Hi there and welcome to another edition of Tuesdays Worth Corey today. We have Kay. Yes again again Yes, well this may actually be the first show we show with you So we don't know which ones first I don't we have two shows with you and this one is about images images. Yes, and so tell us a little bit about what images are What customers are asking for sure so images are a read-only copy of like example is a golden image You want to create so you want to stamp out many VMs from like a one snapshot or got it You're gonna spin up like a hundred VMs, and you want to start them all from that one. So basically it's sort of like Static infrastructure exactly right. It's like a point-in-time capture of the state of your guys that you want got it And how will you want it got a perfect right and it can be one of us This can many data some people use it for to make sure that their infrastructure never changes, right? But some people use it just to make sure they have a starting point Exactly with all the security things that they want set up exactly okay great exactly right So we've had images in Azure for a long time. Yes, and we recently introduced something called managed images Okay, so whenever I talk to customers the kind of questions I get is I want to share this image across different subscriptions, right? Or I want to take this image and I'm going to take it to another region sure I have my infrastructure in the region. Yeah, right. So there it's always been very hacky to do Yeah, you gotta like copy exactly this kind of weird stuff some brilliant people have come up with like very clever solutions Okay, but we think it was harder than it needs to be built in the flat Yes, so what we didn't say that for years. Yes We finally have a solution. Yeah, that's a nervous laughter There were so many other Azure features that were in front of it as you know So it's my fault. I got it. Okay. Go. Okay, so if you introduce shared image gallery Which allows kind of answers all these questions and gives customers all these capabilities. Okay. Got it, right? So for example, I Wanted so let's quickly take a look at what the existing landscape is. Okay, right? So we have platform marketplace images. Okay, we have a custom managed image Okay, and now we're introducing is something called shared image gallery. Okay, and so customers ask us What is the relationship between managed image? Yes, so the shared image gallery agreed? And so the way to think about this is that shared image gallery is built on top of managed images Okay, so shared image gallery can do everything managed images can do so a shared image gallery its own object It is yes, so it is its own resource It is since at the same level as the virtual machines and VMSS. I see and discs Okay, so you can that you have direct rest apis to interact with these objects And then other custom-managed images are inside this gallery. Yes, but they're like associated with it So is this separate so you create? I should know this before we have this session Maybe so I'm not asking dumb questions, but that's all right.  I see Okay, got it got it. So So the the advantage is you can stop working with managed images or you can delete them to save costs and only work with shared image gallery okay, okay, okay, so Going back to this like you can you do and everything with a managed image or should we the shared image gallery that you Can do with a managed image, okay? And the way to think about this is so we have three new resources Okay, something called a shared image gallery sure you have something called a container exactly and then we have another container inside It called the image definition, okay, and we have another containers inside it called the image version. It's just like nesting dolls Nesting yes, I like the Russian dolls. Yes, so kind of the inception going on. Yeah, right, okay? And to show this in like a picture form So what you have is like the shared image gallery here Outside container you have multiple image definitions inside it right and then you can have multiple versions inside it Okay, and we follow semantic versioning okay the name of the version. Yeah, so you can define whatever makes sense for you Yeah, of course and you can define 1.0 or 10.0 even for sure. It is completely up to you There's no order. It is whatever order makes sense for great for you Okay, got it. So the version is completely up to you. Exactly. Yeah, and so how do you get to shared image gallery? So Looking at the history of Azure everybody had an unmanaged image, which was just one of us just right bunch of data disk Yep, like VHDs that have their each own like your eyes. Yes, that's right Yeah, so you have to convert that into a managed image Yes, right, and then you can create a shared image version out of it And and then you can create a VMSS or a VM and you can build from build actually instantiate exactly Yes, I got it. Exactly. Okay. Got it, right? So the next piece I want to talk about is so the first one you create the image definition the version or one in the same But then you can add additional versions on that as you update things exactly so people like patch it and exactly Each version is its own independent deep copy. I got you and you can we have the concept of latest So you can say I want the latest for this image definition Sure, great and it'll let you deploy and it does it awesome just like with our marketplace images You can say pub offer skew and yeah, yeah, and you'll always get the latest That's awesome. So we're bringing that concept to this as well. Okay, cool. Yep, and so The other ask was sharing of images, right? So because these are arm objects, you can apply our back access to these Yes, so you can give somebody an our back access to your shared version Oh, cool or the inside of your subscription even exactly Yeah, and then that user can deploy to any subscription. They have they have access to all cool So you don't need to like copy an image into one of those Exactly and like you don't have to pay now. How does this work from a regional perspective? So was this tied to a given region? Yes So so but I will come to the region. I jumped ahead. Yes. Yeah, that is the other problem. We wanted to solve Oh, I know it's a big one. Yep So once you solve this you can share and you can deploy in different regions in different subscriptions within the same region Okay, great, right? But now what you can also do is so oh look, I see you guys Yeah, I got maps like nothing explains like yeah maps are awesome, right? That's awesome So traditionally what you would have is like you would have an image in like this U.S. Yeah, and then like another image in Western Europe, right? And you have to like keep them and exactly Another in Australia, right? Now what you can do is make that image a part of a shared image gallery Yes, go from managed image shared image gallery perfect And then you can replicate that to multiple regions right and you can so it's as easy as specifying the name of the region That's how easy that's awesome and we handle the replication on the back end for you I see so you just everything gets taken care for you exactly perfect yet going back to making it simple for the Yeah, yeah, that's right. It's the goal. So now the other thing what we introduced is We've introduced that you can do versions Replication per so regional replication per version per version you can decide one version Maybe only in one location versus like version exactly Yeah, and the benefit of that is like you can save cost, right? Like you can have the latest version in like many regions Yeah, all the older versions and only in one right so another way to save money. Yeah So right so you can basically make sure yeah, you're bad your old ones that you may have to revert back to you Don't need to keep it. Exactly. Yeah, and the great part is that awesome So the awesome part is that you can update at any time, right? So you can come back and update the region list or you can remove the regions from the list So it's completely cleans it up for you automatically. That's awesome. So it's completely dynamic. You can make changes anytime Yeah, that's really cool. All right, so That's awesome. And so today this is in preview. This is in public today I don't want to go do this, but it's only available in command line and PowerShell, right? So it's not it's not available today yet in the portal and not in the portal But it's coming soon, but I just had a chat with the portal team and the experience is looking awesome It looks really good at least in that little map. You click on the button in the mock-ups Yes, good. Good. So the other thing we're introducing is something called scale.  There is still like a storage hardware Of course and you can only read so much like if thousand VMs start reading from it It's gonna run into storage It's just spinning too much or the SSDs are chemically firing too much. Yes Is that right? Go on. All right So what we're allowing you to do is specify the number of replicas you want okay image Okay So one it offers high availability because it'll put it on different storage accounts in different hardware Sure, and if the storage hardware goes down you you can access your images Yeah, right and the second thing it offers is it allows you to Spread these VM calls to all these And you have to pay for that though. You have to pay for that I see so you if you say five you pay for five stores that size then and it is obtainable Okay, and so we just like one great story. I like to share a lot of the customer feedback Yeah, so originally in the original design. We had the same replica count for all the target regions So in in our preview one of the customers came back and said I really don't like that because I don't need it Right, so we went back only one region needs this many. Yeah, exactly So the team went back and kind of made some changes for the public preview and now you can do replica count per region Oh, very cool. Yeah, so you can say I want this region and all very very configurable Exactly. That's really cool. Yeah, so it's really customer centric And you can say global replica count if you don't want to configure per region We'd rather not do it and you can override per region as well. Oh, very cool So it is completely up to the customer all kinds of power here. Exactly. Let's see some actual. Let's see some actual How would you do this? All right, so What I have is commands in the Let me just zoom in here a little bit. Okay, so the first step you would do is create a research group Create research of course all this. Oh, yes Fortunately, that is very fast. That's it's very fast. Yes. Yes. So the net you so in the CLI what we have is the azsig So share six times will share it in which gallery very clever We realize typing that would be really some so we came up right right, okay? Azsig create pass in the group name pass in the gallery name. So I created one specially for you, right? I agree. I appreciate that so the next thing what you would do is you would create the image definition Okay, look at that so many options so many options, right? And I'm going to talk about these customer configurability, right? I'm so just going back. I'm talking about the publisher persona here, and then we'll talk about the consumer sure sure So what we have is the image definition create pass in the group name gallery name Perfect and the image definition name So I pass them to Susan as an example if you were publishing images for your Tuesday show Or if you had decided to do a like a Wednesday show or a Thursday show, right? So glass So you could have an image definition for each of you got those okay, and then publisher offers queue So if you if you're working with our marketplace images, you know every image has a publisher offer skew Yep Now what we allow you to do here is fine your own define your own that's awesome So it allows you to keep track of these in a much simpler way exactly So so when I show you did playing a VM with it right now It's gonna be with a resource ID yeah on resource ID But in the future will allow you to do publisher offer skew and latest and which is defined by you Yep, so so to be publisher offers skew and then version exactly So we're the image definition is defined by those three and then version is a layer on top exactly now multiple versions within those three exactly Yep, I got it. So So you get it so publisher offers always type and then it was type and then description images for Tuesdays, right? And this you can do this this allow this type so making sure that it's either the best or the worst of both Exactly. So as a publisher if you think this OS or this image does not need to run on premium storage Don't allow you don't allow you can put that in exactly and you can also define end-of-life dates So as a consumer I can look at the image and I can know if these images are out of end-of-life Oh, cool. So I don't I don't deploy those. Oh cool by accident, right? Yula Yula privacy so this is like if other if you shared it with someone else You maybe you can put that in and then they'll see those before they'll see those if they do a get on these images They'll see all these end-of-life dates and end-of-life date the Yula and the person and release notes. Yeah, and release notes like really Important right you want to know what's changing. Yeah, those are required though. I presume no these are not required But you can put them exactly just very cool. Just nice cities for the publisher to kind of maximum yep round out the publishing experience Right and then what we've added is something called minimum CPU core maximum CPU core So as a pressure you can tell your consumers here's the size you have exactly like kind of aim for this size That's very this ballpark like same with memory and maximum. Yeah, right? So you're done with creating the gallery you're done with creating the image definition. Yes now you need to create the versions Yes, that's the resource you're gonna deploy a VM or VMSS correct, right? So here we now you have to point to an image that you've already created Yep, that's your version. So yep So you pass in the gallery name you created the image definition name you created Tuesday and Great name So gallery image version 1.0.0 higher agent Yep, yep, and I could also have done 10.0.0. It's up to me Whatever you want and multiple regions if you want more multiple reasons I will just come to that so in this example I'm just published in one region and one replica count and I'm passing in the resource ID of the managed image I want to create this product right simple. So next what I want to do is as a consumer Okay, so next what I want to do is I want to publish a new version But I want to publish that to multiple regions, right? So I'm going to pass in the same image definition name version 2.0.0 Sure time run and then if you notice in the target regions I have a central us south-central us and I have this little thing called equals one replica count Exactly within that given location so the overall global replica count is five exactly south-central and east or one exactly So it's completely. I got it. I got it and and you can come back and run this command again I got it update this list. Okay, it will update it for you So what you can also do is while these images are replicating different regions You can also get the replication status of these like who doesn't want to see progress Yeah, so you can see how far there Got it. So you can get that good and the good point is like you can you can see if it is completely replicated into a region And you can start using it there. Yep and So what you can also do is switch to a different subscription and you can just pass in the resource ID of that Diversion ah and now you're creating an actual and now you're creating the VM call. So that's how easy it is So this is the same way you would use an image from a gallery as well Like if you're using managed images today in CLI nothing changes you just pass a different resource ID cool And that's how easy it is that easy. Yep, very cool Well, gosh, just great. So this is in public preview anyone can go use this anyone can go use it They have to sign up for the feature. Oh, they do have to sign up for it. Okay, so do we have we want to put a link? Maybe here we will just or somewhere here. Yes It's gonna be below the video It's not gonna be empty Unless Rick misses it out Anyway, okay. Thank you so much for doing this is great Great show great great cover to a great feature, right? I think that people will really enjoy this It's super powerful. We saw great like like feedback and great interest on any night. Yeah, let's go Let's get let's get people out there using it. Give us feedback Let us know what you think both about the show about K about me about the feature Hashtag Azure TWC. So that's Tuesdays with Corey and we'll see you around next time. So Kate. Thank you so much Have a one you have a wonderful Tuesday. I will and you have it's gonna be really weird I'm having a wonderful Tuesday on a Venice day, but okay. Well, no one needs to know those details And see you later And you're rolling there They go they got clap nice No, it's not it's not it's super annoying. Sorry. All right. This is in front of live studio audience This is not yeah, everyone's laughing in the audience. All right laugh track. Yeah. All right. You ready? Yeah


Where do we meet last time summer jam and what happened there nothing really start the cow Get a good roasting. Oh, yeah.  Thank you so much. You're doing God's work back to the interview.  Why go for the second when you can be on top How often do you handle the ball on the court at least six days a week?  Do you have anything else you like doing? No, that's about it Kind of talentless really just play basketball. Appreciate it. Yeah Thank you


Alright, sorry about that guys. I forgot to randomize the box. So everybody gets a chance at a free XR box This would be in a separate video, of course. So here's a dice. Really? Here's a customer names. Let's roll it and It is a four and a six ten times. Good luck We go one two three four five six seven eight nine Ten times four six ten Michael Johnson. Congratulations, man. You're gonna get yourself an XR football box So I appreciate you guys. Jaspy's case breaks.com


In previous videos, we've seen how one can use Maclaurin series and Taylor series to help us solve many calculus problems in a way that's oftentimes easier than without said Maclaurin series. And so it's gonna be beneficial for us to find Maclaurin series for functions. And therefore we oftentimes need to be able to multiple and divide power series in order to create new Maclaurin series. So case in point, take the function f of x which equals x times cosine of x. Notice that this function is x times cosine of x. It's a product of two functions. Now as x is a monomial, which is a polynomial, it's its own power series representation. You don't have to do anything for a polynomial. It's already a power series. Now for cosine of x, we've seen that it's a Maclaurin series is equal to the sum where n equals zero to infinity of negative one to the n times x to the two n over two in factorial. And so if we wanna find a Maclaurin series for f of x, we can take x and multiply by the Maclaurin series of cosine like we see right here. So we get x times this Maclaurin series. We'll times the Maclaurin series by x when you distribute this x through, because this is just a sum. You distribute it term by term by term by term by x point of rules. When you end up with this x to the two n and you times it by x, that'll give us x to the two n plus one that you see in right here. And so everything else is gonna be exactly the same. You get the sum from n equals zero to infinity negative one to the n. You get x to the two n plus one divided by two n factorial. So this gives you your Maclaurin series right here. You get x minus x cubed over two factorial plus x to the fifth over four factorial minus seven x to the seventh over six factorial, et cetera, et cetera. This gives you a Maclaurin series. And if you just need a Taylor polynomial approximation, we can take something like this in which case this would be t seven of x. This is the degree seven Taylor polynomial for x to the cosine of x, x times cosine of x, excuse me. And therefore we can get away with using this Taylor polynomial to approximate this function because these two things will be approximately the same thing. Great. Well, things can get a little more hairy of course when both factors have an infinite power series. Like what if we want the Maclaurin series for e to the x times sine of x, right? We've seen the Maclaurin series for e to the x. This would equal the sum where n equals zero to infinity of x to the n over n factorial in expanded form. We get one plus x plus x squared over two factorial plus x cubed over three factorial, continue on. We can do the Maclaurin series for sine. It's very similar to cosine that we saw just a moment ago. You take the sum where n ranges from zero to infinity of negative one to the n times x to the two n plus one over two n plus one factorial, right? In expanded form, this will look like x minus x cubed over three factorial plus x to the fifth over five factorial, continue on and on and on and on and on. So these power series are essentially just infinite polynomials. And so if you had a polynomial times a polynomial, like if you took one plus x plus x squared over two factorial and you wanted to multiply it by x minus x cubed over three factorial, if you did something like that, you would just foil this thing out. That's what you would do. You'd use the distributed property and look at all the possible products. You distribute the one, distribute the x, distribute the x squared. You look at all those possible combinations. The fact that the sequence is now infinite doesn't actually change how that works. So what we're gonna do is we're going to take the first power series and write it on top. So this is the power series for e to the x you're gonna see on top. And then I'm gonna take the Maclaurin series for sine and write it down below. And so we're gonna take sine of x right here. Now for convenience, I did put a space right here, right? Because I want like terms to be in common column. So we have an x column. I'm gonna skip this square column because sine doesn't have one. Then we get an x cube column. And you make this thing go as far as you want. Go on and on and on and on and on and on, right? And so now look at all the possible products. You're gonna have x times one, which is an x recorded here. You're gonna get x times x, which is an x squared. You put in the x squared column. You're gonna get x times one half x squared. That's gonna give you a one half x cube. You're gonna write it in the x cubed column. You get x times one sixth x cubed. That'll give you one sixth x to the fourth. We're gonna put in the x to the fourth column. And then keep on going as many times as you need to. Then take the next term. We have this negative one sixth x cube. You're gonna times that by one and you're gonna end up with a negative one sixth x cubed. Okay? Then you're gonna take negative one sixth x cube times that by x. That'll give you negative one sixth x to the fourth, which we get right here. Record the product there and keep on going.  That would give you an x to the fifth term you put in that column. Then you're gonna take x cubed times x cubed, multiply those together. You're gonna get negative one over 36 x to the sixth. You put that in that column and keep on going. You'll notice I kind of do dot dot dots because how many columns do you need? Well, that kind of depends on what we're going for. We'll come back to that question in a moment. You do all those possible distributions. Just do one monomial at a time and go through every possibility from the first one. Okay? So you're gonna get two rows. Well, it depends on how many terms you did. You're gonna get one row for the first one. You're gonna get another row for the next one.  But the thing is if you aligned everything in columns, we then can combine like terms very quickly.  You get one half x cubed minus one sixth x cubed. In this case, we do have to find some common denominators, but the difference will be one third x cubed. So then we get a one sixth x to the fourth minus one sixth x to the fourth. Those will cancel. If we did the x to the fifth terms, we'd combine like terms there. If we did x to the sixth, we would combine some like terms right there. The key thing to make sure though is that you have to make sure you do enough rows so that everyone involved gets included there. Cause it's like, how do you produce an x to the fifth? An x to the fifth would come about by taking x times an x to the fourth. An x to the fifth would come about by doing an x cubed with an x. I'm sorry, I'm trying to do x to the fifth, x cubed with an x squared. You would also get an x to the fifth by taking an x fifth right here and times it by one. And those are the only possibilities. So if you want to get up to the x to the fifth, you might have to do an extra row. So you have to kind of do some extra rows right here. You get this kind of like this matrix of all this information.  For each case, you get an x, you get an x squared, you get a one-third x cubed. The next term, well, the x to the fourth canceled outright, so you just have zero of those. The next one would be an x to the fifth, which we don't know what the coefficient is. We didn't do enough information, but we could find more information and do it if it was necessary. We could recursively find the coefficient of x to the fifth. And we could recursively find the coefficient of x to the seventh or x to the sixth or however many turns we need to do. So generally speaking, we don't have a formula for the product of these two Maclaurin series, at least not a general formula. We do get a recursive formula, and that's typically sufficient because honestly, like we saw in the previous video, when we were trying to estimate an integral using a Maclaurin series, typically we don't use the whole series. We actually use a Taylor polynomial, which is just a partial sum of the series. And so in fact, whatever it is, we need the Maclaurin series for e to the x times sine of x4. We actually might be a good way with just the degree three Taylor polynomial that we see on the screen right now. So this recursive approach is actually sufficient for our purposes. Sometimes we have to do division as well. Like what if we want the Maclaurin series for tangent of x? Tangent of x is the quotient of sine and cosine for which we know the Maclaurin series for sine and cosine. We just saw the one for sine. We've seen cosine previously, but here's a reminder in case we forgot it from eight minutes ago. Anyways, if we're trying to do long division with polynomials, you're gonna take the numerator. It's gonna be our dividend right here. Then the denominator is gonna be our divisor. So here's cosine's Maclaurin series. Here's sine's Maclaurin series. And you do all the possible quotients. How many ways does x divide into one? So you look at the term here, the first term here, the first term here. x divided by one is x. You're gonna record that number on the top, like you see right there. The next thing to do is you're gonna take the entire divisor and times it by x, in which case one times x is an x, right? One half x squared is gonna give us a negative one half x cubed. x times positive 124th x to the fourth is gonna be 124th x to the fifth, like so. And you keep on going as far as you need to go because they might tell you find the degree three Taylor polynomial or in this case, the degree five. And so then once you multiply the divisor by your quotient right there, x, you then subtract these things from above, right? x minus x is zero, so you get nothing there. Negative one sixth x cubed plus one half x cubed is gonna give you a one third x cubed. One over 120 x to the fifth minus 124th x to the fifth. The fractions can be a little bit tedious, but you're gonna get 130th x to the fifth. And then keep on going as many terms as you need, right? Then you repeat this process. Look at the first term, one third x cubed. And so you take one third x cubed and you divide it by one. All that, in this case, it's not super exciting. It's gonna be one third x cubed again, right? That's the number you're gonna record up here. And so then what we're gonna do, erase all this, we're gonna take this number and distribute it onto each term in the divisor. Go as far as you need and we're gonna record those things down here. And so assuming we got this number correctly, these two numbers should be identical. So when you subtract them, they're identical, just like Fred and George Weasley, one third x cubed minus one third x cubed, they're gonna cancel and then do the subsequent division right there. Negative 130th x to the fifth plus one sixth x to the fifth, that's gonna give you two fifteenths x to the fifth. And then keep on going and going and going as far as you need. The next question is then how many times does one divided into two fifteenths x to the fifth that then brings it up here and you can keep on going. And so then we've now figured out the first three terms of the Maclaurin series for tangent. You get x plus one third x cubed plus two fifteenths x to the fifth and you can keep on going on and on and on. This gives us a recursive way of finding the Maclaurin series for division and multiplication. And with a little bit more sophisticated mathematics, we can remove this recursion and actually get a general formula. It turns out these tangent numbers are actually kind of a fun sequence, like one, one third, two fifteenths, what's the next number. There's actually a nice formula that predicts what it's gonna be, but developing that formula goes beyond the scope of our class, a recursive formula will be sufficient for us and so we can find these Taylor series and these Taylor polynomials more specifically by recursion for division and quotients. And sometimes we're gonna need to be able to do these things. And so that brings us to the end of lecture 47, where we're gonna talk about Maclaurin series and Taylor series. As I keep on heading to in our next lecture 48, we're gonna talk about Taylor polynomials and focus more on how Taylor polynomials can be used to approximate problems, like integration like we've seen before. So we'll learn more about the Taylor polynomials next time, but we've already seen how useful they are. The Taylor polynomials approximate the Taylor series and like in situations like this, we don't know the whole Taylor series, but we do know the Taylor polynomial up to degree five. And so we might actually just use that to approximate sign, our tangent here. Tangent is approximately this polynomial right here. And so next time we'll talk some more about that and do let me know if you have any questions in the meanwhile, if you want to, I mean, feel free to post your comment or your questions in the comments below. I'm happy to answer them for you. It really is kind of fun and I hope to hear from you all next time. See ya.


Well, thank you. Oh, it does work good So so thanks to the organizers for for putting this together, you know the workshop and the school last week Which I wasn't at in the thing last year like this is this is great and also I want to thank Craig and Mel for providing this This sort of fun space of mathematics for us all to to work in like it's it's really you know math as Craig said about Mel and that like I'm sure Baren to say about Craig like it would just math would be a lot Poor or feel to be a lot poorer without without their contributions and that's in some sense part of the the theme of my talk because You know Okay, so a conference honoring Mel and Craig of a conference on characteristic p-algebra. This is what the eighth ninth talk and there has been No talk about tight closure theory 20 years ago. That would be unthinkable, right? That would be impossible and I think that that is because of the You know the common wisdom so right so so Craig's periodization by the way, hi Craig Craig's periodization that he did yesterday. The reason I asked that question is he said 1985 to 2007 was smells, you know tight closure period, right and And and what is 2007 to me what that is is the year that the pre-print by Holger and and Paul Monsky went up on archive Showing that tight closure does not commute with arbitrary localizations. I Say arbitrary on purpose because they said carefully in their talk. I'm well in the in their paper That they don't that there's some that they that there are some cases They don't cover and including open localization like localizing at a single element And that's kind of my jumping off Point so I want to say I want to think of this talk as an invitation To kind of reinv you know to have a different perspective on the tight closure operation and similar operations That hopefully will make people want to kind of work in it again So okay, so Brenner Monsky's I haven't written anything yet, but whatever So Brenner Monsky's Paper I think Created the following common wisdom well we wanted tight closure to be a geometric operation right like in a real closure like for being as closure sort of is But it doesn't commute with localization in particular in Brenner Monsky's paper They showed that it doesn't work well with deformation theory and so therefore it couldn't be an Operation of geometric interest so let's so that the stuff that is actually of geometric interest is the The things that come from tight clue that arose in the study of tight closure theory namely test ideals and Certain F singularities such as F purity and strong left strong F regularity, right? But the thing is I think that there are some some geometry to do here That's that that they left that that everybody left on the table, which I'd like to present to you as a meal Now I'm not a geometry So I hope and and there are definitely people in this room And on this zoom that know a lot more geometry than than I do So so like I say an invitation, you know people who who know more geometry than I do like I'm gonna show you how to take the the the the tight closure of an ideal sheaf or you know a sheaf or a sheaf of modules will work will work the same way and and and sort of an invitation to sort of See if there is something geometric to do here Okay, so conventions Oh, and the other thing is Right, so what do I want to say? So our is just gonna be a commutative ring and When I talk about tight closure is gonna be a characteristic P ring P bigger than zero The theory in so so yeah, so I'm gonna basically I'm gonna show you for various closure or even pre closure operations on on Ideals in in in sort of a family of rings that's associated with the scheme I'll show you sort of when you can make a sheaf theory out of it And when that sheaf theory works fairly nicely, so there's sort of varying degrees of Niceness and that the coherence is gonna be kind of the big question at the end Like when one of these like you take the the the closure of a coherent sheaf when is it when is that a coherent sheaf? It'll be a sheaf, but I don't know when it's gonna be coherent assuming that so so properties of Preclosures I'm gonna call them new New in the sense. I mean nothing's new in mathematics, right? It's all there for us to discover But I mean that you know that I've identified that are important properties We're gonna have something called open persistence and something called glue ability Okay, and open persistence is gonna show you that there's a sheaf and glue ability means That You don't have to shrink it down. You'll see what I mean. You'll see what I mean and so Or I'm so I'm gonna show you this sheaf, and then we'll talk about closeness and then at the end and with a new Singularity type I Say new because of course conjecturally because this is gonna be between weak F regularity and F regularity So conjecture at so conjecturally it should just those should all be the same, right if you're because weak, you know We can play strong, you know Ian's gonna talk about some progress on that later in the week, I guess and But But anyway, it's it's new to the extent that if you don't know that that that week of regularity is F regularity then then you don't know then you don't know that this is either of them Okay, so So many words. All right, so definition Okay, so what's a closure? What's a pre-closure? So? Let s be a poset partially ordered set Okay, so a pre-closure is a function cl from s to s You know written You know x goes to x cl Such which is such that for all x and y in s x is bounded above by the closure of x and if x is Under y then the closure of x is under the closure of y Okay, and If cl is item potent i.e. the closure of things are closed We call it a closure So pre means might not be item potent. So this is this is all standard terminology from partially ordered set theory Okay, so I mean examples of that so why am I going to post sets is because You know if you're talking about ideals then you can talk about subsets and things but if you're talking about ideal sheaves then those aren't really sets and so you have to You have to but there's a poset structure Okay So So if If x is a scheme, I mean if you don't like that just think algebraic variety But no a scheme so like spec r then a a pre-closure CL on x For now is going to be on all the affine open You know rings that come from it right is a choice for each affine open you in X of a pre-closure also written CL on the ideals of OX of you So that's so anyway, so if you just have that that you can't really do much So that's why we need these these conditions of open persistence and globality So Definition yes, these are all definitions so far So and so of course I say pre because it might just be a closure, and then you took yeah, so pre-closure CL on a scheme X is openly persistent for any Containment of affine opens UV affine opens in X and G the corresponding ring homomorphism So here what u is spec? s v is spec r Right and G is like the the corresponding ring map going the other way For all ideals of r G of I closure is contained in Is closure so I call this openly persistent because it's persistent on these in the containment of opens And the reason I use persistence is because this is you know from from tight closure theory persistence is is You know you want to say that well, okay, so so for basically any ring map if your base ring is good enough Then you know the tight closure should persist along that ring map, but it turns out open persistence is a much easier thing to satisfy Basically because G of R not it's contained in s not But anyway so So fact if if I'm gonna say if CL is openly persents openly persistent on say spec R and What happens if you then I closure RF is contained in IRF closure for all F and R Right because that's just the basic affine the basic open sets and in spec R Okay, so Proposition I'll call this proposition a Tight closure is Openly persistent right so and here X on any netherian FP scheme And it's just because So basically the reason I won't get the proof I'll give a reason G So if R to s is open in in that sense right the corresponds to a containment of affine open sets Then G of R not is contained s not and In tight closure theory the only problem with persistence is that you know the things in our not might Like go to zero or go inside of the minimal primes of s and that just never happens. It's because of going down So it's it's flat and you know any Inclusion of opens is flat and so flat implies going down and that means that the minimal primes behave Well in such a way that G of R not is contained in s not um, okay So theorem 1 slash definition, so let CL be an openly persistent on a scheme X So we're gonna extend To a pre closure on The ideal sheaves I DSH the the post set of ideal sheaves on X OX I don't know the terminology As follows Okay so for For such an ideal sheaf. What do you do? and for s sorry for you open in X I need to so and s in OX you is in I of you If this is the definition of I of you I'm sorry. I closure of you if what if there if for any X and X there exists a affine open V with X in V and V is contained in you and s restricted to V is in I of V closure in OX of V which makes sense because OX of V is a ring and IV is an ideal in it and and we've defined the closure on all these Rings for all the affine open sets, right? so So anyway, so this at least gives you an assignment for each You of a set I closure of you, but in fact, this is this this actually makes yes No any ideal sheaf. Yeah later on coherence and cause a coherence will come up in a big way, but not yet Then I closure is in fact a sheaf of ideals and CL is a Here I'm going to say is a pre closure on the set of ideal on the post set of ideal. She is of X and if X is an aetherian and CL is item potent on each affine open then CL is item potent in other words you have an actual closure on the ideal sheaves and One more statement, which is part of this theorem moreover given an open you and S in OX you we have s in I closure of you if and only if there exists an open affine cover of you such that Alpha So what meaning that that you is the union of the you alphas and these are all open alphines inside of you for all alpha s restricted to you alpha is in I of you alpha closure in OX of you alpha and the proof is is you know straightforward But would take the next ten minutes if I wanted to actually do it because you know how proofs and sheaf theory are But it really is straightforward. It's just you know applying sheaf stuff to this Okay, but I said exists an open affine cover and I'm saying that you have to like for all these use you have to sort of find enough and for every X in in I guess in you Yeah Yeah, for every X in you you have to find sort of a different V and so on so but what you want is you want some Flexibility right you want to you want to be able to say oh well I just want once I go down to to like some affine open and you I want to say okay So I closure of that V should be I of V closure and that's what we need glue ability for So glue ability right so first of all corollary I star exists you take a Sheep of ideals and in a netherian FP scheme, and then you can take it's tight closure in this way Okay, so glue ability Definition let C L be an openly persistent closure operate pre closure operation on spec on spec R let's say C L by definition is glueable if For all collections F alpha in R not R not if you is the complement of the union of the minimal primes And for all G in the ideal generated by the F alphas and for all ideals I of R and For all X in R if X over one is in the Closure in IF alpha for all alpha Then X over one. I just need G in the radical of this Then X over one is in the closure of I G and these are taken in the corresponding localized rings and then For a scheme X C L is glueable over X if it's glueable on All affine opens My chalk broke, so let's see if there exists another one There's a good one Can I be here? I can be here, right? So proposition B So this is going to show that there's a connection with with commuting with localization at single elements If C L is openly persistent spec R IF closure equals I closure localized at F for all F and R Then it's global man. I spent so much time Preparing for you an example of a of a global one that does not commute with localization at at single elements But there isn't really time to present it. I'll just say what it is So I'll just say fact the converse Fails There exists a counter example arising I'll say arising from Zero local cosmology is it basically you have an idea like the closure is like if you have an ideal in a ring you you you delete all the Zero-dimensional primary components So that's that turns out to be a global closure operation that is not That that is not that doesn't commute with localization at single elements even in a two-dimensional Power series ring overfield Okay, okay So Proposition C Tight closure is global over any netherian FP scheme Yeah, and I wrote out the proof here, but But it is it's I mean it it comes if you've worked with tight closure theory You can sort of imagine how the the proof would go you look like you know G is in your radical of your ideal generator by your F alphas, but you might as well just take F1 through Fn For for each of the localized ones you take a ci that is kind of tests for the tight closure of your of your X in that thing but those that but because of the condition that all your f's avoided all the Minimal primes your ci's also have to then avoid the minimal primes of r when you delocalize and then you just take the product of the ci's and and you Present g as a lean or g to the n as a linear combination of you know some powers of of of f and then there you go And so now g to the n Times c x to the q is an i bracket q and then and then they're for all q and the errands can vary wildly, but that's that's okay Okay, so now I've talked through the proof And I guess that's if you have a you know big test element then there's an easier proof, but you don't actually need it for this so Theorem two and this is where the coherence is going to come in Let's see L be a glueable preclosure on a scheme X Let I be a quasi coherent. I'm just going to call it qc ideal sheaf s Be an o x of you where you is a open s is in I closure of you If and only if for all affine Opens V in you as restricted to V is in I of V closure Moreover For all affine Open V I Of the closure equals I closure of V Exactly what one wants right this means that That this is true of tight closure as well So what about quasi coherence of I closure So let R be a ring Let's see L be a glueable Preclosure on spec R Then the following are equivalent all affin R I f closure equals I closure localized at F B I Tilda closure is quasi coherent and C I Tilda closure is equal to I closure tilde So right so so somehow this this now comes to the to the four now the question of whether tight closure commutes with localization at Single elements for a particular ideal and for a particular ideal is exactly the question of whether when you take the corresponding Coherent sheaf of ideals you take its tight closure. Is that does that remain quasi coherent? Spent a lot of time on math sign it looking up stuff about Criteria for checking quasi coherence other than you know the definition or push forwards and Like I said, I don't know much So Okay, and Then Let's see Here's a new singularity type another you know, we just had one last talk now Let's have one this talk It's kind of a non-local f-singularity type. So let's say that R is Semi-affregular if for all ideal sheaves I I Equals I star and equivalently If X is spec R Is to say that RF is weakly F regular Brawl F and R and I know nobody's defined weakly F regular this This week, but it just means that every ideal is tightly closed So it's you know, so it's equivalent to say that every ideal in in our F is tightly closed for every F and So what do you have so you have? F regularity so just yeah two more results F regularity implies semi F regularity implies weak F regularity It's more or less obvious But Proposition if R is a Jacobson ring for instance finally finally generated over a field Then R Is semi of regularity? It's semi. I'm gonna say semi if and only if weak semi of regular if and only if weakly F regular and So then you might say oh, well, okay, so then maybe it's just the same as weak F regularity in general But if it is Then a big thing would be proved because here's another proposition Let R be weakly F regular suppose That for all prime ideals Such that R localizes P is weakly weakly F regular R localized at P is also semi F regular So for instance, if you knew that semi of regularity and weak F regularity were true for local rings even just local rings of R then R itself is F regular Not just semi of regular but F regular. So what does this mean? This means that for Jacobson rings they're equivalent But if you knew that they were equivalent in for arbitrary And in local rings, then you would know that then you would solve this major conjecture that we get for regularity localizes And I'm sorry for going over time but Thank you for your attention. We could have one question Oh, Craig has a question Do you have a question Craig? I think that was just a clap. That was a clap. Okay, I Guess I mean the only comment would be that last thing. It's it's like Murphy's result your equivalents Yeah, I guess it's the same kind of idea the proof is the same kind of idea well But well, no, it's it's just well, it's just because maximal ideals contract to maximal ideals Is is kind of where it comes from and and since we kept regularity is local on maximal ideals and That's kind of where it comes from. I don't think yeah, I didn't prove it like like Murphy's thing I didn't need I didn't need any accountability argument or anything Thank you. Yeah, I think the interest of time will Post on any other questions to the deep back. All right. Thank you


I'm working away from home at the moment and it's difficult it's difficult to write down just to even remember what what we're spending. I think I'm gonna start recording it on video. Hello Neil, I am the type of dad and I've been vlogging here for a while and I'm about 90 days into sharing publicly what we spend as a family every day online. I think it's gonna be 10 times harder and I think it's probably 10 times more pointless so just make lots of short weird videos just to catalogue how money is my family. Okay I've already hit an instant problem I completely forgot to video massive spend up in the mini supermarket. I bought two ready meals in bulk, two for £4 and to try and get myself off spending on the coffees I've bought a jar of coffee and some milk so I'm hoping that'll save me a few quid. The joy there is 10. The hardest thing is just just remembering to get the camera out when I'm spending. I'm ticket this morning, joy is 10. All right so I didn't video it but I did take a really bad photo of it instead and then there was the works canteen, I had breakfast there, that was joy of 10. I took a video of that. So that was my first day of trying to video everything I've spent I was just one more thing. I do need something to eat. So that was my first day of trying to video capture everything I've spent today. Man it's hard I didn't know if I'm going to do this tomorrow or even more than once a week. If I don't do it every day is it is there any point? I don't know. Thanks for watching and you know what to do with the thumbs and subscribe and everything. Can you please help my daddy get 1000 subscribers? Just click on his face. Thanks! Bye!


Last week, major companies announced that they were pulling advertisements from Twitter following Elon Musk's endorsement of this viciously anti-Semitic conspiracy theory, where a Twitter user claims that white-hating Jewish people are facilitating the great replacement of whites by flooding countries with immigrants. Now, as you can see, Musk responded saying, you have said the actual truth. Now, this is the same conspiracy theory for those who don't know that was echoed by the Tree of Life synagogue shooter. And Elon Musk agrees with him, apparently. Now, on top of that, Elon Musk has unbanned, holocaust denying Nazis, and he still refuses to remove Nazis like this one, who has a literal transphobic slur in their Twitter handle, along with 1488. And, I mean, all of this has been known for a long time about Elon Musk. To be charitable, his sympathy towards Nazis and white supremacists is kind of an open secret, because he constantly communicates with them. But him endorsing that conspiracy theory was sort of the straw that broke the camel's back, because it was one of the few instances where there was no room for interpretation. There's no plausible deniability there. You have to take it at face value because he's just saying it. He went mask off. He said the quiet part loud. And as a result, advertisers couldn't justify spending money on Twitter, and they left. And now he's in full-on damage control mode and a shameless effort to lure back Twitter's advertisers. And one of the first things that he did was censor pro-Palestine speech on Twitter. Now, for ADL President Jonathan Greenblatt, that was all he needed to do because he praised Musk for his, quote, leadership and fighting hate just one day after acknowledging that Musk endorsed this despicable anti-Semitic conspiracy theory. Oh, and Greenblatt then went on to thank UTA for firing Susan Sarandon after he held Musk as a leader.   The Rolling Stone reports, Pariser tells Rolling Stone that while he has historically supported the ADL's work against anti-Semitism and hate speech, unless the organization changes course, I plan to step down from the tech advisory board. He noted, however, that if the ADL acknowledged it had made a big mistake and course corrected, he would keep his place on the board. Though some ADL advisory board members who spoke with Rolling Stone continue to support Greenblatt, others have raised questions about his leadership, saying that the CEO's hardline stance on defending the Israeli government's actions in Gaza, as well as Greenblatt's history of aligning himself with an individual known for espousing anti-Semitic conspiracy theories, are eroding the organization's credibility, and I think that's important. The ADL has done good work in the past, so I want them to succeed, but Greenblatt here is undermining them by going to bat for the actions of the Israeli government, right? And I think that what's happening with the ADL is a microcosm of a bigger thing that we're seeing with Israel's defenders. If support for Israel's government automatically gets all of your accusations of anti-Semitism erased like that, it tells you a lot about the motivations of these folks, doesn't it? It suggests that Israel's defenders are only invoking anti-Semitism to shut down criticism of Israel, and this is a very dangerous game to play because anti-Semitism is very real, and unfortunately it's on the rise around the world and in the United States, and conflating criticism of Israel's government with bigotry against the Jewish people is deeply dangerous, because Israel is a government, and we are allowed to criticize governments without having that criticism be conflated with the people of that government. Nobody questions whether or not we're Islamophobic for condemning Saudi Arabia's genocide in Yemen. Nobody cries racism for condemning corrupt warlords in sub-Saharan Africa, nor should they. But since Israel's genocide on Gaza is indefensible, the go-to tactic is to just shut down criticism altogether with cynical accusations of anti-Semitism.  We've seen this happen with Naomi Klein, Bernie Sanders, and it's just, it's gross, because in other words, it means that any and all criticisms of Israel are illegitimate by default. But if you unconditionally support Israel, their defenders will give you a pass, even if you say the most horrific anti-Semitism imaginable as Elon Musk did. And it's not just Elon Musk. Pastor John Hagey was embraced at the March for Israel demonstration despite blaming Jewish people for the Holocaust and claiming that Hitler was sent by God. And now we're seeing the same thing happen where Elon Musk, who blamed Jewish people, not Israel, for white replacement, is all of a sudden embraced by Israel's defenders because he's signaling support for Israel's genocide in Gaza. In fact, he's even being welcomed in Israel. The New York Times reports, Elon Musk traveled to Israel to meet Prime Minister Benjamin Netanyahu on Monday, touring the scene of a Hamas attack in a visit that appeared aimed at calming the outcry over his endorsement of an anti-Semitic conspiracy theory on X, the social media platform he owns. On Tuesday, after arriving in Israel, Mr. Musk wrote on X that actions speak louder than words, wearing a flak jacket he toured Kaffar-Aza in Israeli kebuts, where dozens of people were killed during the Hamas attack on October 7th.  Does anyone actually believe that Elon Musk has had a change of heart in the week since he blamed Jewish people for white replacement? Of course not. Nobody believes that. Netanyahu doesn't believe that. But Netanyahu is giving cover to this anti-Semite specifically because Musk has the power to control the flow of information on Twitter, and that could be an invaluable asset to Israel's war propaganda effort. And Netanyahu is already facing massive backlash in Israel for a number of reasons. He's facing corruption charges and 76% of Israelis think he should resign, and protesters even chanted jail now outside of his home earlier this month, largely because they blame him for the security failure that allowed the Hamas attack on 10-7. And once again, he's now being criticized over this publicity stunt with Elon Musk. For example, Haritz writer Ben Samuel slammed Netanyahu for giving Musk a hero's welcome, writing, Israel's repulsive embrace of Elon Musk is a cynical betrayal of Jews dead and alive, welcoming such a toxic mogul with open arms and taking him around sites of a massacre that has been belittled, demeaned, and denied on his watch should be a stain on Netanyahu's legacy. And also Esther Solomon, the editor-in-chief of Haritz, tweeted, Blay an anti-Semite and publisher of anti-Semitism, Elon Musk, should be persona non grata in Israel. Instead, Netanyahu, plumbing new depths of amoral sycophancy, gives him a PR visit to the Kubitsom, devastated by Hamas, profane, venal, bilious, both of them.  This man endorsed an anti-Semitic conspiracy theory that is so vile, it was literally used to justify the slaughter of innocent Jewish people. But now Netanyahu is giving him cover and taking him on a tour because it's politically expedient. Because Elon Musk could be a valuable asset to him and his genocide against Gazans.  And Elon Musk is really going all in on his embrace of Israel's war propaganda. And I say this because during the Twitter spaces with Netanyahu, Musk seemingly agreed with the dumbest lies that Netanyahu used to manufacture consent for his genocide in Gaza. Case in point. The one thing you cannot do is give immunity to the terrorists because they're hiding among civilians. Because if you give them immunity, everybody says they shouldn't be doing this. But effectively, nobody's willing to take the action to make sure that this is not an effective tactic. Because if it is, it'll repeat itself again and again and again. By the way, Hamas says we're going to do it again and again. But it's not only against Israel that they'll do it. This will spread very quickly throughout the Middle East and peril the entire region. From there, they'll go to Europe and from there, they'll also go elsewhere to America, whom they call the Great Satan. We're just a little... Yes, America is the Great Satan. America is the Great Satan. And this is an Iranian axis. Yes, yes, yes. It's Iran, Hezbollah, the Houthis, and Hamas. Yes. It's all part of that same axis that goes against Israel, the United States, free civilization, and the moderate Arab states. We're all on one side. They're on the other side. You have... We have a first omission to destroy Hamas. Nothing's going to stop that.  If you want security, destroy Hamas. If you want a better life for the Palestinians in Gaza who've been hijacked by Hamas, destroy Hamas. All of that is a precursor to the question that you asked. You first have to get rid of the poisonous regime. As you did in Germany, as you did in Japan in World War II. These were two... There's no choice. There's no choice. That's a prerequisite. Yes. Listening to that makes it very clear as to why Netanyahu is doing this. 1.2 million people on Twitter heard the Israeli Prime Minister Firmonger about Hamas coming to America because they're supposedly hell-bent on global domination just like the Nazis. I mean, this is why Netanyahu felt the need to give Elon Musk a pass for his anti-Semitism. That's why. Because he's useful. But what's especially gross is these self-interested assholes are both performing for everyone. And I think it's obvious to see that, but a lot of people won't see through their opportunism. Musk, for example, only cares about money and wants to get advertisers back on Twitter and possibly sell Starlink to Israel as well. And maybe to Gaza if Israel approves, of course. But it's definitely not an occupation. Don't call it that. And when it comes to Netanyahu, he only cares about his imperialist and neo-conservative ambitions. But I mean, they're both pretending to care about people when they don't. It's all an act. It's theater. Now, Jewish activist and writer Joshua P. Hill explained in a sub-stack published on October 29th how this game that's being played by Zionists like Netanyahu literally endangers Jewish people around the world. And here's what he says. It is easy to denounce every anti-Semitic attack that has come in the wake of Israel's relentless and genocidal attack on Gaza. I condemn them all. I want my family and my friends to be safe. I want to be safe. And so I also condemn the decades-long Zionist effort to claim my entire religion and claim my entire people. Judaism is a religion, a culture, and in many ways an ethnicity. Zionism is a political and colonial project. In equating them, the state of Israel and Zionist propagandists tie my people to their violence. They link us to their atrocities in the minds of millions. And for them, that is just one of the acceptable prices worth paying in their efforts to get the world to side with their program of ethnic cleansing, occupation, and apartheid. Now, I do not believe that our safety is really the priority of the Zionist government. They work overtime to tie my people, our entire religion, to their violent project in the eyes of the world. No matter the harm and backlash this contributes to. They force Zionist Jews everywhere into the arms of the right, into the arms of people who couldn't care less about us. Exactly. People like Elon Musk, for example. Now, I would highly encourage you to read the entire column. I'll link to it down below because he makes a lot of really important points. And I think that his voice is important here in this discussion. But I think this is why it's important to reiterate, again, that Jewish people are not responsible for the actions of the Israeli government in the same way that Gazans aren't responsible for the actions of Hamas. Again, people are people and governments are governments. So if you say that it's anti-Semitic to criticize Israel's indiscriminate bombing campaign in Gaza, you are necessarily ascribing culpability to Jewish people who have nothing to do with those atrocities. And I think that that conflation and generalization of all Jewish people with Israel is anti-Semitism. It's almost an incitement of hatred and violence against these people who have nothing to do with what the government of Israel is doing. People are not monolithic. Israeli citizens don't even like Nanyahu, and governments often don't have the same interests as their people. So it is absolutely absurd to me to say that if you criticize what Israel is doing, you are anti-Semitic. But defenders of Israel don't care. They'll gladly conflate any and all criticism of Israel with anti-Semitism because it's a tactic that works. That's why they do it. But it's disgusting and a very dangerous game to play, which is why so many Jewish people are speaking out and saying not in our name.  You don't speak for us. Now, with that being said, it is important to point out that there are instances where actual anti-Semites will try to cloak their anti-Semitism and anti-Zionism or criticisms of Israel. And we should call that out when we see it. For example, Candace Owens. She's a good example of this, who lately is criticizing Israel. But at the same time, last year, she was defending Kanye West when he was going on his pro-Hitler ranks. So it's good to acknowledge that there are bad faith people who are using this as an opportunity to gin up anti-Semitism and they're trying to do that under the pretense of criticizing Israel. And it's also important to point out that non-Jewish people should never ever dismiss the seriousness or reality of anti-Semitism just because it's disingenuously weaponized by Israel's defenders. Because anti-Semitism, again, is very real and I think we all have a responsibility to combat it whenever we see it. And one way to combat it is to not let anti-Semites like Elon Musk get a pass after endorsing one of the most despicable anti-Semitic conspiracy theories in existence. Thank you for watching.


Next question is from tax-free mitt. What advice do you have for someone who tends to hop between training programs and diet plans? I get excited to start something new, but tend to quickly get fixated on what I want to do next. Yeah, you're falling in love with the feeling of excitement and motivation. Novelty, yeah. The entertainment of it. Yeah, and now here's the deal. I understand, okay? When I feel motivated and excited, it's one of my favorite feelings. Nothing's hard, everything's easy. I got all the energy in the world to do all the stuff that I need to do. I've never had to get a really convinced client to work out when they were motivated. It just happened. I've never had to convince a client to eat right when they were motivated. It was when that feeling went away that we start to see some of the challenges. So the problem here, what the person asking this question is, you're so in love with the excitement and motivation that when that fades, well, that's it. What do I do next? Because now that's gone. In my training career, what this usually looked like was a person that would sign up for a marathon and then a triathlon and then obstacle course race and then it's like, they always had to sign up for something or to keep themselves working out. And that was just a losing strategy at some point. So what you need to do is develop the skill of discipline and consistency and trust the process. Now, what does that mean? That means that when you follow a program, let's say you follow maps and a bollock, right? Maps and a bollock is broken down into three phases.  It's just the way it is. I love phase one. I like the heavy training. I like the low reps, right? Phase three with the super sets and the faster pace. Like that's my least favorite. But if I follow the program, I go all the way through. I trust the process because I know the value that I provide and ultimately gives me much better results doing that. So I mean, I have a little challenge to that. You necessarily don't have to. You could still continue to do this way, but it really is the difference between exercising and training. If you are going to the gym and you're bouncing from routine to routine and you're trying different stuff out all the time and you seek this novelty every time you go to the gym and you need that for motivation to get you in the gym and exercise, then fine. And if you're content with where your physique is at, where your strength is at, there's nothing wrong with you exercising like this. But if you have goals in mind and you're trying to improve strength, you're trying to build muscle, you're trying to change your body composition, then you wanna follow a training routine.    I guess better than nothing. Right, with no rhyme or reason. And again, if this person is in a place where they are content with their performance, their health, their strength, their aesthetics, if they're happy with all that, then it's actually not that big of a deal.  And you know what the problem is is that they don't tend to, what this tends to lead to in my experience is not every week I'm trying something different. It's I try something different, try something different, try something different, nothing excites me, I stop.  It rarely looks like something new every single week. It starts that way and then it becomes like nothing is sparking this in me and now I'm gonna stop completely. Yes, I guess sort of like to both your ideas, like it's obviously very important for you to be specific at what you're doing in terms of training towards a goal. And so if that's in mind and that's sort of your cornerstone that you come back to, there's plenty of room for you to interrupt that with novelty and adding in like a new type of a program to be entertained by and just keep it fresh and keep it going. And this is something that I do occasionally. And this is why I got into like unconventional tools or methods or learning something that I could then adopt and then incorporate into my meat and potatoes programs. So it's not so stale, but at the same time, this is what's moving the needle. And so for me, like I have to learn how to just be disciplined to come back to that continuously if this is where I wanna take my progress and where I wanna take my body. So yeah, to Adam's point of just exercising, yeah, there is room for that. If you're talking just longevity and staying healthy and enjoying just the overall activity of movement, but being specific and having goals and trying to get somewhere you really do need to narrow it down. Well, that's how I would, if this was a client of mine and they came to me with this problem, that would be my follow-up question. Well, do you wanna change anything about your body? Are you happy where your strength is?  Are you happy where your performance and strength is? And they said, yeah, I just like to, cool, keep doing your thing. It's not a big deal. But if you go, well, I would like to lean out a little bit more and I would like to be a little stronger or I would like to work on my shoulder. Okay, well, then we need to train. And then one of the things that, and this is where like tracking, I think has its advantages, is when I'm trying to train and I'm really trying to achieve a goal, I will focus in on like specific exercises in my program. So let's say you're following maps anabolic, I'll pick like my squat or my overhead prowl, pick a couple of movements and I'm tracking them. And I wanna see week over week progression. And so that'll keep me excited about the next week. Like, oh man, this week I put five more pounds on the bar with my push press or my form got even better or my range of motion was even better on my squat. So I'll hyper focus on a couple of movements within the program that will keep me excited about coming back to that exercise in the following weeks if you have trouble sticking to a training routine and you know that's what you need to do. Well, I think another good mentality with that is as you're going through your program, like you're noticing where the weaknesses lie and where the areas you could address to improve in terms of like seeking out another type of a program instead of going towards something that draws you in based on entertainment or something that you really enjoy like try and focus it a little bit more on maybe what's really difficult for you, you know, to kind of like change your mindset around that in order to improve the overall. Yeah, and again, this is all assuming that this person's always consistent. And again, in my experience, this particular type of person isn't consistent a long term. They're consistent because they're doing all this variety and at some point they just stop. So yeah, if they were always consistent coming up with something new all the time, I guess that's better than nothing. But again, in my experience, that tends to not be the case.


I'm gonna answer in this video a question that I asked in a live stream earlier this year, and that is what is the thinkpad of cars? Okay, for those who know anything about different computers thinkpads are the model to have they are the brand to have Because they don't look like a brand look at this thing. It's ugly. All right. Who's gonna get this? Why are you gonna get this? You're gonna get it. I'll tell you because it's durable. It's long lasting I got this thing for 90 bucks It was like 10 years old and it's still running great. These computers are easy to fix. They're cheap That I don't know. They're just the best that as they come right if you're getting into any other kind of they're like the anti-apple Okay, you're not gonna get you're not gonna see this at you know Starbucks, right? This is not for conspicuous consumption. This is for I want a computer that works I don't care about how it looks and plus oh look I get the best part is right here the little I don't know The the red dot mouse thing that is the best part. You might hate those but that is the best So what is the think pad of cars? This is a question I had I actually at the beginning of this year I did not know anything about cars. I don't I'm not a professional now But you know, I was I was looking for a car because I had this Nissan Maxima The transmission was dying and I'll go ahead and say before people click off the video I'll go ahead and give you a spoiler this car a Lexus ES 330 during this terrible market for buying cars I got this thing for less than $2,000 and it is basically the think pad of cars. I don't want to say just this model But one it when I asked people what is the think pad of cars? They pretty much told me if you want a good car that is going to last long Independent of how it looks now. I think this car looks very nice I don't really like you know the how the I don't know headlights look but in general I think this car looks very nice, but what people said is get either a Toyota or get a Honda and One thing that I did not know is that Lexus is are actually just Toyotas, right? It's the same company same people They they just make the their luxury cars are called Lexus's and actually Honda's You know their luxury Variety is a Acura or whatever so I'm gonna put this in here. So it's not I don't know in my way So how did I get this car? How did I get it for less than $2,000? I'll go ahead and say Once I learned all this stuff about cars I was actually looking for a Toyota and there was a Toyota Camry at some dealer that I was checking out a used Toyota Camry, I want to say there was one that was like 2012 and one that was You know a little older But he had one of these it's a 2004 ES 330 Lexus and I really loved I love the way it looks Let's actually let's get inside Way back in the day I used to have a Hyundai Sonata Which was it was like the luxury brand of the Sonata and it just had this interior that is a lot like this You know you get this kind of fake wood paneling. I like how clear this is And it just drove fantastic and this one drives. I didn't end up getting that Lexus But because the guy there he was like, oh, I want $8,000 for it Or and you know, I tried to get them down to like 6,000 if I traded my car. I was like, oh, geez don't 8,000 I can't do that. But you know, he didn't he didn't want to negotiate on that So I said goodbye and basically what I did is I'm gonna find I like this model of car I like the way it looks I like the way it feels And they pretty much like they have a really good reputation for lasting a long time So I want to get something like this. I mean, I wasn't totally I'm never gonna be totally settled on one model But really what I did is for the next I Don't know. This was like last summer But for the next like three or four months every couple of days I would just get on Craigslist and check for Lexus's Toyotas Acura's Honda's whatever and eventually one day I was settling and doing work Somewhere and I pull I see that there's some guy in a town like two hours away Nearly two hours like an hour and a half who was selling this car for 1,900 bucks. Okay, this Lexus ES 330 in fantastic condition for 1,900 bucks now the poor soul I ripped this guy's You know off so bad just by buying the car that he wanted to sell at the price that he wanted to sell because he literally Like the used cars are so expensive now. He literally could have You know wanted twice as much and someone would have gotten what a bought this thing just as quickly But I saw it like within 30 minutes of the ad being out there I drove I dropped everything went and got my checkbook and drove out here to get this car Or at least see if it's legit and it's been running great for a while now Why did he sell this for less than $2,000? Well, here's here's one reason I think there are little superficial bumps on it And I think his thinking he was well firstly you ever talked to someone on the phone I called this guy up you ever talked to someone on the phone He could you could just sort of tell they're a liberal You know what I mean like I he just answered the phone and he had this really you know like in PR voice and I was like Oh great. This guy doesn't know anything about cars. I might actually be getting a good deal He might not actually be ripping me off But so there are little superficial dents like that. I want to say there's another one back. It's pretty subtle Yeah, so right here if you can see that so there are a couple dents on it and How what's its mileage? Okay, let's look at its mileage Okay, if you're a normie and you don't know a lot about what brands are good This kind of mileage might frighten you. Okay. Look at that 212,000 miles. Okay, that's a lot of miles like if you if you are if you have like a VW that's a lot of miles if you have you know a Nissan it's a miracle your transmission hasn't been replaced five times if you have a Nissan But 200,000 miles on a Lexus or Toyota that's like maybe halfway through its lifespan Or you know at least it has you know a good couple of ten thousands of miles I mean this thing is basically worth two thousand and parts. Okay, that that was my view of it Even if it if this is going to be a total lemon, I better get this thing because it's nice AC runs great Very smooth. It feels like he just brought it off the lot So I think this guy's selling it. He just was like well 200,000 miles. That's a whole lot. Wow. This is oh Man, that's a I feel like this car is gonna break and it has a little dent So less than 2,000 bucks and I think he totally ripped himself off And you know what? I was almost that guy who was like gonna. Oh geez dude him about. Oh, well What about 1500 or something? You know, but you know, I didn't do that because I was like I'm already ripping this guy off just like given in the giving him the money he wants So all right other thing people Lexus people love showing people this but if you hold down the unlock button, okay, watch Yeah, it rolls down all the windows and I don't know opens the sunroof I think that's for like in the summer if you want to cool down your car before you're about to get in it in a Couple minutes. I think that's the purpose of that. Yeah now you can't close them back You can't like do the unlock or the lock button to like, you know undo it because it'll just I I thought it was going to do the alarm, but I guess it's not doing that Oh, man. I my panic button is broken or something Anyway, oh there it goes There it goes. Uh, anyway, uh, what was I going to say? So yeah, this is the car I got Um lexuses toyotas, hondas, acuras. I feel like um, they all have pretty good And now there were other cars. I looked at I did I mentioned vw's. I did try this very nice vw passat Um, and it had way less miles on it. I mean, it was more expensive than 2000 bucks It's like maybe like five or six thousand dollars. They wanted for it. Um, but the thing with those like Especially european cars european cars just do not have From what people told me a reputation of lasting very long Uh, so let's look inside.  You got a little coin thingy over here Um, got the typical stuff parking brakes down there. Uh, that's the trunk. Oh, let's let's look at the trunk Oh, yeah, and you also have typical Oh, so the other thing right there the guy, um The guy who had this car before he left his COVID-19 vaccine passport in here. So I can pretend that I got the uh, COVID-19 vaccine um Um, anyway, that was the one thing he left in the car. I like he searched it a lot before he left But I think it was like in the visor Uh, anyway in the back So here we got uh, pretty spacious Um, it comes with this little net which is very nice I actually have there's my bag of like live streaming stuff and Big ethernet cord which I occasionally use and I actually got this car for like or this book for Detailing and cleaning up cars. I think I just got it at some random book sores like a dollar And uh, I will say in terms of other bad things about the car or like wear and tear You will see There's a little wear and tear on this leather here. It's only on the interior side. The outside is fun This inside is a little worn in torn. Same thing on the passenger side I don't exactly know why that is.  I could probably have this thing looking good as new but honestly It's faults are more obvious in the sun right now Most of the time you can't even like see it like that and it just looks like, you know, perfectly fine You're not going to look at it And think it's a crappy car or anything Um and back here in the back See the typical stuff here. I actually I don't know how I feel about like this this kind of uh, I don't know with the I don't even know how to describe that thing But I don't know how I feel about it Got some cupholders here. Okay typical stuff And you got to pass through so you cannot pull these seats down So this might not be a good car, you know, it's not a pickup equivalent obviously But they do have a pass through which I have used like if you have something really long You need to put through here. You can Um, and it's also nice, you know, if you're traveling and someone in the back needs to get something from I don't know you Store your candies back here or something. They need to pick up a snack, right? There are ways of using that as well Um Yeah, I'm trying to think if there's anything else this thing is just nice Uh, I'm very happy like I'm not I'm not a greedy person in the sense that I like getting money Uh, but I think I am greedy in the sense that I'm totally someone asked in the comments Luke you're one of those, uh How much did I get this for kind of guys, aren't you and I definitely am I love a good deal I am greedy for deals. Maybe that's a vice. Maybe I should tone that down But I love I like this car, but I love the fact that I got it for less than 2000 bucks I'm very happy with it. I actually The month before I got this car. I bought a new lawnmower a zero turn lawnmower And I paid to the dollar twice as much on that than I did is this So the most expensive vehicle I own is actually a lawnmower not a car Anyway, so I'm just rambling out. That's it. That's my think pad of cars Um, you know happy searching if you're looking for one of these I like this model in particular, but um, you know, I would be happy to have any other kind of Toyota Or like a decent Honda my the only thing I care about is is it going to last long? I'm not going to have to buy one in like two three four five years. Um, you know I just don't have to think about it. And of course the insurance is actually lower than the car that I had before Um, I don't know. Maybe I pay 400 bucks in insurance a year something like that. Maybe a little more than that, but Um, I think for my age, that's uh, that's pretty cheap Age and gender, etc. All right, that's it. See you guys next time


Rice is the world's most important food crop, eaten and grown across the globe. With 90% of rice production and consumption occurring in Asia, the continent is at the heart of the global rice sector. Here, each person eats around 80 kilograms of rice every year. In some countries, such as Bangladesh, Cambodia and Laos, this number can double, rising to 160 kilograms per year. With people's lives so intertwined with rice, it cannot be denied that in Asia, rice is life. Rice is the world's most important food crop. For Asians, rice doesn't just stand for sustenance. In almost every country on the continent, this staple food is of great cultural, historical and political significance, a shared reality that brings together all Asian nations. More than that, rice unites Asia as the backbone of its economic development, food security and political stability. Rice creates an economic environment for economic growth. The tremendous economic miracle of Asia over the last 50 years coincides directly with increased rice supplies, decreased prices and stable and affordable rice. At the same time, Asia is also home to around 70% of the world's poorest people, those who are deeply affected when rice prices increase. Rice farmers, particularly in areas in Asia, are one of the sectors of the population that are poor. So with increasing productivity and increasing income, they can now buy food that are good for their health. They will have money for education and for health care. So in effect, they will reduce the poverty that they are in now. For more than 50 years, the International Rice Research Institute, or ERI, has been helping rice farmers grow more rice to reduce poverty and improve the lives of farmers and consumers while safeguarding the environment. ERI's high-yielding rice varieties helped prevent famine in Asia in the 1960s, lifted millions out of poverty and saved countless hectares of natural ecosystems from being turned into farmland. Despite these successes, feeding Asia's ever-growing population continues to demand more of rice production, a challenge that rice scientists are set to face. The people are increasing, but the production is not increasing. So what we're doing is to increase the production of rice so that we'll be using the same area at the same time we could feed the increasing population. ERI works with its public and private sector partners in leading the Global Rice Science Partnership, or GRISP, in Asia. Through collaboration, ERI is tailoring new rice varieties to meet the local needs of farmers and the preferences of consumers across different parts of Asia. ERI is helping farmers prepare for climate change, reduce the impact of rice production on the environment, and use natural resources, such as water, more efficiently. ERI is also finding smarter ways to grow rice and communicate new technologies to farmers, such as providing rice production advice via mobile phone apps. Based in the Philippines, ERI works across Asia and Africa with its partners and donors from all around the world as part of the Global Rice Science Partnership. With the potential to bring people out of poverty, out of unproductive fields, and out of the trap of rising prices, rice research is an essential part of the future of Asia and the world. You can help too. Join us as we deliver a better world through Rice Science.


Good morning and welcome family friends staff and of course fellow graduates as We are here today. I would like to take a moment to thank friends and family in particular I'm sure my family is especially excited since today marks the end of texts asking You'll still love me if I fail analysis, right? And you'll still love me if my GPA drops, right? Jokes aside, I can say wholeheartedly. I would not be here today if it were not for them in One of my courses this fall. We looked at the word comfort Its meaning today is one many of us are familiar with to provide support But the origin of the word has the meaning to strengthen So thank you so very much for being both types of support Comfort to support and strengthen Berkeley was my dream school. I still recall receiving the email that admissions decisions had been released Ironically, I was in the middle of a campus tour of another UC school. I won't say which one It was around four o'clock in the afternoon, which I remember since after calling my parents and sister I called my grandparents. They live in Pakistan, which meant I was waking them up at around 4 a.m However, I knew that this news would have to supersede time zones Picking a major was something I found to be very difficult In fact when my family would hear of a new major. I was considering they would call it my major of the month. I Don't blame them either At one point or another I was majoring in economics almost every type of engineering architecture something pre-law and something pre-med Ultimately, I found that mathematics was the perfect intersection of all of these sets While many who study math are passionate about theorems and analysis For me it was the process and way of thinking Studying mathematics has given me a framework for how to learn The pursuit of knowledge is at the core of our celebration today As I reflect on the experience we have shared One thing has become certain The set of knowledge we have acquired at Berkeley is infinite Studying mathematics. I have learned logic and analysis Similarly, my fellow graduates have learned the details of so many disciplines here at Cal however, this infinite set is not strictly academic knowledge as edgy teens we learned how to make memes as Dance enthusiasts we have learned choreography in parking garages and across campus as As participants in democracy, we have seen the actualization of civic responsibility Consequently on occasion we have learned how to reroute ourselves to avoid large groups and protests on campus as Members of this community. We have understood how to push ourselves in every capacity Sometimes we have even had to learn how to fail and in doing so how to pick up and try again as Millennials we have learned what it means to work as hard as is humanly possible If we feel there is something our community needs we make it While here we have learned about living on our own The process is similar for many within a few weeks our home sickness for family and friends from home sets in Then we begin to mix little things like full-size refrigerators and dishwashers But we have learned to appreciate Treasures like quarters on laundry day We improve our culinary abilities in learning to not set off the fire alarm every time we cook We adapt our meal prepping to an oven that half the time will preheat and the other half the time will be a gas leak We have learned more than we can possibly imagine in physics and chemistry We derive a hydrogen molecule in Biology we find out that there may be more to learn than the mitochondria being the powerhouse of the cell In computer science we take a recursive leap of faith in American cultures we learn to question American society and analyze the issues our Generation will face in language and history courses. We learn about the world beyond Berkeley We have grown by meeting individuals around us who come from all walks of life They may be of different ages and of nationalities Previously foreign to us Yet, even though we are all so different. We have found similarities as well in My first upper division linear algebra class. There was a student who could not have been older than nine At the same time another of my classmates was a retired gentleman who was auditing classes for the sake of learning Though we were different every Tuesday and Thursday at 8 a.m We sat in the Valley Life Sciences building together and were greeted dober utro by Professor Svezda stankova In learning from others we have grown as individuals who are accepting in doing so ultimately upholding those great morals and values of diversity and inclusion that our universities legacy supports and represents This legacy is one that came to be 150 years ago So while it may be terrifying to leave behind our time here We may instead look forward to joining 150 years of history as alumni of Berkeley The world is completely changed from the one it was in 1868 It is everyone in this room's responsibility to shape the world for the next 150 A few weeks ago A friend and I grabbed dinner from sliver and he told me about a quote he heard from the tv show American Horror Story A character says Nothing scares me. I went to Berkeley This resonated with me in a way. I'm sure it will with all of you Nothing can scare us because we made it through Berkeley My dream for my own future is to set up a system of urban cooperative farming in Lahore, Pakistan Thus creating jobs directly tackling the issue of hunger and building what we refer to as Neha, Pakistan Although I cannot be sure this pursuit will be fruitful After Berkeley, I feel prepared for the challenge and know that it is priceless to work on a project you truly believe been As the ceremony continues The time is rapidly approaching when we will no longer be undergraduate students at UC Berkeley Thus, we must relearn a few things We must adapt to our new environment far away from loved ones here We will no longer set our clocks to Berkeley time We may no longer have boba available at every street corner Moffitt Library will not be down the hill for all-nighters We will no longer be surrounded by a group for whom this campus is home Before closing this chapter in our lives, I urge you all to relearn one last thing Revisit the person you were when you applied to this school At some point each of us sat down to write an essay for admission In which we showed how certain we were that there was someone we wanted to be And most importantly That attending UC Berkeley was integral to our ability to become that person We all came here with goals and aspirations that we may not have had adequate time to pursue As of today though, december 15th 2018 We the graduating class of UC Berkeley have an opportunity to become whoever it is we choose to be So now I urge you all to go forward and be that person you once aspired to become For the last few years having been surrounded by some of the most hard-working and brilliant people I have ever met I hope no doubt that each of you has the ability to achieve those goals and become those people As with most endings this one is bittersweet Berkeley has helped shape each and every one of us into who we are today And while it may be sad to leave We are who we are in part because we are Berkeley bears So congratulations to all of the graduates Thank you to all of the family friends faculty and staff who have helped us along the way And last but not least go bears