted there? And, um, one thing I didn't get to mention, just because it definitely would have taken too long, is it turns out actually the set of, um, so not m sub k, but m gamma. So the set of all, equivalence classes actually forms a simplicial complex in and of itself. It has that kind of structure on it, and that's what's called the Morse complex. Not to be confused with the Morse chain complex is the Morse simplicial complex. Uh, or it's sometimes called the Morse, uh, what is it? The complex of discrete Morse functions. Um, but it turns out that currently that that's really trying to be understood because it also tells you a lot about the topology of not just the graph, but if you were looking at discrete Morse functions on simplicial complexes, it tells you about the topology of that original simplicial complex. So that's currently where discrete Morse theory is, and, um, what is it? And these m sub k gammas that we were counting here, um, turn out to be the f factor of the complex. Um, and currently what's being kind of researched there is the homotopy type of that complex. And uh, and yeah, that's everything. Thanks, Andrew. Um, does anybody have any questions that they would like to ask? Uh, I have a question. Thanks for the nice talk. Um, is there the appearance of the Laplacian, I think is really interesting. Do you think there's some kind of maybe discrete hodge theory type reason that this would come up? I wouldn't know. I didn't look into anything that deep, admittedly, and I don't know hodge theory, which is the second reason why I didn't think about that. But I'm not sure, honestly, I was also quite surprised when I saw the Laplacian show up. I admittedly thought, um, this discrete Morse theory had no connection to the Laplacian. So when it started showing, when it started showing up here with just the product of the eigenvalues, we started suspecting it could show up some more. Um, but yeah, we actually don't know. We didn't look into it. We're very surprised. And yeah, we're just highly curious what the other gradients could look like then. So for the critical points, is there a way of associating an index to them? Um, oh, to the critical points in the discrete Morse theory case. Yeah. Um, so the index that's associated to them is just the dimension of the cell, whereas in smooth Morse theory, it's the whole, like the number of negative signs when you look in local coordinates here. It's just the dimension of the cell that's considered to be the index. Okay. Do you have a reference for the topology of these Morse functions? Uh, what do you mean by the topology of the Morse function? I mean the simplicity complex that you mentioned. Oh, yes. There is, what is it? It's called, so yeah, there is one paper. I guess I can type that in the chat. I can type that in the chat afterwards. It's called, what is it? It's by a paper by someone named Charry and it's just called complexes of discrete Morse functions, I believe.