 Okay, perfect. Great. So with that being said, it's my pleasure to introduce on a lease Kaiser from the University of Michigan, and she'll be telling us, giving us an introduction to configuration spaces and great groups. Awesome. Thank you so much. Yeah. So, I'm on a lease. I use she her pronouns. You can follow me on Twitter, since I guess that's where this seminar kind of sprung out of. I just finished my masters at the University of Michigan and I'm starting my PhD in the fall, and this is kind of a talk meant for a pretty general audience about what I studied during my masters. And that's something I did under the supervision of Dr Jenny Wilson at U of M. I had a little bit of weird health week so like I'm not at the top of my game but you know we're just going to, we're just going to go with it. All right. So, let's talk about what the kind of goals are. Nope, that's not what I had next. See, already off to a great start. All right, a little bit of motivation. So, great groups are an object that were first described in terms of configuration spaces which are much older objects in 1891, and they were formally introduced by Arton in the 20s so they're sometimes called Arden's great groups. And both configuration spaces and brave groups have a really, really broad range of applications that you can do. I looked into a couple of these for my thesis research, although I can't say that I could explain every word that's up on the slide right now. And I'll talk a little bit about that, that most exciting one probably which is robot motion planning at the end. Okay. And yes. All right, so we're going to start out by talking about right so I just said we have great groups and configuration spaces where I start out just by talking about what our what our brain groups and trying to get a feel for that. Yeah, here's my guess. Yes, we will define the brain groups will define configuration spaces and then I'm going to describe the relationships between the two. So these actually are very deeply related objects, but it's not obvious. At first at first glance why that's so and so I'll try to give a good justification for that. And then, yeah. So, alright, what are the breakers. We're going to be talking about both the brain group and the pure brain group. So we'll start with the brain group, which in my talk I'm going to write as B sub n. There are many different ways you can denote the brain group, depending on whose books you read. Okay, so I'm going to talk about the brain group on end strands. Okay, and well. All right, so this is a group and its objects are braids. And so what does a brain look like. It's going to be a collection of strands that cross over each other. And the important thing about these about these braids is the kind of combinatorial information it's telling us about what strand crosses over what and when. So here I've got a braid on six strands right so I'd be six vertical strands. And I'm going to read my braids from top to bottom and I'm going to number the strands from left to right. Okay, so for example this red strand starts as the first strand and ends as the second. And this blue strand starts as the sixth strand and ends as the fourth. Okay, so if you've ever you know braided someone's hair you've made a friendship bracelet, you're on the right track right that's what you should be thinking about when you think of a brain. So, right so I've got my six strands, and then let's just kind of break down the information this brain is giving us. So the first thing that happens great because again I reach from top to bottom is my second strand crosses over my first strand. Yeah that's the first piece of information it tells me. Then my fifth strand crosses over my sixth strand. Okay, my third strand crosses over my fourth strand, and my fourth strand crosses over my fifth strand. So in a little bit we'll talk about how to how to write that down in this in this notation I'm using down here, but this is again just an idea this is the information my brain is telling me it's very combinatorial in nature. Um, so two braids are going to be equivalent if we can continuously deform one brain together without shipping the ends of the strands. Okay, so these two braids on the screen right now are equivalent. And I color coded the strands just to make it easier to see. Alright, so what this first braid is telling me is that I'm crossing my second strand over my first third strand. And then on the right, this braid, you know it's much longer it looks more complicated but if I imagine if I just had strings on a piece of table right. I can take that this top part of this blue strand, and just pull that to the right, and I can kind of pull the red strand to the left and just kind of separate those there. So that really all of the cross crossings that are happening if I just kind of pull those together a little bit is I have my red strand crossing over my blue strand. I can pin down the ends of the strands and I can continuously to form one right into another, and the order they cross over each other that does matter. Then two braids are equivalent. And I realize I speak a little fast I'll pause just for a moment here. Alright, so I said that this is a great group. Alright, so we've talked about what the objects are so let's talk about what the group operation is. And that's going to be concatenation. And so here I'm going to take, if I want to take. If I have a braid alpha and a braid beta. And I want to know what alpha beta is, if I apply the operation with the two of them. The way that I would come up with this new braid is I'm going to take the bottom of my braid alpha. Okay, and I'm going to glue the top of my braid beta to it. Okay, so I'm taking. Did I make pictures now I did not. So I'm going to take the bottom of alpha and glue the top of beta to it. Right so I can see right here, hopefully you can see my cursor is kind of that split. Right so first I do what alpha tells me to do, which is to take the second over the third, then the second over the third again. Then I do what beta tells me to do, which is first over second, and then third over second. So I've got these braids, I can concatenate them and I know when they're equivalent. And so let's talk about kind of how we could write this down so you can give a presentation for the braid group, if you don't know what a group presentation is that's okay. But the braid group, we can write down a list of generators for it. Okay, and so what it means to write down a list of generators is that I can come up with some braids where I can build any other braid out of either a combination of generators or inverses right. So here my generators are going to be my generator, Sigma one would be crossing the second strand over the first strand and Sigma one inverse would be doing the reverse okay so it would be crossing the first strand over the second. So here's an example. Sigma two inverse okay so this involves the third and the second strands and this one is going to be where the second strand crosses over the third strand. Again this is a thing where convention on how you label your generators various from person to person so apologies if this doesn't match up with something you've seen before. They, they mean the same thing that's worded a little bit different. And so I can see right if I've got a much longer braid. I can break it down into its generators right so in this braid I take my second strand and I cross it over the first that's what Sigma one does. Then I take my third strand cross it over the second that's Sigma two right there. And then I take my second strand cross it over the first that Sigma one again. Okay, so I can build any braid out of these generators and their inverses. There are also relations on the braid group so you can give a nice group presentation. So if you are in a group theorist maybe that's something you're interested in. So the first relation is that to generators commute if and only if they're sufficiently far apart. Right, so if I have. If I have Sigma one and Sigma three, that's saying Sigma one says I take my second strand I cross it over my first Sigma three says I take my four strand across over my third rate they're not really involving the same strands, you can you can separate them so Sigma one Sigma three is the same braid as Sigma three Sigma one. Okay, so I can picture taking the second strand here and kind of pulling it down a little bit, taking this four strand here pulling it up a little bit on my screen and I would have the same braid as I have on the right. Okay. So that's one of the sets of relations I have the other one is a little bit longer. And so this one says here for example that Sigma one Sigma two Sigma one is the same thing as Sigma two Sigma one Sigma two. Let's take a moment and see what that means. So Sigma one says take my second strand over my first Sigma two says take my third strand over my second, and then take my second over my first. And really kind of what this is doing to my braid is it's kind of taking this third strand and making it end up in the first position. And then taking this first one and crossing it over. But notice that if I take this blue strand here, right and I just kind of deform it upwards on my screen to maybe over here. And I take this red strand I just kind of pull it to the right. Again because of the way these strands are laying on top of each other, these two are equivalent rates. And it turns out that those relations are all you need to give a presentation for the break group. Okay. Right. So that's the braid group. I've got these strands across them over each other. I can do my operations by Academy. So the pure braid group is a subgroup of the break group. Okay, and it's the subgroup where all of the strands and in the order they started in. Okay. So, you know, this is an element of the break group that that I think we were just looking at earlier. Right and I can see that the first strand ends in the third place, the second strand and second place, the third strand ends in the first place. Right, so I can see that I'm permuting the order of my strands. And if I don't permute the order of my strands if I have a braid where each strand and again the color coded to make this easier to see. If I have a braid where each strand ends in the order that they started in. That's going to be an element of the pure break group. Okay. So that's a big difference between the pure break group and the break group pure break group means strands and where they started. You can also give a presentation for the pure break group, although it's much, much messier and so we're not really going to dwell on it. But again, if you're curious, you probably care that at least the fact that we can give a presentation. So here are generators are a little more complicated, because again we need our braids to our strengths to end where they start. So our old generators no longer work. So now we kind of need these generators that kind of cross like twist two strands together and then put them back where they started. Right, so we have these slightly more complicated generators. And if you're, and it's, these are a little annoying to write in terms of the generators for the braid group, especially when you have a great group of lots and lots of strands. Again, as I said there are relations. They're very messy. They're kind of a pain to work through although I promise if you wanted you could sit down and could kind of take some strings and could play with this and you would see that. Oh yeah I believe that this is true. Although I would be, I would not be able to tell you that these are the only relations that would be too hard for me. So it's not a not an easy problem to show. Okay. Um, so that's the pure braid group. So then let's talk a little bit about a new type of object configuration. Right. Again, I'm going to talk about two types of configuration spaces. So I'm going to talk about an ordered and unordered configuration space. I'm going to denote the order configuration space on end points of a space X as comp sub n of X. And I'm going to denote the unordered configuration space on end points of X is the same but with a bar over top. Okay. Again, there are actually a lot of different conventions and I don't think this is the most common, but it's the one that I use. And these named very differently. All right, so let's start with the ordered configuration space and let's talk about what the definition for this is. So, if I have the order configuration space on end points of a space X, what I'm doing is I'm taking X to the end okay so I'm taking tuples of points and X, but I don't want my tuples to repeat the same value ever. Okay. So, if the, if I have two things in my tuple they can't be the same so let me write a brief example. Let's see. Yes, here we go. For example, one to minus one is in the configuration space on and on three points of our. Okay. But one to one would not be in this configuration space. Okay, and again it's because I now I'm repeating one of my coordinates. And topologically I'm going to give this a subspace of topology so I really am I'm taking us X to the end, and I'm taking the subspace where I'm excluding these. You can you can refer to this as the kind of the heavy diagonals what you're excluding or maybe the fat diagonal. I just like to call the heavy diagonal. So that's what the definition is, and I'm going to refer to a whole tuple as a point in configuration space, and each of the little pieces of that tuple I'll refer to as a particle. Right so particles are points in my original space, but I need several of them to come up with a point in my configuration space so these are the words I'm going to use just to try and keep it clear what I'm referring to. Okay, so a point in configuration space is going to be made up of particles. All right. So let's think about visualizing this. So one way of visualizing this if I wanted to think about the configuration space on two points of, for example, the bill line is I can literally take our two. Right, I'm and I want to look at the subspace where X one and X two are not equal. Right so I'm just taking our two and I'm cutting out this diagonal. Right. And so then if I wanted to represent a point in the configuration space I can just plot it kind of in a way that we're used to. But this quickly becomes not very feasible right if I want to look at four points in our. I don't know about you I can't visualize four dimensional space. So that doesn't really work very well. So when we're working with kind of more dimensions. Another way of doing this is instead of take looking at our squared, I'm just going to like draw my real life, and I'm going to label each particle in there. Okay, so I'm going to mark where the particle, the particle one is, I'm going to mark where particle two is and I'm going to distinguish between those. And if I wanted to represent multiple points, I would just need to make sure that I have that I'm labeling them all very clear. Okay, so this method works a little better for things with more high dimensions. I'm going to use both of these for some examples in this presentation but I'll try to be clear which one I'm using when. Okay. Alright, so two different ways that I can think about my configuration space. Then let's talk about the unordered configuration space of endpoints. Okay. So the unordered configuration space of endpoints. It's a quotient space. So I take my order configuration space and I quotient by the action of the symmetric group. And realistically what that means is that if I have x one. I'm going to say a little more expensive. Let's have, let's have one to minus one. Right, this is in comp three of our pay. And another point in comp three of our I could have one minus one two. Right, and these are different points. Okay. Under the action of the symmetric group, they are going to be the same so they're going to be equivalent. In my unordered configuration space. So I'm quite literally what I'm doing is I'm forgetting the order of my points. Right. So we could also write this. I think some people use this as a convention for writing unordered points, I could write the point as, for example, minus one, one, two, using brackets. Okay, so on order configuration space I'm quotient team by the action of the symmetric group I'm going to forget the order of all of my particles. Ooh, I think I said order points a few times there that was bad I should have said we forget the order of our particle. Okay. So, um, let's look at a couple examples right of maybe the interval that's a nice very tidy space that is easy for me to think about. Right, so if I want to look at the order configuration space on two points of the interval. And then I'm just looking at two poles in the interval where the two coordinates are not the same right so this is very similar to the example we did with the real line. So here I'm using kind of that first method of visualization where I'm looking at I, I square. Okay, and so I have the unit square where I'm excluding the diagonal. For the unordered configuration space, I'm now identifying the points x, y and yx. Okay, so that's actually going to be identifying points across this diagonal, and it's going to be equivalent to taking just one of the two triangles we have before. So, just to be clear, order configuration space we have kind of these two connected components, unordered configuration space we now only have one of them, because we've forgotten a bunch of our points we've, we said that they're going to be equivalent. Okay, let's look at a little more fun visualization so what if I want to think about three dimensions about the biggest that my brain can do. So if I look at the order configuration on three points of the interval. Now I'm thinking about the unit cube, but I'm removing three planes right I want to remove the planes x equals y, y equals z, and x equals z. Okay, so my, my configuration space is is again it's this cube with these planes cut through it. And if I want to think about like what that kind of looks like I can actually pull it apart and there's going to be six different connecting components here. Okay, so, so this is the order configuration space on three points of the interval. And it turns out that if I want to think about unordered configuration space on three points. Again, when I take that equivalence relation, I'm going to end up with only one of my connected components. And this is a very cute thing about when you're looking at configurations of the interval. When you quotient by your symmetric group you end up with just one of your connected components. Okay, so here I've arbitrarily chosen one to shade it. Right. Okay. Okay, I did not choose one to shade in I made someone else make this picture for me I cannot make pictures that's good. So my partner on me made these for me and I'm very, very grateful to them. I believe they did it using blender if anyone is curious about how to make very nice 3D images. Okay. All right. Um, so we talked about this configuration space these break groups. And, you know, I kind of get that like break groups have a bunch of strands configuration spaces have a bunch of particles but like it's not really immediately obvious why we talked about these two things at the same time. And it turns out that the pure brain group is the fundamental group of the ordered configuration space of the plane. And the brain group is a fundamental group of the unordered configuration space of the plane. Okay, if you're familiar with the term fundamental group here you might be kind of trying to roll that through your head right now. You don't know what the word fundamental group means that's actually okay. Very imprecisely, please don't nitpick me on this definition. What this means is that if I have a loop in my configurations order configuration space of the plane that corresponds to a braid in the pure brain group. And if I can continuously to form one loop into the other, then the braids represented are the same. Okay, so loops in configuration space of the plane correspond to brains. And when it's the, the order configuration space corresponds to the pure break group, the unordered configuration space corresponds to break. All right, this is all just words. So let's look at an example. So we're going to think about a loop in just to, to, here we're going to do configurations based on four points of R2. Okay, so here I'm again I can't visualize four dimensional space. So now what I'm doing is I'm taking slices of the plane. Right and I'm putting time is zero at the top and time is one at the bottom and I'm just taking four slices here. I've got these, these order these points in my order configuration space so I can see I've labeled four particles as black, red, blue and green. I realize actually that's not very color vision friendly so I'm later I will connect them. But just so you know that there are four labeled points there. Okay. So the loop in this configuration spaces right is it's, I have these particles they're moving around at no point to two particles occupy the same space. Right, so let's go ahead and kind of fill in what this loop looks like. Okay, and here what I'm doing is I'm kind of drawing in the path that each particle takes. I'm going to be observer I'm going I'm kind of observing from this. I'm observing along the x t plane. And when two strands cross over each other. What that means is that two particles have the same x value at the same time, but they can't have the same y value right because two particles can't be the same and so whichever y value is smaller so whichever particle is closer to me. So I'm going to denote that as crossing over the other particle. Okay, so I'm trying to, I am the observer, I'm looking at the x t plane, and I'm looking at this loop and how these particles move and I'm kind of, you know, tracing the lines that they follow, and labeling these crossings like so. And now that I've done that. Um, this is starting to look an awful lot like a braid. Right, so if I if I really take that I'm going to take out the kind of the planes and everything that that extra stuff. This on the left, this is a break right this is a pure braid on four strands. And you know if if the way that I drew it out earlier is like nicer for you, you can think about it in this way. Right, because this is encoding the information of a break right it tells me that my fourth particle goes over my third, and my second goes over my first and so on. Okay, so this is. This is kind of a visualization of how a loop and configuration space really does correspond to a braid. And so the reason that order configuration space corresponds with pure grade group is that if I think about a loop for this to be a loop and not just a path I want to end at the same point I start with. And in order configuration space that means that each particle needs to end where they started out. Okay, so that corresponds to this idea of the pure grade group. So when I'm thinking about unordered configuration space. Now this point is the same as this point, even though my particles are permuting themselves. Okay, so in order unordered configuration space I can commute my particles along that loop. And that allows me to get braids that are actually in the braid group, not just the pure grade group. Okay, so I can see here right if I do the same thing with a different loop but now in unordered configuration space I can end up with a braid in the break group, not just the okay. Okay, so that's kind of the big thing I want you to cross was just like here's the thing called braid. Here's this thing called a configuration space and here's this really cool way they're related. And it turns out that it's it's configuration spaces of if you look at our or if you look at like our cubed or higher dimensions they don't really have very nice braiding. But if you look at particles on the real line they can't really cross over each other if you look at a loop right they can't ever pass by. And if you think about particles moving in 3D space, any braids you can actually just unwind so there's actually not interesting braiding at least in the context of loops in in higher dimensional space. It's really cool that like the configuration spaces of this plane corresponds to this completely other very combinatorial very algebraic thing. So one application just because it's it's fun to describe it's like what whenever someone's like what math video and I'm like oh I do this it's like really cool and they're like okay but like what it was it for. Thankfully, I've come up with a very succinct answer to give them that is usually very satisfying. So say, I, I work for Amazon, I'm a capitalist, and I've replaced all my workers with robots. And I want to. I have these tracks on the floor that I want my robots to take. And I don't want them to bump into each other right so maybe maybe I've got like a station here, no station here and station here and a station here station here station here. There's various tracks that go between my stations. Okay, I'm drawing a graph right now by the way. Right so maybe I've got some some different paths that my robots can take different station to station, and maybe I've got like three, three robots, so let's call this robot one. And I've got robot to, and I've got robot three, and I want them to make their deliveries along these paths and I want them to move stuff between stations. But it would be really bad if two robots bumped into each other. Right, that would. I just have these tracks so if my robots limited to each other that's quite bad. And so if you want to think about all of the different paths that you can have your robots take really what you're looking at is the configuration space of this graph. And it doesn't have to be a graph. Okay, so if I if I didn't have a graph if I just had a bunch of robots moving around in free space. That would be fine too. Right so again my configuration space would be telling me all of. All of the different places that I can have my robots in. And then, when I look at the, the fundamental group of this of this graph for example it tells me all of the different. There are some loops I didn't have robots take. Okay. This also means that like lots of applications here. Another thing is you could, you could have maybe drones that are doing some things that I wouldn't support. And so I don't actually do any of these applications. But this is a nice like, you know when you when you tell your aunt this, she understands what you're saying. So that's really all I wanted to say for today so just again wrap up, we looked at the break group, which were these strands where the strands don't have to end up in the same place they started. We looked at the pure grade group where strands do. And we looked at an ordered and unordered configuration spaces right so these subspaces of x to the n, where no particles can occupy the same space. So that's about how loop in the configuration space of the plane corresponds to a brain diagram. And I just like to acknowledge that my master's was funded and supported by the Marjorie Brown Scholars Program, which is a master's program, a fully funded master's program at University of Michigan for students who tend to come from underrepresented or underrepresented backgrounds. If you're interested in hearing more about that program. Feel free to let me feel free to contact me about it. I'm happy to talk about it. It's a really great opportunity for people who often don't get the chance to go to grad school math. And I'd like to thank my advisor Jenny Wilson, who was tremendously supportive and honestly the best advisor I could ever, ever dream of. Thank you all for sitting here and listening and learning a little bit with me. Thank you very much, at least go ahead and thank you for a round of applause. Does anybody have any any questions. I realize one of my character flaws is that I speak too fast for someone to interrupt me during my presentation so now is a great time to ask a question. I was kind of wondering about something a while back. You had the sort of cube right the planes in it. And when you when you took the quotient, like miraculously somehow you just kind of get one connected component out of that. Is this is just like something that always happens when you do this from unordered ordered or can you end up with like multiple components sometimes. So I know that this is true for the interval in the real line. I'm not sure whether it's necessarily true for other spaces I haven't thought about that which I really should. The other thing that's interesting is that if you're on the end configuration space you have a number of connected components you have corresponds to the number of elements in SN, right so there are six permutations of three numbers and we have six connected components. And if you think about like points moving along an interval, like I have a particles moving along an interval. What each connected component corresponds to is this idea like if I have particle one particle to particle three on a line. They can't cross over particle to right they can slide back and forth, but if they, they can't bump into each other, like they can't cross over continuously. Because the point where they're at the same spot is not actually included my configuration space. So the reason there's like a different content neck component for every element of SN is because they quite literally represent okay, this is the order of my particles on the interval, and here's how they're allowed to, to move. So I don't actually know for other spaces. There's some really cool work that like there's some really cool pictures of configuration spaces of graphs. There's a really great senior thesis by an undergrad that I read that I highly recommend if you're interested in learning about configuration spaces I can find it and send it to anyone interested. So I imagine that the student looks at a bunch of configuration spaces of graphs and I imagine that would be a good way to test like weather. I think I think they draw the order configuration spaces but I imagine you could try to draw the unordered configuration space yourself, and then look and see whether it matches up. Although actually now that I think about it in these graphs. There was just one connect in these configuration spaces of graphs there was just one connected component. Actually definitively then I can say that this this connected component to unordered configuration space kind of correspondence does not always hold. Sorry, I think I interrupted someone also. Oh, I was just saying that if the original space was disconnected surely you would not have only one connected component in the final unordered right it's like the original space started disconnected. Yeah, I would I would believe that that would be rude if you, if you did end up, or should really just sit down and work out some more configuration pieces because they're so fun. Do you happen to know if people look at these things. So it seems like the plane is kind of the most interesting sort of thing but like if you just compactify to a sphere or something or like go on like a, like a flat surface or something. Are these kinds kind of things are. I believe so. I have not but I'm pretty sure that's something that people do. I think my advisor looks at configuration spaces of just manifold in general and kind of what you can say about that. But I think configurations on sphere also. Again they have lots of real world applications right so they're, they're very fruitful things to study. I'd like a small comment just in answer to that question as well like the one thing I don't know much about configuration spaces at all but like I know one super cool example that I've come across is when you want to study a double pendulum the like and you want to you can parameterize this by like two numbers like how much the first thing for taking how much second one is so like the configuration space of the tourists is like the configuration space of the double pendulum and I think it's true that like the bits where the numbers are the same correspond to equilibria. So like there's this fun link between like Morse theory and like homology and stuff and like mechanical bodies and systems. That is so cool. It's so cool. I wish I knew more about it I don't I just know it's a cool story. Like, yeah, me too. I'm so glad I came today so I could hear about that because I, I put a big list of things that configuration spaces and grade groups are related to but I don't actually know how they're related in all of these cases only some of them, where's my big list. I claim that this is just a small subsection of the things that you can apply this to. Does my work did my work ever run into braids on surfaces. Um, no, my, I've mostly been just kind of. So when I did my thesis I was looking at braids more in a combinatorial group theory sense. I've been thinking about, I thought a little bit about like order ability of break groups that kind of thing. So I actually have not thought about braids on surfaces but that's something I would love to learn more about. I'm kind of curious on that side. There's this. Is it like this L space conjecture about like left order ability of fundamental groups or something is this just similar or do you know. I don't know that. I do not know of that, but okay. Yeah, if you send me a longer message I might be able to look up whether that is, is actually. Yeah. Not on my head. Anybody have any other questions. Thank you all so much for having me into exact for organizing. Yeah, thank you. Nice. I think I know it's one more time. And I guess I will stop the recording here.