Holy are you, eternal, free and whole, at peace forever in the heart of God. Where is the world, and where is sorrow now? I am calling you out of the world. Can you hear me? Will you answer this call? I am calling you out of the world. I am calling you out of the roles. I am calling you out of the concepts that the mind has identified with. You don't need to fit in with a person or a group or a family. This communion experience with God, with source, is everything. Ask for your birthright. Ask for the state of mind you were created as and in. Ask for the kingdom of heaven. Ask for your vama to be returned to awareness. Ask and you shall be given. Knock and the door shall be opened. Ask.


Hello, hope you are well. My name is Frank and welcome back to my channel. It is a pleasure to have you here with me today and I'm going to show you what I'm currently eating. I've only had time these past two weeks to have one meal a day and something that both of us never thought would happen. I've been eating vegetables so let me show you guys the meal, go over each food, the nutrition and simply why I'm eating it. So I would say about an hour of prep per day goes into my entire meal. Thankfully I have the help of the Instant Pot, I can throw most of the stuff in there. The protein for this meal is what you guys saw in my fridge last week. This is the petite tender steak from Frankie's Free Range Meat. It's very lean, it's nicely portioned and after its name it is tender, very easy to eat. These are good portions for one or two meals because some of them are like this size which I'll eat the whole day, that's two meals and some of them are a little smaller so I'll have that in one meal. We have these packaged like this right now which is like 2.75, 3 pound packages. We can do them small but obviously with the labor and that type of stuff I have to increase the price. Artichokes and cauliflower, don't don't don't, but I guess before we talk about that we could just briefly go over the rice. You guys have seen this, it's the short grain brown rice. It's from some farm, it's an organic sprouted brown rice and you guys have seen this in my parentry video as well. I just put two cups of rice, four cups of Geralt Steiner bottle of mineral water, some salt and a tablespoon of coconut oil. The reason I use coconut oil over something like beef tallow or pork lard is mainly because the coconut oil is antimicrobial and I do get refined. If you guys saw my video on coconut, you'll know that refined is not that healthy but I really can't stand the taste of the regular stuff and this is pretty easy on my stomach. Just some regular sea salt right now but I will have the land salt available on Frankie's Surrange Meat relatively soon hopefully next week. This is the mineral water and this is what I actually forgot to show you guys in that pantry video. So I've been switching between Geralt Steiner and Mountain Valley Spring right now, who knows maybe on Frankie's farm in the future we'll sell water. I do have a sourdough culture going because some clown in the comments is like, day seven you're asking Frank to make a sourdough pizza recipe as if I'm going to do it but I guess I am. The water kefir here, so I think I'll go more in depth on this next week or the week after because you guys are buying too many of my kefir grains. Loosely I have one fermentation here, this is the initial key for fermentation and this is the second fermentation with the flavorings. Let me know if you guys would be interested in like a one-off key for sale where I made it like this for you guys and shipped it out on a specific day of the week. I think I can manage that but I certainly couldn't manage doing it every single day. So the main reason for consuming artichokes and cauliflower is really fiber, gut motility. You know I've had a lot less gas almost like when I was carnivore you know I'm not really farting at all now and when you have a good balance of you know protein, starch, fiber, cookies in your diet if everything's cooked properly if nothing's overcooked you should have near perfect bowel movements and really gut motility throughout the entire day. These were cooked in a steel pan like this just steamed with some water and salt, garlsteiner water again, avocado same thing except I put the water and salt in the pressure cooker and I throw a lemon in there just for some added flavor and it keeps it a little prettier but I usually just pull the heart out of this and I eat it sometimes I'll have the leaves if I feel like doing it. The mineral content you know vitamin content isn't too significant you could argue hey there's a small amount of B vitamins, there's some manganese, some copper which is good for contrasting the meat but overall it's the fiber it's feeding the gut microbiome and it's very good for gut motility. You know a lot of people are afraid of carbohydrates, plant foods with the carnivore stuff but you know as I've developed more of an understanding of you know the gut microbiome the bacteria in your stomach like the bacteria in this kefir is nom nom nom nom nom nom nom it's eating the sugar you know I put sugar I put molasses in here but the bacteria is na na na it's eating the sugar so it turns the sugar into B vitamins into bacteria so although this is sugar water the kefir grains turn it into bacteria just like our microbiome turns the cauliflower or the artichoke or the brown rice na na na na it eats it and turns it into bacteria for the most part and that's what the enzymes in our body don't break down and absorb but even the bacteria produce specific enzymes to break down starches. Of course we have my girlfriend Hitomi's vegan cookie recipe and I've been making larger batches just so I don't have to do it as often. Instead of using the chocolate chips someone recommended a carob chips which I actually enjoy the flavor of they just don't have the caffeine and I'm not breaking out as much from them and I really cannot maintain my weight without eating these cookies you know I just lose weight don't have energy not good so I usually have to prep all this stuff once a day today I did it earlier so I you know I put the rice in the pressure cooker cauliflower in the pot artichoke in the pressure cooker and now what I'm gonna do is just throw it in the oven warm it up a little bit and have everything hot. Now everything is organic but you know by no means is that the end all be all beacon of health it's just with modern food access when you go organic you do reduce most of those agrochemical pollutant concerns I mean especially compared to the conventional food which is practically poison. Now the artichokes were very hard to find organic I had to go through a local restaurant supplier and most of them aren't doing so well so if there's ever a product that you can't get at your supermarket I would just Google restaurant suppliers in your area and reach out to them because they are kind of struggling right now and they would be willing to do those smaller orders at least I would assume on a weekly basis and by smaller I mean like a hundred fifty two hundred dollars worth of stuff the cauliflower was actually at Whole Foods the brown rice was through the restaurant supplier and of course the meat is through Frankie's here inage meat the ingredients for the cookies I typically get from Whole Foods but while that rice and cauliflower is warming up in the oven I'll show you guys downstairs the rice as well as the artichokes so this is how the artichokes come from Ocean Mist USA organic I think I paid like 30 or 35 bucks and they gave me like 12 artichokes which is actually really good you know I can't eat more than one of these a day so it lasts a while two weeks you guys have seen this rice before Lundberg family farms organic sprouted short grain brown rice now I'll show you guys some nuances of the prep here so since my parents use commercial dish soap on everything I have to like wipe it off with filtered water every single plate every single utensil every single glass and I mean I can't wait until I can afford to you know get a place you know a little further outside of the city where the water is actually clean but for now this is what I've been doing so this is to put the meat away now put the meat on one plate and then I'll cover it with another plate and this is what I've been doing lately basically for anything for the enzymes that I take with most meals I use a couple different ones so I like just putting them on a paper towel and then I'll take them if you guys could please badger me in the comments because I've been delaying making a digestive enzyme supplement for you guys for like six months now I have everything ready I just haven't gotten to it so for the kefir I had one jar fermenting that I strained out earlier and the second jar I'm gonna strain now for this meal and then after that I'll refill both of them I'll add a little bit more sugar as with the plate issue I have to also rinse everything off once so what I like to do I pour a little bit of water kefir through the strainer into the glass I kind of switch that around a little bit pour it out into the sink and then I'll pour the actual glass that I'm gonna drink and I know some of you guys are gonna say oh stainless steel kills and damages the key for bacteria I usually don't use this strainer the other one just in the dishwasher and just a brief contact isn't that big of a deal but you can't use like a steel vessel for the fermentation process or you don't want to use a steel vessel to just continuously drink out of so that's really everything so we can sit down and eat now I mean this petite tender it's definitely an underrated cut of meat it is nice to just have something that you don't have to worry about how you cook it because it's gonna turn out tender every time so I kind of like go around the plate take a little bite of everything the other jokes if I have time I'll actually do the leaves and if you guys haven't had other jokes before there's like the flesh on the inside of the leaf you have to scrape it off with your teeth so it's very laborious to eat the leaves and probably take you a 10-15 minutes to eat one artichoke and what they usually do is you dip the leaves in like olive oil or butter with salt in it it's delicious in the center after you peel all the leaves away or you could just like cut into it through the stem how the top portion of the stem being edible it's the heart of the artichoke which most people are familiar with which is actually pretty big in a decent chunk of calories and these I would say there's probably as much meat in the heart if not more than on all the leaves throughout the artichoke my keeper is a little bit alcoholic because I fermented it when it was closed and it sat a little too long but you could always prevent that does taste better though so when I'm about halfway through what I think I'm gonna eat I go take the enzymes I'll put the enzymes in a handful like this and the combination of the enzymes and the probiotics is basically all you need to perfectly digest the food the enzymes will break down the larger components like the proteins into amino acids the lactobacillus will prevent any bad bacteria not bad but any potential overgrowth of things like Candida and the bacteria itself will also help digest some of the starch the carbohydrates the sugar you know on one hand I wish I could just be like a normal person enjoy my life and eat whatever I want and like go out and stuff on the other hand you know since I did have the liver damage I've been able to help probably tens of thousands if not hundreds of thousands of people with their own health so I'm not super hungry right now and what sounds very appealing to me is not having to cook tomorrow so this meal or the rest of this meal should be enough for tomorrow so I'm gonna save this just leave it on the counter in the summer I will put it in the fridge I don't have the rest of this artichoke tomorrow too but we still have some cookies to eat so I still like regular chocolate more than carob chips but these are definitely better than the white chocolate and they have a very unique scent you know I would say it's it's comparable to chocolate that it adds like more complexity and a certain scent I wouldn't really call carob a similar flavor to chocolate I mean they're both brown chips but so our carob chips to chocolate chips as like chicory is to coffee I haven't looked into either of them that much all I know is I was breaking out over the weekend from eating some chocolate and now my skin has cleared up if you guys missed my Hitomi vegan cookie joke earlier I made a recipe video making vegan cookies if you guys want to know what's in these it's just organic flour coconut oil coconut milk sugar vanilla carob chips sometimes I'll put the natures glucose in it a little bit of salt man I just got a great idea for Frankie's vegan cookies like what I should name them should to get rid of my name on the cookies but if I share it someone's definitely gonna steal it if you're a vegan leave in the comments below because I know you guys watch my videos would you buy vegan cookies from Frank because I don't know if I have another outlet to sell them right now I mean I think I've supported the vegan community more than most vegans that freaky deaky what's called astrology girl went from like 30,000 to 150,000 subs in like a month after I did a video on her some reason my girlfriend Victoria never blew up though she just doesn't make enough videos too busy in Russian la la land hello hope you are well and welcome to my channel my name is Victor it's a pleasure to have you here with me today I wanted to share with you full day of eating vegan delicious nutritious plant-based meals so that's gonna be for today the alcohol in this kefir is definitely not gonna be out a little bit regardless of how little it is so I have to definitely make a new batch of that so it's not alcoholic yeah I literally feel drunk but thank you guys for joining me today you can go to Frank dash upon our comp to see Frankie's rearrange me for the petite tender steak Frankie's rearrange foods for the water kefir Frankie's rearrange foods for the best bar so you don't have to spend three hours cooking your meals every day please drop a like on the video leave Frankie boy comment down below don't bother subscribing because YouTube will unsubscribe you anyway and you might as well not click that notification bell because they're not gonna notify you of my video so I'll see you guys for tomorrow


Hi, let's go ahead and get started. We're going to talk today about how to put a Google Calendar into your Moodle course. Now, there's a few different contexts within all that. But I'm going to launch off with, in today's session, involves how to put a block in your Moodle course that shows one or more Google Calendars. I'm going to go ahead and turn screen sharing on. And here we are. OK, on the left side, I've got Perry Carmichael's CAD2 course that he gave me permission to use as an illustration. And on the right side of my screen, you'll see my Google Calendar. We'll touch on that in a minute. Let me go ahead and show you what we're talking about, though. You'll see this block down here with a scrolling text, kind of fancy. But he's got an agenda view Google Calendar. And I'll define what agenda view is here in a sec. Anyway, you'll see the CAD Lab Calendar. And this is actually a calendar that is assembled from more than one Google Calendar all into one view. And I'll show you how to do that. It's pretty straightforward. So Perry created an HTML block in his Moodle course. And in that, he pasted in some HTML that Google Calendar provided. And he didn't have to know any HTML to put this in there. Google does all the work for you. I'm going to go ahead and go to the tab where I've got the Moodle Instructor Lounge, which you all should be enrolled in. I'm going to turn editing on. And I'm going to go ahead and create an HTML block in that course shell. When you've got editing turned on, you've got the Add a Block block. Click on the dropdown and go down to HTML. And so there is the new HTML block. I'm going to go ahead and get into Configure. OK, well, that's loading up. I'm going to go over to the right side of my screen now into my Google Calendar. Look at that cool, awesome camo. That's a whole another session on how to do that. So if you look at my little list of my calendars on the left side, I want to bring in my eLearning and EdTech calendar along with two others that I sometimes, well, I usually, almost always, will bring in events on the calendars for our little labs over here, Media Services. We've got two little computer labs, one that you know about and one that's a secret. And I put both of those calendars for those labs into one view, and I'll show you how to do that. So I'm hovering over the eLearning and EdTech calendar and in the little dropdown to the right, I go down to Calendar Settings. This is a little step-by-step process here. If it would be more helpful, in addition to the recording that I'm producing here, if it would be more helpful for a step-by-step screenshot tutorial on my blog, I'd be happy to do that too, just let me know. So I'm in the Settings for this calendar and I click on the little HTML button over near where it says Calendar Address, click HTML. Little pop-up comes up. And of course, if I want to link directly to this calendar as a, make it, make it display as a webpage, I could copy this long link and paste it somewhere. That's one way of doing it. But I want a little, little faster than that. And I also want to, I want to embed more than one calendar. Remember I mentioned those three calendars into the block over in my Moodle course. So I'm gonna click the Configuration tool right there. And I get into this view, and here is the default view of what it's assuming that I'm gonna want. So I wanna keep it simple in the Moodle course. Let me just get a quick glimpse again of Perry's. He's got a simple layout and you're gonna try to replicate that as much as I can. I'm gonna go back to my lounge here. Okay. So I mentioned a minute ago that Agenda View is, and this is the next few options here I should say are like my recommendations to start with. So if I look at Agenda View, I think that for the purposes of this Moodle block, it might serve a better purpose. If I start taking some of these options up here, away, I'll keep the title on there. I might wanna change that though. Let me see if I delete all these guys here. And maybe I might make this, maybe I'll just call this, and again, these e-learning events. That's one calendar I have in my Google calendar, but I've also got those two other labs. I'll bring those here in a second. But everything relates to e-learning events, maybe more accurately even, e-learning and edtech events. There we go. So down here in this little list on the lower left, the calendars to display. I've got that little media zone lab right there. Boom, put that check. And I've got our secret e-lab that no one knows about because it's a secret right there. And that'll pull in the events off of the e-learning slash edtech events calendar, as well as the media zone lab calendar and the e-lab calendar all into one view. And again, I'm in an agenda view. Let's just compare again back over to the month and the weekly views. So you might agree that agenda might work best for the context of this Moodle course. I'm gonna make the width because this is a little block in Moodle. It's not a full web page. It's just a little block that appears on the left-hand side. I'm gonna try 200 pixels wide. And I also don't want it to be very long. So maybe about 300 pixels long. It's just about block size. Look at my notes real fast here. That's what I'm starting off with at least. Maybe four. Let's see. You might kind of need to dabble. You might even need to, when you start pasting the HTML and into your Moodle course, it might take you a few times to get it just the way you want it. But 200 by about three or 400 in height. Might be about where you're at. What does 350 give us? Let's see what that looks like. Heavy, medium. We'll try that at first. Okay. So up here where it says, copy and paste the HTML below and blah, blah, blah. I select all that and I can right click and copy or I can control, see your command, see. And over back into my Moodle browser tab, over on the left side, before I paste into here, I don't do that quite yet.  Think of the double brackets as being, putting that text box into HTML input mode. And now I can paste. So I'm gonna go ahead and just real quick, save changes and get a quick glimpse of how it looks. And it should appear on the left side, I believe. I'm gonna move it to the right here in a minute. Yeah, that's how it comes in. Okay, okay. I can scroll up and down. Looks pretty good. Before I forget, I don't wanna forget to remind you, every time you see a Google Calendar, whether it's, it doesn't matter what view it's an agenda or month or weekly view, whatever. Every time you see this little plus Google Calendar button, you can easily subscribe to that calendar so that you see the events in this calendar in the context of events on your own Google Calendar. And so this is useful in there for you, but think of your students who might keep their own personal calendar of classes, other classes they're taking here at LBCC or personal stuff like go shopping tonight for the party tomorrow night, et cetera. So they will see their own personal study calendar, their own personal lifestyle calendar mixed in with events happening due dates for assignments and quizzes and stuff we're talking about within your middle course to see everything in context in one calendar. All they've gotta do is click this plus Google Calendar button. And if they're logged into their LBCC Gmail or any Gmail, it will ask if they'll just verify do you wanna subscribe to this calendar? And once they do that, they'll be able to see all their work and study and personal calendars all merged together in one view so they can plan accordingly. Okay, so I'm liking how this is looking already this 200 by about three to 400. I have 350 member in height in the settings. That looks about right. I'm gonna click that little gear up there again and configure this block because I want it to go over to the right side. I can click this where this block appears and this on this page here. I think this is all I'm gonna need to do. We'll double check it though. I want it on the right region and I want it near the top, maybe. And this is one of those you might have to fudge a little bit and go back and forth to get your block to appear right where you want it. And I'll save changes. We'll see how this looks. Looks about right to me. Yeah. Okay, let's compare that to Perry's. Let's look at them side by side. Tear this off here. There's Perry's with that fancy pan scrolling text. There's mine over here without that. Yeah, that should do. What do you think? So that covers what I was gonna talk about today. Once again, let me show you once again how to get to this, the calendar page and the page in your Google Calendar to get to that configuration tool. Again, I went to the specific calendar that I'm looking for into the calendar settings and next to calendar address, that little HTML button and back to that configuration tool. Okay, so a couple of things before I close out today. A couple of things that I want to offer up for follow-up sessions, following up on this one. I've got a nice workflow where you can, you know how you add an individual calendar event into your Google Calendar one after another after another, that's pretty straightforward.  You can plop those into an Excel spreadsheet or a Google spreadsheet. And if you'd like, I can show you an easy way to collect all those up in a spreadsheet that you can then eventually upload into Google Calendar and then all in one fell swoop, there's all of your events on the specific dates and times and locations. And expanding that concept out even further, another session might be where you can export your Moodle calendar, which would be populated by virtue of indicating due dates again for those assignments and those quizzes and those other activities in your Moodle course that would populate your Moodle course calendar, you'll be able to download, I can show you if you like, download your Moodle calendar and import that into a Google Calendar all in one fell swoop. So there's two potential follow-ups to this session if you like. I think that covers it for today. Any other questions, of course, you can either respond to the email that I'm gonna send this link out to. Any feedback, of course, is appreciated. Also, I wanna make sure I'm trying to hit availability for you as much as possible. So if there's a better date and time of the week that would typically work better for you, let me know. I've gotten way more participation on these Tech Talk webinars in the recordings compared to the live sessions, which is telling me most people, again, our schedules are all over the board as far as being able to meet the same time, but if there's a specific time that might work better for you if you'd like to attend these live sessions, please let me know. Any other questions, again, you're not gonna hold on me. Thanks for joining me on this one. I look forward to hearing your comments on this session and if you like the other ideas for the follow-ups on this Google Calendar stuff, let me know. Thanks, I'll see you next time.


Thank you. Thank you very much for the invitation to speak. I'm really excited to be able to participate I'll be it remotely. I'm glad that that worked out Less glad that it had to work out kind of wishing that the coronavirus were better under control especially here in the US but Yeah Since I'm given the last talk I I get to to also say I think we should thank the Organizers, I mean that they've done a wonderful job taking an An in-person conference and very quickly turning it into a great online conference So please join me. I guess somehow Yeah, thank you Ian and great job Guys, this is seriously fantastic, and I know it was a lot of work. So thank you very much Okay, so as I said, um, I'm I'm gonna talk about techniques of computation in Equivariant and and then some in motivicomotopy Trying to hit on as many of the key words from that tidal of the conference as possible I'm I'm aiming this to be more Introductory than sort of the research side and please do if you have questions Ask away, and I'll try to answer them as we go through Most of my focus in the talk is going to be on the Equivariant computations a lot of them for for Everyone's maybe second favorite group the group with two elements And the reason I'm going to be focusing on this one is is Well several fold First the group with two elements, which I'll call C2 from now on is the Galois group The complex numbers over the reels this ties it to a lot of Geometric and algebra geometric concepts namely if I want to talk about say descent for real vector bundles It's the same thing as understanding a complex vector bundle together with this descent data of a C2 action Second a lot of classical and chromatic Computations can be seen in this C2 Equivariant story and actually I know that that you all saw Dan Isaacson's series of talks earlier in the conference where he talked a lot about the connections between Motivic over R, Motivic over C, C2 Equivariant homotopy and and classical homotopy So I'll pick up on some of those themes as well And then finally and this one really I should have led with because in some ways it's the most important from my perspective We can actually do computations here a lot of the literature about Equivariate homotopy theory tends to suggest that computations are essentially impossible Some of which go so far as to say it's impossible to do some of these I I don't find that to be the case and I hope that by the end of my talk You agree that a lot of these computations are much more doable Then you may have thought initially Okay, so I'm going to start by just saying that we're going to be working in the following context There we go, blossom a bit of a spell. I'm going to be working for Working in what people sometimes called genuine Equivariate homotopy now I I I Hate the word genuine here. It's for two reasons. It's very value laden Especially when you compare it to the the contrast we will talk about naive Equivariate or genuine Equivariate. There's a distinct hierarchy Established there and it's not actually supported in the math so what I would advocate for and I hope I can start to get traction in this is Not to call this genuine, but rather to call it something instead like complete and the complete here means that we have All transfers And this is a theme that I'm going to spend a little bit of time talking about as we go forward But before I do that, I'm actually going to start with just a little bit of a review so how do we talk about Computations and where do the invariance live? So how do we Understand how do we understand? Homotopy groups G spectra and G spaces and Dan also So sorry there is a comment by Yuri Suleyman He says that one version of nice just G spectra is often called borel complete. Yeah, that's true I don't understand the frowny face there Yuri It is the case that that the homotopically meaningful version of naive G spectra is the is borel spectra and here We don't necessarily have transfers. These are things we sort of free up the action historically And I guess the not having transfers is is probably why you had the frowny face So I'm with you on that one So in the equity context the first thing that I run into is I can't get away from thinking about homotopy sheaves instead of homotopy groups and Of course the real way we should be doing algebraic topology and sort of this ideal world where we have complete control over everything Is I would be able to just immediately tell you what maps out of any say finite CW complex were I'd love to be able to tell you what maps out of any finite CW complex are We approximate this instead by restricting attention to maps out of the building blocks of finite CW complexes Namely spheres. I'm going to do the same thing for for G spaces or G spectra So we'll consider we'll look at Functors and these are functors from the category which I'll call fin G op in to say a billion groups and This fin G op. This is the category of finite G sets and Equivariate maps and I'm going to do this just via The yonata lemma and so the ones I'm going to care about are I'm going to take equivariate homotopy classes of maps from Oops, I'm describing a functor from t plus so t together with the district base point smashed with the end sphere Into some fixed G Spectrum E and my superscript to G here is just reminding myself that I'm taking the collection of equivariate maps this is this is a Contra variant functor as written because I'm mapping out of the T plus slot and so in particular it fits into this form and this is what I'm called the Homotopy coefficient system Already, I'm suggesting a way that I should be thinking about my G CW complexes the CW complexes I'm going to build not just out of spheres with a trivial action, but rather out of spheres Again with a trivial action But I allow myself to to take disjoint unions of these and to permute the copies of the spheres in those stacks And that's how the group is going to be acting So I can map out of this and this amounts to picking out maps from spheres into various fixed points So the first thing to notice Is that if I have a district union of things so T district union T prime and then I take the Distant base point and smash this with the end sphere map this into E Then the inclusions of T and T prime into the district union give me a pair of maps Backwards So I get a Decomposition whoops like this So in other words my functor from finite g sets up into Abelian groups isn't any old functor It's one that takes the district union, which is the co-product in finite g sets Which makes it the product in finite g sets up to the product in Abelian groups So in other words this construction gives me a product preserving Functor and I want to stress here that I'm in the algebraic context So saying that I'm a product preserving functor is a property of the functor rather than additional structure Well any g set has an orbit decomposition. So this functor slot plus smash SN into E is determined by the values on Orbits by which I mean Transitive g sets So g mod H as H varies over the subgroups And if you haven't seen this before then I would Suggest that you spend a little bit of time thinking about what what's the geometric content? Maps out of g mod H Equivariate maps out of g mod H into some g space and you can start to see the interplay between fixed points for various subgroups and then g itself Okay, now this is the kind of thing that I didn't need to be working in G spectra. Yeah, I could have made sense of this in G spaces provided n was at least two and If I'm in g spectra any kind of g spectra be it the Burrell ones That Yuri brought up be it the complete ones that I'll be working in or anything in between I still have these homotopy coefficient systems the key feature of the this complete Equivariate homotopy is that I have not only these Contravariant restriction maps, but also the covariant transfer maps so I'm going to define a category the Burnside category of g has objects finite g sets and the morphisms So home in the Burnside category from s to t is going to be Well, I'll be a little glib here and just put parenthetically the group completion of the set of Correspondences s and t so here I have two Equivariate maps f and g and then I'm doing this up to isomorphism So again, I'm doing I'm going to be working in an algebraic context so I'm working up to isomorphism as as you know from Clark's work or Angelica's then I could have instead considered an enrichment of this be it one where I have a two category and Instead of considering this up to isomorphism I remember the isomorphisms as the two categorical part of the data or an infinity category where I build much larger Diagrams again recording isomorphisms and various pullback conditions Oh, and I should say if I'm going to say that I have a category I need to say what the composition law is and composition is via pullback so given two Correspondences, then I can pull them back and I get another correspondence Okay, so In this category since the category is the same as that category of Excuse me the objects are the same as the category of finite g sets I can still talk about things like district union and Cartesian product In this category though Well, the Burnside category it's Canonically self-dual and by canonically I mean it's the identity on Objects and then I just observe that I have my correspondence Which has you know that maps in two different directions and that's just a sort of an artifact of the way I'm writing I'm choosing to read from left to right because That's the only way I know how to read English, but I could have instead swapped it and Gone from right to left then I would be seeing instead Hom from as written right here. That would be the same thing as com from t to s So a is oh, I should have given this a name Sorry script a so a is Canonically self-dual and the district union is Now both the product and the co-product And if you haven't spent any time working with this category sort of thinking through what this might look like I would suggest seeing for yourself how the district union could possibly be the product in other words see How do I write down maps in the Burnside category from t district union t prime back to t and back to t prime? Whereas in general, I'm not going to have those maps just in finite g sets So the the players in the equivariate context in this complete one Are Mackie functors? these are so a Mackie functor is a again product preserving Functor from the Burnside category into Abelian groups and I'm always going to indicate my Mackie functors with an underline So that there'll be a little bit of type checking to contrast these with Abelian groups Again any g set can be decomposed into orbits and if I use that orbit decomposition I get that a Mackie functor is determined by a much smaller amount of data So let me spell that out For G is CP and I'm going to name a generator CP. So choose a generator to be say Gamma Okay, a CP Mackie functor is the following data of first an Abelian group M of g mod g which is a point second I have a CP module M of CP and Third I have maps a Restriction Which goes from M of a point To M of CP and to transfer M of CP To M of a point and I'll often write this as a little diagram That that people call a Lewis diagram after Gauntz Lewis My restriction goes like this And my transfer goes up and then I have an action of My group CP on the on the CP module And then these satisfy a few axioms so first the restriction The restriction lands In the CP fixed points and the transfer Factors through The co-invariance and then second I Have a condition called the Mackie double coset formula Which says that the composite of the restriction with the transfer is The sum over the elements of the group which is items called the trace If you use the gawa theory names Okay, so that's it and how am I supposed to connect these two Correspondences, so how am I supposed to see this as something coming from the Burnside category? Well, remember that I have in the Burnside category. I have Or excuse me in the category of finite CP sets. I have a map CP to a point That's just the crush everything map And I have a map from CP to itself. That's multiplication by gamma And this gives me a little commutative diagram because point is terminal Now when I think the category of finite g sets, I can embed that co-variantly as the forward direction map in the Burnside category or I can embed it Contravariantly as the backwards map in the Burnside category and if I embed it Contravariantly that gave me my restriction map This one and if I embed it co-variantly that gave me the transfer map So both of these two maps here the restriction and the transfer arose from this quotient map from CP to a point and Then the first conditions this one about the image of the restriction landing in the fixed points or the transfer factoring through the orbits are Exactly summed up in the commutativity of this little triangle. So it's it's actually just the functor reality condition And finally this Mackie double coset condition. This is what you see if you pull back CP over a point with CP over a point and the pullback is CP cross CP and then I want to write that in terms of CP sets so I want to break it up into its orbit decomposition and when you do that you get exactly this condition Okay, let me make it a little more concrete Because I will actually do a couple of computations later and I want to be able to to use these so There's the There's the representable functor the burnside Mackie functor the value at a point is given by Z direct sum Z and the value at CP is Z and Then the restriction and transfer maps I'll just write them as little as little matrices this one sends the first thing To one and a second to P and the second is Zero one The vial action here is Just by the identity So as I said, this is actually the the Functor I get by mapping out of a point in In the burnside category again, this is product preserving because Distant Union was the product and so it's it's that it's literally the universal property of the product to say that harm Out of a point is product preserving Okay, if you've also seen the burnside ring the burnside ring is the Groten deep group of finite g sets so I should also be able to connect these two summands to finite g sets and I can this summand is the g set Point and the district union of copies of point this one is the g set CP as A CP set and every CP set breaks up into a district union of points and CP and then my restriction map is just forget the CP action and just remember the set and That takes point to a set with one element and it takes CP to a set with P elements and that was this map Okay, so the other that I want is the constant Mackey Functor Z and this one is The value at point is Z The value at the at CP is also Z the restriction map is the identity The viral group whoops, sorry While group action is also the identity and then that forces the transfer to be multiplication by P Because since the restriction is injective then I can compute the transfer by computing the composite of the restriction with the transfer And I see I have no choice here The These two Mackey Functors are pretty closely connected The target is what sometimes called a co-homological Mackey Functor Now at this one, I should pause and connect this already to what we see in the motivic story Often when we talk about pre-seves with transfers in motivic homotopy, we're referring to things like this that are close to co-homologic Mackey Functors and There I see the same kind of condition that the composite of the transfer and the restriction is multiplication by the index of the group and That's the condition that I'm writing down here In a covariant homotopy we allow these more general kinds of transfers Which you should think of as actually also showing up in the motivic context This is analogous to the transfer along say a finite at all now, okay So Before I continue questions about this so far. I Know a lot of this is review But that doesn't mean the questions won't have come up you said you said a and the line is a is a Bernstein Mackey Functor. Yes. Yes. Okay, and it's the usual thing in math where proper names become Become adjectives and so you end up with long strings So Sean asks why co-homological is this an important distinction These do these do show up a lot and and they're the kinds of of Mackey Functors that you see with group co-homology and That's that's one of the reasons why I might Describe them that way So from that perspective they it is a very natural class of Mackey Functors that arises So there's been a lot of work in this for us The the constant Mackey Functor Z is a fairly easy one to do computations with as I'll as I'll show you in just a minute And it also arises naturally in the Equivariate context Yes group co-homology does take values in these Group co-homology naturally has an extension to a Mackey Functor and And when I do group co-homology, I consider it in in one of these contexts and they're always coming from Modules it's always something that's a module over the constant Mackey Functor Z No, thanks for asking okay, so The reason that we talk about Mackey Functors in Equivariate homotopy is that Mackey Functors play the role of Abelian groups in in Genuine G-Spectrum in other words all of our usual algebraic invariance Actually Mackey Functor value So for example normally I might talk about Homotopy groups of a spectrum and In the Equivariate context, I have the homotopy Mackey Functors An Equivariate spectrum I can talk about the a Generalized co-homology theories value on a space or Spectrum and in the Equivariate context, I have a Mackey Functor It's worth of the co-homology of X in some E theory So I have a richer structure that I could be working with For those who might Worry or wonder about such things that they're the category of Mackey Functors is an Abelian category We have enough projectives and injectives so we can do homological algebra the way we normally would and then In a little bit. I'll also talk about how the category Mackey Functors has a symmetric minoidal product So we were really exactly like with Abelian groups as Reflecting what we saw in spectra. We build a model in DG Abelian groups In we can do the same thing in Equivariate spectra. We take Equivariate spectra and we compare it to to DG Mackey Functors or DGAs in Mackey Functors So one of the things that I want to be able to do is talk about ordinary Homology and I find it easier when I'm talking about ordinary ordinary homology to just show you how to compute this in some examples so How do we compute homology? Remember, this is supposed to be a Mackey Functor, but I'll just tell you the value of this At at some point with coefficients in something and now just for simplicity for myself. I'll start at this point switching to the group being C2 So I Had I had I planned I Had to use some of that newer technology. I would ask via a poll. What's your favorite way to compute? ordinary homology You know, it's like we would do in a calculus class do like a quick spot check But if I were to do that, I would guess I would guess you would say cellular as opposed to singular although singular is certainly nice for sort of Formal reasons. Yeah, thank you. And yeah cellular is the way that we actually can compute things easily We write down a small chain complex do it so let's do that here and let's start with an example, so We'll just do this via cellular homology and my example is going to be Let's look at a representation sphere So I'm going to take S to the C by which I mean the one point compactification of C and Then C remember I said earlier My C2 is also the Galois group of C over R And so this this naturally has an action of C2 as the as the Galois action So if I were to to draw this well, this is the this is the Riemann sphere So I have the real line sitting inside the Riemann sphere and then I have the two hemispheres And here was my S to the R Sitting as the equator and S to the R. Well, this is just S1 and When I think of the two Hemispheres in my Riemann sphere Well, I could put them in as As showing up and actually I'm doing a different projection than you're probably thinking of I'm gonna have the positive Complex part being the upper hemisphere excuse me the the positive imaginary part being the upper hemisphere and the negative imaginary part being The lower hemisphere so my group acts by swapping the two hemispheres and Leaving the equator fixed So I can build this as an equivalent cell complex. I have two copies of The one sphere and they're swapped. So I'm gonna have a C2 cross S1 Because that's two copies of the one sphere I'll draw a cartoon as I go through. It's my one sphere and my one sphere and they should have been the same and the group acts by swapping them This is a C2 cross it and I'm gonna map this to the one sphere Where I just fold them down. So it's via the identity. It's a twisted version of the fold map And then I can include these into the corresponding C2 cross discs and that amounts to just putting in a little disc on each of these and When I pushed this out So actually let me Oops When I push this out Now I've exactly built my Riemann sphere With the two hemispheres that are swapped so here's my cell structure and If I want to take the cellular homology, well, what I need to do is figure out What am I supposed to do when I evaluate? What's the homology of one of these G mod H? cross a sphere or G mod H plus smash a sphere So the building block is I'm going to take the homology Hn of Whoops, you know, so H star of G mod H cross An end sphere with coefficients in some Mackie functor M Well, this is going to be I'll do reduced. This is zero if star is not n and it's just evaluate M at G mod H If star equals n Now I can start to write down what what my homology is going to look like notice that this map here this one this is the same thing as C2 to a point Crossed with the one sphere and remember my Mackie functors are exactly built so that they know what I'm supposed to do to Maps between Orbits so a map C2 to a point. This is something that I can evaluate my Mackie functor on Now I can write down the chain complex using that so in degree zero again, I'm doing the reduced theory I have nothing Here's my degree in degree one Well, I had my one cell. There's only one one cell and it's the one sphere So I have M of a point And in degree two I had a single Equivariate to sell It was the one coming from M of C2 and the cellular boundary map is just M of C2 going to a point So notice this is the covariate version of this. So this is the transfer From M of C2 to M of a point And this tells me how I can write down the this Homology for any of these so I get H1 is the co-curnal of the transfer and H2 is The kernel of the transfer and it's a little more work. You can get the These as instead Mackie value to things it amounts to thinking about G mod H and and putting in another slot Where I crossed with some fixed T okay So since the this is a talk in the broader context of a summer school, maybe I'll say as an exercise for you whoops as an exercise Figure out homology groups of S K times C with coefficients in any Mackie functor M For all K and M you use the same idea that I talked about here You have to think a little bit about what happens in the co-homology version So namely when K is negative, but it's a it's fun to to work through Okay There's one other thing that I want to point out here and that's actually right here. I Am going to give a name to this map From S1 into this C sphere. I'm going to call this a sub Sigma I'm going to call this sometimes the Euler class of The sign representation and if I'm being super pedantic, I'd actually call this the suspension of a sigma because a sigma is a map from the zero sphere into Now instead, it's just the one-point compactification of that imaginary axis in C Where that's swapped because well, we know how complex conjugation works and here I'm seeing this co-fiber sequence C2 goes to the zero sphere or C2 plus goes to the zero sphere and the co-fiber is this sine sphere And this whole part that I'm writing down in this case is the suspension of that co-fiber sequence So it's something to keep in mind all right, so I I brought this up because Just doing these computations Understanding the co-homology of these spheres the K times C spheres as K varies Allows you to get a lot of mileage equivariately, so I'll start with a theorem And this is due to lots of people Individually, I'm gonna say that the first parts of this is due to Duggar It's due to who crease and it's due to me Hopkins and Ravana, and that is that there's a filtration on The C2 spectrum of real boredism So M. You are or Again, if you're coming from the motivic context You should think of this as MGL and then the theorem is due to different people so in motive motivically, it's due to Hopkins and Morel and We're walk and there's a filtration on M. You are Ding with associated graded gr of M. You are is I'll just write it as the Eilenberg McLean spectrum Associated to that constant Mackie functor Z and then I'm putting in a bunch of formal indeterminance where the degree of each of these indeterminants is Just like in in dance talks my indeterminance are going to be graded by representations or in this case, they're by graded actually have two irreducible representations And this is just I times C or again. I'm using the complex conjugation action on C What this means is if I want to compute the M. You are homology or M. You are co-homology of some space or spectrum then I have a spectral sequence and the e2 term Is given by well say the homology, but I'll write it this way I'm going to do the homotopy again homotopy Mackie functors of the function spectrum From say X into Each Z I join these indeterminants Well, this is just the homology oops, sorry the negative co-homology of X with coefficients in Z and then I join a bunch of it determinants and this spectral sequence converges to the M. You are co-homology of X So it's like in a tea. Here's a brook spectral sequence But I'm using this different filtration and the filtration is the slice filtration Named after the motivic slice filtration Waivotsky that Was was done by dugger initially Okay, so what I want to focus on and I'm seeing that Time quickly passes. So what I want to focus on is that this is a spectral sequence of Mackie functors So this is sort of the the first Order approximation to understanding the way I can do equivariate computations Mackie functors form an abelian category Which means I can talk about spectral sequences of Mackie functors And in this case, what does that mean? We have two spectral sequences There's the fixed points The value at a point And there's the underlying And then they're connected. I have a map of spectral sequences That's reflecting my restriction map from the fixed ones the underlying And I have a map of spectral sequences from the underlying back to the fixed points And the underlying was actually a spectral sequence of c2 modules So I have all of this added structure that comes in It's it's a lot of added structure But it's not an insurmountable amount of added structure The biggest thing I can do is I can use the fact that since these are maps of spectral sequences If I have a class that's a cycle or maybe a permanent cycle Then the image of that under any map of spectral sequences is a cycle or a permanent cycle If I have a class that's the target of a differential Then I know that under a map of spectral sequences It's still the target of a differential. So I get a lot of additional constraints on this So as a as just an example In the in the spectral sequence Computing the homotopy Mackie functors of M. You are The ideal generated by two is an ideal Of permanent cycles and so why it's just because Two times any class x or remember I'm looking at something where I started with the constant Mackie functor Z and in the constant Mackie functor z two was Was the transfer of one in the underlying And then I have this Frobenius reciprocity condition. Let's me move the x inside. This is the transfer of one times the restriction of x So this is the transfer the restriction of x And in the underlying spectral sequence the underlying spectral sequence is just the ordinary attia Here's a book spectral sequence computing the homotopy of mu out of the homotopy of mu So it it collapses with no extensions And the restriction here Is a permanent cycle Oops, it's not how you spell always So that's giving me a huge amount of information about the structure of the spectral sequence that I don't know how I would have known otherwise I needed that this This large number of classes namely twice anything could actually be written via the Mackie structure as As a permanent cycle Okay, so in the time remaining I need to push into bigger groups And I need to talk a little bit about the the The added structure. I've already started to dance around some of this first You'll notice I used a different wild card here than my asterisk I used a five pointed star here here Here and then here I just used the ordinary asterisk. So this one this wild card was Following notation of who increased. This is the roc2 grading So I actually have more information That I have at my fingertips And second I talked about an ideal here Which says that I should be thinking about this actually as a spectral sequence of rings And that's true, but I won't go too much into it. So in fact The slice spectral sequence is a spectral sequence of commutative monoids in Mackie functors And these are called green functors Sean asks If there's a that there's a result saying differentials are power operations Yes, yes The Maybe the the best way to say what the differentials are In the classical a t here's a spectral sequence is that they're cohomology operations Because they're maps connecting between Onlandberg-McLean spectra and here in for For something like m u r the The fibers are again suspensions of onlandberg-McLean spectra. So the initial differentials are exactly Co-homology operations in this case from Co-homology with constant z coefficients to itself and then all of the higher differentials can be expressed as secondary or or higher order operations Just as we would see with the ordinary a t here's a brick spectral sequence In this case, it's a it's a consequence of knowing the form of the spectral sequence knowing that the fibers are all These generalized onlandberg-McLean spectra But yeah, I can't think of them in exactly that way Okay So I'm In that the time remaining and I've said that already I want to do one last added bit of structure So here I've used that the mackey structure shows up and it gives me a way to produce a bunch of permanent cycles And to transport differentials Then I know that this is a spectral sequence of these ring objects these green functors So I understand that at each page I have a ring and for each g mod h I have a ring the restriction maps are all ring maps And so I can use all of this to To continue to bind classes to other classes and simplify the problem The last part is to use the norm so we have Also multiplicative transfers And these are actually arising from functors quite generally on on the complete Spectra, so I have a norm functor from h spectra To g spectra And this is a symmetric minoidal functor That's going to take some e and you should think about it as going to I'm going to smash together G mod h copies of e i.e This is a tensor induction And these norm maps have the property that since the tensor product is the co-product on commutative rings I have canonical maps We have canonical maps for any Commutative ring In g spectra I have a map from the norm Of the restriction of r Back to r this endows the Homotopy Mackey functors of r with these external norm maps And here I have to use the grading by the representation grid So just as earlier when I talked about the sum over the Vial group or the sum over g being the trace I'm using the galwa theoretic language there here. I'm also using the galwa theoretic language You should think of this as being as being heuristically the product over over g mod h Of some element And since my ring is commutative If the group is acting by permuting now the tensor factors around but the multiplication is actually commutative So it doesn't care what order that they were in this gives me a way to take an element that's fixed by h and produce an element That's fixed now by g The this structure was first studied by Tambara who looked at these and called them t and r functors these sort of mackey functors together with multiplicative transfers And the last result is the slice spectral sequence Is a spectral sequence of tambara functors This one I don't know how to show this and in the Motivate context the analogous operations the analogous norms and normed motivic spectra were done by Bachman and hoi wa they described How you can think about the norm maps and how to build these added norm External norm maps on commutative monoids in this context And I would expect that the slice spectra's even should have this property In the equivariant context The somewhat surprising feature is that the slice filtration is actually the universal filtration That has the property that it takes commutative monoids in spectra to a spectral sequence of of Tambara functors. Um, and if there are questions about that afterwards, I'll answer it So I I I saw you pop in Frederic, so I know that I'm That I am almost out of time.  So means that we have so I was popping for questions, but you can you can take times Oh, okay. I reserve my question all right So we have a twisted Version of the Leibniz rule and let me just spell that out in one case So I have the I have that the differential On some class that I write as the norm Of x again Churistically, so I'll do this in a different color because this part's a lie This is supposed to be x times the conjugate of x And then I know from the Leibniz rule how to compute the differential on a product. This is the differential on x times gamma x plus x times the differential on gamma x Remember the differentials were maps of mackey functors so I can pull The vial action out So this is dx Times gamma x plus x times gamma of dx And this is the same thing then as one plus gamma on x times oops not the one I want to do dx times gamma x And now I can make this true statement, which is the transfer of dx Times gamma x So this is the the version of the Leibniz rule that shows up in this case The differential on the norm. It's just like on a product But I'm supposed to remember that the norm was a kind of product where the group permuted the factors around And so the differential is going to take that to a sum where again the The group permutes now the summands around And that's exactly the role of the transfer You can do better though This is saying something just about d of n of the norm. It's just the usual Leibniz property And so this is the very last thing I'll say And You can do better if d n Of x so now I'm in my slice spectral sequence is y Then I actually get a longer differential d 2n minus 1 of That same classroom before a sigma times the norm of x is The norm of y For this, I don't know a classical antecedent. This is saying that instead That once I put in this a sigma it actually the differential almost behaves like a ring map at the expense of shearing it from That d n to d 2n minus 1 So it's like saying If I know the differential on x then I can is y Then there's some kind of differential on x squared that looks like some kind of y squared and these these Collections of properties I've described them are the way That people are doing computations with these spectral sequences So I think I'll stop there Okay, so first my thanks a lot for the nice talk And so we We are so let's fire the first question So can you read it or do I read it? Yeah, so Sean asks couldn't I view a sigma n and n is two different power operations? And then it would look like some of bruner's work. Yeah, I think that's exactly the way that I want to do this They it's I should be able to connect a sigma n to some kind of of actually in this case Diolashov operation because I'm looking at an operation on homology um, but I don't I don't quite know how to make those work. I'd love to talk to you more about that um Yes, they Sean also asked do these norms prolonged the category of filtered equivalent spectra. Yes, they do um Uh, Yuri asks is the universal property of the slice filtration written down anywhere? No, um Yes, I don't think Maybe but I don't I don't recall Oh, wait, maybe in the handbook of homotopy in the chapter. I wrote for the handbook of homotopy I believe I talked about the universal property of the slice filtration There so thank you for Make me remember that and then um Anonymous attendee asks could I mention some of the applications to chromatic homotopy has promised in the abstract? also, yes So the Applications of some of this And I'll be quick The applications are first. Let me recall a theorem of Han Chi And this says that the lubin tate spectra En for any n are real orientable in other words, I have a map Of ring objects in the homotopy category M. You are to en knowing this if n is two to the k minus one times anything so times m Then the Hopkins Miller theorem says that c two to the k acts on En Which gives me then via just the that the Norm forget a junction I described above a map from c2 to c2 to the k of m you are into En That's again a map of ring objects, but now in the homotopy category of c2 to the k spectrum And then recent work of of me She and ming kong jeng Says that you can use this to build a model for e theory en as the kn localization of Of some quotient of and you are And to spell out exactly what stuff is would take me a little far afield But the important thing is the slice spectral sequence Here has a describable A more understandable He too and then et cetera terms Then the corresponding lube and tape theory had we knew the c2 action On en but being able to describe the c2 to the n action on en in a way that we could write down the homotopy fixed point spectral sequence that was that was sort of the the Bloody edge of the state of the art using these sorts of Equivariate methods and stepping through the norms and these sort of quotients of the norms of mu The slice e2 term is very easy to write down and to describe And then you can use the techniques that I was describing over the course of the talk to Sort of bootstrap differentials inducting up over the order of the grew And use this to get a lot of information about the homotopy groups of Of the hopkins miller spectra In ways that we never were able to before Okay, so so I have a question What's the link between tambora and green functons? So There's a forgetful functor every tambora functor has an underlying green functor So a tambora functor you can think of as a green functor together with these additional multiplicative norm maps And then there's actually Just as there was a hierarchy That started with coefficient systems and it ended in mackey functors where I start to put in more and more transfers There's a hierarchy between green functors Which are tambora functors with no multiplicative transfers all the way up to tambora functors which have all multiplicative transfers And this hierarchy I can it's exactly analogous to the additive hierarchy for the mackey functor case and This is this is an important Feature, so thank you for bringing it up Zorisky localization does not work well In tambora functors. So for equivariate commutative rings Zorisky localization doesn't preserve the property of being a commutative ring It does always preserve the property though of being sort of the spectral version of a green functor so an algebra over an e infinity operat But it's it's a very particular kind of e infinity operat one in which the group doesn't act And so that's a a subtlety that shows up and makes some of the computations a little trickier Okay And also I have a kind of maybe vague or broader question so So you describe mackey functor and they are defined for finite groups, but is there a theory for other groups? So first example would be profile night groups like yeah Yeah, um, yes, um, and There are several several versions of these for the um The cases that were most studied in in classical homotopy theory were compact lee groups Where again we have a good notion and they're and you have mackey functors for a compact lee group and they're describing the homotopy groups of a a genuine g spectrum for g compact lee There the multiplicative version of these only shows up for finite index so pairs of finite index subgroups There's no sort of degree shifting part that can show up in the in the compact lee for profile night Dresden's even ochre have a a bit vector construction for For sort of profile night groups that's generalizing the ordinary bit vectors and they're Describing the profile night version of the burn side ring in that case because the bit vectors The bit vectors of z is is where the the truncated bit vectors are exactly giving me the various burn side rings as I look for Like c and or whatever my truncation system was Barwick also in his spectral mackey functors has a really beautiful approach to understanding the profile night case of a mackey functors as well Beyond this you can nothing nothing that I was writing down really depended on On the group being finite. Um, I could still talk about finite G sets for g not finite I start to run into pathologies like if g is divisible there aren't any interesting finite g sets And then I'm going to start to run into trouble but aside from from those cases You can talk then about You can talk about mackey functors. You can do all the same thing Last question. I was also thinking about I don't know if you know this The rust cycle modules. So it looks really like mackey functors, but there are two operations that are added So it's kind of multiplication by unit and residue maps for valuations and it It it made it could met you we could see that as mackey functors for the so-called motivic galois groups where you have transcendental extension So I've you seen something like that and I guess Um in I haven't but but that's a That's an interesting thing. I'll think about that I think I mean one of the reasons that I wanted question Yeah, yeah One of the reasons I wanted to give this talk is I think a lot of the techniques that we've been using in the Recently in the equi variant context should port through without change Into the motivic one all the stuff that we've been seeing with the multiplicative transfers anything showing up in the in the bakman hoi wa Normed motivic spectra. We should have analogs of these two kinds of conditions on differentials in certain spectral sequences And this last one the one that changes degree It's allowing you to lift differentials multiplicatively in a way that's That can actually be pretty surprising To get new ones. So I'd love to see the analog of that Motivically okay So it seems we have no more questions. So again, Mike, thanks a lot for a nice talk


What's up guys welcome to today's video today. We're gonna call this cut a spiky textured quiff haircut We're gonna start off sectioning around the head in a u-shaped pattern So you can see right below that parietal ridge area and I'm gonna go in with my clipper fully open a long Foreguard and I'm gonna work my way through just to get the bulk out of the hair And then we're gonna go through and do a little scissor over comb So let's get this hair off and then we're gonna work the rest of the haircut So you'll notice right at the end but as I'm going up with that clipper I kind of rock out just to leave that weight line and now I'm gonna go through and do some detail work doing scissor over comb Best way to do scissor over comb is notice that I have a stationary blade right there And I work my way up the head and I'm basically going Parallel to the side of the head. I don't go around that curve I want that little bit of a weight line towards the top so you can see right there. That's a good angle I'm just going straight up not straight into the head So that's gonna build up that weight in the crown area But I'm just fine-tuning everything using scissor over comb And then I really want that kind of lived-in organic feel to the edges of this haircut so I just go through detail it and then we're gonna go into the top All right, so now I'm gonna let down the top. I'm gonna comb it all forward I'm gonna part it right down the center and we're basically gonna create three sections There's gonna be one on the left one on the right and then in the back We're gonna separate the front and back to create one big section right in that crown area All right, so you can see how the sectioning breaks down one two three in the back And now we're gonna start in the back working vertical sections pulling them straight out from the head And now I just want to focus on that elevation elevation is key to getting a successful cut in the back I don't want to flatten anything. I want to build up that weight So I connect it to the bottom where we did that scissor over comb and clipper work And then I just use a point cutting technique to still keep that texture Keep kind of a lived-in line and I work the round of the head So I pull my guide over and then I cut with the round of the head and I keep working all the way to behind the ear So that side section right there where I finished off my cut That's where that disconnection is gonna start So this is gonna have a disconnected top. It's gonna have a nice disconnected weight line around the parietal ridge That's kind of the beauty of this cut. I wanted it to have that feel So I'll just continue working around the round of the head You can now see that my fingers are pointing down my thumb is pointing in the direction I'm going and I just keep pulling that guide into my hand and following the round of the head to start Pushing that disconnection to the side So you can see that build up of weight in the crown. That's the goal Now we're going to bring everything back We're going to travel across the top of the head until we start to curve in that fringe area And then I'm going to over direct everything back still using a point cutting technique Fingers parallel to the floor and I just continue across creating my line all the way across the top of the head So you can see another section bringing it back just to the previous So we're traveling across the top That disconnection is going to happen in the tip of my fingers that hair is much longer It's going to fall over everything. We're going to cut that later So I just keep continuing to cut that line across the top Point cutting guys because there's no reason to put a solid line into this cut. It's going to be textured It's going to be spiky.  You might as well just do it all at once. So we're cutting a line It's got a purpose But we're using point cutting to soften that line and to make sure that it has that texture built into it So this is the part that i'm over directing back. It's only a couple inches back I want to push a little extra length into the front. We're going to do some more cutting into that later I'll show you some dry cutting techniques, but this is the overall picture of the haircut We're going to do the same thing on this opposite side Just working horizontal to the floor all the way across Traveling guide up to about two inches from the hairline and then we over direct everything back Into the sides and what I want to do is I want to draw a horizontal line all the way across I'm going to connect That weight line that we created in the scissor over comb and the clipper work So there's still going to be a nice heavy weight line But it's still going to feel like a disconnection So you can see how it disconnects in the front still I continue pulling those horizontal sections down to a stationary guide Connecting it all together. It's getting longer towards the top middle So I just continue pulling that over till I run out of hair and I'm going to do the same thing on the opposite side You guys will see the effect that this creates when we get into the dry portion of the cut I'm going to move into the blow dry. I'm using my manerva gold blow dryer If you want one of these things, it's super powerful. Go to manervabeauty.com I'm using my ergo paddle brush and I'm just working my way around the head using that tension to smooth everything out And really look at my weight and assess it now blow drying is so important in men's styling We're going to blow dry it different for the finished result But right now i'm blow drying just for the cut So I blow dry everything smooth and forward and now we're going to go in and do some detail work I'm using the new matbeck velcro clip that you can get on our shop shop fse You can get a two pack of them. They hold the hair so well So you can see I take that parting and I just slide the clip up in holds the hair nice and tight And I continue working up the head. I'm using a texturizing scissor To just do some slide cutting add some texture to it remove some weight from that heaviness Um, and that's really the technique guys. It's pretty simple All right, so now we're going to move to the opposite side. We're going to do the same exact thing So just sliding uh that texture scissor and this is a really good look at the texture that it's creating And the softness that it creates using that textured blade Now I'm going to finalize the top taking horizontal sections across the top Parallel to me and just working that blade vertically in point cutting Not to disturb the line, but just to take out a little extra weight The last little detail we're going to do I just scoop up the hair on the sides in my comb And I just do a deeper point cut just to add a little more texture a little more movement to the line So now let's take away some length in the front if you like this length you can keep it What I'm going to do is a twist cutting technique So slide cutting but twisting the hair up in the fringe pulling it straight out from where it lives The twisting over directs certain pieces of the hair so it creates a broken line And I just take off some of that length So when I go to to style this hair to be up, it's not too long This is a great technique you could use it on any short hair that you're doing a shorter fringe than this a side fringe Make sure you guys keep an open mind to all these techniques that we go over You don't have to do this exact haircut, but if you like the result of it Then it's good to use this in every haircut that you do Just going to finish up with a little detail work with the scissor So just using the tip of the scissor working around the edges. I don't want to fine tune everything too much That's why I use that foreguard in the beginning because I wanted some length. I want that texture around the edges This is a more lived in look, which is great for the time period that we're in now It's something that will wear a little bit better a little bit longer So now here's a tip for you guys if you noticed I use the heat from the blow dryer to make the hair stand up in the air What that's doing is it's breaking down the bonds and allowing it to form in that direction So then when I go in and put the product in I'm just detailing the hair adding texture to it But the style wants to stand on its own a lot of guys try to take product and form and make their hair stand up Just using the product. Uh, I go the opposite So use the heat from the blow dryer as your advantage and then go in add the product and do your detail work through it And you can see all that texture and all that movement building up I just really really love this style If you wanted the top a little bit shorter, you can absolutely do that I go around the edges and clean everything up and here is our end result. I hope you guys like this cut I hope you guys can use it in the salon. Let me know if you have any questions in the comments below Thanks for watching