Time to start the next lecture. All right, thank you for setting the recording. It's my pleasure to introduce Thomas Poulstra. I give the next set of lectures applications a big cone McCauley modules and algebras. I first met Thomas when he was a starry eyed undergraduate looking at graduate programs. And I'm happy to report he came to Missouri and was even my student. And in spite of that, he's had a lot of success. So Thomas is currently finishing up an MSRI postdoc and will be starting a position at the University of Alabama. They'll finally, for those of us in the United States, Alabama will finally have something besides their football team to be happy about. Yeah, it's a very good school. We're very glad that we'll open up a community of algebra area there. And Thomas has been working in characteristic P in a variety of areas related to to tight closure and Hilbert Kuhn's multiplicity and many other related areas so thank you Thomas and we're happy to have you here. All right, well, wonderful. So thank you for such a kind introduction. And thank you to the organizers for inviting me to speak and and also invited me to speak on such a wonderful topic. One of the more rewarding and interesting topics and community of algebra to dive into. So, and, and I'm fully convinced this topic of big coma colleague models and algebras will always be in the future of community of algebra and study. So it's a remarkably important topic and a subject and it's a great responsibility to talk about it to you all. So, so yeah so yes this is the summer school on high closure and I get the pleasure of giving a couple of lectures where most of it in prime characteristic. We are gone. So what I will be doing in my two lectures is I just want to talk about the applications of they call Macaulay modules and algebras, as it relates to specifically the homological conjectures, at least some of them. But I also that we won't get to this so much today but definitely on Wednesday. I also want to talk about how big calling Macaulay modules and algebras are used or specifically algebras are used to define and study this is this is talk about work at that but so we'll talk about the homological conjectures and how to study singularities that resemble tight closure singularities but in a way that's independent of characteristic, and it's allowed us to So, so, so, so yes are will be commutative no theory ring, not necessarily a prime characteristic I'll be careful to say when we do need prime characteristic or when it doesn't matter. All right, we're going to be talking about big calling Macaulay models, particularly these are modules that do not have to enjoy the property being finally generated so we'll be careful about specifying when something is finally generated or not. So the definition to start with so let's start off the local ring curl dimension of the ring is a d take a system of parameters x1 through xd, then a module is going to be called say big Cohen Macaulay with respect to this system of parameters if couple of two things happen. One, we want that this sub module of the ideal generated by the parameters times m is not the entire module itself. And so right so the module m doesn't have to be finally generated so this is a non trivial condition to impose a module is non zero if it's not finally generated. And two is we want that x bar the sequence is a regular sequence. Right, so what I mean by this. I that Xi is regular is a non zero divisor on him mod x1 through Xi in or a different way to say this so we don't have to work with quotients but we can instead just talk about colon sub modules condition to is the same thing as saying that. This should be an Xi minus one down here. It's going to be the same thing is that x1. My screen froze. Excuse I don't know if y'all can hear me I'm having cool issues on my end. Yeah, you're audible. I don't know do you want to try on sharing or sharing. Well no that's the thing my entire screen is just frozen. I don't even know the mouse or anything. Yeah, I mean your audio is coming through find your videos also. Okay, thank you. This is my computer. Okay, I hope I can, can you, can I be heard. Okay, I see some thumbs up. All right, yeah. All right, I think I think the writing program I was using is going to be trouble so we're going to use something else. Okay, so we'll just go this online. So I believe, right so we talked about what a big Cole Macaulay module is with respect to system parameters so keep our ring local for a little bit, or keep our ring local so we're going to say them is balanced. Okay, or what you'll, what I'll just call this or just, I just like calling these big Cole Macaulay, or you'll just see the abbreviate this even further as just BCM. If it is Cole Macaulay with respect to every system of parameters. And so, a little bit of history here, some history. So, so the study of these things, and their importance and relevance was due to Hoxster's work in the 70s, which I realized the importance of these things in the subject of commutative algebra and really preach about these. So, but what Hoxster was able to establish is that if so just to make life easy, RMK is a complete local domain. So if, if R contains a field. So, either a field a character CP if we're in the character CP setup or a copy of the rational numbers if we're in characters to zero. Then are in that's a big Cole Macaulay module, a balance. So if it's characteristic at this point, then, let's see, some other very significant advances around this so Hoxster unit key. So this would be in the tight closures as this 90s. This would be that if, if characteristic of ours prime, then are plus is a big Cole Macaulay algebra. Our plus so our plus is the absolute integral closure of our, and a choice of the algebraic closure of its fraction field. But, am I going out with this right so big Cole Macaulay algebra is our algebra that is big Cole Macaulay is a module. So but, but the, the amazing thing here is that if your characteristic P there's this canonical natural given. Big Cole Macaulay algebra that always exists and then. And then along these same lines so something much more recent, but so I believe this is, this is recent, this is 2022 21 or no no excuse me maybe this time when this first appeared on the archive for 21. So if, if our is mixed characteristic, say, zero P, then the patic completion. So, our plus is big Cole Macaulay. And then I'll remark in the echo characters zero case our plus is never calling the colleague in dimension three and larger, and that's actually part of the exercises. But in the characteristic P in the mix characteristic setup there's is really canonical choices and so so a couple remarks about these these theorems. So the the Hoxley unit keys result. This was after their result came out a simplified proof of this appeared by work of unity and loop as Nick and. We'll be talking about this in a couple of lectures after I finish up my lectures can I will talk about our plus being Cole Macaulay in case he and exactly what they're able to achieve using relatively simple techniques. So, but the thought result is cannot be described as a simple result at all it's depending fundamentally on the theory of perfectoid spaces of shoal say that, you know, arguably aren't him fields metal. It's depending on the development of prismatic homology and so forth. So there's a comment in the chat. Yes, when I say big Cole Macaulay point out, it always is going to mean balance. I don't want to talk about on balance calling the colleague modules. But there are, but there's exercises on on on how to construct modules that are big Cole Macaulay but not balanced. But from here on out, we only care about the balanced ones because those are what are important for proving these conjectures, most of the time. And the more serious ones. But the greatest achievement and all of this and the existence of big Cole Macaulay modules and algebras hands down goes to Andre. So, so the most powerful theorem at our disposal, and arguably the greatest, one of the best contributions to the subject to commutative algebra up to this point is the following theorem. And I believe this is published in jams in 2019. So Andre team. So if say our to s is a map of say complete local domains. You can do much lighter than this but let's just keep it simple for the time being. And then there exists a diagram to be our to us. We are. Yes, such that are to be our S to be so best are BCM algebras over RS. And in fact, Andre does more than this is it's given a map artist and there exists these BR and BS is so for example characteristic P. You could always put our plus and s plus that would be fine if aren't as a characteristic P there's always a diagram with our plus and s plus so we'll just put it up there. You can just take our plus s plus. So this would work. Characteristic bars P. But I still want to emphasize that this map right here does not have to be unique. And so, so the terminology that gets used here is, is what Andre has proven is there's the existence of what are called weekly functorial big coma college algebras. And this theorem is the greatest achievement of this and we'll be able to prove some interesting things with it. So. Okay. So, so the first one, say let's focus on this focus on, I mean, let's focus on tight closure first. So some applications is just these theorems are just going to be fundamental tools for proving these theorems I'm not to discuss. So, so, so the first one, say let's focus on this focus on, I mean, let's focus on tight closure first. For a little bit so some applications to tight closure theory. There's a couple of theorems here so theorem. And so once again let's just take RMK complete domain and characters. So theorem. If let's say X bar. Hi, is a parameter ideal. Nice, I believe flooring and rescue would have talked about this. And the tight closure this parameter ideal can be realized by extending the ideal I to our plus and then contracting it back to our. But of course when it comes to proving theorems are very powerful and interesting statements using the techniques and what's going on in the background is way more useful and powerful and interesting as things to extract. So that's going on in the background of this theorem. So let's just call it another theorem this is rightfully due to Karen Smith, which will motivate an analogous definition of a rational and rings at art of prime characteristics so which will get to more on the second lecture so theorem. Following our equivalents. One, or is that rational to for all BCM our algebras say are to be the top map of what we call model biology modules is injective three. HDM art. HDM for us is injected. But the key one to focus on here is number two that ends up being a useful way to define what went one long Carl Schrie call what are called BCM rational singularities and these are going to end up mimicking f rational singularities and characters. Thomas. Yes. Sorry, this is a little bit poorly phrased but can this property being a big code Macaulay I'll probably be detected just from the co kernel of RTV. Certainly no. Well, I mean, certainly the co kernel had that D minus one but I mean that's that's a lot to ask. Right. I mean it's equivalent. Yes. Yeah, sure. Yeah. Thanks. We said, actually hold on maybe I'm just understood your question to ask it again. Well, your answer about the depth of the controls is pretty much what I was. So, so those are just some applications. All right, and then let's see, and then I want to mention a conjecture here, I believe this talked about an inch on Mars lectures. So conjecture is the following our equivalent on our is strongly a regular to our to our plus is here. But it is I just want to write this conjecture out this is conjecture. A different way to say this is that every ring of prime characteristic that a splinter should be strongly a regular and you've seen in the lectures at every strongly a regular radius splinter. And, you know, the hope is is this conjecture is true, and we give more meaningful insight on how to extend the notions of strong F regularity in the mixed characteristic world and just studying say splinter rings maybe. But, but this, but I do want to mention that mention on my cross we do have ways of extending the notion of strong the F regular to the mixed characteristics setup. In terms of big column of collie algebras. But it's a, you know, it's a little bit of work to be done to really understand what's going on there, but, but the point I want to make here is if you. I wholeheartedly believe that if you want to study this conjecture of splinter implies strong or even the weaker conjectures of, you know, every weekly F regularity is F regular like the property of week F regularity passes to localization. So I've become increasingly convinced that the study of that problem is equivalent to studying properties of big column of collie algebras over our. So, and it looks that's what's going to be necessary to solve some of these big open problems. So the some other applications so the homological conjectures which will we will be focusing most of our time on. Let me just directly show up here so the ones we will prove are the menomial conjecture and then bit as a corollary effectively we can derive the direct summer and conjecture. Amazingly enough is completely trivial and characteristic zero. Three. The vanishing map of tour conductor. These are all theorems now. And then one of my favorite things. Direct summands. Regular rings are calling college. And so hopefully I believe for has shown up in the lectures, at least in prime characteristic right if you are a direct summon of a regular ring, or even just a weekly F regular ring, then you actually have to enjoy the property of being weekly F regular. And then that's enough to imply Colin McCauley. And so the proof that direct summands are regular rings are Colin McCauley is a relatively, or thing in prime characteristic and equity characteristic zero and especially the mixed characteristic case. There's a much different beast and it requires these, this theorem of Andre on the weekly functorial big Colin McCauley algebras will use the true force of the homological or excuse me the of Andre's work to get number four. But for one and two. We will, we will not need nearly as much. So, and then the other application that we won't spend any time on, but the, the study of big Colin McCauley algebras shows up very significantly in the study of test ideals or multiplier ideals. So the Lynch one line Carl Schreed have done these mixed characteristic test ideals, and they developed this theory in an appropriate manner to, to, to establish the uniform symbolic apology property and regular things. So, what does this theorem say, so it says that if, if RMK is an excellent regular local ring. And this is for all R so no singularities. Then of say to crawl dimension D, D, then for all P and spec R, the D symbolic in power is contained in regular and power, and this is for all natural numbers. So we won't be going into this but we say where this comes from so I'm not sure Smith and the equity characteristics zero case, and they use the theory of so called multiplier ideals to make this happen. Which ended up being the inspiration to finish off the proof and mixed characteristic. Yeah. So you have Hoxton Heaney key around the same time as the ELS result. Prove the same thing in character CP and actually recover the iron laser fell Smith theorem by a reduction to prime characteristic. And then, then my street in the mixed characteristic case. But I will say that the ma street stuff was certainly inspired by the multiplier ideal theory of iron laser spiff or analogously the test ideal theory of work to Kagi to prove the Hoxton unity results and characteristic P in a novel way. And should also mention there's, there's an analogous version of this property for not necessarily prime ideals or non reduced ideals and that solution in the in the mixed characteristic case was settled by a to Kumi Moriyama somewhat recently. All right. Okay, that's just a maybe very partial and incomplete list of reasons to care about this subject. So that being said, I want to just start diving into some math. So, so we're going to be you. We're like I said, it's just, we're just assuming that they call Macaulay modules exist. Balance they call Macaulay alters exist and we're going to assume weekly functorial they call Macaulay alters exist as well. We're going to assume the whole thing and derive some application so subject one. So applications to the homological connectors. All right, so the first one will go for is the monomial construction. All right, so it's going to say the following. So we're going to take a local ring of curl dimension D. Let's make positive dimension. So it's an interesting statement. Let's say X one through XD, a system of parameters. And for every natural number and we're going to have that X one times XD raised to the power has no chance of belonging to the monomial ideal generated by X one to the N one. And let me put a remark here like what a more useful way to maybe a very useful think about what this is saying. So what is the same. So, so I recall that we can identify the top local module of R as a direct limit system involving these parameters. And so how's that natural identification. So the top local co homology module of our is isomorphic to a direct limit. Let's say over the natural numbers index by cities of well it's going to be the teeth piece will be our mod X one to the T, XD to the T. So that's the teeth piece of the direct limit system. The T plus one piece would then be X one to the T plus one X D to the T plus one. And then we're going to multiply by the product of the excess here. And so, all we're saying so. So notice. So let's do a converse of the statement so X one through XD to the end not being inside this monomial ideal. Okay, so what let's suppose that element wasn't. So notice X one dot XD to the end is an X one to the N plus one, XD to the N plus one is equivalent to. So what is that if and only if if and only if, well, not an if and only if, but it in particular, it implies that X one dot XD to the end, plus the N plus one piece here of the direct limit system. So I'm looking at the image of the element X one times XD to the end. And the N plus one piece of the direct limit system up here is zero and HDR. Okay. But this this element right here is right if you go backwards in the direct limit system you see that this is naturally the image of one plus X one dot XD. Right, I'm saying right. Right, these two out these, these two elements are the same because their representatives are of this, you know, hit each other in the direct limit system. And so, and so what I'm saying here is so therefore monomial conjecture is if and only if. Monomial conjecture is true if and only if one plus X one, XD, are does not equal zero as an element of the top local. Right, so what the monomial conjecture is doing is it's just pointing out particular elements of top local homology that have to be non zero points amount. Okay, so let's do a proof. Let's do the proof. There's going to be a couple of limits so in the notes I put a lot more in the notes that I'm going to put on this. In particular is a couple of limits and I believe have been covered in other lectures but if not, please just go to the type notes that you should have available to you to read the proofs and in fact the proofs that I've written out will help out with some of the exercises. So in the notes, there's a lima 3.4 that we're going to use. And so what is it going to say. So, okay, local, local arts, a map. Oh, wait, I'm getting ahead of myself. We'll need this for the direct summing gesture I'm getting ahead of myself. Excuse me, we're ready for the proof of the monomial conjecture. Proof, we'll use limit 3.4 for the next year. Proof of the monomial conjecture so. So suppose for contradiction, we can multiply these parameters. Raise it to the power that we can land inside x1 to the n plus one. Okay. This is going to be if and only if the R linear map are to our mod x1 to the n plus one xd to the n plus one, sending the element one to x1 xd to the end is the zero.  And we're going to tensor with him. Okay. So that's going to tell you that the map him to ill mod x1 xd to the n plus one and plus one is the zero. But, okay, so what is this really saying so this says give me any element of them multiplied by x1 through xd to the end, and I have to land inside this sub module x1 to the n plus one up to xd to the n plus one expanded down. So this is if and only if using like these colon ideal language. Now, it's equal to x1 to the n plus one up to xd colon. All again. But there's going to be an exercise so let's see that exercise number is two. So using the fact that m is calling McCauley. It's not a big call McCauley module. It's not difficult to prove that that colon ideal gets quite trivialized. Just using the fact that x1 through xd is a regular sequence. And all of its permutations are regular sequences on him, right that's important that is balanced big call McCauley here. But this is a contradiction. Okay, this is exercise two. But this is a, because it's not equal to the sum. All right now, let's move on to the infamous direct sum and conductor so. And arguably, the reason, I mean, from what I can tell from context is that the direct sum and conjecture is the motivation behind this, this all the study was Mel's motivations was the main focus, in terms of all these homologic injectors, direct sum and conjecture about to talk about was the main focus. Common in chat. And he just said, yes, yes, you have to reduce to the complete case. So yes, let's back up just a minute. Yes, so there's a comment here I should have made some reductions to use that there is a big call McCauley module I need to get to ours a complete domain, and I didn't do that reduction so so yes. So, suppose you knew that, you know, we're going to prove by calculation. So if x one through xd to the end was inside that in primary ideal. Then you can that same containment has to hold in the completion.  And you can just replace your ring with that. So yes, you can reduce the complete domain case and so therefore you can use the call McCauley modules. Thank you for the comment. So we're going to get to direct summoning conductor and to do that, we need a couple lemma so so we're going to be talking about things splitting so we're going to need some criterion for splitting. And this is lima 3.4 in the notes so arm KB local me finally generate a module. And R to him a map, or linear map so send them the one to some element, the module. The following are equivalent. One, R to M splits. And then there's a direct summon of them by this map. And then to is that I take the injective whole of the rescue fields and tensor with this map. Objective, where the RK was injected whole right so splitting is a property that can just be checked after tensoring with just one module. So we're mapping to a funny, funny. And then lima 3.5 will also use is that if we have very explicit descriptions of injective holes for certain classes of freeing so. So if arm K is a Gornstein, and we'll use, for example, if your ring is regular. For example, regular. Then the the injective whole the residue field has a very explicit description. Namely, is a topical co homology module, which of course we have this very nice way to think about these is certain direct systems, and then so one of the exercises. Here so I did so if you haven't seen how something like lima 3.5 is proven. And there's an exercise right up in the notes. And there's an exercise immediately following this lima about how to describe the the injective whole as a similar looking direct limit system that looks like local co homology. But isn't. It's a case that are is not going steam but only assumed to be calling McCauley and has a canonical ideal. It's a very useful exercise. Right. You take a look up. All right, so that being said, we can do the direct something conjecture. So I'm going to say the following so let a be a regular ring. A to R a finite map. A to R splits. And let's do the proof by the monomial conjecture so So the so there's a lima. I didn't write out. So by lima 3.2. You may assume a is local regular in which case s is going to be a finite extension array. Or excuse me ours a finite extension. And then let's see so. Okay, so we're a is a regular local ring so alright so that means this maximal ideal is minimally generated by the dimension elements. If D equals to mention our or dimension a, which will be the mention of our Let's say the maximal ideal of a is generated by a regular system of parameters. Then the injective hole by limit 3.4 The injective hole of a k is isomorphic to top local homology model of a which can be realized as the direct limit system with this with respect to the regular system of parameters x12 xd. Moreover, the the soccer k isomorphic so k say is a my maximal ideal of a I can realize as the stuff inside the top local homology module. Annihilated by the maximal ideal, which is then generated principally by the class generated by this element. Right so this this single element one plus x12 xd of a principally generates the soccer. We also know that the right so this is the injective hole the residue fields so the way k sits inside this injective module, it's essential is what that means so zero colon hdm. To this in this local homology module is essential. And so what does that mean so that's if which just means that if we have a non zero sub module of the top local homology module, then the soccer actually has to be contained inside of it. So in conclusion, so by based on the dilemma 3.4 a to R splits. If and only if HDM a to HDM a are so this is isomorphic to our tensor. So if this map is one to one, but that map is one to one, if right so the point is is that the kernel of that map is a sub module of HDMA of a. And so that map is not one to one if only if the kernel is a non trivial, but that's if it only if that sub module contains the soccer. But that soccer is principally generated. And so that's going to be if and only if the image of one plus X1, XD. So, let me just spell it out very carefully here. So that's if and only if, well, if and only if one plus X1, XD. Okay, so the soccer generator, which would then be mapped to one plus X1, XD are right so you're thinking this would be inside the direct limit system. Hey, X1 to the T, XD to the T. You're thinking about this element inside the direct limit same exact looking direct limit system but with our array and say, and right so what we're crying is is that map is one to one if and only if the image of the soccer was non zero. That's what lima 3.4 is going to buy the fact that splitting can be just checked after tensoring with the injected hole. So, meaning we only have to check that the soccer the injected holes map to something on zero. But we can very explicitly point out what that soccer element is, and where it's going to inside this map and the monomial conjecture says precisely that that element can never be zero. And yeah, that's, that's 950 so. So next time, we'll talk about the bandishing maps of tour conjecture. We'll talk about a very nice application of that which is one of my favorite theorems is that direct some ends of regular rings. They are always Colin McCauley. And then, after that we'll talk about these so called big Colin McCauley rational singularities as they were introduced by Lynch one month Carl Schweed in a fairly recent publication. So, thank you. Thank you Thomas. Do we have any questions or comments. I do have a question. So, it is always true in a coin McCauley ring that every system of parameters is a regular sequence. So what's the difference between coin McCauley and the coin McCauley I think I'm missing some very big Colin McCauley, big Colin McCauley means it doesn't have to be finitely generated. Oh, I see. Okay. Yeah, yes. So yes, so the joke is right. All right, so the definition you'll see is that a module is called a small Colin McCauley module if it's finally generated and big Colin McCauley. Right so every small Colin McCauley module big. It doesn't sound right but big is not small. Small is big, big not small. Sounds good. Thank you. Do we do we have any other questions. All right, well let's thank Thomas for his fine talk. And I guess we will reconvene on Wednesday at whatever the appropriate time is for your time zone. Great.


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The Private Teacher Only folder allows you to add posts to CESA that are only visible to teachers. This feature is available with CESA Plus and CESA for schools. First, I'll show you how to add work to the Private Teacher folder while creating a new post. Let's make sure you have folders on for your class. Tap the wrench icon in the top right corner, scroll down, click Show Add to Folder Step, and make sure either Teachers or Teachers and Students is selected. Then, create your post. Tap the green check, select Students, and tap the green check again to go to the Adding folder and Skills screen. From here, tap the small blue lock icon at the bottom of the screen to add the post to the Private folder. Make sure the Teacher Private switch is green, then hit the green check to finish uploading the post. The item will not be visible to students or families, and no notifications will be sent to students or connected families. You can also add student work into your Private folder from the Approval Queue and Notifications area in your inbox. This might be useful for assessments your students complete that you don't want shared with students or families just yet. Tap on the folder icon below the post, then tap on the Private folder to add it to the folder. When you approve the post, it will not be visible to students or families. If you have already approved a post and then later decide to add it to the Private folder, it may have already been seen by students and connected families, but it will be removed from the Family and Student Feeds once you tag it to the folder. If you want to remove something from the Private folder, just tap on the folder icon and deselect the folder or click the three-dot menu under your post of choice, choose the Edit Folder option, and remove the check mark next to the Private folder. The post will now be visible to students and connected families. Thanks for using CSAW!


Classical liberalism, liberalism, libertarianism are all forms of political philosophies. They're questions about what the political legal order should do, okay? That's what they're about. Liberalism is not a metaphysics, it's not a worldview, it's not an ethics, etc, etc, etc. There may be all sorts of accounts of how liberalism came up, the origins of liberalism, etc, etc. But the interesting question is how do you justify, how do you defend that? And there are all sorts of resources that one can appeal to. There's a social contract theory, there's a Kantian tradition, the particular theory or approach that I'm interested in is a neo-Aristotelian one. Not that I'm contending that Aristotle was a libertarian. Indeed, he was a part of the statecraft, his soulcraft point of view. But there are insights in a neo-Aristotelian view about the nature of the good life and the importance of self-direction that help us understand why the state can't be the legislator of morality. Well, it would be Aristotelian in the sense that the account of the good life is sufficiently individuated and unique, that you can't just generalize from what the good life is to laws for everybody. And so, but that account of the good life is Aristotelian, but it gives rise to the problem of integrated political diversity. How do we solve that? And that requires a solution that looks at self-protecting the possibility of self-direction. And that's what's interested. There's a neo-Aristotelian basis for a problem that requires a classical liberal or libertarian solution.


Fans of the written words from all over the world know that London has been the home to some of the greatest authors and playwrights that has ever worked upon this earth. From Shakespeare to Dickens, from Dr. Johnson to Virginia Woolf, London is full of places with literary connections. And for those who may want to share their love for literature with their love for a good drink, this is going to be a very special video for all of you. In addition to the great bookshops, the city is scanned with landmarks such as pubs, museums and libraries, all associated with some of the greatest writers in history. You'll be surprised to know that there are plenty of pubs in London that are tied to some of the greatest novelists, playwrights and poets in British literature. There's even a London literary pub crawl that will lead you on a drinking tour to share pints with Charles Dickens or Virginia Woolf in the very places they're haunted in life. As we begin this vlog, I want to introduce you to an area of London called Fitzrovia. And guess what? This area where I'm standing right now was named after a pub. The area that we call Fitzrovia today is named after the famous Fitzroy Theven, which was one of the main social hubs of this area after the First World War. It is basically an old Victorian pub, historically popular with famous artists and writers, with its heyday probably being in the 1930s and 1940s. Named after the Fitzroy family, who were ennobled by Charles II, the pub is credited with giving London District of Fitzrovia its name. Not just this one, London has many more such clubs and pubs with popular and proud literary histories. Let me see how well read my audience is. Comment below if you know the club or the pub associated with our beloved barred Mr. William Shakespeare. The Theven was originally opened in the year 1833 as a coffee shop and it was only opened as a pub in the year 1897. This Victorian pub is right in the centre of an area known as Fitzrovia or the edge of a more well-known district known as Bloomsbury District, which is very famous for its literary connections. If you know the famous writers associated with Bloomsbury District, then don't forget to show off your literary knowledge in the comment section below and put their names in the comment below right now. All pubs are equal but some pubs are more equal than the other. I am sure this is what George Orwell must have said while walking inside this pub for the first time. Yes, you got that right. This ambient pub in the Fitzrovia district was once the regular drinking spot of the Big Brother creator George Orwell and our very beloved George Bernard Shaw. One of the regular customers at this pub included Dylan Thomas who used to scribble poetry on the beer coasters for the ladies in the bar. The Fitzroy Tevan was once such a famous meeting place of great intellectual minds of 1930s and 1940s that Augustus John the famous painter and one of the regular patrons of the pub said, in 1927, if you haven't visited the Fitzroy, you haven't visited London. The pub even has a wall of photographs of all these prominent people who have visited here. Among all other photos, you will find the past photos of George Orwell and Dylan Thomas who drank here when they were working nearby at the BBC. The Tevan's wall is full of photos and artwork. It is a way through which they remember the literary and artistic geniuses of all these people who would drink and socialize here making it arguably the most well-known pub in London. Places like Fitzroy Tevan makes my belief stronger that without any doubt London is the best city on the planet for lovers of literature and bookworms. One of the world's greatest literary cities, London streets, pubs and clubs have had some of the beloved names in the history of English literature. If you want to explore any other amazing literary site around the globe, then don't shy away. Comment down below and I'll be more than happy to incorporate that in my next vlog. I would want to end this video on the streets of Fitzrovia by quoting the famous Bohemian painter Augustus John who once said, Fitzroy is like the classroom junction of the world. Everyone goes in and comes out at some point or the other. That's it from my side for this video lecture. I'll meet you very soon in the next lecture. Till the time we meet next, happy learning, keep loving literature and stay tuned to arpitakarva.com.


our press conference. I'm joined here on stage by President Lagarde, Vice President Degindos. My name is Wolfgang Preusel. We are also joined by journalists on Webex. As always, if you want to ask a question, please turn on your camera and also the microphones. And with that, I'd like to hand over to President Lagarde. Please. Thank you, Wolfgang, and good afternoon to all of you. The Vice President and I welcome you to our press conference. Inflation continues to decline, but is still expected to remain too high for too long. We are determined to ensure that inflation returns to our 2% medium-term target in a timely manner. The governing council therefore today decided to raise the three key ECB interest rates by 25 basis points. The rate increase today reflects our assessment of the inflation outlook, the dynamics of underlying inflation, and the strength of monetary policy transmission. The developments since our last meeting support our expectation that inflation will drop further over the remainder of the year, but will stay above target for an extended period. While some measures show signs of easing, underlying inflation remains high overall. Our past rate increases continue to be transmitted forcefully, financing conditions have tightened again, and are increasingly dampening demand, which is an important factor in bringing inflation back to target. Our future decisions will ensure that the key ECB interest rates will be set at sufficiently restrictive levels for as long as necessary to achieve a timely return of inflation to our 2% medium-term target. We will continue to follow a data-dependent approach to determining the appropriate level and duration of restriction. In particular, our interest rate decisions will continue to be based on our assessment of the inflation outlook in light of the incoming economic and financial data, the dynamics of underlying inflation, and the strength of monetary policy transmission. We also decided to set the remuneration of minimum reserves at 0%. This decision will preserve the effectiveness of monetary policy by maintaining the current degree of control over the monetary policy stance and ensuring the full pass-through of our interest rate decisions to money markets. At the same time, it will improve the efficiency of monetary policy by reducing the overall amount of interest that needs to be paid on reserves in order to implement the appropriate stance. The decisions taken today are set out in a press release that is available on our website. The details of the change to the remuneration of minimum reserves are provided in a separate press release to be published at 3.45 continental European time. I will now outline in more detail how we see the economy and inflation developing, and will then explain our assessment of financial and monetary conditions. Looking at the economic activity first, the near-term economic outlook for the euro area has deteriorated, owing largely to weaker domestic demand. High inflation and tighter financing conditions are dampening spending. This is weighing especially on manufacturing output, which is also being held down by weak external demand. Housing and business investment are showing signs of weakness as well. Services remain more resilient, especially in contact intensive sub-sectors such as tourism. But momentum is slowing in the service sector. The economy is expected to remain weak in the short run. Over time, falling inflation, rising incomes and improving supply conditions should support the recovery. The labour market remains robust. The unemployment rate stayed at its historical low of 6.5% in May, and many new jobs are being created, especially in the services sector. At the same time, forward-looking indicators suggest that this trend might slow down in the coming months and may turn negative for manufacturing. As the energy crisis fades, governments should roll back the related support measures promptly and in a concerted manner. This is essential to avoid driving up medium-term inflationary pressures, which would otherwise call for a stronger monetary policy response. We welcome the recent Eurogroup statement on the Euroarea fiscal stance, which is consistent with this assessment. Fiscal policies should be designed to make our economy more productive and to gradually bring down high public debt. Policies to enhance the Euroarea's supply capacity can help reduce price pressures in the medium term while supporting the green transition, which is also being furthered by the next generation EU programme. The reform of the EU's economic governance framework should be conducted before, sorry, concluded before the end of this year. Let's look at inflation. Inflation came down further in June, reaching 5.5% after 6.1% in May. Energy prices fell again, dropping by 5.6% year on year. Food price inflation continued to slow but remained high at 11.6%. Inflation excluding energy and food edged up to 5.5% in June with goods and services following diverging trends. Goods inflation decreased further to 5.5% from 5.8% in May. Conversely, services inflation rose to 5.4% from 5% in May, owing to robust spending on holidays and travel and also reflecting upward base effects. The drivers of inflation are changing. External sources of inflation are easing. By contrast, domestic price pressures, including from rising wages and still robust profit margins are becoming an increasingly important driver of inflation. While some measures are moving lower, underlying inflation remains high overall, including owing to the persistent impact of past energy price increases on economy-wide prices. Although most measures of longer-term inflation expectations currently stand at around 2%, some indicators remain elevated and need to be monitored closely. Turning now to our risk assessment. The outlook for economic growth and inflation remains highly uncertain. Downside risks to growth include Russia's unjustified war against Ukraine and an increase in broader geopolitical tensions, which could fragment global trade and thus weigh on the euro area economy. Growth could also be slower if the effects of monetary policy are more forceful than expected or if the world economy weakens and thereby dampens demand for euro area exports. Conversely, growth could be higher than projected if the strong labor market rising real incomes and receding uncertainty mean that people and business become more confident and decide to spend more. Upside risks to inflation include potential renewed upward pressures on the costs of energy and food, also related to Russia's unilateral withdrawal from the Black Sea Grain Initiative. Adverse weather conditions in light of the unfolding climate crisis may push up food prices by more than projected. A lasting rise in inflation expectations above our target or higher than anticipated increases in wages or profit margins could also drive inflation higher, including over the medium term. By contrast, weaker demand, for example, owing to a stronger transmission of monetary policy, would lead to lower price pressures, especially over the medium term. Moreover, inflation would come down faster if declining energy prices and lower food price increases were to pass through to the prices of other goods and services more quickly than currently anticipated. Let's look at the financial and monetary conditions. Our monetary policy tightening continues to be transmitted strongly to broader financing conditions. Risk-free interest rates over short to medium term maturities have increased since our last meeting and funding has become more expensive for banks, in part owing to the ongoing phasing out of the ECB's targeted longer term refinancing operations known as Teltro. The large Teltro repayment in June went smoothly as banks were well prepared. Average landing rates for business loans and mortgages rose again in May to 4.6% and 3.6% respectively. Higher borrowing rates and the associated cuts in spending plans led to a further sharp drop in credit demand in the second quarter, as reported in our latest bank lending survey. Moreover, credit standards for loans to firms and households tightened further, as banks are becoming more concerned about the risks faced by their customers and are less willing to bear these risks. Tighter financing conditions are also making housing less affordable and less attractive as an investment and demand for mortgages has dropped for the fifth quarter in a row. Against this background, the annual growth rate of lending continued to decrease in June, falling to 3% for firms and 1.7% for households, with annualized growth rates of 0% and minus 0.2% in the second quarter respectively. Amid weak lending and the reduction in the Euro system balance sheet, the annual growth rate of broad money fell to 0.6% in June, with an annualized growth rate of minus 1.1% in the second quarter. To conclude, inflation continues to decline, but is still expected to remain too high for too long. The Governing Council therefore today decided to raise the three key ECB interest rates by 25 basis points. Our future decisions will ensure that the key ECB interest rates will be set at sufficiently restrictive levels for as long as necessary to achieve a timely return of inflation to our 2% medium-term target. We will continue to follow a data dependent approach to determining the appropriate level and duration of restriction. In particular, our interest rate decisions will continue to be based on our assessment of the inflation outlook in light of the incoming economic and financial data, the dynamics of underlying inflation, and the strength of monetary policy transmission. In any case, we stand ready to adjust all of our instruments within our mandate to ensure that inflation returns to our medium-term target and to preserve the smooth functioning of monetary policy transmission. We are now ready to take your questions. Thank you. Thank you, President Lagarde. And the first question goes to Janna Rando of Bloomberg News. Janna, please. Good afternoon, Madam President. Good afternoon. Two questions, as always. Does the absence of guidance for September, I assume you expected such a question, mean the Governing Council is equally open to another hike, to a pause, or even to the conclusion that rates have already reached their peak? That will be the first and the second one. I'm wondering how much of your decision on lowering the remuneration of minimum reserves has been, or how much of that was driven by concerns over central bank losses? Thank you. Thank you very much for your two questions. You know what, let me just remind you what we have been doing. And I'll come to your interest, which is what happens next. But let me just remind you what we're doing and what we have been doing. We are raising interest rates, all three rates, by 25 basis points. And this is the ninth decision to hike that has been taken in a matter of one year. So invariably, one meeting after the other, we have raised rates. The DFR is now at 375. And this decision that was unanimously agreed amongst all members of the Governing Council is driven by one key determination, which is to bring inflation to target, our target, which is medium term inflation. There is unanimous determination to do so. What have we decided as well? And I'll come to your minimum remuneration because that was your second question. We have also decided, and I'm going to quote because it's a sentence that you might hear over and over, not this time, next time and future times. We say, our future decisions will ensure that the key ECB interest rates will be set at sufficiently restrictive levels for as long as necessary to achieve a timely return of inflation to our 2% medium term target. So some of you may have noticed a slight change of verb. And that is not just random or irrelevant. And I think it's predicated on our determination to be data dependent. So I have said on many occasions that we had ground to cover or we have more ground to cover. What I'm saying here is that data and our assessment of data will actually tell us whether and how much ground we have to cover. So we are deliberately data dependent and we have an open mind as to what the decisions will be in September and in subsequent meetings because this determination based on data might vary from one month to the other. So we might hike and we might hold. And what is decided in September is not definitive. It may vary from one meeting to the other. So I hope it's very clear that we are not in the domain of forward guidance. But we are very strongly rooted in our determination to break the back of inflation and to take inflation back to 2% medium term on a sustainable basis. And to do that we will be informed by data. We will analyze the data and we will then decide at that point in time. We will have the inflation outlook. We will have two readings of inflation with all the underlying dissection and understanding of it. And we will have even more understanding of our monetary policy transmission. And that will be repeated every governing council after every governing council so that we can make sure that as I said we break the back of inflation. Now on your second question. Our primary job is to set monetary policy so that inflation stabilizes at 2% over the medium term. No dispute about that. But we also have a public duty to make the implementation of our monetary policy as efficient as possible. So by cutting the remuneration on minimum reserves only, that's the basis that we will use, cutting it to zero, we will reduce the amount of interest that we pay, which means the implementation of our policy is going to be more efficient. At the same time it is going to have no impact on our policy stance, which is currently determined by the remuneration, not on minimum reserves, on excess reserves, which as you know are far plenty than the minimum reserves. So that's the decision that we made. More efficiency and that's also of course predicated on the principle of proportionality. Thank you, President. And the next question goes to Annette Weisbach of CNBC. Annette, please. Thank you very much for taking my question. I have a question on the discussion concerning the time lag of monetary policy decisions and the effect it will have on inflation because you have of course tightened at a record pace now and there are discussions that one first has to wait a little bit until to really see the materialization of those rate hikes. So how are discussions going on that front? And then I would like to know whether you have discussed accelerating the shrinking of your balance sheet and if not, whether this could be a topic for the so-called second half of the year. Thank you. Thank you very much for your two questions. I'll take the second one. We have not discussed the reduction of our balance sheets and we have obviously acknowledged that the balance sheet has reduced and it has reduced as a result of number one, the reimbursement of more than what was expected to be reimbursed under the teltro reimbursement in June and also by the fact that we have not only stopped net asset purchases but we have stopped any reinvestment. So the balance sheet of the ECB is reducing as a result of that but we have not discussed any further reduction at this point. On your other question, as you know, transmission of monetary policy is one of the three metrics that we use in order to determine our monetary policy stance, the strength of monetary policy transmission. And we look at it that way. We see it as essentially a two leg process. The first leg is transmission to financing at large. What is the impact on lending, lending for banks, lending by banks to both corporates and households? And we try to measure that as accurately as possible by looking at volumes of loans, by looking at rates, by looking at obviously the bank lending survey that many of you have commented yesterday, which informs us about what banks anticipate and in which area and which directions they expect to either increase or reduce the volume of loans and reassess their risks in relation to borrowers. So that's the first leg and there are clear indications that our monetary policy is transmitted through that channel. The next one, which is the second leg, is transmission to the economy. And from financing to the economy, how quickly does that move? How efficient is it? And we are really beginning to see now transmission materialising whether you look at investment in housing, whether you look at investment altogether, there is obviously a decline on those two accounts. Still too early and there is certainly more in the pipeline, but we are definitely seeing monetary policy being transmitted and being transmitted strongly. That's obviously the case for the first leg and it is also now beginning to be the case in the second one. Thank you, President. And the next question goes to Francesco Canepa of Reuters. Francesco, please. Good afternoon. So at the last meeting you said that you were not even thinking about pausing. Would you repeat that today? The second question is, would you agree with a line that has been repeated by many of your colleagues on the governing council that the risk of doing too little outweighed the risks of doing too much in terms of rate hikes? Mr Canepa, you'll have to understand something which is that I'm not repeating things if prompted. I decide to say what I want to say, okay? So what I just said in response to the first question, I hope was clear, but I'm going to repeat it for you just to make sure that I'm as clear as you would expect. We are moving to a stage where we are going to be data dependent. And we will take the new projection by staff. We're going to take two new readings. We're going to take more information about how transmission is taking place. And on the basis of that, we will determine whether we hike or whether we pause. What I can assure you of is that we're not going to cut. That is a definite no. But on the other side, could be a hike, could be a pause. And if it is a pause, it would not necessarily be for an extended period of time because as I said, it will vary from meeting to meeting because we continue to decide on a meeting by meeting basis every time informed by the data. All right? I hope it is as clear as I can be because we are and that was absolutely endorsed by the governing council which has validated this decision unanimously. We are determined to operate on that basis. Voila. Your second question I've forgotten. You always ask something that I should repeat or that I should endorse, but I'm not doing that. Sorry. You know, if my colleagues take this view or the other, that's fine and they can certainly be free to expand in one way or the other. But I can assure you that on the occasion of today, it was a unanimous determination. And the determination is rooted in one very simple direction, 2%. We want to break the back of inflation. 2% is the goal and we will get there. Come what may. Thank you. Now turning to WebEx. I'd like to give the floor to Fabrizio Goria of La Stampa. Fabrizio, please. Over to you. Thank you. Welcome again. Thank you for Madame Lagarde for this opportunity. I have two questions. Actually, one is on the financial stability. Do you consider the European commercial real estate as a matter of concern? And the second one, can you elaborate more in detail the spiral between wages and prices? Thank you. Thank you a lot. I'm sorry. The second question, you say the spiraling. Yeah, can you elaborate more in detail the dynamics about wages and prices and inflation, basically? So the risk of second-hand effect? Thank you. Yeah, that's correct. Thank you. Will you take the first one? So good afternoon. With respect to commercial real estate, it's quite clear if you look at our financial stability review that we have paid a lot of attention to commercial property and to the declines in prices. These declining prices started even before the tightening of monetary policy. And, you know, the different agents and players that are exposed to commercial property that are mainly non-banks. So it's something that we pay attention to that we have analyzed in the past. And there is a clear, let's say, divide between commercial property and residential property. In the case of residential property, we have not seen any declining prices. What we have seen is a slowdown in the evolution of prices. But commercial property is something that we have paid attention and that will continue paying attention in the future. And on you second question, I think you raised the question of potential risk of second round effect. Let me just take you back for a second because it's a matter that we have reviewed that staff is doing a lot of work on and that we have discussed. It's the role of wage growth and the role of profit margins as two key drivers of inflation, which are gradually eliminating the external causes that were prevalent before. The wage increases is definitely playing an increasing role as a driver of inflation. And so have profit margins. On the profit margins front, we are seeing a slight decline in the first quarter of 23. And as you may remember, our projection is also predicated on the fact that profit margins are wide enough to accommodate the squeeze that will be needed in order to allow for wage increases. But if we look at inflation expectations, if we look at how wages and profit margins are evolving, we are not seeing a second round effect result. And for the moment, at least nothing suggests the risk of strengthening any second round effect. But we are really monitoring that very carefully because as, you know, service inflation is the sort of hard not to crack in terms of inflation, given that on goods, it's going down, energy is coming out and it's negative. Clearly, we're focusing on services that are labor intensive and that are also not as sensitive to interest rates. But we're looking at that very carefully and we'll continue to look at that extremely carefully. Thank you. Madame Lagarde. And the next question goes to Andres Stumpf of Expansion. Andres, please. Thank you. Madam President, regarding the September meeting and the possibility of hiking or housing, as of today, does the burden of the proof fall on an improvement of the inflation outlook? Or would it be enough if it's just the situation does not worsen? And a second question, if I may. Spanish headland inflation is now in line with the mandate and the labor market is as hot as it has been for more than a decade. Is it something like you would like to see for the whole eurozone or is it necessary for the unemployment rate to go up for the ECB to stop hiking? Thank you very much. Do you mind me asking you to repeat what you said about the unemployment rate in Spain? No, it's at the lowest point for more than a decade. Including for young people. Yeah. I was just saying that you would like to see that for the whole eurozone. I mean, the inflation going to go up by 20% and unemployment being low. Or is it just not possible and the labor market has to cool down? I'll start with that one. That new question about Spain. Because it demonstrates the beauty of Europe and the heterogeneity that we have amongst the 20 countries that are members of the euro area. Obviously, the numbers that we see now for Spain with inflation trending towards 2% and hopefully sustainably so. Plus, unemployment numbers that are as low as they have ever been is a good set of numbers for the country and for the economy at large. It is not the same in all member states. And there are member states where inflation is still very high and has been high and is expected to remain high for longer. So we have to be very attentive to the aggregate number. Those are the ones that are driving our inflation outlook, that are helping us determine our policy. But we also have to look at each member states and the characteristics of each member states. We shall see. Data will actually inform us better. But the hope that we have is that inflation will come down to 2% medium term. That is our primary mandate. As you know, we don't have a dual mandate unlike the Fed. And that is what is going to drive our determination. If we can do that with unemployment at the lowest level and staying there, that would be very, very, very good. It is funny because you ask the question of the lawyer. The issue of the burden of proof is very often the lawyer's issue. I think the burden of proof is going to be on the data. Because we are determined to bring inflation back to target. But we are also going to be in the hands of data and the assessment of data. So the burden of proof is going to be on data. Thank you. And the next question goes to Martin Arnold of the Financial Times. Martin, please. Thank you. Madam Lagarde, two questions for you. First of all, how much do you think tighter monetary policy is contributing to the weakness in the Eurozone economy that you identified? And secondly, what would it take in terms of reduction in your growth forecast and a reduction in your inflation forecast in September for you to pause interest rate rises? Thank you. Thank you very much, Martin. What we are working on and what staff is trying to be as detailed, specific and conclusive about is the impact of our monetary policy on inflation. So this is really what is what is the focus and what is driving us. What impact our interest rate hikes have produced and will produce on inflation. And obviously the exercise of producing the inflation outlook that will be conducted in September by ECB staff will be very informative for the governing council to determine how much has been transmitted and what is expected to continue to be transmitted. As I said earlier on to Annette, in the financing leg a lot has been transmitted a lot. We know that in the economy at large, not as much yet. And your second question is totally hypothetical and I understand, you know, why you would want me to answer that question. But it's there are so many data and so much to take on in terms of the projection that will be produced in September that I cannot answer your question in advance. We will really have to see. Thank you. And next question goes to Jean-Philippe Lacour of Agence France Press, AFP, Jean-Philippe, please. Hello, Madame President. You're suggesting a possible pause in rates in the near future. Does that mean that during this period of possible pause, we could see an acceleration of the QT and then it would be like a trade-off among those and hogs at the governing council. Thank you very much, Monsieur Lacour. Our key tool, our key instrument in the current circumstances, given the level of inflation that we have at the moment, is interest rates. So there will be no trade-off between either interest rate or QT. Interest rate is the main tool and the most efficient one. So it's the one that we will be using. There is the possibility of a hike. There is the possibility of a pause. It's a decisive maybe. But, you know, don't expect me to go one way or the other. As I said, the burden of proof is going to be the data. And the ultimate point that we are determined to reach is the 2% inflation. Thank you. Next question goes to Jan Malino for Handelsblatt. Jan, please. Thank you. Good afternoon, Madame Lagarde. You've mentioned that several economic indicators recently have been weaker than expected. And there's some economist warning already that the ECB is overdoing it with rate hikes. And they think that more patients would be necessary. So what's your response on that? And my second question, climate activists criticized that the green bond purchases of the ECB under the new scoring system have largely stopped. Have you perhaps promised too much in terms of what the ECB can contribute to climate protection? Thank you. So we would be doing too much and promising too much. Look, doing too much is certainly not something that we are considering. What we are focusing on is the inflation numbers. We are committed to deliver 2% headline medium term on a sustainable basis in a timely manner. This is our commitment. This is the drive of the governing council. When I look at headline numbers, when I look at core inflation number, when I look at all the many measures of underlying inflation, trying to really isolate the most robust element and removing the most volatile part of it, we are seeing inflation that is still too high and that we project to remain too high for too long. So we have to do what we have to do, which is currently to hike. We know that we are getting closer, but we also know that the options are not on the table to continue to hike or to hold. And we also know that it will vary meeting by meeting because we want to be determined by the data and the analysis and the assessment that we make of the data. It's not surprising that it is transmitting to the financing channel. It is not surprising that there is less investment in housing. It is not surprising that there is less investment all together. This is the intended impact of the monetary policy stance that we have determined. And this is precisely what will return inflation to target. On the other point, have we promised too much? We have promised in our strategy review and subsequent decision, we have promised to be agreement aligned. And obviously, as noticed by all of you and as noticed by Greenpeace as well, we stopped any asset purchases under the asset purchase program because we are no longer reinvesting. So as a result of that under the CSPP, we are not reinvesting. We are letting that portfolio into runoff mode. What we have decided is that we will stick to our commitment to be Paris agreement aligned. And we will in the course of 23 elaborate the means and the ways by which we will be Paris aligned. I just want to remind you, and this is not to distract me for a second from the fight against inflation, but I just want to call your attention to the massive amount of work that is being done in the supervision arm of the ECB in order to really raise the level of awareness and the level of concern by banks and through banks by their clients. And this is really beginning to work. Thank you. Turning back to WebEx, I would like to give the floor to Tom Fairless of the Wall Street Journal. Tom, please. Thanks so much for taking my question, Madam Lagarde. You say that the burden of proof is in the data for future decisions. What data is it that gives you more confidence that you may be able to pause? And what data will you be looking at, especially over the next months, that would stop those rate hikes? Is it core inflation? Is it more that you're looking at the downturn in the eurozone economy and concerns about that? My second question is about the US and the comparison with the eurozone. The US has raised rates significantly more than the ECB for a similar core inflation rate, or even the eurozone core inflation rate is slightly higher. The IMF, your former employer, said this week that the ECB might have more ground to cover. Do you agree? Do you think that there is a suggestion that there might be more work to do in raising rates? Thanks. Thank you, Mr Fairness, for you two questions. What data will be important? All data. And obviously we will be looking at all inflation data. Starting with headline, we will have two readings before our next September Monetary Policy Governing Council. We will look at core inflation, but we will go deeper into underlying inflation numbers to really appreciate and understand what is moving and what is steadily, sustainably moving. We know that some indicators are showing, particularly in goods, it's pretty obvious. You look at producer prices, it's pretty obvious. But when you start looking into services in particular, you have to really dissect those that are energy intensive, those that are interest intensive, those that are, sorry, interest sensitive, what share of labor there is in various segments of services. So we have to look into all that to understand whether the action that we have determined in the last year and a half now is really having a long standing impact that will take inflation back to 2%. Don't forget also that we will be looking at the inflation outlook that will be produced by the ECB, which will give us, again, the expected path of inflation and the readings for 24 and 25, 25 will obviously be very informative for us. We will look at labor numbers, we will look at investment numbers, we will look at just every single parameter of the whole economy of the euro area that we can look at in order to inform ourselves. I know it would be helpful for those who are using the same models that are playing the same exercise to understand from us that there is one particular specific indicator, one particular set of data that prevails over the other. No, it is not the case. We're looking at all of it and expectations are part of the picture as well. The institution that I used to lead, the IMF, has updated its wheel April numbers and I don't think that any of its terribly talented staff would argue with me that inflation in the United States and inflation in Europe are two different beasts that are fueled by different drivers and in a context that is vastly different both in terms of fiscal, in terms of labor market, in terms of energy. That certainly has something to do with explaining why figures are where they are and you know that better than anyone else. The Fed started a little bit earlier than the ECB and started from a base which was not negative interest rates either. And the next question goes to Isabella Bufacki of Isola. I wanted to add for Tom, you asked do we have more ground to cover? At this point in time I wouldn't say so because as I said the data that we just discussed and the assessment of data will actually tell us whether and how much ground we have to cover in September and at subsequent meetings. And as I said in the early part of this press conference it may vary from one month to the other. Thank you. Isabella please. Sorry. Good afternoon and thank you for the opportunity. President Lagarde I have two questions. One is on this message that you're conveying of the next meeting that it could be a hike or a pause. The next meetings with the next meetings. And so I mean compared to what we had up to now that is 25 basic high coming to the next meeting. Well do you have a message to the European citizens that the fight against inflation you are getting closer to break the back of inflation. I mean there is some good news in this announcement or these comments that you're making. And then the second question if I may is on your assessment of the ECB Euro area bank lending survey because it had some data on demand that was I think never seen the past 20 years. And so whether it was discussed at the governing council and it had an impact also on today's decision. Thank you. Thank you so much. Yes we are making progress and I think the good news for European compatriots is that inflation is declining. You know we had a reading of 10.6 percent in October. We are now at 5.5 percent. But is that enough. Is that good as a result. Not quite. We have a target which is medium term 2 percent sustainable and we want to get there. But I'm sure that you know people when they do their shopping when they do when they fill up the tank when they pay the gas bill they see that the increase is not quite the same as six seven months ago. But are we satisfied. Are we claiming victory. No. We want to go to the end of the game. The bank lending survey. Yes. We looked at it carefully. It was debated with governors and if you look at the monetary policy statement in the last I think it's in the last paragraph that we have. It's it's it's very very significantly described. We say the annual growth rate of lending continued to decrease in June falling to 3 percent for firms and 1.7 percent for households with annualized growth rate of 0 percent completely flat and minus 0.2 percent in the second quarter respectively. So not only are we seeing it very mildly mildly growth year on year but we're seeing a decline in the last quarter and the same holds for both corporates and households. So yes of course this is one of the one of the element of transmission that we have we have seen discussed and the fall in demand is also obvious particularly on the part of corporates a bit less so for households but on the part of corporates absolutely absolutely yeah. Thank you and this brings our press conference to the end. We wish you a very nice and happy summer and see you back here for our next press conference on the 14th of September. Thank you very much.


The sentence is the highest ranking unit of syntax, since there are constructions that can hardly be defined as standard sentences. A distinction is drawn between two sentence types, major sentences and minor sentences. Major sentences have a fully developed subject predicate structure, as in a cat sat on the mat, where a cat is the subject and sat on the mat the predicate. The predicate in turn consists of a verb and an object. Major sentences allow operations such as substitution or transformation. Minor sentences, like Merry Christmas by contrast, behave as fixed units that cannot be rearranged. They often have no fully developed subject predicate structure. They do not allow the substitution of elements even if they are of the same type, and they cannot be transformed into different sentence types. Thus, in syntax, the focus is on major sentences. In present day English, there are four types of major sentences. Declarative sentences or statements, which generally have a subject that precedes the verb. Imperative sentences or commands that typically involve a verb in the base form and often have no overt subject. Interrogative sentences or questions, which either involve subject operator inversion or a WH element as a question word, and exclamatory sentences or exclamations that are introduced by a WH element but do not involve inversion. Note that major sentences can be elliptical, that is, some normally obligatory element of a grammatical sentence may be missing, but can be recovered.


Ladies and gentlemen, the President of the United States and the Secretary of Education, William Bennett. Thank you very much. Thank you very much. Well, good afternoon. It's a pleasure to be with you today as we recognize and honor 287 elementary schools. Of course, education has been a part of our culture from its very beginning. Moses, for example, was known to his people not as a king or as a prophet, but as a teacher. And when I had him, that's what I called him, teacher. Boy, was he strict. If you made a mistake, he'd do that trick where he turned his cane into a snake. All of his students did very well, you can be sure. Well, seriously, though, I can't think of an event that I'd be happier to attend.  Bill, I know you'll be leaving next week, and I want you to know that your legacy is in America more committed than ever to improving the education of our children, and we all thank you. Now, replacing Bill after the Senate gives its okay will be Laurel Cavassos. Both Bill and Laurel are examples of what education can produce. Laurel is a zoologist and physiologist, Bill, a philosopher and a lawyer. They've taught kids, and they've managed institutions of higher learning. And it was their hard work, determination, and the drive they had when their hunger for knowledge was fulfilled that brought them so far. They don't simply speak well about American education. They represent American education at its best as students, teachers, and administrators. But let's talk about your schools, the schools we honor today. They are very important places because they provide the children who attend them a solid foundation for literally everything they will have to know to be good and useful people. William Wordsworth said that, the child is father to the man. It's a profound thought, not as straightforward a statement as it sounds. It means that the experiences we have as children have a great deal of influence over what we become later in life. And when a youngster's early educational experiences are not good ones, if the youngster goes through childhood baffled because nobody answers those basic questions about life and the universe, his thirst for learning will decrease. And when we lose our desire to learn, the world around us begins to shrink. Opportunities shrink and our natural desire for self-improvement deteriorates. Tragically, we see this happen time and time again in schools that seem to serve more often as places that kids go during the day to kill time, rather than places where they learn to go to learn and grow. While the children that attend your 287 schools are in little danger of being unfulfilled in their educations, instead they arrive in school bright with promise and they're met with a concerted effort to make sure they fulfill that promise. And that means principals and teachers taking a firm hand and telling their students, this is what you must know. This is how hard you must work to know it, and your reward is the knowledge itself. The K.R. Smith Elementary School in San Jose, California, it's a school with an ethnically diverse student body of Asians, Hispanics, Blacks, Whites, and Native Americans. And why is this school among the 287, we honor? Well, let's take a look at the expectations Superintendent James F. Smith and Principal Will H. Ector have for the kids and how those kids do. There's an hour a day of mathematics instruction, instruction that emphasizes problem solving skills, and what do the students have to show for it? 80% of the school's students are above average in math proficiency. Students have to take two hours a day of reading and language arts with a curriculum heavy on literature and light on textbook formulas. And what's the result? The scores of K.R. Smith students on the 1987 California survey of basic skills for writing were the highest in the San Jose school district. And that's in a school whose kids are predominantly immigrants or from families that move around a great deal. Kids whose first language isn't English. And how is this possible? Well, Ector put it best when he said that education is our business, excellence is our goal. Excellence is not simply an A grade. Excellence is a philosophy. It says do your best at all times because you owe yourself no less. Well, that kind of philosophy spills over into everything and helps children to develop not only good study habits, but good character as well. The demand for excellence is a booster to the all-important self-esteem that kids must have to sustain hard work and resist the temptations to sloth or self-indulgence that the world and the devil place in their path. It's the best preventive against the scourge of drugs that I can think of because if you give children a sense of the past and the future, they won't be so tempted to obliterate the present with drugs. All American elementary schools can learn from your example. And to make that easier to accomplish, the Department of Education has just released a report called James Madison Elementary School, A Curriculum for American Students. It's a portrait of a model American elementary school complete with a full-scale curriculum in all relevant subjects. And how did Bill Bennett come up with the curriculum we propose in this volume?  You know, some would want to judge America's commitment to education solely by counting the number of federal dollars going into education. All of you here today have proved that excellence cannot be bought. If it could, some of our most troubled school systems would be our best. No, excellence is the product of dedicated people like you. Working hard, relying on a rigorous curriculum, and always demanding the best from your students. People like Nancy Jude, the principal of John Marshall Elementary School in Glendale, California. Facilities there are crowded, and Nancy says we try to work with what we have. What they have are committed teachers and enthusiastic students, and that's what makes Nancy Jude's school one of the 287 that we honor today. All of you have done well by your communities and done credit to your profession.  I don't think any of you are ever going to be characterized by that little story about the little boy who came home with his report card, and when he got back to school the next day, he said to his teacher, you better watch out, because if I don't get better grades my father said he's really going to smack somebody. Well, thank you, and not just for that, but thank you for all you've done, and God bless you all. I got my orders from my boss, he said I better start talking, so I will. Terrible thought just passed through our minds though, all of you here, who's out watching the kids? We hope you left them covered. Mr. President, we have a very special award for you, and to help me present it, I would like to call on Christine Duggar and Andy Suggs, Christina Duggar, excuse me, is from New Orleans, Louisiana, Andy Suggs from Rosswell, Georgia, and I'd like two of them to come up here and help me make this presentation. A collection of photos of some of my happiest moments in this office. They are pictures or photographs of me visiting some of your schools, and the schools that are represented here. I'm going to leave it here for in hope that somebody will bring it to me, I am not a good person.


piecewise functions are those made from pieces of other functions. It's kind of like Dr. Frankenstein making his monster by taking the spleen of a radical function plus the pinky toe of an exponential function plus the liver of, you know, a power function or something like that, right? We piece together and make this monster function. So as piecewise functions are made from pieces of other ones, we have to specify which domain applies to which part, right? So you look at what's on the screen right now. This is a piecewise function with two parts. f of x is given as the square root of x minus four and eight minus two x. Well, when does it look like the radical function square root of x minus four? Well, when x is greater than equal to four. Okay. When does it look like a linear function eight minus two x? Well, that happens when x is less than four. When it comes to a piecewise function, we have to be very specific about the domain. So finding the domain of a piecewise function is relatively straightforward because they're telling you what the domain is. It's like, I wish they always would do that, but they have to tell you that the ambiguity cannot survive for a piecewise function. But questions of the range can be very challenging because when it comes to these pieces, when you look at eight minus two x, this would look like a line that is decreasing with negative two. So it looks something like this. Its domain and range are all real numbers. But when you look at the graph of f over here, its domain is not all real numbers. So how do we pull this apart? Well, you're going to have to look at it by pieces, right? And as you get good at graphing these functions, we can start graphing these pieces, right? So let's start off with how would you graph the square root of x minus four? So in my mind, when you start off with that piece right there, it's like, okay, I'm going to start off with the graph y equals the square root of four, right? Its domain is going to be zero to infinity. Its range is going to be zero to infinity. I know that because it's one of the standard graphs. Then you replace the square root of x with the square root of x minus four, that causes a shift to the right by four units. That'll change the domain to be four to infinity. But a horizontal shift doesn't change the range, so the range would still be zero to infinity. And so if I ever graph that, so when your four, greater than or equal to four, hey, that's the domain of this whole piece, I'm going to grab the whole chunk. And so by taking x to be greater than or equal to four, I'm actually graphing the entire square root function, which you can see right there. So you put that on the screen. So if I were to do this kind of side by side, let's see if I can get a way of doing this. So if I try to graph this thing over here, for which let's say that the y-axis is right here, and the x-axis is right there. And so counting this thing off, one, two, three, four, you're going to get this picture like the square root over there. That's what's happening, okay? So coming back to the main picture over here, what do you get from this other chunk? So this was our square root of x minus four. The other chunk came from 8 minus 2x, when x is less than four. So in this situation, our basic function we could think of is that's just as a line, right? It's just y equals x. What transformations did we do to it? Well, you're going to get 8 minus 2x, you know, that's going to be some type of shifting and reflecting and such. As it's just a line, I might actually prefer instead of thinking of transformations per se, just think of it as like mx plus b, the slope intercept form, where my y-intercept is going to be 8, which although that's off my screen, that would be somewhere up here, right? One, one, two, three, four, five, six, seven, eight, something like this. So that's your y-intercept. And then as it's a line, a line is determined by two points. So if I was graphing this thing, I include the y-intercept, which I did, then what happens when x equals 4, right? If I plug it into the function 8 minus 2 times 4, that will give you 8 minus 8, which is 0. Oh, 4 is the x-intercept of this line that gives you this point right here. And so kind of connecting the dots, you think of this is what your graph looks like right here. And so for our function, because they tell you the domain is less than 4, that becomes the domain of this function, right? So that the domain is just negative infinity up to 4, because that's what we said it has to be. The range, though, can get a little bit tricky. As we think about this thing, well, I'm going, going, going, going, going, going off towards infinity in that direction. So the range of this function, it's going to go off towards infinity. But it goes down until we get to almost 0, right? We don't quite touch 0, because since we don't include the number 4, but up to number 4, we don't actually get 0. So the linear part of this graph right here, its domain will be negative infinity to 4, its range will be 0 to infinity. Then when we look at the square root part of the graph, its domain is going to be, I'm going to put these together here, its domain is going to be 4 to infinity. So if you glue these things together, the domain of f, this is going to look like negative infinity to 4, union 4 to infinity. When you put those things together, that gives you all real numbers, negative infinity to infinity, right? The range, though, is a little bit more interesting, okay? When you take the range, we're going to take 0 to infinity, that's the range for the square root part, and then we union that with 0 to infinity. And so putting those together, 1 goes from 0 to infinity, includes 0, 1 goes from 0 to infinity, it doesn't include 0, we see the range is going to be 0 to infinity, like we saw on the picture right here. In which case, if we were trying to draw our picture right here, we would have then connected the dots with our line, like so. Let's look at another example. This one's a little bit more involved. There are three parts to our picture this time, but we'll just take it step by step by step. In terms of our function, we're going to have our blue part, we're going to have a green part, and we're going to have a yellow part. So I guess yellow is the picture of the graph on the screen, so I'm going to do a red part to help us see this part. So the blue part, when x is less than 1, that's going to be the line x plus 1. So if we deal with that, we look at the function y equals x plus 1, its domain is given as, well, negative infinity to 1, because that's what we're told. What is its range? Well, that we have to think about for a second, it's a line. So if I was to graph this thing, I'd be very interested in, well, what's the y-intercept? I know that the y-intercept is going to be y equals 1. I can see that. I'm also interested in what happens at the point, the switching number. Notice how it switches from one function to the other at 1. What happens at 1? So if you look at that, you look at y equals 1 plus 1, that's equal to 2. So I anticipate the point, the point 1 comma 2 should be on that graph, but notice the domain does not actually include 1. So I wouldn't include the point 1 comma 2. I would actually go up until it. So it's actually open point 1 comma 2. And so then we connect the dots because we get a line. And so we see that the line as we go to the left, since it was an increasing line, it goes all the way down to a negative infinity, that's its range. And then it goes up until 2, although 2 is not included. And that gives me the domain and range of the blue part and the graph. If we move on to the green part, this is a parabola, right? It's the parabola that goes from 1 to 3. And so that's going to be the domain of this thing, y equals x squared minus 3x plus 4. And its domain is specifically given 1 to 3 right here. What happens to this parabola? Well, if I was to graph it, some things I'd do is first, what happens at 1? What happens at the switching number? So if I compute that y equals 1 squared minus 3 times 1 plus 4, you'll notice that adds up to b, 1 minus 3 plus 4, that's going to give me a 2, right? So the point 1 comma 2 is on the graph. Now, 1 is included in the domain here. So this actually tells us that when I look at this point right here, 1 comma 2, I actually can fill in the dot now. And so this is a connection, right? These things touch each other, 1 comma 2 from one side, 1 comma 2 on the other side. This means there's this continuous graph. There's no breaker hole right there. On the other hand, what happens at 3, when x equals 3, you're going to get 3 squared minus 3 times 3 plus 4.  This would give us the point 3 comma 4 on the graph, like so, like so. And as I'm trying to graph a parabola, I probably care about the vertex. Remember, the vertex is going to equal h, that is h is equal negative b over 2a. So this would look like we're going to get a negative, negative 3 over 2 times 1. So we end up with a positive 3 halves or 1.5 if you prefer. And so that's going to be about right here, where the vertex of that thing should lie. Then the k value is just going to be evaluate the function at 3 halves there or 1.5, whichever you prefer. So we're going to get 3 halves, squared minus 3 times 3 halves plus 4. That gives me a 9 over 4 minus a 9 over 2, which I'm going to rewrite that as 4. So that's going to give me 18 force. So we're going to write the 4 as 16 force like here. And so 18 take away the 16 is going to give you a negative 2 force plus the 9 there. That should give us 7 force, which would equal 1.75, which you can see that right here. This is our vertex, 1.5 comma 1.75. And so then connecting those dots together, we get a parabola that looks something like this. In terms of its range, because the range of a parabola is determined by its vertex and its concavity, we see that the range of this thing is going to look like 1.75 going off towards infinity. But we have to kind of stop, right? We don't go all the way to infinity. We're bounded by this point at this point. So we actually only go up to 4 like so. But we get all the numbers between those. So for the last sector, we're going to do another line right here, right? This is a line which notice it as its x intercept 5. That's helpful. Then I'm kind of curious what happens at 3 when you plug in x equals 3 into the function there. You get 5 minus 3, which is equal to 2. So that gives us the point right here, 3 comma 2. Notice this point is not on the graph. We can connect the dots to make the line, which we do right here. But for the red function, we see that our domain is given to us. It's going to be 3 to infinity. But in terms of the range, the range is given us to us by 2 towards negative infinity. I should write the other way around. Negative infinity to 2 since it's decreasing. And so we want to put this all together. When we put all the domains together, the domain is pretty easy. The domain of g here is going to equal all real numbers. Negative infinity to infinity. We can see that listed here. We can go up to 1. We go from 1 to 3. We go to 3. No big deal. The range, though, we have to put this all together, right? What do we get on the range? Well, the blue part, remember, it goes all the way down to negative infinity. So does the red part. So we can go all the way down to negative infinity. There is this gap right here. There's this jump discontinuity at happening at s equals 3. But notice we actually do get all the values between y equals 2 and y equals 4 because the parabola will take care of that as well. So we get everything up to 4, for which then in the end, we would say that the range will be negative infinity to 4. And 4 is included inside of the range. So it can be a little trickier to find the range of a function. But if we can graph it, particularly piecewise functions, we can find the range. And that really is the best strategy to finding the range of a function. Think of the graph.