The best and worst fruits to support your health. Okay, as you might already know, I have three rules when it comes to consuming fruit. Number one, stick to low sugar fruits. Number two, only eat them when they're in season. Check out episode 149 or 110 to learn more about this. And number three, consume fruits in moderation, even when they're in season. Okay, the truth is the main purpose of fruit is to help us put on weight. We actually have a deep rooted desire for fruit. It turns out that great apes suffered a genetic mutation that allowed them to convert the sugar and fruit fructose into fat. In fact, great apes only gain weight during fruit season and fruit season occurs seasonally. Fruit does not ripen year-round in the jungle. We've inherited that genetic mutation so that fructose, which is the sugar in fruit in high amounts leads to all sorts of weight gain issues. Also, fructose is the main cause now of fatty liver disease, which most people that I see in my practice have. And it's one of the main causes of insulin resistance and mitochondrial dysfunction that I've talked about in previous episodes. So I do have a saying, give fruit the boot. But that's just to get your attention that there are certain fruits that we really should avoid pretty much at all costs. So what are some of these high sugar fruits to avoid? Well, some of them might surprise you. Apples, believe it or not, an apple a day does not keep the doctor away. These days, a glazed donut has about half the sugar content as an apple. Now, interestingly, a new study from Tufts University in Massachusetts suggests that ice cream is a healthier choice than a multigrain bagel and other foods like salty crackers. In fact, in the research which was just published, experts developed a food compass to rank any type of food from one to 10 based on nutrition, the higher the number, the healthier the food. When comparing foods, the study gave an ice cream cone with nuts and chocolate ice cream a 37 while a multigrain bagel with raisins received a 19 and saltine crackers got a 7. Now, ripe bananas, believe it or not, one ripe banana contains 15 grams of sugar. But if you love bananas, you don't have to give them up. Eat green bananas. Green bananas have not changed into sugar. The green banana is basically one of the best prebiotic fiber bombs there is. So eat green bananas, throw the ripe ones away. How about grapes? Now, sorry people, grapes are probably one of the worst fruits out there. They have a cup of grapes, has twice the sugar content as a Hershey's chocolate bar. Twice in one cup of grapes. There's very little fiber in modern grapes. They have been bred for sugar content. Remember, the benefits of grapes long ago were in the seeds and the skin. We've genetically modified the grapes to no longer have seeds. And now the skin, which on a small grape was most of the grape, is all stretched out. Instead, it's been replaced by all that sugar in the grape. So leave the grapes alone. Okay, with all that doom and gloom, let's get to the fruits we should eat. Now berries, particularly raspberries and blackberries, really top the list. Blueberries are getting lower and lower and lower on the list. Why? Because blueberries have been hybridized for sugar content. When I was growing up, blueberries were these little bitty things that were so bitter, you had to put a ton of sugar on that, which was wonderful. But now blueberries have been bred for sugar content. That's why you can go to the grocery store and buy organic blueberries that are the size of grapes and taste sweet. If you're going to get blueberries, get the frozen wild mountain blueberries or the small ones and you'll be quite safe. But again, remember, blueberries only ripen normally once a year, usually in the late summer and early fall. There were no 747s bringing blueberries to Costco in January from Chile. So 365 days of endless summer, even if you're eating lower-sugar fruits, is not how you were genetically programmed to handle fruit. Now there is a fruit that you can eat year-round, and that's an avocado. An avocado is actually a single-seeded berry. And the good news about an avocado is that it's mostly mono-unsaturated fat and fiber, soluble fiber. So you get all the benefits. So avocado is the one fruit you can have every day. Mulberries. Mulberries may be one of the best high-polyphenol fruits there are. They are low-sugar. They are prized in Middle Eastern and Asian foods because of their potent health benefits. They once again only last for a very short period of time, and during that time they're perfectly safe to eat. Probably the lowest commercial sugar fruit that's available year-round is a kiwi. Now a trick is eat it with the skin.  You will actually get tremendous polyphenol benefit out of that skin, and it'll take you just one time to get used to the hair, but it will become an enjoyable experience. So kiwi is actually one of the lowest sugar-containing fruits. But again, don't eat it year-round. Try to eat it during the season. Finally, pomegranate seeds. Pomegranate seeds contain a really cool omega fat and have tremendous ability to uncouple and protect mitochondria. But here's the trick. Eat the seeds, not the juice. You're actually throwing away a lot of the health benefits that are in the seed, particularly the omega fat that's in it. So again, pomegranates only occur once a year, usually in the late fall, perhaps in the early winter. That's when to eat them. Okay, now remember most of our fruit that's brought from other countries is picked unripe. And if you remember from the plant paradox, unripe fruit is actually loaded with lectins, and they do not get the message from the mother plant to stop those lectins because they're no longer attached to the plant when they fully ripen. So what happens is they're brought to America unriped and then exposed to ethylene gas, which starts the ripening process, but they no longer get the reduce of lectins. Also remember, really ripe fruit is a sugar bomb. Fruit was designed by the plant to tempt you to eat it and to take its seeds elsewhere and deposit them with a generous dollop of fertilizer, your poop. So the plant is trying to get you to eat its fruit for its own nefarious purpose, and don't listen to that siren song. All right, to conclude, stick to low sugar fruits. Number two, only eat them in season. We have never been exposed to fruit 365 days a year in the past, and please remember that fruit is nature's candy and approach it as you would any other candy. Also remember, a cup of grapes has more candy than a Hershey's candy bar. Choose accordingly.


Good morning, everyone. I'm very excited to be here. I had a very good intro, but just so you know, I run the mobile product team at American Express. In fact, I've been looking after the AmEx mobile app for the last 10 years. I'm here to tell you a bit of a story, and the story is really about how we keep evolving the experience of the AmEx app. But before that, I want to see hands up from American Express customers who use the app. Let me see. OK, I think we've got a good audience today. Actually, let's get into it. First off, where we are today. So we've really been on a bit of a quest to set the standard for mobile in the industry. And I set the standard because we always want to be the better app out there. We don't really look at our traditional competitors when it comes to our experience in the mobile channel. In fact, we're always looking for what is the latest, best-in-class pattern and experience out there. And that's why we feel that what we provide to customers is really a bit different from what our competitors do. So for example, if you use the app, you will notice, and you can see here, that we've got an activity feed of few with American Express. And it's very much more the wrong word. Twitter, Instagram, UNIMED, all the major apps out there do it to really drive engagement with the customers. Now, we've really come a long way. So we've got literally millions of daily actives. The app is the main interaction channel for American Express. These days, we are also a growth engine for the company. So if you like, we've really ticked the box when it comes to driving business model. But I think what makes them even more happy is how satisfied our customers are with the app. So 75%, 4% of those are very satisfied. If you were going to see actually very satisfied and satisfied, that would take it up to 95%. And additionally, we've won a bunch of awards both in the US and international. In fact, in the US market, we've won the best credit card app for three times over the past few years. Now, you may be wondering how we go here, right? So really over the past 10 years, the app has been in continuous evolution. And every single time, we're trying to address a different opportunity, what different problem that we have in our hands. So what you can see in the screen is how the app used to look like back in 2010.  If you like, but that's kind of the whole evolution over time from 2010 all the way to today. Now, that is only really possible by building a number of blocks among the years. Starting with, we've got a very scalable global architecture. So we've got the app available in 22 markets. So you need architecture that really allows you to move very quickly globally. We now have a very comprehensive feature set. The reason why I've got that over there is if you've been around long enough in the industry, you may remember the days that actually mobile apps were playing catch up with websites. That's no longer the case. I'm very thankful about that. We've also been very careful when it comes to balancing the focus on the enterprise need and customer needs. So as a product manager, you always have that tension that you've got to resolve. And I'd say that we've done a fairly good job with it. I know that people don't necessarily like processes. However, a good enough or a solid process actually enable success. I'm going to talk a little bit about experimentation and the importance of having a bit of a framework for it. And last but definitely not least, we are very proud of the team that we've got. And not only the product team that I'm very privileged to lead, but also the partnership that we've got going with design, engineering, and analytics. Just really a few of us here. So I think we can probably see our little secret if you like. So we believe that our success really comes from, we are always looking for what's next, search for growth, et cetera. We are not really afraid to fail, make mistakes, et cetera. And we do learn from that. So the framework actually is very simple. So you just need to keep getting at it. And I think that's kind of the key thing here. So at any point in time, you're going to have either a problem that you've got to solve. Maybe there is an opportunity that you want to go after or has been presented to you as a team. So the first thing is like, OK, IDA, come up with your hypothesis, et cetera. How are you going to work around that problem? How are you going to go after that opportunity? The next bit is, OK, put that into action. You're going to have to try to grow those ideas and really drive results. Obviously, I wish this was linear. It is not. And probably most of the time, you're going to keep going back to the drawing board. So that people that's needed is really key here. But once that you make it work and you have driven growth and you have driven those results, et cetera, the next bit is, OK, now what? So don't stop there. Repeat all over again. And I think that discipline is going to take you very far. Now, we believe that the reason why this works for us is because you've got a bunch of things going on in the team at the same time.  So we've got our own interpretation of the squad-based model. I know that some people have a consensus about it. But I think the premise of having people with a clear remit of focus and they've got that autonomy to drive value over time, it actually works. And at least for us, it has worked out quite well over the years. The other thing is, product is never alone. You're going to really do anything on your own. So let's not forget about that. So I really love the analogy of the three amateurs, product, design, engineering, even analytics. So this mantra of one team, one dream, we live it every day. And the other thing is, you need to have passion, desire. You want to be evolving all the time. So don't forget to keep looking for what is the next opportunity? What is the next problem that you want to either solve for your customers or maybe solve for your company or maybe solve that for the market at large? So I thought it would be good to look actually at a bit of an example of how that framework, if you like, or those guidelines work for us in the team. So this is going to be about how, over the past year or so, we've been evolving the app to our next iteration. And I was starting point as a problem or opportunity. And I think I can probably use both of them interchangeably, is at some point, you may get to having actually too many features.  So we had 50 plus features, all of them with competing objectives in terms of what they were supposed to drive for different parts of the enterprise, what they were supposed to solve for customers at large. And that was creating a lot of pressure on two things, this probability and engagement. So on the left-hand side of the screen, what you've got is, OK, when it comes to engagement depth, you see how Tabba used to look like, five main sections, et cetera. And underneath, the relative engagement for those sections. And what you will notice very quickly is the majority of engagement just happened in the home tab of the app, which is to be expected. I mean, we learn people there anyway. But the point to make here is, sometimes your biggest asset, or the thing that has made you very successful over time, can become a challenge. And what we realized is, look, by giving you everything you kind of need as soon as you arrive in the experience, you have very little of an incentive to just keep going beyond that. So I think the learning here is sometimes a little bit of friction actually helps to drive certain results. The other important point is, this mantra of build it and they will come, I'm sorry, but that doesn't simply doesn't work. So you need to have tools in your tool care to really drive engagement, awareness, to work with your product marketing team, or even the same product team, et cetera, to think about not only the experience that you want to have, but how you're going to get people together, how you're going to get people to be aware of it, et cetera. And if that was like an enough large problem, at the same time, I'm going to express, decided to enter into the banking foray. So now we offer checking accounts for both consumer, commercial customers. And what that really meant for us was, well, we had a framework that was really designed for you having a particular card in wallet with American Express. And we had to start moving from that to go beyond the card. And now it's just not really designing or thinking about the product more in contact of customers rather than card members. So how we went about it. We did a couple of things. The first one is we run a parallel discovery truck. And the small digression I'm going to make here is I'm a big believer in dual truck agile delivery. So if you have really come across, look it up, let's run today. But really the idea behind it is at any point in time in your product team, you're going to have two trucks. One that is very much focused on BAU delivery and kind of driving value and implemented backlog that you've got, I guess, already. Are you going to have another one that is very much about prepping for what's going to come the next six months, the next 12 months, the next 18 months, and et cetera. So we started in the discovery truck AT18 and really envisioning in terms of what could we do to try to solve some of the problems and leverage the opportunities that we had. In fact, what you see there is some of the very early wire frames that we created. So that's how the thinking of the team started. Obviously that evolved quite a lot. It went through a lot of testing, et cetera. But the general idea was we wanted to do four new things. Or we felt that there were four big ideas that we wanted to go after. The first one was, OK, the information architecture that we had didn't work. So let's make it customer-centric. That makes sense. Secondly, we felt that probably the layout of the app required to be a bit of streamlining and also create power shortcuts, or really ways in which you can really help people discover some of the additional features of functionality that the app got for them. Because we felt that we're going to embark into this general of really evolving massively and changing structurally the app, we figured that was probably a good opportunity to do a design overhaul. Why not? More scope. And on top of that, we were like, OK, let's also try to solve for financial clarity, which is an area where we have struggled a little bit over the years. The other part of the journey was about how you test all of these areas or how you test your hypothesis. Another thing that's like one single answer here, but what is clear to me is you do need, in your toolkit, at least a couple of ways in which you can experiment. The path that we took was, OK, we've got the app in many markets. So why don't we use the UK app? That's a bit of a testbed. So we've got the UK app with all of these changes in the testbed. And we'll keep developing as BAU, DUS, and many other markets. So we did a foundational build in the UK. In fact, the images that you see, that is exactly what we implemented in the UK market. And how we went was, OK, UK market as a launchpad.  We also had a very, let's say, tight analytical framework. So we kept looking at data almost on a daily basis. So I don't know how many different p-boards, adjustments we made after results, et cetera. But in the end, some of those ideas, actually, we felt that they were ready to just be rolled out across the whole lot and therefore to grow them. And when it came to the US market in particular, we figured that maybe rather than doing a big banner approach where people will feel like, oh, wow, someone has moved my cheese, et cetera, let's try to do it in a more iterative manner. That leads me to actually results. We are very pleased that not only transformed the overall experience with very little disruption, customer satisfaction remained very high. We improved depth of engagement. So we really ticked the box when it came to those hypotheses. You can see here how it started, how it's going. Obviously, it looks very different. But the interesting thing here is, even though it just feels like a bit of a very massive visual change, there's a lot more that is happening behind the scenes. And what that really does is it really enabled us for future growth. And future growth for the app, and we expect when I keep growing the app, but also for future growth, us AmEx try to bring more products, new services to the marketplace. So we really have a very solid framework for the next three years to keep growing alongside the enterprise. That brings me to actually back to the framework. So now what? We completed that evolution. So what do we do now as a team? We believe the next leg of evolution, among many other opportunities, and I guess I'm going to really share all of them in a public space. We feel that we've got a very good opportunity to create what I would call a coherent family of apps. So until very recently, you've got a standalone app, one app to rule them all. And that's the app that my team runs with. When we went to a model where we're going to have a family of apps at American Express, a family of apps basically mean a group of apps with related functions or functionality. They usually have the ability to integrate with one another, and they are often owned by the same company. So think about the meta, family of apps, the Google, suite of apps, et cetera. And probably the key takeaway here that I would like you all to take from here is interoperability is key. But additionally, we feel like you need to have a very clear value proposition on customer segment when it comes to the different apps that you've got in the family. So for example, the core AMEX app, I'm going to call it All Customers Membership. We're going to have an app that is going to have a greater focus on, let's say, specialized use cases when it comes to commercial customers, so SME CatFlow Management. AMEX also has a standalone business very much focused on dining. And that's where the rest of the app comes into play. We are very excited about this next leg evolution and many other things that we are working through at the minute. And that's really all from me. Thanks a lot.


Now, we've done a lot of theory up until now. I think it's about time we actually do something. Let's look at measures of central tendency and dispersion. Now, central tendency is just taking all of your data and representing it by a single value. And dispersion tells us a little bit about how spread that data is. Let's do a few calculations.


Glenn Clark after Stevenson falls 76 61 the Albright here today at Owings Mills gymnasium now joined with John by John Yeah, curry who finished with 20 points today John. Yeah, it was I know it was a good game for you But it's tough to enjoy it after a loss like that Can you take me through the effort that you guys had in the first half you played one of your better halves of the year? I guess why weren't you able to sustain it? And how do you try to recreate that effort moving forward that you had in the first half? I think we just came out. We knew all rights a tough team. We were hyped in the locker room We just came out with a lot of energy.  So Coach was talking about someone need to step up and say hey, I'm not going to be the one I'm not going to let them defensively get past me and score as many points as they do How much are you guys talking about that and how important is that moving forward to sort of have the personal Accountability of it's not going to be we're not going to allow that moving forward I think our captains and the two seniors Imani and Christine are very tough on us They know like neck things I would need to work on in our They just make sure that we're always in it and we hold each other accountable in practice, you know We're a very defensive team So practices are tough and we try to just get on each other and keep that up Especially like in the locker room and warm-ups. So one area pride for you eight for eight from the free throw line tonight I know that's big for you How important is that that when you get those opportunities? You're not leaving any points at the free throw line?  So when you take them, you got to make them We shoot a lot of free throws in practice So I think it's just a given that you should make those once you get to the line That was a good thing today. Johnny. I know it was a tough loss But I really appreciate coming over to chat with us afterwards. Of course. Thank you. She's Johnny a curry I'm Glenn Clark for go mustang sports comm


And at the end of last class, I was asked a question that I want to share in the whole audience. And that question was, didn't you say you were going to be talking about high-dimensional versions of black hole uniqueness in the last class? Yes. So why are you saying it's not true this class? So maybe I need to clarify what I said by the end of last class.  It's, in fact, true. And we'll see that next tomorrow. But Israel's proof of static black hole uniqueness is not known to generalize to higher dimensions. So while the claim is still correct, the proof doesn't generalize. That's what I was trying to say yesterday. And so the proof we're going to see today, in the way we're going to see it, also does not generalize to higher dimensions. But we'll see tomorrow how you can alter it so it does generalize to higher dimensions. So today we're going to talk about a multiple black hole situation. So what was question two and answer two yesterday. And we're going to discuss a proof by Banting and Masur Alam from 1987, which is 20 years after the proof of Israel that we saw yesterday. And it's much more geometric. It's one of the most beautiful proofs, actually, I know, in mathematics, which is why I wanted to talk about this topic because I really like this proof. And it has had a tremendous impact on mathematical relativity throughout. So many proofs are based on it, or imitate aspects of it, or actually use the results in the way you get it from this theorem. Hugh Bray's proof of the Penrose inequality, Romanian Penrose inequality, for example, wouldn't go without this proof. So that's why it's important for anyone who wants to do mathematical work in general relativity to know this proof. So with this motivation, let me state the theorem. My Banting and Masur Alam, 1987. And let me say up front that the key ingredient, sorry, the key ingredient in this proof is the Romanian positive mass theorem that you've seen last week. So again, we start with the static system, three-dimensional, the static, vacuum, asymptotic to Schwarzschild, meaning that the lower order terms of this form, this is the decay terms are like one over R squared, faster decay faster than one over R squared. The first derivative is like one on R cubed and the second derivative is like one over R to the four. This is what this notation means. This is just for those people who want to know the precise decay of the lower order term of mass M. And again, they assume positivity of that mass to begin with, but there are more modern methods where you don't need to assume it to begin with, but today I'll gloss over those with inner boundary, which is now potentially disconnected. So it consists of finitely many sigma two I, sigma two I connected static horizon and I bigger or equal to one, then I is one. And the corresponding spacetime, this isometric to Schwarzschild of the same mass. So we can potentially have multiple black hole horizons to begin with, but eventually we'll only have one and everything will be Schwarzschild as we discussed last time. Oh, I'm sorry, this would be an S, of course. This was gonna happen. Okay. And I'll write the remark. One can drop the assumption that M be positive by a maximum principle argument, which I'm not gonna explain here because I'm not sure everyone knows maximum principles. Okay. And recall from last class that we already know that sigma two I is totally geodesic, has mean curvature zero by assumption and also the stress free part of the mean curvature is of the second fundamental form vanishes and also the normal derivative of the lapse is a constant for each I, not necessarily the same constant for each I and actually is a positive constant, again by the maximum principle for those of you who know what that means. We discussed this last class for a single black hole horizon carefully. Okay. So now comes the idea of the proof and I'll do a lot of it in pictures. I'll only do those computations explicitly where I think one can learn something from the computation or otherwise the proof is not clear if one doesn't do the computation. Okay. So here's the idea. The first step is that we double the manifold. So we have our asymptotically flat, oh, there's not enough room here. We have our asymptotically flat manifold with potentially several horizons. So this will be sigma two one and this will be sigma two two and this is where S goes to infinity here as well and this is M three. So the first step, what we're gonna do when we did this explicitly in the case of Schwarzschild before is we're gonna reflect it through the horizon and have a second copy and call this M three tilde and we're also going to copy the coordinates that we have near infinity and calling them X i tilde and S tilde, the radius. So here we have coordinates X i tilde and radius S tilde that goes to infinity. Up here we have our asymptotically flat coordinates X i. Okay and because the surfaces are all totally geodesic so the second fundamental form vanishes completely we can do this here without having a crack. Okay, the first derivatives actually align. This is due to this condition, this assertion here. So the regularity at the gluing here will be C one one where C one one means it's C one. So with the continuous first derivatives and the second comma one means the highest order derivative is Lipschitz continuous. If you don't know what that means it's not so important for the rest of the proof. If you do know what it means, that's the regularity we have on all these surfaces. Apart from the surfaces, we're smooth. I mean the manifold you can think of smooth anyway and in addition we have the metric and we also just copy the metric down here and then everything has this nice regularity here. The second derivative could jump but the first derivative is actually continuous, okay? Which is because the second fundamental form vanishes so in some sense this could be used as a reflection symmetry plane or surface. So that's the first step, we double M three and G and then we glue them together so we have M three hat to be M three together with M three twiddle and of course we identify the boundaries and we only count them once and we have G hat to be G on M three and G twiddle on M three twiddle and now we want to do something similar for the lapse function. Here we also have our lapse function N and down here we want to have yellow the lapse function N twiddle so that together we can combine them to a lapse function N hat and our definition of N twiddle of a point P twiddle will be minus N of P. What does that mean? If we have a point P twiddle down here because this is a double this means it's a reflection copy of a point P up here. This we can plug in into N, take minus of it and call it the lapse of this point P twiddle and then we combine them to get N hat to be N on M three and N twiddle on M three twiddle. So that means now on the whole thing we have a manifold, we have a metric which is smooth except on these surfaces where it has this regularity and we also have a function and now let's think about this function. Of course the function is smooth up here and smooth down there. What does the function do across these? Well it becomes zero there so changing it from plus to minus doesn't matter makes it continuous and if you look at the condition that new event is a positive constant here you can actually make this equivalent interpret this as being equivalent to being able to merge this function N and N twiddle C one one here as well. So the conditions here are exactly what we need to merge these two pieces together along the horizons in the C one one fashion which is exactly smooth enough for what we want to do later. And this is also exactly what happens we didn't talk about the left function in detail in the isotropic Schwarzschild picture where we double Schwarzschild this is exactly the same thing that happens there on the new on the copy we have a negative left function in the Schwarzschild case as well. So if we hadn't started with Schwarzschild in isotropic coordinates with half Schwarzschild in isotropic coordinates we would now have complete Schwarzschild in isotropic coordinates. If we had Schwarzschild just a second if we had started with Schwarzschild in different coordinates which are only asymptotic to the isotropic coordinates we would still have double Schwarzschild just in funny coordinates. Yes. It could be see infinity on the boundary also I think that's something one typically assumes yeah but here all we need is that it matches C one one so then we could have assumed that from the beginning. We don't know anything about the second derivative across the surface in this reflection procedure end of story we just don't know a priori. A posteriori we're gonna know it's Schwarzschild and then of course it's smooth but a priori we don't. But that's all we're gonna need to prove it's Schwarzschild. Okay? Good. So now let's think about the asymptotics for a second. Here we know that it's asymptotic to Schwarzschild as S goes to infinity then the metric here is going to look exactly the same as S twiddle goes to infinity and the left's function is gonna look like minus one plus M over S instead of S twiddle instead of one minus. Okay? So that's natural and the next step what we're gonna do now I'm gonna erase this here hoping that everyone already copied it so I have room for more pictures. Now we're gonna perform a conformal change from this scenario where now we have here M three hat G hat and N hat to scenario here where we'll have M three I don't know how to call that wedge maybe. G wedge and N wedge. No N wedge M three hat wedge and G hat which will be conformally related to this in the same sense that if I started here with Schwarzschild then double it to double Schwarzschild and then perform this conformal change I would get flat space only I would get flat space without the origin in Schwarzschild, right? So I want to do something that if I started if this is Schwarzschild then this would be R three without the origin and the Euclidean metric. So what was the formula in Schwarzschild? The formula was that this metric here was a function phi to the power of four delta. So to go here I want to multiply by phi to the power minus four to get delta, okay? So I want G wedge to be some function phi to the power minus four times G hat. So in case I started with Schwarzschild and I picked phi to be the function I wanted in Schwarzschild then I get the delta back. But now I need to say what phi is in general. And actually even if I was in the Schwarzschild case I don't know if my Schwarzschild is written in isotropic coordinates. So then I don't know if this function takes exactly the same form in the variable S as it did in Schwarzschild. So I need to find a way to write this without appealing to coordinates if I can. And the idea is the following. Look at Schwarzschild, maybe I'll do that here. And this is a cool idea that's been used in other scenarios as well. But I think it has been tremendously underused. Is there a question? Do you have a question? There's something you can't read? Or what is this one? Oh this is the trace free part of the second fundamental form. So if we were in Schwarzschild we would know in isotropic coordinates and we would know that phi of S is one plus M over two S and N of R was one minus two M over R. And R was phi of S squared times S. And this allows us to look at N of R of S. Smart Seller suggested to call it. And you'll call this N hat of S in the Schwarzschild case just as over there. And this you can compute to be one minus M over two S divided by one plus M over two S. So the lapse function that reads like this and the R variable reads like that and the S variable in conformal coordinates, isotropic coordinates. And now the cool idea that they had was to introduce a function phi of N hat. Not of a variable, not of a coordinate, but of this function as over one plus N hat. And if you do that, then what you get is that in Schwarzschild phi of S is nothing than phi of N hat of S. So this is rewriting the same thing over and over again in different ways. So I'll repeat, we have this conformal factor that we computed for Schwarzschild. We have the lapse function that we wrote down for Schwarzschild. We write this lapse function in isotropic coordinates. It's the same physics, it's just written differently in different coordinates. And then instead of expressing it in terms of S or R, now we express this function little phi in terms of a function of N hat, which then is a function of S. So then we can express this phi of S in the Schwarzschild scenario as this capital phi applied to this N hat. And this we can do here, where phi is this capital phi applied to N hat. Or in other words is two over one plus N hat. This is just an algebraic trick in the Schwarzschild example, but in general it allows us to write something which completely is independent of coordinates. So even if we worked in Schwarzschild but in different sets of coordinates where the phi would take a different form, this would still be true. So we've peeled the physics out of the coordinate expressions or the math or the geometry or whatever you wanna call it. We've peeled it out of the coordinate representation of everything. Now we managed to write this conformal change in a Schwarzschild scenario like this. And of course no one stops us from just performing a conformal transformation like this and see what happens, okay? So that's what we do. We define our G wedge to be phi to the minus 4 G hat and actually our M3 wedge to be just M3 hat. So we don't change the manifold at all, we just change the metric on this manifold and in that in a conformal way with a conformal factor that we can compute from the lab's function. Okay, in order to do that of course we need to verify that this phi never becomes zero which is obvious but it also never becomes infinity which is fine because N is between zero and one and then N hat is between minus one and plus one and that's okay. Yes. Because I want, this is the manifold with boundary and this is another copy of the same manifold with boundary. Now I glue them along the boundary so I want to have a name for the manifold without boundary for the total manifold, okay? And in fact, one thing you can observe about this manifold is that it's geodesically complete, okay, this here is geodesically complete. Not important for the proof but, okay and now I perform this conformal change so I don't know how to draw it yet because I don't know the asymptotics yet so I'm not gonna draw it before we've figured out the asymptotics. But before we figure out the asymptotics or let's figure out the asymptotics first, why not? Asymptotics. So as I said here we just have the asymptotics of M3G and N has the asymptotics we assumed, yeah? G and N, G, Gij is like one plus M over two S to the power four delta ij plus lower order terms and N is like one minus M over S plus lower order terms as S goes to infinity. Now let's look at P to the minus one because we're gonna need P to the minus four so I'm starting with P to the minus one. That is one plus N hat and in this case one plus N over two. One plus N over two with this asymptotics is like one minus M over two S plus lower order terms. So then G wedge ij is five to the minus four G ij is like one minus M over two S plus lower order terms to the power four times one plus M over two S to the power four delta ij plus lower order terms. Okay, I've just multiplied the formula I found for this to the power four with this formula. Now, if you have something like this you can compute it like a binomial and then you get one minus M over two S to the power four plus lower order terms but one minus M over two S times one plus M over two S is also binomial and gives you one plus something of the order one over S squared, so no one over S term. So this completely reads like and I'll write it in a suggestive notation one plus zero over two S to the power four delta ij plus lower order terms. Okay, so the G wedge, the new metric I get up here in this infinite end looks like asymptotic to Schwarzschild of mass zero. So I can now try to draw that. So after this conformal change M three well the northern half here G wedge I'll not write anything there, I'll write it on top. So it would still be asymptotic to Schwarzschild but now of a different mass namely mass zero. Recall if this was Schwarzschild that would be Euclidean space so mass zero is very consistent. Yeah, so in some sense asymptotically for any such metric the same magic happens as in the Schwarzschild case only just asymptotically we kill the mass. Okay, so what happens at the other end? So this is what they call the northern half. So now let's look at the southern half or the double. Of course the metric has the same asymptotic as before because we just reflected it but the left function now has the opposite sign so it will be like minus one plus M over S. So G twiddle ij looks like one plus M over two S to the power four delta ij plus lower order terms oh S twiddle and N twiddle will look like minus one plus M over S twiddle plus lower order terms as S twiddle goes to infinity. Now then if I compute my little phi minus one that'll be one plus N twiddle over two. This doesn't look as nice anymore because the one and the minus one cancel. And instead you get M over two S twiddle plus lower order terms. That means that this conformal factor goes to zero not to one anymore in this end. So it will do something tremendously different. So here in this end the conformal transformation just pulled everything even more flat. In this end it will do something different it multiplies it with something that asymptotically becomes zero. That's not so surprising either because if we started with Schwarzschild then the lower end would fold in and would be R three around the origin. So we can expect that something like that happens too. And in order to observe that it's a useful idea to introduce a different set of coordinates that allows us to study something near a point rather than near infinity. Okay but let's first write down what would G ij look like. G wedge ij would look like V to the minus four G twiddle ij. And that is it would look like M over two S twiddle to the power four which is what we've just computed delta ij plus lower order terms. Okay so it looks like something that goes to zero. That's not good for a Romanian metric, yes. Because that's what happens in Schwarzschild. If you look at the isotropic or the conformal Schwarzschild in isotropic coordinates then the left's function is positive up here and negative up here. We call the Kurskal diagram I drew last time, right? On the inside the black hole on the other side on the other exterior it has the opposite sign because time changes, I mean. Yeah? That's why I'm doing it here because I want to do exactly the same thing that happens in Schwarzschild because I want to recover Schwarzschild, right? If I did anything else I'd be doomed. So here we are faced with the problem that the metrics looks like this. If we were exactly Schwarzschild there just wouldn't be any lower order terms. They would also look like this. So now we introduce new coordinates and like this we take the coordinate x i to be m squared over four times x i twiddle over s twiddle squared. So what does that mean? Here we had the coordinate x i with radius s and we decided to call them x i twiddle and s twiddle down here. Okay, that's what x i twiddle and s twiddle are. So I take the i th new coordinate is the i th old coordinate divided by its length squared and multiply by this number. Geometrically what this means if you're in Euclidean space and you have a sphere of radius m over two, we call in isotropic coordinates that's the radius of the horizon. Then this is an inversion in the sphere in the following sense. If you have the origin here and you take a ray and you take a point here then there is something called an inversion in the sphere which is like a reflection in the sphere that you can take a point from the outside to the inside and vice versa. If you perform this twice, you get the identity. So it has all the features of a reflection in some sense. And if you then instead, if you know this then it has this formula if you take the unit sphere but now we want to take the sphere which is the horizon. So the sphere of radius m over two and then you get this formula. So this is the inversion in the sphere. And if you think of this as being exactly Schwarzschild in isotropic coordinates then we said the northern half and the southern half are isometric and there is a reflection and this is exactly the coordinate change you would perform to the isotropic coordinates to swap the southern and the northern half. So that's where this idea comes from. And then we do a computation and namely that AI and we introduce capital S as capital X I squared plus capital X two squared plus capital X three squared. So the Euclidean radius in this new R three. So now we wrote R three without the origin in different coordinates and this is our new radius. It's and we can relate the old and the new radius by seeing that the new radius is m squared over four times S twiddle. So this is how the length of these vectors changes under this inversion.  Wait, there is an S squared missing here and S four X K D capital X K. This is chain rule, okay? This is chain rule applied to the inverse of this formula. And now that I have this, let me write, sorry, I want to understand this metric. So I write D wedge as what we've computed m over two S to the power four delta I J plus lower order terms D X one I twiddle D X J twiddle. This is what this really means to write it like this. Yes. And now I've got a formula that allows me to take this S and transform it into a capital S via this formula. Oh, this should be an S twiddle, sorry. And take the D X I twiddle and D X J twiddle and replace them by this, okay? To transform this metric into these new coordinates. And I'll need to erase something before I can write this anywhere. So you do the computation and what you get is delta I and J plus and now it's not lower order terms anymore about higher order terms because now we're inverting the radius. So instead of taking S twiddle to infinity, we're taking capital S to zero. So it's higher order terms D X capital I D X capital J and there's a D wedge when we started. In other words, if I take G wedge and plug in D capital X I D capital X J, then I get delta I J plus higher order terms as capital S goes to zero. Okay, so let's look at this formula once more. There was an S twiddle in here. We take this S twiddle and replace it through this formula by a capital S. So it will be like capital S to the four times some M's and so on.  Okay, if I did this precisely exactly in Schwarzschild and isotropic coordinates I wouldn't get any higher order terms and I would have recovered the flat metric. So let's draw that. So what happens is that this asymptotic Schwarzschildian end gets bent in the same way, the southern end of the Schwarzschild band gets bent in to something which looks like almost flat space with one missing point. If it was Schwarzschild, it would be just our three without the origin and we can glue in a point P infinity to close this off geodesically completely. So we glue in this one point and in fact one can do this glue in as was verified by this computation if you track the higher order terms more carefully also so that the regularity here is C11. And now the new thing, so then M3 wedge union this P infinity together with the metric G wedge extended by the flat metric into this point is geodesically complete. Just like R3 which is Schwarzschild R3 minus the origin after you glue in the origin is geodesically complete. Yes, it is capital S going to zero, yes. So the important thing we've achieved by this gluing in of this one point is that our resulting thing is geodesically complete and if we were Schwarzschild, we had started with Schwarzschild it would just be R3 delta. Any questions for this computation before I raise it? No linear I think. This magic with the one term that's missing that's happening in the other end doesn't happen here because of this cancellation of the one and minus one. So we lose one order. Okay, so now we've produced a geodesically complete manifold which has the same topology as before the conformal change except we glued in this one point and it has an asymptotic one end it's asymptotic Schwarzschildian with one end that has mass zero. So M wedge three united with P infinity G wedge is asymptotic to Schwarzschild or asymptotically Schwarzschildian of mass zero. We've already computed that and I want to write it down in words and now if we were Schwarzschild we would also be flat completely, yes? Of course in general we won't expect to be able to show that a priori that this is flat because then we would be pretty done. We need more steps. Instead what we want to do is we want to use this magical positive mass theorem in the Romanian case for positive energy theorem or Romanian positive mass theorem whatever you want to call it that says if you have a geodesically complete manifold we do which is asymptotically flat of mass zero we have even more than that asymptotic to Schwarzschild of mass zero and it has non-negative scalar curvature then it has to be Euclidean space. There's a little bit subtleties because it says ended a smooth but let's discuss that later. Okay so now let's compute the scalar curvature of this to see whether it's non-negative. BmT is the positive mass theorem. Okay, so this is the really the rigidity case the positive mass theorem. So how do we compute that? Well we have a formula for G wedge, right? So G wedge and G wedge was phi to the minus four of G hat. And G hat was really just G in two copies. So we can compute our wedge and so this is a conformal transformation and we can use we can use the formula for conformal change of the scalar curvature to deduce exactly what is the conformal what is the scalar curvature, okay? And the formula there is such that I never remember it is R of G hat. And this way round turns out to be more useful it's phi to the minus, let's be typo I think seven or something like that. R G wedge minus eight the plus in G wedge phi over phi. So the scalar curvature of the old metric can be computed or expressed in the scalar curvature of the new metric and this is the Laplacian of our conformal factor phi in terms of the new metric G wedge. But this we know is zero from the static equations. One of them was scalar flat. Okay, so now that means we know that our new scalar curvature our wedge is eight times the Laplacian of phi with respect to the new metric over phi. And that doesn't tell us much yet about the sign of it. Okay, but we can compute this because we know what phi is. We call that phi was really two over one plus N hat. Now the same way that there are formulas for conformal change of the scalar curvature or the transformation of scalar curvature under conformal change, there's also formulas for the transformation of Laplacian upon a conformal change of the metric. And the formula reads like this for any function F, which is smooth enough. We have the Laplacian with respect to G wedge of F can be computed as phi to the four, Laplacian G hat of F minus two over phi Laplacian, oh, no, sorry, D phi applied to the gradient of F with respect to the metric G hat. Okay, so the Laplacian with respect to the new metric of any function F is the conformal factor to the power of four. The Laplacian with respect to the old metric of F minus two over F, D phi applied to the gradient in the old metric. Now we need to apply this to phi and then we get Laplacian of G wedge applied to phi, is phi to the power four, the old Laplacian of phi minus two over phi D phi squared in the old metric. Then another nice formula is, in fact, that this part here in the bracket by a straightforward computation is nothing else than the Laplacian with respect to the old metric of phi minus one, which is one over phi and not the inverse function, okay? If you compute the Laplacian of one over a function, what you get is Laplacian of that function minus two over phi times some power, phi to the minus two, sorry, phi to the minus two, okay? So now what we got is that Laplacian of phi with respect to our new metric, which is what we need to understand the scalar curvature of the new metric is minus phi squared, Laplacian of phi to the power minus one in the old metric. Recall that phi was two over one plus n. So that means that phi to the minus one is one plus n over two. The one this is is minus phi squared, Laplacian of g hat, sorry, n hat, one plus n hat over two. Now this is really nice because of course Laplacian of a one is always zero, yes? Where's the minus sign coming from? Probably I missed one here, thank you. Everything is going to be zero in a second, so it's not that bad, but thank you anyway. So if I have the Laplacian, Laplacian is a linear operator, so then I can compute it separately for the one half and the n hat over two. The Laplacian of a constant is of course zero. So what I remain with this here is really one half of the Laplacian of g hat of n hat. Now recall on the northern half g hat and n hat are just g and n, so this is the Laplacian of n and n was harmonic, so this is zero. On the southern half g hat was g and n hat was minus n, so the same argument applies. So by the static equations, this is zero. I'll go through the computation once more in a second. So we get that the Laplacian of phi with respect to g wedge is zero. Now look at this. The scalar curvature of the new metric was proportional to this Laplacian. So that means the scalar curvature of the new metric vanishes as well. So it's scalar flat.  And now I've lied to you the whole time because everything is not smooth on the gluing surfaces and then this point p infinity. So now I'm writing where smooth enough. So away from this one point where we may not have enough regularity and away from these two surfaces were finitely many surfaces where we may not have enough regularity to even compute the scalar curvature which involves second derivatives. Here and here, so to say, we're scalar flat. So we do satisfy the assumption of the positive mass theorem except being smooth. Now if we were smooth, which if we had started with Schwarzschild we would be smooth. The positive mass theorem would now tell us, okay then this is globally isometric to Euclidean space. There is a weak version and this positive mass theorem as you've heard last week is proven by Shane and Yao in 1979 and 81 and by Whitton in 71. 81, sorry, 81, little bit ahead. And there was a version of Whitton's proof for weak regularity manifolds already in 1986 by Robert Bartnick which is one year prior to this. Still they gloss over this and they just say, okay, now we apply the rigidity case of the positive mass theorem by Shane and Yao and by Whitton and then everything is Euclidean space. But in fact, they could have just cited Bartnick with his low regularity spinner proof, so Whitton style proof of the positive mass theorem and said, okay, this is enough regularity to apply Robert Bartnick's result of the rigidity case of the positive mass theorem and conclude that this is actually really globally isometric to Euclidean space with the isometry being smooth here and here and of C11 regularity here and here and at this point. So I'll write some of this down.  So I'll write it here. Rigidity case positive mass theorem in weak regularity by Robert Bartnick, 1986. And I wrote down the citation somewhere like this here. This is the mass of an asymptotically flat manifold and it appeared in communications on pure and applied mathematics 1960. In 1986 on pages 661 through 6931. Okay, so let me walk you through this computation once more just in case I lost you previously. We've computed before that the manifold is asymptotically Schwarzschildian of mass zero up here and that here we can glue in one point to make it closed and geodesically complete. And then we looked at the scalar curvature. We used a formula from conformal geometry that tells us how the scalar curvature of a manifold is related to the scalar curvature of really conformally changed manifold. This allowed us to conclude because we knew from the static equations that the original manifold was scale flat that the new scalar curvature is proportional to the Laplacian of the function phi with respect to the new metric. Then we computed or I told you a formula how to transform the Laplacian of a function with respect to a conformally transformed metric in terms of the old metric. And we plugged in F equals phi and realized that the term we get here is nothing else than proportional to the Laplacian of one over phi, which is very nice because one over phi is a constant plus something proportional to our lapse function. And because the Laplacian is linear, we can compute it as just the Laplacian of the lapse function or Laplacian of minus the lapse function on the southern end of our thing. And this according to the static equation is zero, which means we've concluded that it's scalar flat. The new metric is also scalar flat, okay? Yes, I would think so. And I think it's better to discuss this after tomorrow's lecture. I'm not saying I'm gonna prove that tomorrow, but I'm not the weak case. But so maybe it's worth a comment now that you asked. So we've got this equation and all that we've done, that we're doing now using this equation is to conclude that the scalar curvature vanishes of the new metric. Tomorrow, I'm going to show you an alternative proof of what's going to come now of the closing argument of the Banting-Masrula-Lam proof where instead of using only scalar flat, we also exploit this equation much more, okay? Which allows us to generalize it to higher dimensions. But today, somehow Banting and Masrula, they don't explicitly write it like this. They somehow use it implicitly just in the computation of the scalar curvature. And then they say, okay, now we know it's scalar flat and they forget about this fact. At least that's how I read it. Okay, so now, thanks to Bartnick and Bartnick's generalization of Witten's proof of positive mass theorem rigidity. We know that our resulting manifold is precisely R3. So positive mass theorem rigidity in lower regularity. Let's say low enough regularity. I'm not claiming that you can't prove it in lower regularity. Regularity tells us that M3 wedge union P infinity, G wedge is globally isometric R3 delta. Just for those of you who are interested, so far we have not made use of dimensions except that we're using here the positive mass theorem, but actually Bartnick's version is end dimensional. So up to now, everything works fine and dimensionally. Yes, did he? Really? Okay, I'll check for tomorrow. Remind me after the class please. I don't know, but definitely there is higher, by now we have higher dimensional versions of Witten's proof in low regularity by Lee and the flock for sure with no more than the necessary assumptions. And also of course we have a high regularity, Shane and Yao proof in higher dimensions, which I'm sure you've heard about last week, or I think so, but so we do have the theorems and I thought that this was fine, but I'll check again. So this globally isometric, so now we're globally isometric. So now this weird thing turns out to be flat Euclideans, okay, so this guy is in fact globally isometric to R3 delta where the isometry could potentially be of low regularity here and here as well as here and it's smooth everywhere else. So now we need to recover it, why is that helping us to know that we've been schwarzield to begin with? Okay, now we're in the same situation where we are in schwarzield, but why does that help us? And there are actually a number of different arguments how to do that, and I'm not gonna go into much detail, but in the original, so Bunting and Massoud, at this point, the site work by Robinson were actually already present in this paper of Miller, Thimhagen and Robinson and Seyford that I cited yesterday, Robinson and Seyford, should be 1972, where they use the cotton York tensor, which I'm gonna write down in a second from conformal geometry to conclude plus the static vacuum equations to conclude with schwarzield that we started with schwarzield, okay? So this argument is not due to Bunting and Massoud, and it's been there before. Just to give everybody proper credit. So the cotton York tensor, and this is for n equals three, this is the point where this becomes a three-dimensional proof, okay? The cotton York tensor is a tensor defined as a derivative of a Ritchie, I'll write it down in a second, so it's a third-order differential operator, and it sort of plays the role of the vial tensor in higher dimensions. So you know probably that from dimension four and higher, the vial tensor is zero if a metric is conformally flat. In three dimensions, the tensor that plays this role is the cotton York tensor. And in fact, there is a theorem by Eisenhower saying that it's conformally flat if and only if the cotton York tensor vanishes, but we're not gonna need that. So the cotton York tensor C is defined at that easy, most easily defined in abstract indentation. So it's two nabla Ritchie j k plus one half d k, of course not like this, k i nabla. R is the scalar curvature, rick is the Ritchie curvature, and C is the cotton York tensor, and the brackets are anti-symmetrization with a factor of one half, okay? Okay, that's the cotton tensor, so it's a third order differential operator, and the cotton tensor C vanishes if G is conformally flat. And actually, I forgot to say that if you didn't know any of these conformal geometry formulas, it would of course be a good exercise to verify them, the scalar curvature. And also this is a nice exercise to do if you want to get more familiar with conformal geometry. So what Millard, some Hagan, and Robinson, and Zyford do next is they take the cotton tensor, which involves the Ritchie, the scalar curvature in the metric, and of course covariant derivatives, and plug in the Ritchie equation from the static equations. So we had n times the Ritchie was the Hessian of n. You can write this as Ritchie ij is one on n, Hessian ij of n, and we know that the scalar curvature is zero. This was our static equations, and you can plug them into the cotton tensor. And then you get a formula that the cotton tensor is such and such and such in terms of the metric and the labs and derivatives of the labs. Okay, so C is then C of dn, probably up to third derivatives, and G, and a zero, because we already shown that we're conformally flat. And then what they do is go back to this Israel idea of looking at the level sets of the labs function. Of course, the tensor that zero everywhere is also zero on each level set. And then they conclude from this formula here that is lengthy, that each level set has constant n to begin with by definition, and constant mean curvature and vanishing teres free part of the second fundamental form. So it's ambilic. It has constant Gauss curvature and so on and so forth. So we cover all the identities we did yesterday in a different way, and then close the argument like Israel did by explicit computation of Schwarzschild. So we have the cotton tensor. This implies that each n equals constant, which yesterday we called them sigma two n is as yesterday, I put it in quotes round and mean intrinsically and extrinsically and then the same explicit computation as yesterday and it shows you that you're Schwarzschild, okay? So this is where this proof became three dimensional because we use the cotton tensor and the cotton tensor doesn't have the same nice conformal property in higher dimensions. However, there is an argument by, it plays the role of the vial tensor but the vial tensor has a different formula, completely different formula, yeah? It's a part of the Riemann tensor, right? So it has two derivatives. This has three derivatives. So it has the same spirit, but it's not the same thing, okay? So but you're thinking in the right direction because one, and I'll write down the reference next class, I forgot to bring it or I can look at it. No, I cannot. I'll bring it next class. Wang in, I think in 1980, in 1998 or something like that proved the same thing in higher dimensions. So all the first part I said it was immediately higher dimensional except all the numbers four get replaced by four over n minus two and so on. And he found a way of using the vial tensor here instead of the cotton tensor to make it higher dimensional. I'll mention this again tomorrow, okay? So there is a version of this recovery of Schwarzschild argument using the vial tensor which is a little bit more involved than the original cotton York tensor argument but has the same conclusion in higher dimensions by Wang. And maybe I'll just close here for today. Thank you.


you know, draw some parallels to the MSI initiative. So Dr. McCambley is interdisciplinary scholar of higher education. She studies a role of organizations in their production or reproducing of systemic racial inequalities. She draws upon a large range of both analytic and interpretive methods that to study the influence of aspiring change agents or institutionalized racial inequities within higher education. So with that, you know, she focuses on constructs related to racialized organizations, institutional persistence and change, racial brains, political development and racial backlash. And then lastly, organizational sense-making. And so she has a larger research agenda but the research that she'll kind of speak to here, you know, sort of examines the conditions under what conditions do equity agendas address racialized inequalities rather than operating as new labels for older practices. And so with that, I am incredibly excited to have Dr. McCambley join us here this afternoon. And with that, I will turn it over to her. So thank you, Dr. McCambley. Thank you, Edmund. I'm so happy to be here with my old colleague, Edmund and I'm so happy to receive this invitation. Let me go ahead and share my screen and get started here. All right, so Edmund, if you don't interrupt me, I'm gonna assume you can see my slides and you can hear me. So I'll go from there. So again, good afternoon and thanks again for having me and a huge thank you to the rest of the folks at Rackham for welcoming me into community with you today. I'm coming to you from my home in Pittsburgh and as such, I and my university occupy the ancestral lands of the Haudenosaunee, Lenape, Osage and Shawnee peoples. I recognize in my work and the ways I raise my children and my relationship to lands and waters that displacement and erasure are not just histories for native peoples, but they are the present. And I honor the legacies of both the land and the many indigenous people who call this land home past, present and future as an exercise in preservation and reconciliation. So my talk today, I'm sorry, little thing. My talk today is called More Than a DEI Label, Engaging and Racialized Change Work. So by way of introduction to myself and my work, my projects, at the core of my research program is a refusal to center the settler colonial gaze that demands proof of either harm to a worthiness of historically marginalized communities. So instead the underlying question guiding my research is by what mechanisms does white supremacy as a structured set of processes and outcomes get reproduced over time? Under what conditions are those processes weakened or replaced? So put another way, we all know at this point that equity initiatives in higher education are and have been under different names, pretty ubiquitous. And yet many of us bring a healthy dose of skepticism of their likely success, born out of years of seeing only one of many such initiatives produce measurable results. So rewarding this kind of research program to focus on hope rather than skepticism, we might ask, what do we know about the conditions that make equitable change as systemic as the causes of educational inequity? One of the foundational frameworks I build on for asking and answering these questions is Victor Ray's 2019 theory of racialized organizations. And I identify the field of higher education as a collection of these racialized organizations. In Ray's theory, it's organizations rather than individual intentional acts of bias that routinely privilege one racial group over another. We can identify such organizations by noticing and attending to the normed and accepted ways they deliver unequal resources and magnify privileges of the dominant group. This is the institutionalized or ongoing outcome I identify there on the right. And the core tenants of this theory are as I discuss them, the modes of reproduction that routinely create inequity on the left include first the differentiation between white and non-white organizations that legitimates the unequal distribution of resources. So we're talking about MSIs today and I think this is kind of the crux of my connection to this program, right? As we think about how MSIs are routinely and legitimately meaning we think it's normal the MSIs are routinely given less access to prestige and resources than predominantly white institutions, right? And second, the way that whiteness is used as a credential or a call toward legitimacy and bureaucratic processes. And third, the racialized decoupling of formal rules in ways that benefit whiteness at the expense of minoritized communities. And we'll get into in the course of empirical example kind of what these more theoretical statements look like. So while this framework is a useful lens for coming to spot and understand how racial equity is so persistent it is not necessarily intended to help us find our way out of these racialized conditions. So to this end in my work I'm putting forward the construct of racialized change work. Racialized change work refers to the purpose of action that organizations take to build new equitable organizational arrangements or tear down old inequitable ones. So more specifically racialized change work is the focus attention on dismantling racialized organizations and the routines within them that magnify the agency of dominant groups at the expense of minoritized racial groups. And I'll give an example and I believe Raman is on the call. So in a recent piece that I really loved for Raman Lieta and his co-author Teresa Hernandez in the review of higher education we saw a vivid example of racialized organizational practice in the routines involved in faculty hiring processes. So think for example of hiring processes where you have been a part of where greater scrutiny has been placed routinely on candidates of color based perhaps on skepticism of the relative quality of their PhD institutions or perhaps a perception of them as activist researchers that might discredit their scholarly contributions. So racialized change work in this domain then might look like strategic action to permanently invalidate those types of routines and introduce new routines in their place. So I wanna ground today's talk in an empirical example to give us sort of a concrete jumping off point for thinking about how the policies and norms and routines we have all engaged in with regard to MSIs as racialized organizations and our own work can be sites for renewal for change and for healing. I hope this example helps us find some common connections to your work as a graduate school as faculty members, as staff and students to help us engage in the discussion at the end of my talk. I will then use key findings from this empirical study to give a closer look at this idea of racialized change work as an analytic and an actionable tool for change and use that model for kind of a segue into questions and discussion about connections to your own work and our collective work as a field. So my empirical example today comes from a study that centers the work of the fund for the improvement of post-secondary education. So it's a federal agency founded just after the civil rights movement designed to encourage institutional changes within higher education that would transform the way colleges and universities served what they called, quote unquote, new students that is returning adults, women, low income and racial minorities to bring about equitable college completion. So FIPSI has ostensibly been charged since its beginnings in 1972 with doing something about inequitable kind of completion and higher education at large. But under the Obama administration, we saw a stark change that newly centered matters of race, class and equity in the agency's expressed policy. So the shift ties back to the big puzzle I presented. If we see this move as an equity policy intervention successful, we should see some meaningful effects on how this organization delivers resources and agency to minorities, minoritized communities. And if we don't, by what mechanisms does that occur? So while this case is a federal agency at its core, this is a case no different from the functional units we all work in. It is an organization shifting its policies from being race neutral to more directly centering minoritized students and the colleges that serve them. But how does kind of the rubber hit the road once we start to implement this new frame? So making my research questions specific, I asked, does equity framing as a practical intervention affect the distribution of resources? And how did this organization's implementation of this new kind of commitment to equity weaken or not racialized patterns of affording privileges to predominantly white institutions? So in order to answer this question, we also need to kind of tap into how would we know practically if a policy or an organization or a department's actions are more or less disruptive to racial inequity? So I measure, sorry. So I measure this organization's impact and we could just as easily say a graduate school's impact, a hiring committee's impact on the status quo in terms of the degree to which they dismantle or adhere to rules that disproportionately benefit white populations. In higher education, this looks like the histories and taken for granted rules that maintain inequitable resources and legitimacy afforded to the most prestigious and predominantly white institutions of higher education at the expense of the broad access institutions, the tribal colleges, the HBCUs that serve a majority of our nation's students of color. By awarding preference to these types, organizations reproduce the conditions of a racialized field wherein non-white organizations like MSIs by formal or demographic designation are also the most poorly resourced. In this way, the treatment of MSIs via policy and inter-organizational relationships as a critical lens on the degree to which an organization or actor is perpetuating or diminishing racial inequity. So I began my work on this case with kind of a beginning assumption that perhaps the change I came across under the Obama administration was just one in many changes over time in terms of how this agency kind of took up issues of equity. From a bird's eye view, I found via archival analysis that the guidelines in the pre-change period from 95 through 2011 were remarkably stable. These guidelines, which I quoted as equity evasive fit a really common pattern in U.S. racial meaning that references diversity broadly but does not center specific identities in its problem diagnosis, nor does it commit to action on specific injustices. Whereas in the post-period after Obama appointees intervened in the program, the agency's priorities shifted to using what I quoted as an equity conscious frame which targets and names specific identities and the need to remedy an inequity. I also discovered other important policy design features of this era that are material to my findings that I'll walk through in just a moment. So I wanna show you kind of quick exemplars to illustrate what I'm saying in practice, right? So here in this text exerpted from Fipsi's 2006 guidelines, we see the agency urging the field to develop education reform proposals for a changing world. They go into a list of eight priorities, three of which I have here highlighted in green or yellow, including the dramatic rise of information technology, the increasing diversity of post-secondary learners and the rise in competition among providers. So here Fipsi urges its applicants to anticipate these dynamic forces for change and develop bold new project ideas. The text in yellow represents the only reference in these grant guidelines to specific student identities or needs and it's just kind of a quick reference to diversity. So while some of the equity evasive guidelines that I analyzed actually don't mention identity at all, I selected one that does just to emphasize here in this talk that this era is not about being openly anti-equity, right? However, this is just one of eight invitational priorities, which is the equivalent of something that's a good idea but you don't actually get extra points for taking it up, okay? By contrast in green here, I've highlighted that in the post-period Fipsi has moved from eight invitational priorities to a set of five absolute priorities. The uniting theme of these priorities highlighted here in yellow is that each goal, access or completion or transfer rates must focus on underrepresented, underprepared and low-income students. Maybe not the language I'd use but we could impact that later. A refrain that references the extensive text in the grant guidelines dedicated to problematizing unequal college success outcomes specifically in terms of race and class. In short, applicants were required to attend to specific educational inequities directly to even receive a full grant review. Every applicant was also required to disclose their MSI status and the income and racial demographics of their student body. So this gives you a sense of what this change in framing looked like and indeed this shift looks similar to many shifts towards equity we see or initiate in our own organizations. But as I conducted these analyses, I noticed that the equity framing was not the only change in this policy. The enactment of the policy, that is the rules, the guidelines, the benefits and the burdens of the work also changed. So to this end, I analyzed the policy designs to understand the relative benefits and burdens assigned to target recipients. These types of policy features often pivot around power and social beliefs about the deservingness of those the policy will serve. So this is sort of something like a trap that policy folks often fall into, which is we can center minoritized communities in a policy, but then how do we treat them and how do we treat them differently? So first I found a shift from unrestrictive to restrictive project design standards. So while applicants in the pre-period were welcomed to propose nearly an infinite variety of innovative project proposals on topics like civic engagement or engaged or global learning, in the post-period, each absolute priority had a tight set of potential project types, mostly but not all focused on discrete and often remedial interventions that would fix a specific gap in a pipeline. I also found a shift for more to less generous funding plans in terms of the total dollars awarded in this funding period and in the concentration of even fewer but larger grants to fund the extensive evaluation expenses that the new surveillance requirements made necessary. Which brings us to in perhaps the most striking shift of this period, applicants were faced with much higher evidentiary standards up front in order to be considered for funding. So gone was the emphasis on kind of innovative teaching practices. Funders were looking for either RCT or other causal evidence that the proposed project would change student outcomes. As well as a fully developed evaluation plan comprising of an RCT or intensive quasi-experimental evaluation that met the metrics set by the what works clearinghouse. Indeed, they required involvement on the project by experts specifically trained in WWC methods, a distinctly white group of research and evaluation professionals. None of these requirements were in place in the pre-period. And finally, there's also a shift in how project were evaluated. So whereas 50 in its pre-period evaluated projects in terms of institutional change, that is they were looking for programs that had become institutionalized meaning they were continued after the grant ended and diffused, meaning other colleges picked up the programs. In the post-period, as mentioned, evaluators were hired using grant funds to report causal effects on individual student outcomes. So referring back to the racialized organizations as both contingent on not only the inequitable distribution of dollars or other resources, but also the benefits and burdens that come along with those resources as indicators of community agency. So we see a shift here from an equity evasive era that offered high agency and low burden to an equity conscious era that offered low agency and high burden. But then I had to ask, okay, put that aside for a moment, did these changes in frame and policy affect how dollars were delivered? To this end, I conducted a triple difference causal analysis using trio student support services as a comparison group. I won't get into any depth at all about this analytic approach, but for the data nerds out there, I'm happy to talk or to share the full paper. But what I will do is share a few visuals. So let's begin first with the obvious, most obvious of my measures. So the effect of this policy on dollars distributed to colleges as measured by their student enrollment features. So first on the left, I show the effects here of this change in relation to grant dollars delivered to minority serving students on the left or minority serving institutions on the left. The basic gist of this figure is that for years prior to first in the world, this change in in frame, MSIs were at a disadvantage for receiving FIPSI grant funds. This line here is shown in black and could expect to see at least $2,000 or more dollars less in any given year than a predominantly white institution, even as MSIs did receive more funding from trio student support services than PWIs. And similarly, FIPSI had long underserved high Pell serving institutions relative to other non-high Pell serving institutions prior to this change, a trend that flipped above and beyond the control group in the post-period. So if we think about the question, does an equity conscious frame have material effects on how we spend money, right? As a higher ed organization, whether it's a grant maker, whether it's institution. And in this case, an organization that was always technically focused on inequality and higher education. These figures give some compelling evidence that this is indeed the case. For at least two decades leading up to this policy change in question, colleges serving higher proportions of underrepresented minoritized students and poor students were actually at a relative disadvantage in grants competition. This changed markedly on both measures above and beyond the control group that had consistently tracked positive. But we might ask, what about other measures? There are many other measures of colleges and universities which correlate in important ways with these constructs. I use these measures to more deeply interrogate the different facets of this change. And I'm just gonna present a couple of those here. So first, we can look at this in terms of how grants were awarded to different institutions in relation to their relative field level or field measured prestige. So you can see, for example, on the left that FIPSI has long favored R1 doctoral institutions. A trend that remained constant and even grew slightly compared to the control group in the post period. Similarly, while TRIO, Student Support Services has almost never favored US News Top 100 schools, FIPSI long has. And again, a trend that slightly increased in the post period. On the flip side here, low endowment or low instructional spending institutions on the left had long been disadvantaged for FIPSI funding, a trend which did not change in the post period. And while FIPSI did not have a track record for giving a great deal of preference to community colleges, this track record actually dropped off a bit in the post period as seen on the right. Given what we know about the relative traits of MSIs, community colleges being the most frequent MSI organizational type, these findings are a bit counterintuitive. For example, if we look just descriptively at the proportion of the college going population served by doctoral granting universities, almost 22%, while a total proportion of underrepresented minority students enrolled in college served by doctoral is only about 15.1%. So the pattern here is that MSI status aside, the types of organizations that show persistently higher funding from FIPSI over time see the three X's in that first grade column are the very institutions that do not serve a higher proportion of underrepresented minority students overall in both the pre and post period. Indeed, less than 10% of doctoral universities are MSIs, while over 30% of associate granting colleges are. This presents something of a puzzle given the first figures we went through in this presentation, how can we understand this trend? So to better understand this contradiction, here are figures in which I intersect the college type with MSI status. As you can see, these figures help to clarify where the effect I showed you on my first slide came from at MSI doctoral and not these other types. So from this perspective of service to the most underserved US populations, this finding is not at all intuitive as the very institutions that on average serve the most minority students do not see benefit and even some negative effects from FIPSI's new racial frame. And these are also the MSIs where the kind of highest need and lowest resource students attend. So integrating this information in the equity evasive period, we know that money was delivered to predominantly white, wealthy and elite institutions. Under these conditions, which centered all students in its inequality frame, funds were delivered awarding high agency to applicants and recipients to pursue innovative projects including broad civic, technological and learning concerns. By contrast, under the equity conscious frame, we find a much more arduous application process with a significant portion of grant assets going to funding complex accountability and evaluation designs with a focus on projects more limited and remedial in scope with an emphasis on individual level rather than institutional change in effect delivering lower agency and higher administrative burdens to a subset of institutions. Still mostly PWIs, but with a greater proportion of the relatively highly resourced minority serving institutions. So looping back to my starting questions, did framing matter to fund distribution? The short answer was yes. Under equity evasive conditions, white and wealthy institutions received disproportionate benefits. And we see a large statistically significant effects on the number of poor and underrepresented minoritized student populations served after decades of under service. These effects were accompanied by new application requirements and priorities that replaced rather than eliminated processes that maintain funding inequity. What's notable about this case is that while the funding pattern changed drastically, so did the parameters of grant making related to requirements for intensive grantee evaluation, surveillance and accountability which essentially put non doctoral or lower resource colleges at a huge competitive disadvantage. So just to drive that point home, these kind of intensive evaluation surveillance and accountability requirements, we have to pause and think, in many ways we think of this as kind of a neutral or positive or good requirement, but it is also highly coordinated with predominantly white institutions with high access to resources. So even these types of neutral decision making processes, apparently neutral or apparently neutral requirements, made it difficult, if not impossible for minority serving community colleges to even compete for the funds. These changes also potentially made inequities more durable by moving from an implicit set of processes that legitimized inequitable grant making to legitimated organizational routines and processes that structured this outcome. And by that I mean in the first kind of in the first equity evasive condition, it was sort of a latent or hidden process by which the things that we consider legitimate in the field of higher education in terms of who's pursuing or wants to pitch projects about global learning, about learning communities, undergraduate research, all of these things. It was really an implicit delivery of dollars favoring white and wealthy institutions. But in the post period, we got really kind of structured and structural requirements put in place that put minority serving institutions at a distinct disadvantage even without naming minority serving institutions in a particular way. So in this way, this program further entrenched old patterns while creating a small window for an emergent set of relatively rare MSIs to receive greater funding. But historically racialized categories like the HBCU's tribal colleges and community colleges that still serve the lion's share of minority students were left systemically underfunded. So under the new condition, a traditional source of legitimacy among academic elites, quantitative scientific evaluation of interventions on individuals of color begets a new mechanism that protects the preference for higher ranked and resourced colleges and universities. And similarly, rather than emphasizing learning and institutional transformation, FIPSI's enactment emphasized individual level interventions built on the assumption of student deficits as the driver of post-secondary inequity. This outcome codified in deep in modes of reproduction that limit the relative agency of minoritized organizations and their students and the types of education that they're expected to pursue. An outcome I mark as the process of institutionalizing inequality anew. So this finding points out that we must be attentive to help policy designs embed our biases. A majority of social scientists might agree that accountability and evaluation probably matters or can matter. But how are we applying this standard inequitably to whom and how does it place the highest need students at a disadvantage? So while Ray's 2019 theory of racialized organizations gives us a sharp lens with which to identify the operations of racialized organizations embodied in this figure that I showed at the beginning, which shows the connection between core modes of reproduction and the institutionalized outcome of inequitable use of dollars and agency towards minoritized groups and organizations by bringing in this lens of kind of institutionalization as a process by which we can either support or break down a racialized status quo. I'm putting forth a novel construct of racialized change work. So in the spirit of using this framework to both critique inequity, right, but also to build towards more just futures. So racialized change work is the focus attention as I said at the beginning on dismantling kind of those connections that magnify agency of dominant groups at the expense of minoritized racial groups. So in the full paper forthcoming, and I'll give a link at the end at the Journal of Public Administration Research and Theory, I, my co-author, Jeanette Kalevis develop actionable and testable propositions for the conditions under which racialized change work is likely to be taken up, how it attempts at racialized change work either maintain or weaken racialization and the mechanisms by which this work has material and lasting implications. So as a means of illustrating the application of our framework, I wanna go back to this example I just offered you. So while 50 deployed a new equity conscious frame, one potential mechanism for weakening the racialization and organizations, which then legitimize greater resource distribution to minority serving institutions which had a measurable impact on delivery of dollars. The organization also developed new policy designs that built up new and more deeply embedded modes of reproduction via burdens and formal rules that significantly limited the agency of both MSIs and their students. So racialized change work as a framework and a set of testable and actual propositions is thus intended to help us think as practitioners engaged in historically racialized processes like admissions, hiring, funding and grant making about the myriad ways our status quo practices implicitly and routinely preference whiteness and white organizational types. This framework challenges organizations to give renewed attention to how our own ontological attunements toward things like educational quality, scientific rigor, the purpose of higher education and best practices for our work in admissions, retention or funding may reinscribe the very inequities they set out to diminish. We can also use this framework to push ourselves to see and understand how our commitment commitments to DEI on paper are valuable but they must be more than a label to change embedded racism. And we can think concretely here about what that means. So with that, I wanna thank you and open up the space for discussion and questions. I'll leave you with a few kind of parting practical questions you can ask regardless of your personal rules and responsibilities to help think about enacting racialized change work. So how are you framing your work as disruptive to racial inequity? How do you use as markers, what do you use as markers of quality, rigor or best practice as criteria in your own decision-making processes? Which of these criteria correlate with the norms or resources of white institutions or students in which do not? And how can you re-imagine those criteria? I put my email here and also these two yell key links which for the next 24 hours at least give you access to the two working papers that I am referencing throughout this presentation. One is the empirical paper and then the other is the paper specifically on racialized change work. And so I was only able to give a taste. So if you wanna get deep into it, you can go to yellkey.com backslash point or yellkey.com backslash sing. And just thank you again. I look forward to hearing your questions and also examples in your own work of how you're seeing these things show up.