 I'm for coming and and for staying up later for me because it's early very early for me in California. So, so, so today I want to run the first exercise session on hypercaler manifolds. And before I start before we start doing the exercises I just want to motivate why I wrote the exercises I did and what what I want you to get out of doing these exercises. So, let's start with maybe goal. So, if you've been to Professor is that his first lecture, you, you would maybe think like that there's a very differential geometric point of view from that lecture. And I want to do something so more orthogonal to that and go from a very algebra geometric point of view. My second goal is to like, think about, you know, algebra geometric motivations for hypercaler manifolds. And my second goal, and I think this is a good goal for any exercise session is to get get you ready to dive into the literature. I'm going to drop my exercise sheet in the chat right now. Okay, so it should be loading into the chat right now. And if you click on it there's two sections and there's a lot of problems within each section. So, so exercise fun is about K three surfaces, and they're a geometry and topology. And so K three surfaces are sort of the simplest example of hypercaler manifolds of a compact hypercaler manifolds. And it turns out, actually, this was somewhat surprising to me, but it turns out that if you just do this exercise, and then you look at the further things that I put, you will actually have all the material, like all of the background knowledge necessary to dive into this literature about topology, some Betty number calculations for hypercaler manifolds. And so I thought that was a pretty cool exercise to start with. And the second exercise is more challenging. I think it's quite fun. The theme of the exercise is given a K X trivial manifold. So manifold with trivial canonical divisor. How can you tell if it's hypercaler. Okay, part of this exercise. I'll introduce the the theorem of Boville, which is called the Boville bloga mall of decomposition theorem. And that's a major theorem in the subject is kind of the starting point of the whole subject. I have a question. We assume that taxis also okay there. Yeah, yeah, yeah, yeah, sure. So, I'm just going to be vague here but if you can assume it's caler, if you're more comfortable, you will lose basically nothing by assuming that everything is projective. For these exercises. And yeah so so the theme, the theme of this exercise is how can I tell a given manifold, a caler manifold is hypercaler. Exercise two I think is is is harder than exercise one and I've my hints are maybe less detailed and I did that so that people are more familiar with subject could have a little bit more of a challenge. But of course, feel free to ask me for more hints and I'll be really happy to discuss all the various exercises with you. In addition to the two exercises, I understand that many of you might have questions about the lectures, possibly. So I wanted to make a third room for you to ask to talk about what happened in the lectures. And, and I'll be there to, and hopefully I can answer some some of those questions if you have. So I'm going to make breakout rooms and I'll say, go to breakout room. Room one for exercise one. Same thing to for exercise to and three for lecture discussion. So let me make the rooms now. Okay. They should be open and you should be able to select the room. And I'm going to send the file one more time just in case anyone came in after the chat. Okay, so please select your breakout rooms. And please let me know if you're having trouble doing that also. And I'll be cycling through the rooms, of course. Okay, I'm going to head to the rooms.