Good afternoon, everybody. You're very welcome to the Institute of International European Affairs. We're delighted to be joined by Gilles de Kercov, EU counter-terrorism coordinator, whom I will introduce now. Just to remind everybody that to switch your phones to silent if you would, feel free to tweet. The comments at the beginning will be on the record and the Q&A afterwards will be subject to the Chatham House Rule. So we would encourage you to ask questions in due course. So let me introduce the speaker, first of all, who I understand learned English in Ireland some time ago. So we'll let you pass judgement in due course. So Gilles de Kercov is currently the EU's counter-terrorism coordinator position he's held since December 07. This position was established in 2004 after the adoption of the Declaration on Combatting Terrorism. In this capacity, he coordinates the work of the European Union in the field of counter-terrorism, maintains an overview of all the instruments at the Union's disposal, closely monitors the implementing of the EU counter-terrorism strategy. Previously, Mr de Kercov served as the director for justice and home affairs at the EU Council General Secretariat in 1995 to 2007 and served in various capacities in the Belgian government before then, including his Minister of Justice and Minister of Economic Affairs between 93 and 95. He is also currently a professor of European law at the Catholic University of Bluven, the Free University of Brussels and the St Louis University of Brussels and has written numerous books on European law, human rights, security and counter-terrorism. So the floor is yours. Thank you very much. Thanks a lot for the invitation and for the nice introduction. It's not only an honour to be invited by the institute, but it's a real pleasure for me to be back. As you said, I spent more than a month here in Dublin in 1971 to try to learn English, and if my English remains poor, it's not because the teacher was not good, it's just because I'm not... That's so gifted for foreign languages, but I do my best. And the second one is when I started working for the General Secretary of the Council, one of the first challenges we had was to answer a concern by the then Prime Minister, I think it was John Britton, of the murder of a journalist here in Dublin. And we asked him what to do, and we advised the then presidency, Irish presidency, to set up a high level group on organized crime. And that has really been the start of a lot of our work in justice and home affairs. And then I had the chance to see John Britton a lot more when he was chairing one of the working group of the Convention, which drafted the Constitutional Treaty. And I remember that I tried my best to convince him to move justice, what remained in the so-called third pillar into the first pillar, to move security into the community competence at the time. It took me a bit some efforts, but then he was completely convinced, and it changed completely the way we are working. And I will try to illustrate that even if the Lisbon Treaty is a bit... The mini-treaty, which keeps most of the content of the Constitutional Treaty, has made internal security a shared competence between the member states in the European Union. We've always, in recent years, described the role of the EU as being in support of the member states, quite modestly, not taking the lead. We are modestly providing money, adopting legislation, creating legal framework, setting up agencies in support of the member states. And I will try to explain in the 20, 25 minutes I have that in the last three years, we're no longer only in a support capacity. I think we have decided to be much more proactive for a reason, a simple reason that if you ask in the polls to EU citizens where they would like to see more Europe, they always put security, counter-terrorism, border control on top of their concern. One of the most recent Euro-Bahá methods was 82% of the people who are asked where they would like to see more Europe. And that message went through and was well understood by both the President of the Commission, Jean-Claude Juncker, and the President of the European Council, Mr. Tusk. They have decided to launch a new concept, which they call the Security Union, when you look at the monetary union, that is quite ambitious. And the President of the Commission has appointed a full-time commissioner for security, Julian King, and I'm impressed by what we have done the last three years. It's quite impressive in terms of new legislation, new concept with boost or agency. Europe is now really involved, now in CT, counter-terrorism. So there is quite significant change, and I will try to illustrate that. But let me first start by a quick snapshot on the way I see the threat. No need to tell you how much it remains serious. In many member states, they still have a quite high level, even if Belgium reduced from 3 to 2. I think it's still serious in the UK, in France as well. And if you look at the most recent European report, the T-Site report, on 2017, its member states reported a total of 205 failed or completed terrorist attacks, i.e. a 45% increase compared to 2016. 68 people have been killed, 844 injured, 1,000 people arrested and so on. I could continue. It's really important and significant. But it's not just serious. What is worrying is that the threat is much more complex than it was on the day after 9-11. On the day after 9-11, in a way, life was quite easy. We had one enemy, EQ, struck like a multinational company. A few people, they were in the hundreds. No, we have a much more diverse threat. Many more groups with franchises and affiliated groups. Many more people were not in the hundreds anymore. We're in the thousands. This morning, going to the airport, listening to the radio, the chairman of the International Crisis Group was asked to comment on this and said we have to accept, we failed in a way. We have not been very good. We have not reduced the threat of terrorism. It's a bit of collective failure. And we have to ask ourselves, what did we do wrong? So the threat is much more diverse. And if you look at the source of threat we have now, you have homegrown, which we believe is the most serious threat. We have the legacy of the caliphate, the so-called returnies. We have still people detained, Europeans detained in Syria and Iraq. I'll come to that. We have more and more prison leavers, people who have been either radicalized in prison or have been convicted for terrorism and they have served their sentence. And we have, and I want to be clear on that, I don't believe it's a major problem, but some cases of refugees who are getting radicalized and attracted by these ideas. So let me say one or two words on these four or five categories. The first one, the homegrown in recent speech, Andrew Parker, the head of MI5 said, I see here a real shift. It's not just a spike after the four terrorist attacks in London and Manchester in 2017. It's people who are not linked to AQ or Daesh, who are not traveling, who have not traveled to Syria and Iraq. They just get inspired by this ideology and they are self-radicalized. We know the role of internet, we say word, the role of prison, the two major incubator of radicalization, but we have Salafi organization, radical preachers playing a role as well. The challenge here are first to the number. If I take only the three largest member states, the German, the French and the British state, they have identified something like 20,000 radical individuals at several stages of the process of radicalization, from the early stage up to the more violent stage. The challenge for the security services and the police is, of course, early detection. As early as we can detect signs of radicalization, we can provide non-security response. How do we do that? Many people in the room know that much better than I do, but community policing, we discuss at lunch, plays an important role. Train frontliners to detect these signs that may be social workers in schools and so on. Hotlines, when the French Ministers of Interior decided to have a hotline, I said, we'll never work. What surprised me, a lot of mothers called the state because they had exhausted all arguments so that the kids were not travelling to Syria and Iraq, and they were turning to the state. No and no, I understand our security services are even using big data analytics because by processing billions of data, you may discover or detect changing behaviors. So that's a challenge, but even more difficult, as I understand, is not just to detect some signs of radicalization, but more important is to determine what I call the tipping point because being radical is not a crime in itself. What is the problem? It's when someone radicals start preparing an attack. So he downloaded a tutorial on how to build a bomb. He goes to a shop to build the ingredients for building the bomb. And that is the big problem because in most of the recent terrorist attacks in Europe, all services, they had the data. It's not that they had not identified that someone was a radical, but they missed the tipping point. And the last one to criticize, because I know that it is extremely difficult to do that, but that will have some consequences that I will touch later on. And it's interesting when you read the latest version of Contest, the UK city strategy, there is a major shift, conceptual shift, where now they have decided, based on proposal, the Anderson report and so on, that MI5 should start sharing a little bit more the wholly intelligence at local level, up to, I don't know, even the private sector or NGO, because that's the only way to determine the tipping point. I find that quite interesting. So, homegrown. The second one, returnees. I call the legacy of the caliphate. You know, we've had up to 5,000 Europeans who went to Syria and Iraq. 2,500 have not returned yet. We have 1,100 who were killed and 1,500 who have already returned. So the 2,500, it's a lot. Maybe many of them have been killed but were not aware. The Battle of Mosul was very, very violent and there are many bodies that have been buried and they've not been identified. Some are still fighting. There is still a small dash of 2,000 to 3,000 fighters along the Euphrates Valley. Some managed to already leave for another hotspot, Afghanistan, the Sinai, probably other places, Southeast Asia. And so many detain. We believe we have a bit less, I would say around 2,000 fighters, 666 women and 1,400 children. And this is quite a challenge. So again, the challenge is first to spot them when they return. I think we have improved a lot in Europe. Starting with a systematic check on all citizens where they get out of the Schengen zone and they re-enter the Schengen zone before it was purely random. We have done a lot of progress in biometrics, in feeding the different database. I won't go into the detail. But I think we are in a better position to spot those who are returning. The second one is to bring them to justice, which raises the issue of getting access to evidence because you have to prove that someone was fighting alongside Daesh and not with the Free Syrian Army or whatever group. And most of the data, most of the evidence we have are digital. And getting a speedy and easy access to digital evidence is not an easy exercise because most of the data are stored, a lot in Ireland, by the way, but a lot are still stored in California, in a clothe somewhere there. And using MLA requests takes time, a year, in the best case.  It would be jihadists sending a WhatsApp message to another jihadist, an Irish, sending a WhatsApp message to another Irish to plan a terrorist attack in Ireland. If the WhatsApp message is stored in Palo Alto, you will have to abide by the false amendment, which requires what they call probable cause, which is much more demanding than what we know in Ireland we know on the continent. Three, assess dangerousness because when you cannot secure a conviction, you need to monitor the person and you can only monitor those you assess to be the most dangerous one because I'm told that to monitor someone 24-7 takes, what, 20 to 25 staff and at least in my country, I think we can monitor two or three maximum. And five, and we'll come to that, disengage. Whatever the term, it's jihadicalization, resistance, as they say in London now, rehabilitation or reintegration, we'll come to that. So that's the second category. The third one is, and I mentioned that, those who are detained in Syria and Iraq. What do we do with those held by the SDF, the Kurdish fighters in the Kurdish part of Syria? There is a war, it's not an official state, but they hold a series of Europeans and they won't keep them forever. So that's a question. Do we want to bring them back in Europe? Many of them say they are not very keen. All the more, especially when they don't have enough evidence to secure a conviction because they don't want to monitor them 24-7. And those who are detained now are the most dangerous ones. You can think that the first wave, some among the first in the first wave were probably driven by different ideas and fighting against Assad was probably one of the ideas. But the latest contingent, it's probably more hardcore jihadists with bad ideas. So what do we do with this one? And the second category I just alluded to is women and children. Women, our members say, are more and more reluctant and they see women like men because what we have understood from investigation is that more women than we thought were engaged in religious police, in logistics, in support in any forms, on internet and so on, propaganda, were not that nice. We don't trust wives locked in an apartment waiting that the husband was coming back from the battlefield. And children. And children, it is an issue. 1,400 children, it's not peanuts. Many were born there, nearly 600. We have to protect children. They have nothing to do with the caliphate and they have the citizenship of one of our members says. So what do we do? The Convention on the Right of the Children foresees that a child should remain with the mother. It's in the chest of the child.  So this is an issue we'll have to address at some stage. Four categories, the prison levers. Because of lack of evidence in many cases or because they've just got low sentences, many terrorists will have served their sentence in the coming months and will be released. And most likely, they will be very radical. Unless we have developed effective de-radicalization, disengagement program in prison but very few member states have really found the silver bullet here. But on top of those who were convicted for terrorism we have a high number of people who just got radicalized in prison. They were convicted for small crime or petty crime or serious crime and we've seen this process because in prison it's a quite tough environment. Many people just, they are born again. So they convert in a way. They were probably slow, low Muslim, low practice and they discovered that radical Islam helped them in prison. First because you joined the dominant group and that helps to get violence or because it helps you to regain self-esteem. And this very black and white ideology may help you to rebuild yourself. And so if you look at the figures in France the Minister of Justice said that they will have 450 people who will leave prison, 400 convicted for normal crime and 50 for terrorism by end of 2019. Quite a lot. So how will we assess the risk and what to do with someone who has served his or her sentence? Normally they are free. So should we impose additional measures like on sexual offenders, some regular reporting to the police when you want to go abroad, when you want to change your dummy sign or should they have an electronic bracelet or we probably need to invent something. If we believe we have assessed that those leaving prison will be quite dangerous. And finally refugee. I said very, very marginal number but we see some Salafi organization try to recruit in some of the member states refugees because they are vulnerable. You don't integrate in German nor in Belgian society overnight. It's a long and painful process. So that's a bit the variety, the diversity of the source of threat. On top of it we still have the two organizations. What will happen to Daesh? Will they be something like Daesh 2.0? They will have to reinvent themselves. They, I think, still quite some money at the conference hosted by President Macron a month ago in Paris on terrorist financing. The French intelligence service said that they've assessed they were, they collected up to 2 billion but they've probably invested half a billion in the real economy. So they have a flow of regular money be it in farms in Iraq or real estate in Istanbul or whatever but they have money. And when you have money you can do a lot. And they still have franchises and affiliated groups. So they may evolve like Al-Qaeda after the operation in Afghanistan, after 9-11, a weak center and still very effective affiliated groups. We don't know. And they have groups in Libya, in the Sinai, in Afghanistan, in Yemen, in Southeast Asia and still in Nigeria with Boko Haram. That's first. The second one is that we have a bit forgot AQ. And AQ is still around. It's not just because we've seen the rise of Daesh that AQ had all of a sudden disappeared. In Syria, HTS, Ayat-Tahir Al-Sham or a new group called, I don't remember, it's quite complicated. It's a formal franchise of AQ operating in Syria. And they are attracting a lot of fighters disillusioned by the geopolitical development there. AQEP in Yemen, AQ I am called Jinnim in the Sahel, Al-Shabaab in Afghanistan, less than some people suggest, but still active. And I am interested by an ID floated by one of the best experts of Al-Qaeda, Ali Soufan, a former FBI officer who said probably we could see a merge between the two. Why? Because the three reasons why they split when Al-Qaeda in Iraq split into Jabhat al-Uthra and Daesh have nearly disappeared. The first one was an ego problem between Baghdadi and Al-Zawiri. Al-Zawiri is older and older and Baghdadi is still alive. We heard this week that his son has been killed but he's not in good shape to say the least. Second, we have the problem of the caliphate. For Al-Qaeda, the caliphate is the long-term goal. For Daesh, it was a very short-term objective. But there is no caliphate anymore. So it's not an issue anymore. And the third reason is the use of violence to promote and impose your ideas. I think Al-Qaeda learned from their mistake. It's not by beheading people that you will win the hearts and minds. And I think Daesh might go around that process as well. And that's why you see Moe Moe Hamza Ben Laden, the son of Usama Ben Laden, Moe Moe Kuminik on the propaganda. And that would be a good idea for them to go that way. My last, the ultimate point is what I call the Black Swan. How is it possible that when the Americans left Iraq, when Obama decided to withdraw the troops in 2011, AQI was nearly destroyed. And the money they put on the head of the leader of AQI at the time was nearly nothing. Three years later, Baghdadi in Mosul proclaimed the caliphate. And they attracted 40,000 fighters from 100 countries in the world. How is it that in three years time we have seen the rise of the most powerful terrorist organization ever and nobody has seen this? It's like the Black Swan. We have not been that good to see that. It may repeat. That's the problem. Are we prepared to identify much earlier the Black Swan? A point that I will develop quickly later on, linked to the threat in a way is the unfortunate possibility for a lone actor to use high-tech tool to mount an attack with catastrophic consequences. And I don't want to scare you at all. It's probably science fiction for the time being.  Drones. The nightmare of the security service is to see returnees bringing back the expertise they gain in Iraq and Syria with chemical stuff and drones. They use drones. So you put chemical on a drone on a football stadium and don't need to develop more. Cyber. There is nothing for the time being as cyber-terrorism.  By the way, they strange to see that when you put someone on a sanctioned list his bank account is frozen but a person can keep a Facebook account to do crowdfunding and collect money, Bitcoins and so on. That's another. There is nothing like cyber-terrorism but if you have a lot of money it should be possible to go to the suburbs of Moscow or other places and hire a geek or several good guys, expert in cyber or to go in the dark web and buy what is called a zero day and a zero day it's a vulnerability in a software like WannaCry which was detect by the NSA and leaked through WikiLeaks and used by cyber hackers and artificial intelligence and again there is a huge potential for these disruptive technologies and we should use them one idea I have for instance is to use blockchain to design much less expensive more robust ways for the diaspora to send their money back in their home country instead of using Havala or paying the fees of Western Union. So voilà and finally there are other groups than Daesh and the EQ, we have on a terrorist list organization like the PKK which is a very serious concern for the Turks. We put the military branch of the Hezbollah on the terrorist list as well. And it's a question I want to raise here and I raised during the lunch in some of our members says there is a concern on the rise of far right as well and in France they arrested a group of far right extremist last week and they were planning to kill Muslim refugees and so on. And the risk is to see the two feeding each other in a way far right and on the other hand radical Islamist each other. So one word on the response and then in a telegraphic term I know I have to speed up on the challenges. As I said the last three is after the attack Charlie Hebdo in Paris we've really boosted our work and the heads of state and government ask us to work more in three main direction more on the repressive side which means a definition of terrorism, how to improve the collection of evidence especially digital evidence I mean fighting against the financing of terrorism protection of soft targets so that Berlin to lose and so on does not repeat protection of critical infrastructure reducing availability of firearms was still possible two years ago to buy Kalashnikov in the centre of Brussels in an hour and a half for 300 bucks nothing so that's in border control border management more on the repressive side and I would say a word because that's a real challenge how can we in a way understand the process of radicalization if we ever will and what can we do about that can we interrupt this process and find the right policy it's a lot about radicalization in prison it's a lot about internet important for Ireland it's a lot about probably the ideology itself we're sometimes a bit afraid to raise the issue of radical Islam because we don't want to criticize a religion and link up a religion and terrorism but there is a sort of hijacking of Islam for terrorism purposes and we have to address that and develop and mainstream more prevent in a lot of other policies and during the lunch someone said we need a whole of government but a whole of government approach how can we mobilize more education culture sport access to employment and so on and the final basket of policy where we have been asked to do a lot more is on the external side how can we develop very strong contenders in partnership with all our neighbors why concentrate on the neighbors first because they need a lot of assistance and second because it's our first line of defense from Morocco to Turkey including the western Balkans they are all faced with serious problem of radicalization they need to enrich their city policy which is often too much repressive and not integrating enough a whole of society a whole of government approach not enough prevent and we do a lot and so I spend more than half of my time visiting all these countries one by one and trying to see what they need and how we can help maybe that's I know but just in telegraphic terms that could be the Q&A in a way the challenge that I see the first one is just the one I mentioned prevent can we design effective disengagement program for the time being we've not developed enough this we've built on internet we've done a lot through a voluntary approach with internet companies so that they remove unlawful content should we move to a next step which would be a legislation three do we want to address the issue of Salafism in Europe again, sensitive issue but it's on the rise and that is a concern in several of our member states the second one is information sharing we've done a lot in two of the three aspects of what the Americans call an information sharing environment collect more share more but the third leg equally if not more important say analyze better so collect more there we have the issue of privacy and both the European Parliament and the Court of Justice are more and more restrictive on the type of data that we can collect with collected data on passengers the PNR financial data, internet data and so forth but we probably need a lot more but we are confronted with the data protection limit the second one is encryption the internet company after the Snowden leaks have developed very sophisticated encrypted system which makes the life of our security service much more difficult so that's on the collection on the sharing the good news our member states have done a major improvement and are now feeding populating and using the different database much better than they were three years ago so that's a very good news we still have an issue about the link between the security services as a group which is called the CTG and Europol, agency for law enforcement cooperation and if we want to build some links between the two but the third leg is equally if not more important is how to analyze the data that we have collected and we have shared because as I said in most instances the service had the data but they didn't analyze the data properly. President Macron launched the idea of creating an European Intel Academy which I think is a very good idea and the French are now the DGSC the French security service recruiting other profiles ethnologue sociologue, linguist IT specialist because on question as complicated as this one you need to have different angle different sort of analysis and I think we need, all of us need to do that a lot more so that's the third the third one and I stop I take we need to assess better the huge potential of all these disruptive technologies but also use them positively but assess the the interestness for a culture of public private partnership we're not doing enough on that side and finally with all external partners we have a lot of issues Brexit what will the consequence of Brexit be I don't want to discuss it it's in the hands of a very able negotiator the US we've developed a very strong transatlantic relationship but we have a lot of tension in many respect NATO trade and so on will it expand to security and foreign policy as a whole because probably if and I come back to what I said at the outset if we have not been that successful it's probably because we have not done enough on several aspects development solving conflict promoting the rule of law, promoting human rights and so on so I'm sorry I've been much beyond the 25 minutes but I'm more than happy to answer any question you may have


Vijay Lakshmi Trivedi, who is from the Tata Institute of Fundamental Research in Mumbai, India, and she brought numerous contributions to combative algebra and algebraic geometry on topics like Hilbert-Samwell functions, Hilbert-Koz multiplicities, Frobenius splitting, and more recently introduced the concept of Hilbert-Koz basic function. With that being said, Professor Trivedi, please go ahead. Yeah, thanks, Florian, for a kind introduction. And I would like also to thank the organizers for giving me opportunity to give this talk. So this is the first time actually I'm giving a talk on a pen tip, so please bear with me. I'll be a little slow. So now today's talk, I'll be talking about the existence of HK multiplicity. And in fact, the proof is pretty self-contained and I'll be following the proof given by Monsky. So before that, let me start with the hypothesis which I'm going to presume throughout the lecture. So here, R is a local ring with a maximal ideal M and the residue field K. And the emission of R, actually you have seen in various lectures and of course the characteristic of R is prime P, which is positive. So M is a finitely generated R module. Now, this you have seen many times before, but just to be very clear, I'll repeat the notation. So E-theterated Frobenius denoted by F, this is no, sorry, there is no star here right now. It's the map, which is endomorphism given by R goes to R to the power P. And this makes, gives the another modular structure to the R, which is denoted as flora star R. And the model, I recall the elements of this particular ring F flora star E. I'll denote always this for just following the previous talks like this R that means it's an element R in R. But it's denoted F flora star E R.  So this you have seen many times before, but I'll be repeating it. Similarly, there is a model structure on the model denoted as F flora star M. And everybody's familiar with that. Now, another notation which I'm going to use, this is for a given ideal I. Okay, as defined as the set of ideal generated by set of elements X in X to the power Q. Yeah, this is, I'm sorry, this is total messed up. So here it is the set of ideal generated by elements of the type X to the power Q, where X belongs to I. So Q for me, it always denotes like a power of a prime. So let's say P to the power E or P to the power N. So it's very easy to check if I have a generator given by say elements X1 up to XM, then I to the power I Frobenius Q is given by ideal generated by the elements X1 Q XM Q. So this also you have seen earlier. So I'll recall the following theorem of which has been actually recall several times in the past lectures. So, but it's the fundamental theorem which begins this, I suppose the study of characteristic P methods for commutative algebra. So here, your R is a Noetherian ring. So note that I'm not saying it's a local ring, it's a Noetherian ring of characteristic P positive. And then R is regular if and only if F or lower star E of R is a flat R module. So that is for some E you can say for some E greater than one or for all E greater as equivalent. So note that this is so very funny the regularity assumption has nothing to do with characteristic, but it's translated in terms of a Frobenius map. And he's a one is the first person, I suppose, one of the leading ones who realize the importance of. So in this proof, be none of us approved this theorem but he, the ingredient one of the ingredient in this proof is the importance of looking this length function where M is a maximal ideal and relate to the regularity of the ring. So I'm going to state that part of the thing regarding the length function in the next lemma soon. So, before that, let me say make few important remarks. So, one of the thing you have noticed in many of the lectures, you require the assumption that R is F finite, because you'd like F lower star R which is a model here to be a finite R model you would like to basically would like to stick in a category where the all the models are finitely generated or R. So, that assumption was made, of course, many of most of the general things to satisfy that, but in this today's setup actually we can get by without assuming the F fine net assumption, and I'll explain why. So, this is one remark. So the thing is what you do, you take R and replace it by replace by another ring s. So what is the property of this ring is Sir s is complete local. So, advantage of R is like we correct the statement. So of course it is contains a field. So if I take a, for example, completion of our the it will, it will contain the residue field. That's one thing. And of course the maximum I might ideal of R if I extend to s is going to be the maximal ideal for s. That's the hypothesis and you can further you can assume the residue field is algebraically closed. So that you have seen in mass talk also so for example if I take a polynomial ring I completed I take a power series ring and I take the algebraic closure. And of course, taking replacing power series ring over K replacing it by K bar over power series ring over K bar is not exactly you are tensoring K bar over K, but you are further taking completion. But nevertheless, that operation is a faithfully flat extension so R to s is a faithfully flat extension. So this ring R by another ring s which is complete local, and it has a maximum ideal is generated by the same elements and the residue field is all directly closed. So that way you have assume your ring is actually a finite. Okay, wait a minute. So, so, but this doesn't. The English remark are you assuming R is local. My ring is the local unless I say that is. Okay, okay, okay. Okay, because previously in the country is all you do not assume that R is local for the reason I am asking. Yeah, so the, I'll assume my ring to be local. This exactly this hypothesis unless I mentioned something. And this faithfully flat extension maybe regarded as some have a particular case of what is calling burbaki conflemon or something like this right. Can I say, can you repeat it? Yes, in burbaki there is a construction called conflemon in French, which more or less is what you are doing here, which you construct a faithfully flat just in case someone knows what this means. Okay. All right. So here. So, why this is sufficient. Why can you do this so because I'm looking at the length function so here. So let me be careful here, because I can't talk of. So, suppose I take the ideal I, which is M primary so it makes sense to talk of length function. So, now what happens if I take a length of model. M, though I'm writing this elaborate structure for model M I'll reduce everything to the ring thing soon, but look at this length. Now, suppose I have replaced R by S so why so it doesn't length wise it doesn't make a difference. So I'm just writing it here. M mod you take I provenance Q M tensor S or R is a faithfully flat extension so I can take this as M tensor S or R modulo IQ M tensor R or S. I'm writing this is S. So, so basically, instead of looking at this length I'll be competing this length or the S. Okay, here I already have written it. So, so this is actually just now so already in my file. So, suppose if I replace R by S. So what I also have is my field is algebraically closed. So, look at this length function this is so I'm claiming the second one I mean this one. So, this one. So, so I'm saying that this length function F lower star EM and I extended ideally same as length of M mod IQ M. So, why is it so this is this is very important equality for us because this we are going to repeatedly use and So, how do I put this so look at this one so you have a F E lower star M. I dot F lower star EM F lower star ER as an R module and this is a length of as an R mod I can take it R which is same as taking FK module now F lower star EK. case the recipe for remember and this is came. Now, this one. Now, once you have gone into Froben yes so basically this one more step I should write it here. So, this is. Okay, perhaps here I should have changed it. So, this I can. So, here, this one I can write as F lower star E of I Froben is Q M. Right. This one always. So now everything is like a module applied Froben is over it so it is same thing as length or R M IQ M. And this one this is algebraic closed so nothing is happening here the length is one so this so this equality is this what so board work is not very nice here. So, this is. This is now. So, this is going to be my star two. This I'll be looking at this length function certainly reminds all of us the ordinary length ordinary power and corresponding length which is Hilbert Samuel polynomial. So today the aim is to study this length function. This one. Now, so before that we'll assume this property of this length function so which is suppose if you have a local ring and I'm primary ideal finally generated module. So here, D one is a dimension of the model M. That's important. So length of M mod I and M is a multiplicity plus and plus D one minus one choose D one and so on. So it's a polynomial of degree D one and all the coefficients are rational. So when when you look at looking at this length function very much would like to have something like this some property. So, so aim of the talk is to study this. So this one. Hilbert Samuel polynomial is also telling you something that if I look at just the limit of I and M upon and to the power D and take the limit as n tends to infinity. This does exist with EMI upon the factorial. So, so we would like to study the length function this one. But, or at the most can I say something about the limit. So before that we come to that why we should be interested. Let me give a recall the lemma of the quiz. Since he's the one who started this so it's say suppose if I have a local ring now or dimension of the R is D. Length of R of M Q Frobenius Q is greater than Q to the power D. That's the first one. And the second one is says equality holds if and only if your R is regular. So this this can be for all for some e greater than equal to one or for all greater greater than equal to one it's the same. So, for a greater than equal to one. So this is this is the significance of length function key you know the length of length of the arm or Frobenius Q is the least. It implies the ring has a nice property. Okay, so, so I this is of course the second part is harder so we'll just look at the easier part just to get a feeling of the. This Frobenius powers. So, let us see the first one, which is this one will put the second one, the proof of the one. So here, we're looking at the length function. So you assume, again, R is complete. And K is K bar without loss of generality.  If I replace R by R mod P, if at all, if, if I prove that the length function of this for R mod P is greater than equal to QT, then of course it is going to be sold for R also so we can assume R is a domain. So is this reduction clear that I can assume my R is complete local domain. Now, you have this, then what you do, you choose elements xt inside the maximal ideal giving a minimal reduction.  So, I just call this ring to be a, this should be x, I suppose, xt. This ring is contained inside your ring R. Now, I'm claiming rank of F lower star E R as an R module is same as rank of A F lower star E A module. So why is that? So let us see. So you look at the map F lower star E A. So this is injective. Everything is a domain. So Frobenus exists inside the Frobenus.  It has two Frobenus. Now, so this is, everything is, this is finite over this, this is finite over this. So everything is finite over A actually. Okay. So, if I look at this map, and if I look at module like this. So rank of this module as a module via this map is same as this one. Okay. But rank of F lower star E or R or this is same as R over E. Nothing is changing. So that means rank of this should be equal to rank of this. So that's what it is. And this is equal to Q to the power D. That is easy to check because it's a poly power series ring. It has a basis. So F lower star E A has a basis. I think we have seen it before is a F lower star E X1 I1. F lower star E E XT ID where IJ 0 less than equal to IJ is less than Q. So we got rank of this module is equal to QD. However, rank of F lower star E as R is a certainly. I want this mu of F lower star E. So this, the minimal generator of this module as a R module is going to be greater or equal to the rank of F lower star E and R. This is just the minimal number of generator of F lower star ER localizing further at the quotient field. So obviously it has to be dropped further and which we have just now seen is equal to Q to the T. However, now this one is nothing but by NACA, am I using or this thing is F lower star ER. This is R module and which we have put by our star condition is length of R of MQ. So what we have proved that this is greater or equal to Q to the power T. So this is always true. So that was one thing. So and this part that when equality when actually exactly it is equal to QT implies R is regular. It's a harder and I'm not going to prove, but it tells you something just that looking at the length function tells you about the nature of the ring R. So, in fact, you could have taken a little cruder invariant and and this is what actually we are going to do is the length of R of in fact slightly more general form of this modulo Q to the power D where Q is changing to infinity. So he is saying for one Q the this equality is equal to one, then your ring is regular. But what about the limit? Maybe it does have some it tells you something about the ring. And this in fact turns out to be our candidate for the day. I mean, we are going to study this and this limit function in fact was considered by Kunz, but he thought that this limit doesn't exist. So he dropped drop it there. But later Ramon ski again rediscovered and in fact he put this limit exist. And this is what today we are going to follow his proof. And this. So this is our aim of the day is exist. So in fact, we'll prove it for any model and all. And why by the way, why after so many years, once you looked at this, that's another story will come to that later. So let me go to the sums. Let me set up some notation. So this setting used more in number theory, this particular kind of notation. If F and G are two functions. F and G, I hope you can read this one. We say FN is of order G of N. If there is a positive constant says that the norm of F is less than equal to C into norm of GN for all N. So for all N or for all positive, it doesn't matter because we are free to choose our C. So, okay. So we begin towards the theorem of Monsky. So first thing, what can you see first of a length of the function? So this is what we are going to do. So we'll do it full generality like Monsky did. He looked at the module and looked at this length. So recall my M is finitely generated and ring is a Noetheran local ring and I is M primary to make sense of the length. So this function is of order Q to the power dimension of M. This is important. So this is actually not very difficult. So what you do? Suppose mu of I is given by M elements. Then I to the power M Q contains I for Venus Q for all Q. I mean, it's easy to see. Like if I have I given by generator X1 up to XM, then I M Q will be given by Monomials summation X1 I1 X. I'm sorry, my X keeps turning into capital and a small both actually I M where summation of I J is equal to M Q. So all the I J's cannot be less than Q. So one of them has to be greater than or equal to Q. And so in particular it belongs to the Q. So, so of course then length of M upon I Q M is less than equal to length of M upon I M Q which is equal to P of M of I of M Q. Where P M Q I have defined as a Hilbert-Sammel polynomial for the maximum for the year. So here this is. So it's a polynomial of degree equal to the dimension of the module. We all know that. So here. So suppose so this is equal to suppose your PM PM I of X is given by something like C naught extra the power D1 plus C1 extra the power D1 minus one. I'll just write in the simplest form C of D1 remember D1 is a dimension of M. Then choose your C to be maximum of CI where I zero less than I less than D1 multiplied by M is a fixed number. So I can do that to the power D1. So this implies that PM I M Q is less than or equal to C into Q to the power D1. So that proves the lemma. So are there any questions on that? So at least we have some bound on the length of this function in terms of the dimension of the module and that's going to be very important. So all the arguments actually involved here are pretty basic but you had to just arrange them carefully. So this is my notation and in fact I think it's a pretty common. So lemma to be the set of all elements all primes of the spec of R says that dimension of R mod P is dimension of R. Now in the whole game you'll see the only components of R matters are which are which are the maximal components giving the dimension of the ring. Others one don't matter in the competition in this limit. So let me see. So this is the first instance of that. So here suppose you have M and N. So here is a comment that the inclusion should be in the reverse manner. What what what? I think in the previous one. But actually the first line in the proof. One minute. Yeah, tell me. The first line in the is the inclusion.  Yeah, this one. I don't know why I'm getting this. One minute. This one. Just I don't know why is I'm sorry. Hello. And I apologize for this. I'm in my office. I don't know why I'm getting phone call here. I this is correct. I am Q continue. Yeah. That's okay, right? Yes. Otherwise the second line inequality. We will not get. And the next. This one. This one. Second line in the proof. The inequality we will get if the inclusion is I mean the contentment. I don't get. So IQ is a bigger ideal, right? So when I go model a bigger ideal length will decrease. Is that did I answer your question? Okay. Yeah, thanks for the correction. Yeah. So. MNN a finitely generated model. So such that. MP is isomorphous isomorphic to NP. So this is for all P in a spec of R. So I'm sorry. This is for all P in this. So I'm not asserting that MNN have any map between them. But suppose there is a map when I localize at this minimal, this minimal prime of special type MP isomorphic to NP. Then. I can compare their Rubenius length. M minus length of N of IQ M. This is of order. Q to the power D minus one. So this is saying that if on the maximal dimensional component. Modules are same. Then their length differs only by a smaller number which is Q to the power smaller order which is Q to the power D minus one. Okay. So. In fact, this way of reasoning you have seen it before also, but let me repeat it again. So, you take S to be R minus union of P where P belongs to lambda. Then S inverse R has only the primes in its spec of S inverse not only the minimal primes of this. So that means all this ring is an Artean ring S inverse R. And where maximal ideas are S inverse P. So it's a product of RP by Chinese remainder theorem. Okay. Now what is happening? S inverse M I look at. So of course this is going to be product of MP where P is in lambda, which is same as there is an isomorphism for each MP. In fact, into NP P in lambda and which is by definition is same as S inverse of N. So, using this piece of maximal dimensional component, you have you have a isomorphism at least that from S inverse M to S inverse N. And it's an S inverse R linear morphism. But we have seen several times this. That M comma N because it's a because M and and both are finally generated. We can it will come in with localization and this is what is happening. S inverse M S inverse. So you have a map here so you can there exists a map here corresponding map and call that map to be free belonging to home. I don't know why this says that S inverse fee. Now, you look at the map. Now, which is given by this map fee. So this map is telling you that at each prime coming from maximal component, they are isomorphic. So they have no co-carnal on those primes. So, so in particular, the mention of secret dimension of the module C is less than or equal to D minus one. This is okay. This this part. So now, if I look at this one, this function M upon IQ M. So M you go again go from M to M of IQ M by tensoring with R mod IQ. So, so tenser isn't right exact functor. So I have the sequence C upon IQ and C to zero. So this implies N of IQ and this length is less than equal to length of M upon IQ M. Plus now C is of dimension less than or equal to C is a module of dimension less than or equal to D minus one. So it's this length should grow to the order Q to the power dimension C, which is less than or equal to D minus one is for this. Similarly, I can there is nothing special about ordering M and N so I can reverse the argument replace interchanging N and M and I'll get the same thing. So it will give the theorem. Is this okay. I'm presuming that I'm not going very fast because I'm writing it very slowly so it is not possible for me. I think it's just perfect. What did you say. I think it's just perfect. Okay, thanks. Very kind. So here, so this is the one thing now we are approaching the proof into small small steps. So, but each of them are actually some of the very nice arguments. So you have a short exact sequence so. Okay. So here, the length of M of IQ M is actually equal to length of M prime IQ M prime. Plus length of IQ M prime. So actually, I forgot to mention one thing when I mentioned the Hilbert Hilbert Samuel function. So there the one thing was very important in the proof if you remember is he you can consider graded ring the length function because let me just write it here. So, you want to look at the R mod I n for example, for each end so you had a ring structure on this like R mod I direct some I mod I square like this. And this graded drink structure has been used to prove the existence of Hilbert Samuel function if you look at the proof. If I replace n by Frobenius Q Frobenius n, then it is, there is no ring structure because simply here like I n into I m there's a multiplication is actually I n plus M, but I can't say like that IQ into IQ prime is equal to something it is that is not correct. So this is this is a problem. So we are going through this laborious argument. So, okay, so let's come to the proof.  Now suppose R is reduced this is begin with this one. So suppose R is reduced. Then, what is happening. Then S involved, do I need to serve then RP is a field for all P belonging to this thing to for any pre minimal prime. So you have. So when I localize the sequence. This one at P such a P then it is just exact sequence of vector spaces over RP so of course they will split and there will be. So, this kind of isomorphism let me write. Direct sum and be double prime is equal to MP. This happening for all P in lemma. So by previous lemma we have we have the assertion so let me not write it here so we have this thing. Okay, now suppose this is this is very nice argument actually are is not reduced here you are using that. And there is an endomorphism given by Frobenius. This is this is beautiful. So, suppose R is not reduced, then it has a nil radical and not call it to be a nil radical of R. So, so there should exist you can always choose a q naught large enough say P to the power e not such that and not to the power. I mean q naught is zero. On the other hand, if I have any module, and I look at its e not F e not M this one and multiply this by this. So it is going to be F lower star e of n naught of q naught into M. Right. That's the multiplication but and not q naught is zero. So this is zero. So this is telling me if I take e not. Maybe you need the q zero between brackets right. And when when you write the axis q zero such that and zero raised to the q zero a zero maybe is q zero between brackets, in order to use afterwards. No, no, no, no, no, no, no. No ordinary power. Okay, thank you. But, but maybe you're right. I can do this also Yeah. Yeah, sure. Yeah. Thanks. Yeah, I can choose this right. So, you agree with that right. So, yeah sure. So this is like, yeah, thanks. Yeah. This is telling you if I take E0 iterated Frobenius then your module F lower star E0 annihilated by N0. So it's an R mod N0 module. I mean F E0 lower star M is now you had a short exact sequence between modules. So remember your Frobenius is exact functor actually. I mean you're not doing anything here. So it's a F lower star E0 R module. It's in particular it is R module in particular since it's annihilated by N0 it's R mod N0 module. So it's an exact sequence of R modules. Hence R mod N0 module because N0 is annihilating all of them. I thought I read. Here I have written actually already this I'm going very slow in fact. So here actually I have already written to save the time. So here so you have this exact sequence. I'm writing it as N prime N and N double prime. So it's a sequence of R mod N0 module. So that means R mod N0 is a reduced ring. So by the previous assertion when we dealt with the case R is reduced you can always write length of this thing as kind of additive for the radius case. So now but this module this module is same as if I open it. So perhaps I should write it here. So here because so this module is a F lower star E of M iq dot F lower star E M. So when it goes here F lower star E M into F lower star E of iq into q0. So you got the length function here. So this is same as the length of this here. Now so this is this M mod q q0 M is equal to this into this plus or qd minus one. But remember q q0 d minus one and then this same as order of qd minus one. So in fact you're getting for all q large enough. So for all q greater than equal to q0. So you have the additivity of the length function. So okay is it okay this one. So now we more or less prove the result about the existence of hk multiplicity for the ring. So here I'm assuming RMK no ith and local but I'm also adding the assumption it's a complete domain. I'll just explain in a moment why am I suddenly also complete domain and dimension of r is d that's important. Then there exists a constant which I so my i is always m primary if I don't say anything. So half r i I said it's a greater than one. This actually already I have proved it. If you look carefully I'll just explain again such that here it is. So first time you see that something like a Hilbert-Samuel function you have obtained not quite a polynomial but up to the lower order. So that's the first thing you assume. Okay so we'll give a proof here. So I say assume k equal to k bar. So here why didn't I assume that it's a complete domain.  So I start with case when r is a complete domain. It's important for me. So now we have already seen and that's the first lemma we have seen. F lower star of r as a rank of this thing is actually p to the power d. In fact using the power series argument what we had proved was the thing if you remember is actually p to the power e to the power d actually is q to the power d in fact we had assumed. So this we know. Now r is a domain so what we have is that means so many copies of r of pd are sitting inside F lower star of r. I suppose it's okay right because I am saying if I have S inverse localized the quotient will I just say it properly is isomorphic to F lower star r tensor S inverse r or r.  So then you can choose a last S and some element in the S such that there is an injective map from here to F lower star a we are choosing appropriate S here. So you have a map like this. So here you are using as a domain. Okay so this is isomorphic at the generic at the minimal prime. So if I look at its a co kernel so dimension of C is less than equal to d minus one. So that means now additivity of the length function says p to the power b length of r mod iq minus length of F lower star of r iq my floor is star. This is important is of order qt d minus one. Okay so now multiply the whole thing with the p to the power d q to the power d. So what you get is a length of r of iq into q to the power d minus so this length these are all over r of course when I don't say anything it means over r. So this is this length was actually if I can change it right here it's actually r of iqp because you have taken one probinus it will get multiplied. So here you have one upon qp d length of r mod iqp so one upon q. So if you look at the survey article of you know he said that there are four basic facts about why the probinus works. So one of the thing he said was summation one upon p and converges that is and this was actually very useful for me several times I have used it and other thing he said the map r to F lower star of r essentially is the map you take e of r to F lower star e plus one r. So the way one looks at it you want to compute some comparison with this or this or so you better see the at this level and see what amplification it has when you take the morphobinus. So so here we have c actually that was the obstruction and this is the amplification we have so so we have this one now. So what you do now q just denotes pn actually. Yeah so this I'll denote a sequence of numbers cn or to be maybe I should not talk like that it's very confusing so cn where n belongs to n I just define cn p to the power nd equal to length of r mod iq which is iq is pn actually. So n I'm not saying as cn is a constant it very much depends on q I'm right now I'm saying so if I look at this way so this this star one is telling me just that there is a set of numbers real numbers of course in fact rational number here cn plus one is less than equal to some constant which is here coming c0 into one upon pn d something pn. So this is a convergent sequence so this defines limit if I look at length of r mod iq upon qd q tends to infinity is same as limit and tends to infinity cn which converges so this limit is called e hk of ri. So rewriting everything from the equation you just telling me that length of r iq is equal to e of hk of for this new constant q to the power d plus of order q to the power d minus one. So for a complete local domain we we have already found the thing the limit exist assertion. It might be a good place to stop here because I think we're really over time. Oh I'm so sorry. So what is it? It's over the time. My understanding is that the lectures are 50 minutes but I'm okay okay then I'll stop here actually. Yeah so maybe next time I'll take up this result lemma. Apologize I completely forgot. Oh no it was a good place to start anyway. So let's see if there are any questions for Professor Trivedi. All right I hear no questions I don't see any questions in the chat either so thank you very much Professor Trivedi. Okay yeah thanks so and we'll meet again next Wednesday on the regular schedule for the remaining two lectures. Yeah yeah okay yeah have a good day everybody. Bye.


Hey guys, it's Nathan, aka ThePhotoCount, and welcome back to the channel. Another week of team this season brings us Bundis League of Tots, and it is here today in FIFA 23. What I want to do in today's video is get us ready for a big time promo Friday after a big week of Prem. We are now looking forward to Bundis League of Tots with some pretty big names leaked, and of course, those stats on the loading screen now showing that kind of like we hope some of these Bundis League of Tots cards might be really juice. So I want to talk about that, how the market might move today, what kind of content we could see, and of course, yesterday, all of the craziness with the level 20, 84 times 25, and or whatever you chose, the 1 of 3 90 plus e-fixed Tots, that whole situation yesterday with objectives in general, is making this game kind of go crazy, and it did a lot of things that we did not expect, especially with prices on the market. So we're going to talk about all of that today, and more if you're excited for the video, drop a thumbs up on it, and of course, subscribe if you are new. Now let's talk about yesterday really quickly to start this video off. First of all, we had two SBCs yesterday, the first one being Marquis matchups, which was expected, nothing crazy there. Of course, XP is involved because it ties to the objectives here in the objective section, and EA dropped basically what we would call a double XP weekend, kind of like if you played Call of Duty back in the day, double XP weekend was a fun time to get on and level up, right? Same thing here, every single XP boost in here is a double, right? Instead of 1250 or 2000, it's now 2500 and 4000 XP for each of these objectives. As you can see, I went through and I grinded out some stuff yesterday on stream so that I could get all the way to level 20. And that was the biggest thing yesterday on this game is you could get the XP to get to level 20 and claim either Death's Eye, the 84 times 25, or most importantly, and all of a sudden, very popular, the one of three 90 plus e-figs, tots or tots moments player pick. And this is what really did the game in yesterday in so many areas. We did not expect this to happen because we didn't expect the double XP boost, but as you can see inside, if you're right, it's really easy to get XP. I'm sure a lot of you guys are leveling up super fast. You can do all of the objectives right now except for the tots swap 5 and the Bundesliga premium upgrade. If you play enough games to get the win eight, the play 15, a lot of people are going into friendlies with each other and quitting out to kind of boost, if you will. That's what I did just to get the objective done really fast. The foot moments is working. It says something about the Bundesliga moments here, but that's just releasing today. It's not actually requiring you to play in that Bundesliga mode. So that's why a lot of people are talking about this player pick, man. And this player pick was one of the two things that absolutely went crazy on this game yesterday. Since there are so few tots cards that are available from it, right? We've only got tots cards from the community and the Premier League team this season that are available in that player pick. You have got a ton of really good cards that are popping out of that player pick with really, really high chances. So if you are questioning that player pick, it I think is a really, really good value one to do because a lot of the popular cards we are seeing from that are Alexis Sanchez. We're seeing Rodrigo. We're seeing a lot of, you know, Hyunmin's son, Tot's moments card. I actually got insanely lucky in doing mine and I packed Veran Tot's moments yesterday out of that player pick. It was a very big W from that. I mean, there is a little bit of risk, right? Because you're turning in a one of three player pick where you could get 25, 84 plus rated players. That's a lot of SBC crafting that could take place there as well. So I got very lucky. A lot of people are packing Son, Rodrigo and Alexis Sanchez and that's even impacting the market in its own right as a lot of those cards are going down in price and everybody is going and doing that player pick. If you want to check out a video, we did do a bunch of those yesterday. So I'll post that video link right up here. We open a bunch of those. If you kind of want to see what that's about and make sure kind of I guess to see if it's worth it for you, you can go check it out there. But I will say this, if you're going to open that player pick, be very careful because if you wait to open it until after content today on Friday, the weight is going to be completely different because Bundesliga cards will be available in that player pick and it may not make it as OP as it has been. So that's something to kind of watch out for today. But that was something that really went crazy on the game yesterday. And then also yesterday, we had an absolutely incredible value. La Porte SBC 221,000 coins for this SBC is insane value for the links that he provides for the stats that he has. He's pretty comparable to Saliba, maybe not quite on the level of Saliba's card. But considering Saliba was 350k yesterday and this card is 224,000 coins, a very easy craft Manchester City links. Basically right now in this game, everybody's running prime teams too because of the permanent league team this season being very cheap, good value and what we've all been kind of grinding and working towards this last week. He fits right into so many people's teams. Centerback prices on this game are down horrendous and I would be very careful buying centerbacks from any league right now or any part of the market because this card is taking all of the hype up right now in this game. It's very similar to what we saw about a month ago when the Benucci SBC and the Carlos Alberto SBC dropped on this game. When centerback SBCs drop a good value, you can just tell what it does to the market. It absolutely makes the market go crazy. So be very careful buying centerbacks, but this was much cheaper than expected. It was a really good card and that of course is why you see the market going down as much as it is especially on those guys like Ruben Diaz, Saliba and Ake. Will centerback prices rebound? I don't think they're going to rebound that well. I'll be very careful buying any centerback card today on this game because La Porte is still the brand new card that is the most hype and also the new cards that are coming out today in Buenos Aires, they might come onto the market today with cheaper prices because the hype is all around this La Porte SBC. So we'll talk about the market a lot more later in the video but that's kind of a roundup for the content that happened yesterday on this game. Let's talk about today, right? When this league of team of the season is dropping on this game today and it's a different Friday, right? The first two weeks of team of the season are always really exciting because you've got community and you've got prem and there's a lot of saved up packs for it. Now today begins the next couple of weeks of team of the season where yes, there is hype for these major leagues and the big boost juice cards that are going to come out but it's not quite the same as the first two weeks and that's where we're relying on EA to juice these cards up a little bit and with the loading screen that was updated yesterday, same three players that were on there from Wednesday but EA showed us the stats and some of the kind of, I guess you could say nationalities and positions of these guys, right? We knew that was Alfonso Davies guessing by the left back, 99 pace and 90 passing with five stress skill moves. That's going to be a cracked card. This is of course, Sadio Mane with 96 pace and 90 shooting which is a solid upgrade over his flashback SBC by the way and then Enkunku with 97 pace and 97 dribbling. Yes, once again Enkunku is not an SBC. He is going to be a card that has impacts. This was an incorrect original leak but I want to look through the leaked cards because there are a lot more official leaks as we are heading into today on Friday. Enkunku going to be a card that has impacts today. We think Royce is going to be an SBC. That's the only SBC leak that we have and that could be a really big part of today's content because EA is going to have to release content today that people actually want to take part with because Prem Tots was such a really good week and we just came off of all this grinding for the objectives like Metoma. These guaranteed Tots packs where we have a lot of brand new Tots cards that we want to try out and maybe use in Weekend League. How is EA going to get us to change our teams today? I think an SBC would be the first part they could do that. I think Royce might be our SBC player today and if it's good value with good ratings and maybe a five star skill move boost on that card you might see some people changing over with some German and some Dortmund links. So we'll watch those for sure on the market today. But moving through some more leaks, of course, we have Goretzka moments. We've talked about that. Sadio Mane from the loading screen. We've talked about that. And then here are some newer ones. Davies from the loading screen. Not surprising there. Of course, he's going to have 90 passing and 99 pace as shown there. And then some Bayern cards. Leroy Sane is getting into team of the season. I mean, again, these are voted on, guys, but based on the Bundesliga vote that was put out to the community. So this is not necessarily who deserves it. It's more voting based. But Sane is getting a card today. We are getting a Mussiala card as expected and as leaked before and we are also getting delict. So a lot of Bayern links today. A lot of Dortmund links today. That's expected. There's a lot of cards from those two teams. So definitely watch that part of the market today. But those are nice cards. We're really excited for those. I think they're going to do a really good boost on Mussiala and delict. I think delict might be like the Marquis centerback in this team of season squad, which is probably deserved. And also that's going to be a really, really overpowered card in this game. But again, like we said, maybe pretty cheap because of how the centerback market is right now on this game. Now moving up and forward, we've got another centerback with Schlatterbeck. I think he's going to be pretty cheap. Last year his card was pretty cheap as well. But that could be a very, very nice card there. We've got Bellingham, right? We don't know. These are all predicted stats. Again, this is one of the most looked for cards today. Just people so curious about are they going to upgrade him over the team of the year? Is it going to stay the same rating as the team of the year or maybe be below? We'll have to see on the Jude Bellingham today. But again, these are all predicted stats. These are not final. So we'll probably get some confirmed stats later on today. Maybe some leaks right before content like we had last week. Julian Brandt is going to be a card in this team as well. This one to me is probably going to be one of the cheaper cards. I would imagine the cheapest player or one of the cheapest players in the teams. And then Kobel is going to be a card that we're all going to be packing just like Ramesdale this last week. But he's going to have some nice Dortmund links that will be very sought after as well as a lot of Dortmund cards in this team. Now, some other ones that are very exciting, full crew, very deserved. Definitely think this guy deserved the team this season. Probably one of the cheaper cards in this team as well. Hopefully making pretty fun to try out and pretty usable. Fran Pong with the 99 Pace. A lot of people are going to be looking after this card today because of the link to Diaby, right? And a lot of us did the Diaby SBC. And that's going to be a card that a lot of people will be using in their starting 11 in this next week potentially during Bundesliga team in the season.  He is getting a card as well. Diaby is the lead to be in team of the season. He's going to get an even further boosted card above his SBC card that's been out now for a while for road to the final. But I'm really curious to see if this is going to be sort of the situation like we had with the Alexis Sanchez card where he had a live card, of course, but they didn't make the team of the season like anything super spectacular. They just did a couple of upgrades. They didn't upgrade a weak foot or a skill move. I'd be very curious to see if they upgrade a weak foot or skill move on Diaby because if they do, that would almost make it seem like the road to the final card would not have as much like hype because that one's not upgraded yet. And of course, they're still trying to progress for the upgrades, right? So I think they might keep this Diaby the same and not upgrade his weak foot or skills. That's just my take, but we'll have to see. But that's a hype card nonetheless. And I'm really excited. I'm sure a lot of you guys are as well. For Colomani, I hope they give him a really big boost today. He needs a big boost in some of those special like stats like the reactions and the composure from his special cards this year. But that could be a really nice linkable card with the France and the Bundesliga links and he could be pretty meta as well. So hopefully they give him a really big juice there. Those are all of the leaks that we have right now for Bundesliga. And we do have a third moments player, right? We've got Goretzka. We have got Mane and we've got Le Cois another center back and a French center back a kind of FIFA legend from last year in FIFA. He is getting a Tots moments card as well. It's giving me kind of like the Varan vibes and I'm so curious to see how expensive he is on the game today compared to like Schlotterbeck and compared to Delict. He might be more expensive than all those guys just because of the links and of course the height that he has as a player in FIFA. Now, I will say once again, these stats are predicted and none of these stats are final just yet. So watch out for that. But again, we look over those players. It looks like it could be a pretty fun squad of players today. We don't know who the objective is going to be as of right now. I do not believe there's a specific objective leak, but Royce is that SBC that we might be seeing today. Keep an eye on your Dortmund players. I've also seen some leaks for like Rafael Guerrero. That could be another card that's in the team. That might not be a full list of all the players that are going to be in the team today. There could be a few other ones of course as well. So just watch out for that team that's going to be dropped today in this game. Now price-wise for these team of the seasons, the last two weeks, we've seen team of the season prices start very low and then go up a lot in a couple hours after the content drop. I don't think that's going to be the case today. I think these cards are going to start out more expensive and then drop lower as we head throughout the hours into tonight on Friday into Friday night, maybe into Saturday morning as well, just because there's not as many saved up packs, right? And that's the biggest thing about the market today is there's not as many people going to go out on the market today and want to buy these TOTS cards like there was last week with Premier League TOTS specifically and two weeks ago with community team of the season. There's going to be hype today for sure, but it's not going to be quite the same because the first two weeks are always the most type of team of the season. So watch out for these cards in the first couple of hours because there still could be some fluctuations with cards that maybe start a little bit too cheap and there are some saved up packs today, but it's more like maybe me and you guys have the same situation where I've got a couple of Bundesliga-specific packs that I'm going to be saving for today, but I'm not really saving like any of my rewards from the last week that are tradable packs or packs from multiple weeks beforehand of team of the season that I've been saving for TOTS. There's a lot less of that today than there have been for the past two weeks. So I don't think they're going to start cheap. I think they'll start more expensive and then kind of come down at price, but it could make them more rare today. There'll still be lightning rounds. There will still be packs in the store and there's some pack code we have to look at in a second, but I think that's going to make the cards a little bit more rare today.  Today is day 22. We are all going to be getting an 85x5 pack and also a TOTS player pack from Insati here, which I believe this TOTS player pack, I think it might contain any team that's using a player. So including like EFL, Air2Visit, Primarily Community, and Bundesliga.  Don't expect anything massive, but a 585 pack could bring some unshakable Bundesliga TOTS cards into our clubs today. It could be a good chance to pack one. And then again, that could be maybe where EA tried to get as hyped about trying a Bundesliga TOTS player or two.  And I've got a couple Bundesliga-specific packs, but really, there's not many packs we can save for today, right? All the upgrades that we have are Premier League-specific at the moment. So that really is not a thing. Like we could do last week. We can't really save the 81-plus player picks because there aren't any in this game at the moment. So that's kind of what's all coming out there. I want to talk about this pack code really quickly as we're kind of on the topic of packs. There are some big packs that have been added to the code. First of all, Bundesliga-specific, we've got all the store packs. Right? We got like the Foundations pack.  They've got the Rewards packs that were added right, the Bundesliga TOTS, the Bundesliga-slash-EFL. That is what is in Rewards this weekend, guys. If you're playing Weekend League, you are playing for Bundesliga and EFL TOTS cards in Reds and also the Community TOTS Player Pack. If you guys are awaiting compensation, which a lot of you are, this is the pack that EA is going to be giving out compensation through. So that might be happening today. It might be happening sometime during the weekend. So watch out for that. But a lot of Rewards packs that are inside of here, some store packs like the Foundations, as we mentioned. One that I'm really interested about, two of them, actually. The first one is 83-plus Bundesliga Player Pick. It's a one-of-three player pick, but it's tradable. And they went out of their way to signify that with tradable in brackets there with all players guaranteed to be from the Bundesliga in 83 or higher. See pack probabilities for full details. With a description like that, that's a pack that's going to be in the store. But a player pick pack that is going to be tradable in the store. I'm so curious to see how EA value that in terms of FIFA points and in terms of coins unless that description is going to be updated. Watch out for that today because that could be very interesting. Not that it would impact the market a ton unless it's super-juiced for tots. But that's going to be really curious to see that there. And then, of course, this 90-plus pack. We're going to be looking at this into the weekend. This most likely is our next Icon Upgrade SBC, 90-plus Prime Team of the Year foot-birthday or Trophy Titans. It's actually, it sounds cool, but it's not that amazing of an SBC. It's not that different from the 90-plus that we've had the past couple of weeks. It just adds in the Trophy Titans, but you still have all those Prime Icons that were the most popular to pack from that SBC. And yeah, we're going to be talking about that later in this weekend for sure. So watch out for the store packs today. There's still going to be lots of lightning rounds and a lot of supply there. But again, just maybe not as much hype as there was for the Premier League. Now, let's talk about the market, guys, because with that being said, is there going to be a bump up in prices today or is this market just going to stay mudded? Kind of like it was yesterday. Now, really, the content yesterday that made the market not really rise as much as I thought it was going to was number one, the La Porte SBC being such a great card and super duper cracked. And number two, I really believe, guys, that that eFix player pick is hurting so much of the market right now because people are going and doing that right away. It's all about grinding that and getting that done. And there's not as much demand for buying teams, weekend league teams and stuff like that. And the cards that are moving up the most on the market right now, take a look at Team of the Year Courtois and all of your Real Madrid links with Rodrigo being one of the most popularly-packed cards from that 90-plus eFix player pick. That's kind of why everybody's doing it right now, right? So you have the chance to pack a card like him, Alexis Sanchez, Shunman Sun. Real Madrid links on this game are absolutely flying. He has 390,000 coins. Remember, he was down at 300k flat after we had the Alisson SBC. Look at this, yesterday at content, he was 315k. He went right to 350 and now he's 390,000 coins all because of that player pick, right? You might wonder, why is Rodrigo, Varan, Alexis Sanchez and Sun at the top of the foot been most popular? It's because those are the guys that are getting packed the most out of that player pick and everybody's doing it because they know that it's juiced, seemingly almost glitched. Although I don't think there's anything wrong with it. It's just there's so few players that are in the pool. The chews from and you know, the same players kind of pop up over and over, but since that pool is so small, those are the players that are popping up over and over and over and people are buying players to link up with these guys in their teams. That's where the market's doing the best right now. The market is not doing very good in terms of the prem tots, right? I still am invested in some of his cards. I still have the Salah, but he's down because if think about it, if you had Salah, but you're going and doing the player pick for Rodrigo, you would rather use the Rodrigo that you just packed instead of a 2.7 million coin Salah card, right? That's why he's down and you've got a lot of the rest of the market down besides center backs, just because honestly guys, I think everybody is going and focusing on that and they're trying to link up guys that they maybe have packed from this player pick. So that's why the market is really in kind of shambles right now. I was expecting a quieter content day yesterday and I do still believe that there is some price rises that are going to happen. I still have my Salah like I mentioned. I still have my Bukayo Sakas. I sold one for a lazy sale yesterday at 660,000 coins. Saka is still, that was an undercut right there. He's right around 615,000, 620,000 coins. So he's not down that bad. But yesterday after content, all of these Premier Tots cards definitely dropped in value, but they just didn't rebound, right? And I was expecting a little bit of a drop, but I was also expecting a quieter day of content yesterday and a rebound on prices. But that didn't happen because of all the craziness with the player pick and with the La Porte SBC being such great value. So these prices are still kind of down. Now, am I going to sell all my cards today? I'm definitely going to hope for a market rise after content because I still think there's a lot of hype for these Premier Tots cards because people want to try these out. They're still brand new to them. They want to play weakening with them. They want to try them out and weakening starting today. There's going to be a lot of people buying these cards in my opinion still. So not all of them are going to go up today, but I don't see a big crash happening on this game. Like we saw last week when these cards dropped because, I mean, these cards are in the game now. People opened their packs. They packed all these guys and now they want to use them this weekend. I think their prices on most cards will still do okay throughout this weekend and you'll still see some cards go back up in price. So I still believe in this market in that aspect a little bit. And let's be honest, the fact that some of these cards are still like so cheap like Bruno Gamerash. I mean, he's stayed pretty stable right around the $192,000 coin range, which I know some of you guys invested in him there. Like this is a card that I still think can go up this weekend just because of how great value that he is. So I wouldn't say all hope is lost on all these investments. I mean, the market is still very, very low. Again, like I said, I have a lot of these investments. There's a lot more, I think potential for a rise here than there is for a fall in price. Now, there could be prices that do fall today for sure, but overall with weakening demand, people wanting to buy cards for teams and especially Premier League being the hype that is kind of still in this game today. I expect the cards to do, card prices to do okay. I also think that based on today's SBC content, if we do get Royce, Royce is a big time FIFA name. There could be some Dortmund links like Jude Bellingham's older special cards. I have one of them from earlier on in the week. I'm hoping that he can rise up today if the SBC is really, really good for Royce. Other Dortmund links could be rising today and other links other than that, like just Bundesliga links in general. You saw I have some open Makatos. I have some fantasy foot Ricken, right? Some of the lower tier cards from the Bundesliga, people might be spending some coins on here in the next couple of days, especially if we do get a Bundesliga cup. I think that's the key where the low tier cards once again is we need that cup mode so that people have to go buy Bundesliga cards and use those in their teams. So we're going to be closely watching out for that today because Bundesliga cards should be in a little bit more demand today as well. Now, one of the last things I want to talk about in today's video is a SBC that might be dropping before content today and that is a Player of the Month, if I can find it, Player of the Month, Erling Holland. Yeah, there it is, Player of the Month, Erling Holland. He is leaked as the Player of the Month winner. This SBC should be dropping early today and it should be dropping on this game and it will be live. I think there might be another Player of the Month today as well. Sometimes they drop two Player of the Months on Fridays. Sometimes it's just Premier League. But watch out for an Erling Holland POTM. That will be even more hype for Manchester City players and Premier League players in squads on the game today. Just be very careful with Holland's Man of the Match card. Holland's ones to watch because those cards could take a bit of a hit in value if there is an Erling Holland SBC. Of course, again, even with the SBC we had yesterday, a lot of people might go do a Holland Player of the Month because it is just fitting right into their teams right now at this time with Premier League kind of being the hype. So the market today is going to be a very iffy place. We're going to be covering it all on stream, but it's going to be a fun day of Bundesliga team this season, hope they juice these cards up with the five-star skill moves and weak foots being consistent among a lot of these cards. And we're going to maybe try to see if we can pack a last-minute Premier League Tots card from a couple of these upgraded packs before they go out of packs early today. So if you want to drop by the stream today, link is down below in the description, but it's going to be a fun day of content on FIFA Day. So if you're excited for it, drop a thumbs up, comment down below if you have any questions and subscribe if you're new. I'll see you guys in the stream today. It's been Nathan with Accountant. Catch you guys there. Peace out.


Mark, with your background, I have to ask you, why do you want to be a CEO? Yeah, to be honest, it wasn't top of my list after what I've done previously being a CEO of a junior company. But the real simple answer for this is that our opportunity, wedge mounts opportunity in Texas is the lowest risk, highest upside opportunity I've probably seen in my 25 years in capital markets and around the oil and gas sector. And I think everyone at Investor Intel in the audience should be hearing what we're talking about here. So you've taken on this role. You're the visionary of the company. You have this amazing opportunity. What's happening right now? What are you prioritizing? Yeah, two things we're prioritizing right now is number one, adding additional assets to the portfolio and number two, optimizing and exploiting and deploying capital on the assets that we have acquired to date. Those are the two overriding strategies and the best way that we think we can create value for shareholders currently. So if people look at what we've done today, we've closed two transactions, Willowbend and Millikin. And again, these opportunities are distressed opportunities that we've taken advantage in buying. They're either distressed operationally or distressed financially and this is the model that we're rolling out across central Texas. So those two assets, we acquired them and last week we did a complete field operations program there where we treated 16 of the 18 wells there and we'll be turning those wells on later this week and we'll have production update numbers for investors at that time. The second point I just made again is about additional opportunities, the focus. We have a pipeline of additional acquisitions we can make in central Texas. Again, same focus. We're looking for distressed assets financially or operationally where we can add capital and add our technical expertise in which to organically grow production at incredibly low rates. So I'm going to ask you the next obvious question is what makes you so good at picking the right distressed projects? Great question. It's really about our team's relationships on the ground. So relationships with industry people, relationships with vendors and those relationships are very difficult to, well, it's not like you can just come in and establish these relationships. These are like 20-year relationships with these types of individuals. So because of those we get an idea of deal flow and because of the technical expertise of our team we can prioritize as deal flow comes to us which has the best low hanging fruit upside for wedge mount to take on. So relationships and technical understanding of what we're looking at. Okay. So you've just closed your Willa Bend Permian Basin asset acquisition with Byron King as you pointed out did an excellent job at that interview. I urge others to go take a look at that and now you're really, you've just closed on your Millican project. Is that correct? Correct, yes. Okay. So you're moving fairly swiftly here with your timeline. What should we as shareholders be looking forward to in the next quarter or two? Yeah. The next piece of news is a production update from the company. As I just mentioned we were doing a big field program. It's actually our biggest field program to date on both assets. That was last week in Texas. We'll have some updated production numbers which I think again we should have those in about a week should be pretty exciting for us after a big program. We should be able to bump up production again as we did earlier in January. The second thing, stay tuned for more deal flow as again I said our pipeline is pretty full right now. So I would expect wedge mount has a good chance to close at least two more acquisitions by the end of 2Q 2023. And for those of you interested in finding out more please go to the Wedgemont website and I urge you to take a look at their market capital market structure. Thank you Mark so much for joining us today. Thanks Tracy. Have a great day. Appreciate it.