 The next speaker is Courtney Gibbons, who I'm from the UFP, from Hamilton College. Can you hear me? Okay, good. So today I'm really excited to share with you one of the things that I use a lot as a tool for the problems that I study in outbreak. So the way my story starts is with a weird mathematical object, and for the purposes of this talk, for right now we're going to think of it as this weird green blobby thing with a hole in it, which I don't understand really well, but I'd like to. So what I might do is take some things that I do understand, like vertices and edges and faces and simplices, and start putting them together in some way until I get something that kind of looks like this green blobby thing I don't know much about. But I'm a mathematician, so I want to make sure that you can put things together in the same way I did to understand how we're going to approximate this green blobby thing. And that's what we're doing when we build a complex. So I'll give you some way of putting your vertices down, and then I'll tell you how to connect the vertices with edges and how to fill in the faces. And what I'm doing here is I'm building this thing called a complex, which are maps, like it's a sequence of maps. Alright, now we've got some Russian dolls here. These represent the kernels of these maps. They're all the stuff that go to zero. So how does something go to zero when I map edges to vertices or faces to edges? And you'll see why these are Russian dolls. It's because these maps also have images. And when we build a complex, we want to make sure the image of each map fits inside of the kernel. So these nest really nicely together. And when I look at these images and kernels, they tell me a lot about this object. In fact, what they're doing is calculating homology. A really, really loose explanation of homology is that when you take the image and the kernel that nests together and compare their sizes, how much bigger the kernel is tells you how wholly your object is in some dimension. So that's homology in like two minutes. Congratulations, your experts. Now, I want to talk about sysgis, so I need to talk about astronomy for a second. In astronomy, when heavenly bodies align collinearly, we say we're experiencing a sysgis. And I know you all like sysgis because recently there was a solar eclipse and everybody went outside and had stupid glasses on. We all looked at this eclipse because sysgis are beautiful. And that's what algebraists think too. So we try to construct sysgis in order to understand weird mathematical objects. So in terms of complex, when the kernel and the image align like heavenly body forming an eclipse, we call that a sysgis. It's literally where we stole the word from. So for me, I'm not interested in green blobby things. I just like to draw them. What I'm interested in is something called a module. Now, a module is like a vector space except their relations among the generators can exist for one thing and they can be really weird. So what I want to do, think of that as my green blobby thing, I want to take something I understand better and map to it and start approximating it. So I take something called a free module. Free modules are great. They have a basis, no relations among the vectors. So I send the basis of the free module to the generators for the module and I figure out what goes to zero. So we start getting a kernel. As you might expect, we're going to have a little misted Russian doc here. I'm going to call her Betty for reasons that will become apparent later. So Betty is called the first sysgie of this module. There's only a kernel up here, but what I'm going to do is take Betty who's another module and approximate her by building a map from a new free module to Betty. Now the image of that is Betty and we have a sysgie. Okay, but I still don't really understand Betty that well. So I need to now get a new kernel from that map and I'm going to make another Betty and I'm going to keep iterating this process. So what I'm doing is I'm building a free resolution. It's called free because all of these modules are free and it's a resolution because it's sort of resolving what's going on with my module M. And each Betty along the way is a sysgie module that tells me something about the relations among the relations among the relations of the generators. So if I want to know something about the free resolution, I ask Betty and say, hey, I'm Betty, how big are you? It tells me something about the rotations. Now, if one of the Betty's is zero, we're in a great spot, that means there are no more relations and our free resolution stops. But you could start asking questions like, when do free resolutions stop? When are they periodic? When do the same things keep appearing? Are they predictable at some point? And these are some of the big open questions in commutative homological algebra right now. They're really about asymptotics of these Betty's. And so we get Betty numbers. If we throw away, okay, well, the algebra was important, but how much can I recover if I throw away all the map stuff? Well, you get all of this kind of stuff. All these module invariants, you can rule out certain types of things. I care about these words, but you probably don't because you're here at harmonic analysis boot camp. So let me convince you that you might want to look at Betty numbers from an analytic point of view. We can take the Betty numbers, throw them on as coefficients to this formal power series, and then start measuring things like curvature or complexity to get answers to those asymptotic questions about Betty numbers. Like, how do free resolutions grow? Do they stop? Do they find it? Are they periodic? All right, that was five minutes on Betty on scissor jeez with a lot of weird pictures. If that piqued your interest, you should look up this article from the 2006 April notices that has a little bit more math and no Russian dolls. Thanks.