Ladies and gentlemen, welcome. This is now the beginning of the Huawei Young Talents program launch ceremony. I am Claire Lenz, Director of Communications and Development at IHES, and on behalf of the Institute, I thank you for attending the ceremony. The format chosen for this online event is that of the webinar, meaning that for all of you attending, you can only see the panelists, but not the rest of the audience. And you cannot speak directly during the ceremony. That being said, feel free to post comments at any moment using the chat button down your screen. And you can ask questions using the Q&A button. And we will try to address those questions as we go. In particular, you will be able to use this Q&A button for the scientific talks if needed. Without further ado, we will now start the ceremony with Mr. Marwan Laoud, Chairman of IHES. Thank you very much, Claire. Mr. Zhou Xiang, Vice President, Executive of Huawei European Research Institute. Mr. Wei Liang Xu, CEO of Huawei Technologies France. Mr. Cheng Bai, President of Paris Research Center. Mr. Yu Ming Su, Minister Counselor of the Scientific Department of the Chinese Embassy in France. Mr. Jutang Roo, Second Secretary at the Chinese Embassy in France. Mr. Xavier Asfeld, Science and Technology Attaché at the French Consulate, who are all watching this event today. Ladies and gentlemen, dear colleagues and friends, it's a pleasure for me to introduce a ceremony that marks the kickoff of the Huawei Young Talents Program at the Institute. Even though, as you may imagine, I would of course have preferred to welcome us all at the Institute. At the time when the global pandemic has brought much uncertainty and made it more challenging to plan ahead, Huawei's generous support to IHES is ever more precious as it allows the Institute to take the long view and make plans to sustain a thriving scientific activity. Indeed, the Institute's model relies on the support of private and institutional partners following the vision of its founder, Leon Mochan, who was himself a businessman. Let me take a moment to tell you more about the Institute's very unique model and the vision of research that it carries. IHES was created in 1958, 1958, and it rapidly moved to Bure-sur-Hivette, now at the heart of the Paris-Saclay University, of which the Institute is a founding member. IHES offers exceptional scientists a place where they can devote entirely to research without any obligation of teaching or administrative tasks. The scientific activity of IHES revolves around a very limited number of permanent professors recruited for their exceptional skills and talents. And those are not just words. Out of 11 permanent professors in mathematics since the creation of IHES, seven have been awarded the Fields Medal. Those professors attract some of the very best international scientists to the Institute. Every year, about 200 invited professors, guests from across the world, benefit from our visiting program. This year, the number of visiting researchers has decreased due to the sanitary crisis. But fortunately, the collaborations and partnerships that the Institute has with the scientific community worldwide are very strong and stay active despite the limited physical interactions. The scientific activity at IHES also consists of regular conferences, seminars, workshops, and a yearly summer school. Those events have also been strongly impacted by the pandemic. But the researchers are testing new forms to present and discuss their work. The summer school, for example, was held entirely online, which allowed the number of people enrolled to be multiplied by almost a factor four. Clearly, the relationships between participants during those digital events cannot compare to those built on site through informal conversations when sharing a coffee or discovering Paris together. But at least they allow the scientists to stay involved with their community and cultivate their network. Indeed, the Institute's unique model relies on its excellent scientists, but also on its exceptional environment, both at the local and global scale. IHES is at the heart of a scientific hub with top-level institutions close by. I mentioned it earlier. The Institute is a founding member of Paris-Saclet University, officially created in January 2020. And most of you have probably seen that it made its appearance in Shanghai Xiaotong Academic Ranking of World Universities for the first time this year with impressive results. It is ranked four teens globally for all disciplines. And it is number one worldwide in mathematics. The Institute is very proud of this achievement to which it has contributed. Above all, this ranking attests of the important international recognition of IHES and its surrounding scientific community. This is actually at the heart of the Institute's identity. IHES scope has always been international. As every day, the Institute brings together scientists from across the world as part of its mission. Internationality also characterizes IHES financial model. The founder of the Institute solicited the support of individuals and companies abroad as well as in France. About 30 years ago, the then director of the Institute, Jean-Pierre Bourguignon, who's with us today, who is probably watching, had the visionary intuition to diversify the sources of funding, even more by developing a fundraising model that relies more and more on philanthropy. It's also Jean-Pierre Bourguignon who's transcended the connections between IHES and Asia and more particularly China. Thanks to his networking efforts, the Institute enlarged its international reach and benefited from the contacts that allowed the initial connections with Huawei. I will now let Emmanuel tell you more. Sorry, he will tell you later more about the history of the relationship between Huawei and IHES. In the meantime, we are lucky to have with us one of the key representatives of Huawei to whom I would like to extend my gratitude for their support for our model. And it is my pleasure to introduce Mr. Zeychang Xiang, Vice President, Executive of Huawei European Research Institute. Good morning. I'm Mr. Maui Lai-Hung and Mr. Emmanuel Ulmer and Mr. Xavier Asif and Mr. Sun Yumin, Mr. Ru Zitao and all the guests who attended the conference online. Good morning, everyone. It is a great honor to attend the launch ceremony of the IHES Huawei Young Talent program today. First of all, on behalf of Huawei European Research Institute, I would like to express my warm congratulations to both parties on our cooperation as scheduled. Fundamental research is a large part of the ICT industry guiding the direction of the scientific on the technological progress on the providing theoretical fundamental foundations. In 1948, Dr. Shannon put forward the famous Shannon Norr in the book Mathematical Principles for the Communication. Shannon Norr has become the basic theory of modern information theory and has been widely used in the field of communication and data storage. And has laid a solid theoretical foundation for the development of the communication today. 70 years later, the development of the communication industry still depends on classical mathematical theories. As Shannon Norr approaches the limit, what is the basic theory of post-Sharon area is waiting to be exploited? This theory may lead us to technical innovation in the next 50 years or more and take him civilization to a new height. Huawei is dedicated to technical innovation in the ICT field and willing to bring digital to every person, home and organization for a fully connected intelligent world. Basic or scientific research is a large house for ICT development. Therefore, Huawei attaches great importance to basic or scientific research, especially investment in basic mathematical. We have set up a mathematical research institute in Russia and France with the aim of strengthening basic research, especially the research on mathematical algorithm. In August of the line 2020, Huawei launched the Lagrange Mathematical Computing Research Center in Paris. The aim is to cooperate with France's mathematical professors and experts to provide an open platform to explore new theoretical breakthroughs in the computing field. France has a long history of mathematicians and the generations of mathematicians and the physicists have emerged that have influenced the progress of human civilization. In the contemporary area, a number of outstanding young people will imagine France to make a far-reaching contribution. We hope to support this talent through the IHS, Huawei Young Talent Program. Let them have a better environment to focus on research under development of their talent. Huawei has long-term cooperation with IHS. In the past, IHS academic spirit has impressed us. As a member of the University of Paris, IHS has been dedicated to basic mathematical and physical research for a long time and has made an important contribution to the scientific progress of mankind. It has attracted the scholars for over 60 countries under the category of the many major field winners. We are deeply impressed by the long-term dedication and the achievement of the IHS in the field of basic science. At this moment, the IHS, Huawei Young Talent Program is launched. I hope that the cooperation between the two parties can be further dependent. We are willing to share our insights on the forecast, on the future of the ICT field with the academic, and we are also willing to listen to the new ideas from those young talent. I hope that young researchers can develop themselves to the research in the world's most outstanding mathematical research towards the IHS. Under Light and Lighthouse for the future of scientific on the technical development of the mankind. Finally, I would like to appreciate the directors of the IHS and the directors of the Paris Secondary for your strong support for our cooperation. I hope that COVID pandemic will be over as soon as possible. I'm looking forward to having face-to-face communication with all of you. Thank you. Thank you everyone. So thanks a lot for your very nice words. Mr. Shizang Chang, Vice President of Executive of Huawei European Research Institute. Mr. Wei Liangxi, CEO of Huawei Technologies France. Mr. Sheng Bai, President of Paris Research Center. Mr. Yu Minxu, Minister Counselor of the Scientific Department of the Chinese Embassy in France. Mr. Sitao Roo, Second Secretary at the Chinese Embassy in France. Mr. Xavier Hasvedel, Science and Technology Attaché at the France Consulate. Ladies and gentlemen, dear colleagues and friends. It's a real pleasure to hold this ceremony. Although I would have obviously preferred to see you in person rather than do this online. That being said, this event is very important for us as we acknowledge today the renewed support given to the Institute by Huawei Technologies France. Our gratitude to Huawei is multifold. As you have believed in our research model and have transferred your support to IHES consistently over the years. Let me take the time to retrace the main steps that have been taken so far since the first gift by Huawei. This first gift was made in 2010 under the direction of François Contain. That first donation marked the creation of the Huawei Fund at IHES to finance our visiting program. The first tone of a very successful partnership was led. Already at the end of 2014, Huawei Technologies France turned from donor to major partner at IHES by making a generous gift of 1 million euros which incremented the Huawei Fund between 2014 and 2017. During that period of time, we at IHES had the chance to get to know the R&D team at Huawei on the occasion of our yearly workshops that I had the privilege and the pleasure to co-organize with Merwin Débar. I know that Merwin is probably watching this ceremony and I would like to salute and thank him. The four editions of the Huawei IHES workshop that took place between 2015 and 2018 were a great success and a wonderful occasion for both IHES researchers and Huawei R&D teams to exchange and discuss about topics in pure mathematics and possible applications in technologies. As we know so well at IHES, seminars as well as informal discussion can carry treasures. That's what happened when Laurent Lafogue accepted to present his work to Huawei Vision Forum in the summer of 2017. I will let Laurent tell you more about how his relationship with Huawei developed and led to another chance of collaboration between IHES and Huawei. First through a two-year project on toposis that just ended and then through the creation in 2019 of the Huawei share in algebraic geometry whose first order is Laurent Lafogue himself, thanks to a generous gift of one million. Huawei Technologies France has recently pledged another six million euros to support the Institute over 10 years. Part of this generous gift, more exactly one million euro, will extend the Huawei share in algebraic geometry. The remaining five million euro will finance the Huawei Young Talent program which aims to support the work of talented researchers in mathematics and theoretical physics at the beginning of their career. Every year, the Huawei Young Talent program will fund an average, on average seven post-doctoral fellowship that will be awarded by the Institute Scientific Council only on the basis of the scientific excellence. The fellows will collaborate with the Institute's permanent professors and work on topic of their interest. Five brilliant young researchers have already joined the Huawei Talent programs starting in September of this year. And I have now the pleasure to introduce them to you. They are Yading, a mathematician working on complex algebraic geometry and hot theory. Vasilisa Nikiforova, a theoretical physicist interested in classical gravity and cosmology. Se Zun, whose research is focused on higher titular theory. Katie Vox, who works on geometric group theory and low-dimensional topology. And Yue Wang, who is currently working on application of mathematics to developmental biology and applied stochastic processes. Three of them, namely Yue, Se and Vasilisa will give a glimpse of their work later on this morning. The Huawei Young Talents program also give IHS the possibility to create a prestigious five-year position reserved to a particularly gifted young researcher. Through this competitive post, the first of this kind at IHS, the Institute intends to reward exceptional talent and attract in France outstanding young researchers that might otherwise begin their careers elsewhere. Early stage researchers constitute an invaluable resource to IHS. Their dynamism and diverse research experience inject new life into the Institute's research activity. Through the Young Talent program at IHS, Huawei endorses once again IHS research model by standing on the side of free and disinterested research and taking part in the development of fundamental research in France. Furthermore, it is important to say that our partnership with Huawei has also been made stronger by regular interaction and conversations about our research model and the importance of independent science. We have thus been very happy to see the creation of a new Lagrange Center in downtown Paris and to attend to its inaugurations last month. We are very proud to have inspired some of its features, such as the space dedicated for tea and coffee break, an essential custom to encourage exchanges and informal conversation. Above all, the vision of this new center, as well as its mission, resonates very closely with that of our Institute and proves once again that we share some important values with Huawei. Those shared values and the mutual trust we have built over the past decade make for a very strong relationships between our two institutions. And indeed, the Institute's relationship with Huawei began 10 years ago and this new agreement announcing another 10 years of cooperation, at least. This confirms the durability and strength of our collaboration. We are looking forward to this new decade of partnership. We wish all the best to the new Lagrange Center and we would like to congratulate Mr. Sheng Bei as well as Mr. Xi Zhengxiang and Mr. Wei Lingxi for their strategic vision, their belief in long-term fundamental research and their coordination of such projects. We would also like to thank them for their renewed support as well as all those watching today whose role has been essential in this collaboration, including William Xu, Director of the Board, President of the Institute of Strategic Research and also François Quentin, Benjamin Kouloumer, Mehrouane Débat, Sheng Li, Jean-Claude Belfure and Wang Xu-Sheng, among others. It's a pleasure and an honor to now leave the floor to Mr. Wei Lingxi, CEO of Huawei Technologies, France. Thanks. Thank you. Thank you, Mr. Rumeau. Does everybody hear me? Thank you very much. Yes. Mr. Marwan Laou, Mr. Xavier Asper, Mr. Sheng, Mr. Rumeau, Mr. Rumeau, thank you very much. It's a great honor to attend this KTOC ceremony, meeting for the signature and partnership between ICS and Huawei Technologies in France. This year is not easy, I think. A lot of people say that there are two things that will reshape the world. One is the coronavirus and another one is the American election. But I think it will not stop our cooperation. And even it will accelerate the digitization. So I think at the meantime, that we have a very challenging situation in front of us. But at the meantime, at the meantime, that they give us the opportunity to held the meeting like this. Because I see from the screen that we come from everywhere, from Paris, from Shanghai, from Luxembourg, and many, many places. So I think this is the best proof evidence to show that the digitization is necessary. So on behalf of Huawei France, firstly, I would like to thank you all for this welcome. And congratulations everyone to put a successful event and help building a closer collaboration strategy. Continuous and strong investment in R&D is very important of Huawei's strategy. It's not dependent on the political issue. Every year, Huawei is committed to invest more than 10%. And the last year, it's arrived 15% of his sales revenue, not profit, but the sales revenue in R&D. That became the major model of Huawei to increase our advantage and innovation in front of the customers. Since 2014, Huawei has invested 200 million euros in R&D in the French, only country in the French. For example, in 2019, there were over 96,000 R&D employees and scientific accounting around almost 50% of the whole company workforce. Huawei has been operating in France since 2003. So this year is our 17 years. Up to now, we have more than 1,000 employees directly and 12,000 employees indirectly. And we have more than 300 supplier partners working in the Huawei ecosystem. So this is a very important part because our strategy in Europe and in France is not just do the business, but we won't put our footprint in the France market. That means put the French company in Huawei ecosystem. And this 300 supplier that every year, for example, 2019, we expense almost 1 billion US dollars to buy the French technology, French services in Huawei global market. So 2020 has presented many challenges, as I said at the beginning, both in our personal and professional lives, even though we faced a more complex external environment this year. So Huawei remains side to side with French organization for investing in new research. Our local R&D teams has grown continuous over the years and we just inaugurate the new R&D center. As Mr. Wu just said, in France, that shows another evidence that we continue to invest our R&D strategy in France. Lagrange mathematical computing center with now 300 employees in the R&D in Huawei France, we have fully confidence in the future of French innovation and the open environment. Hovered by this country. We are willing to further create our R&D investment and further strengthen the implementation of our strategy in France. And for France. So Huawei France is driven by local needs and serves the interests of Korea, such as Orange, SFR, Buiga and FRIP, as well as the enterprise customer in the GAC count. We are also planning to build a 5G factory in France end to end. The reason we select France to build this 5G factory is because we want to also share the risk of our logistic provisionment because this is today important in front of the political issue. And we believe also a strong engineering power capability in France. Huawei hopes to attack the local talent, serve local customer and contribute to digitalizing France and Europe. Mathematica is the foundation of the ICT industry and provide a theoretical basis for research and innovation. Maybe you may not believe that the 5G technology people talk a lot is created also by the part of the contribution by our French R&D teams. So France is a world leader in the field and has many world-renowned scientific research institute and many world-class mathematicians. IHES has been committed to fundamental theoretical research in mathematics and physics for a long time and is the best representative of French research institute. Huawei has a long-term partnership with IHES sponsoring several IHES visiting scholars, jointly holding workshops and conducting fundamental theory exploration project with Professor Laurent Laforte through the Young Talent program with IHES. We hope to continue to build a long-term partnership with IHES and the French Academy supporting fundamental science research and innovation aiding in their effort to further contribute to the progress of nature science. Last but not least, I would like to extend my best wishes for this cooperation between our two organizations and I'm looking forward to the great and continuous contribution in a field of fundamental research in IHES. Thank you, thank you for your time. Thank you very much.


Now, can you tell me how difficult has it been for you to get around? That's hard to say. That's real bad. Real bad? How bad? Have you been able to walk? No, no. I might have walked some, but you can't walk all the time. So what do you end up doing? So you will surely end up on the floor. On the floor. And she has calluses on her.


So, let's recall the setup from yesterday, no, not yesterday, Wednesday, so we got G or QP a reductive group and at some point I was assumed for simplicity that it's quasi-split. Also, I fixed some algebraically closed base field. Work in a geometric setting. I was considering the stack bun G. I defined on the category of perfectoid spaces over K, taking any test object S to the group point of G torsors on XS, where this was a relative fact from 10-curve. And so, as they had last time a theorem to the effect that in the suitable interpretation of these words, this is whose R-team stack of dimension zero. Okay, and so then I fixed my coefficients. So, coefficients were OE mod L to the N, where E over Q are also finite extension. Of course, L is not P. And so, we had a vergeduality function from this category of chiefs of lambda modules on bun G to itself. Can't believe that Konsevich asked this question how it governs. So, it naps it into the dualizing complex of bun G. And then we had the definition that F is reflexive if F naps isomorphically to its W-reduced rule. And this was our substitute for the notion of constructible chiefs on the sky, where the usual notion of constructibility would not be the good thing to consider. And the main theorem I stated last time, which still had a star because it still depends on one conjecture, says the following that you can completely understand when a chief is reflexive. So, if and only if for all points, and as I said last time, points on the stack are classified by Kottwitz's set B of G of isocrystals and all comological degrees I. So, this B sum of corresponds to a geometric point XB bar to bun G for such guys and all integers. If I look at the stock of this complex really at XB bar, then it's a complex so I can look at all the homology groups. These are all representations of the corresponding automorphism group of this B, which is this well at least of G's quarter splits it's this inner form of the levy of G, GB of QP if this isn't admissible representation. Okay, and so today I want to explain the proof of this theorem. And so the proof is by by induction on bun G, which is somehow bounded by some fixed B. Okay, so you start proving a similar result just on the same unstable locus and then you go further and further out into the non-same unstable locus, one stratum at a time. And so, okay, the question is whether there is some special case of a more general theorem applying to more general sex and I don't think so. I think it's really something very special for bun G. Yeah, so I proved the same theorem for all open subsets of bun G. And what about things which are in some sense atarable bun G? Things which are atarable bun G. So for things which are very close to bun G, it might still be okay, but okay, I can't tell. Certainly the proof makes use of very special properties of bun G. So you start on the same stable locus. And so what I've already said last time inclusively is that it is following theorem, which for GLN is 2, 2, 3, and U. For general G, 2 to 5, it says that if you look at bun G and then you look at the same stable locus, then this decomposes into a disjoint union overall basic B of a classifying space for the corresponding inner form of G. And so we have to understand what this D add of such a guy is. And for this, let me state the following proposition. So in fact, it holds in the following generality that if you have H in some GLN QP, close subgroup, so for example G of QP for any linear group G or an open subgroup thereof or whatever, then you can try to understand what this D add of the point mod H is. And so the proposition is that the D add of the point mod H to coefficients in lambda is really just the derived category of smooth H representations on lambda modules. The RER was late. Smooth H representations on lambda modules. And under this equivalence and with a fixed choice of high measure, the idea duality gets identified with smooth duality and which takes any pi in absolute to the homomorphisms from pi into lambda, but then you take the smooth part of this and then this passes to the derived category. I mean, this sounds very reasonable, but actually verifying this is a little bit is some trouble, but okay, you can do it. In particular, the fact that you really get the smooth part here only is related to some finiteness results for the homology of smooth diamonds. So there is something hidden, some work hidden in the statement. Also it looks very reasonable. Not true that the obedient categories are identified. Which obedient categories? I mean, this is not a priori the derived category of an obedient category, although somehow posterior it is here because here and we're not in the setup where, so if this was some diamond, some locally spatial diamond, then this n-test some pharmacological dimension, then this d-ad would automatically be the derived category of etals sheaves, but here we're not in the setup. In particular, sheaves on this are somehow, oh, you have to first find a smooth cover of this guy by a diamond and so on. I mean, a priori unraveling what this really is, this is a bit of a trouble. Anyway, once you've checked this, you find that in particular, if you try to understand what reflexive sheaves are, then those are the ones where passing through the double dual or double smooth dual is an isomorphism. So this reflexive, if and only if for i and z, I locate h i of f at a somewhat geometric point, which then becomes a representation of h. That's because it can translate to the same question about smooth representations. And this factor is exact, so understanding the radiality, you can understand it in each degree individually. And then the condition that the double smooth dual is the same thing, means that on all the fixed factors under open subgroups, it must be finite dimensional, it must be reflexive London module, which means it's finite. There's no restriction on degrees. There's no restriction on degrees, right? Okay, so that's good. What do I want to say next? So that somehow deals with the same unstable locus. Okay, so now for, so, so finishes the same stable case. So now we need to, so for the induction, let's fix some b and b of g, not same as stable, not basic. And so then we have this inclusion from some b bun g, which is less than b, into bun g, which is less than equal to b. And that may also already give a name to the complementary closed. We have some i bun g, b, meaning equal to b. And so we assume that the theorem is known, I mean really the obvious variant of the theorem, known for bun g less than b. Okay, so, so what do I have to do? So we need to prove the following results. One is a finiteness result, which says the following. So assume you have some f, which lives on bun g less than b, which is reflexive. Then if I take, push this forward to the whole guy and then restrict to the new closed stratum, so then I get a sheaf on bun g equal to b, and we need to know that this is still reflexive. And you know that it is a tau, in d at tau, by general single experiment. No, because all functions are by definition functions on d at tau. So you modify them to be d at tau if they are. Right, for non-partic compact immersions, I said that I define this rj lower star to be the edge joint of j upper star. So rj lower star, if I did it on v sheafs, would not lie in the d at tau, but I just reflected back into the d at tau. They're using the right adjoint to the inclusion. This means it's complicated to calculate it. This means it's complicated to calculate it, yeah. But you must do it this way. But still commutes with smooth space change. And the second thing you need to prove is a duality result. So again, if f lies on this d at tau, by general single experiment. Okay, formally the statements look very similar now, but I will try to explain that the first thing really is a final result, whereas the second thing is some kind of local duality result. That will become, I think, clearer when I go into the proofs. So why is this what we want to know? So does this imply why does this give the induction step? Well, you already understand reflexive shoes on the two strata. So we understand reflexive shoes on the strata, bungee less than b by induction. And this bungee equal to b is this point moduloose group, which are these kind of automorphisms of eb, where this was an extension of this periodic group gb of qp by a connected group. And so you can prove the usual thing that connected groups don't actually interfere with the analytic sheaf theory. So actually the d at tau on this bungee equal to b lambda is equivalent to d at tau of just the point mod qb of qp lambda. And so this means that on the stratum you again can apply the results it says that these are given by smooth representation. Do you mean the connected group has no homology or something? Right, the connected group has no homology. I thought yeah, I connected kind of some kind of unipotent group here. Yeah, so it has no homology. All right. You don't use the semi-direct water structure, you just need an extension. Do I need the semi-direct product structure? I would think I just need an extension. Okay, I have a semi-direct product in case I need it. Okay. Okay, so so we understand on the stratum. So then say for the theorem we need to prove two directions. So which direction should I do first? So assume first that say g in d at tau bungee less or equal to b lambda has admissible stalks in the sense of theorem. And then we have a triangle like so and this is reflexive by induction and then the whole thing is reflexive by two. This is reflexive by the case of bungee equal to b. And then the whole thing is reflexive as i is proper. It's a closed immersion. And we have seen that inclusions by proper maps preserve reflexivity. And so extensions of reflexive sheaves are reflexive and so we see that g is reflexive. And so conversely assume that g is reflexive. Okay, so what do we want to see? We want to see it has good stalks. Right. And so you do the same thing. So the left thing, so okay, so same exact sequence implies that this thing is reflexive, the same triangle. But then because this is a closed immersion, this actually implies that i opposite of g is reflexive. I mean, so on the open part, we already know that it must have good stalks, admissible stalks. And so this means the admissible stalk. So for the argument of finiteness didn't quite appear, but it will be a step in getting to the duality. But so because of this identification of reflexive sheaves here again with admissible representations of the Jb of Qp, this statement about this reflexivity is really just saying that something is an implicit representation. So in this sense, it's really a finiteness result. Whereas this reflexivity on this stack here is a much more geometric statement. It's much more like an actual reality statement. And so single the wrong page here. So we want to do these things. And for this we use a chart. So to compute anything, charts, mb, which was an mb tilde, and so where mb, so b, so I said this somewhere, but let me now make say it explicitly again. I assume that g is quasi for simplicity. So then b is actually the image of some bm and bm basic, where m is contained in p, it's contained in g, so this is a levy. And this is a parabolic encoding some slope filtration. And what is mb tilde that classifies p-torsors e plus an isomorphism of the reduction to m with the bundle corresponding to this basic element. Then this maps wire the gb of qp, which is the same thing as the mbm qp torsor to mb, which is a set of p-torsors. e such that e times pm is fiber-wise isomorphic, this point-wise isomorphic. And then this maps to bun g e while sending e to the pushout. And the conjecture, which is this conjecture that explains this star, its main theorem is that this map from mb to bun g is equal to b, which is open in bun g is l-chromologically smooth. And we can prove this for geoland. And mb itself is also a smooth outing stick as a consequence. And because the map is smooth and bun g itself is smooth, this guy needs to be smooth. But this is actually unconjectural, so you can directly show that this is smooth. Okay, so now we have to find diagrams that this maps to bun g is equal to b. There's some j tilde. Let's call this j prime here actually. The pre-mature I will denote by mc, mb-circ, here you have i, bun gb. Let's call this i prime. What you get here is a fiber is actually just a point modulo gb of qp. So effectively what you're doing over this stratum is you fix the splitting of the hard aneurysma infiltration. And then you get this as the automorphism group. And you also have on top of this a similar diagram. You can call this point xb. It's just a copy of spark k. The open star two is always on the left and the closed part to the right. Is that conflicting with standard usage? Yeah, this is the open part, this is the closed part. I mean I use i and j in the usual way I think. All right, so we have this picture. And so let's start with the finiteness. So by using the conjecture smooth space change plus some unraveling of the formalism. Plus, why is this part of the unraveling of the formalism, that this mb tilde is some kind of strictly Hanzillion space. So it behaves a little bit like power series over an algebraically closed field. What you can prove is the following, that if you look at the sections on point mod k where I say k and gb of qp is an open propy subgroup. Propy just for simplicity essentially, but it makes things slightly easier. So I mean I fix here my f and d et al bun g less than d lambda and I should put the pullback under pi b here. So if I pull it back under pi b here I get a shift here and then whether I do push forward here and pull back here or upstairs this doesn't matter. And so I want to understand what happens if I push forward here and pull back here and then I want to know that this is an admissible representation here. So what I want to know is that these kinds of global sections if I go to the point mod k that this is finite in each degree and you can compute this just as a cohomology of this open part mod k. So we call that mb tilde over mb was this torsion in this group so you can in particular divide by this group. That's still some kind of stacky object anyway. With coefficients where this is the reflexive. A different way of writing the same thing is actually to write the cohomology of this guy. This calls pi b tilde if the star takes a k variance. Where pi b tilde is a composite. So this description is better because here we still have a reflexive sheath. Here we're pulling back too far this is not reflexive anymore. But what's nice about this formula is that it tells you that it's actually if you do this thing you push forward and you pull back and you know that what you get is just some smooth representation of the script jb of qp. The smooth representation is just this thing here because we've identified the k invariance for any k with the k invariance here. So what you want to prove really is that this is that this cohomology here is an admissible representation. What is the thing with the not smooth just you write the conjecture that the map is well okay mate well pi b is smooth yeah not pi b tilde. Ah so the the torso is not smooth. Yeah so pro pro project quotients on the profite group are not smooth because profite sets the fibers are not smooth. Profite sets are not smooth. There was this other thing that if you have something smooth and then you divide then the quotient is still smooth over the base but that's a different statement. Okay so the torsos and the profite groups are not. No okay so you this gives you a way of computing what you want to compute and so now we need to prove so we need to show that this is finite in each common logic degree and so this uh there are two two ingredients. So by the way the when you write k acting on this r gamma there is required some foundation so in general if a group acts on the top it's a non-descript group acting on the torsos so you need some to know that it takes the other definition. Okay so I will use two things well actually no I claim that in each common logic degree the map is an injection this I can just say and the image will always be just the invariance. Forget about any topology of this group acting on this set I mean I think it's just and is it also a coherent couple so everything inside is okay let me say what these objects are. So the open part of this mb tilde is actually a quasi-compact separated spatial well spatial implies quasi-compact but diamond whereas this quotient here would again be a stack in this sense this is nicer here and so this means that you can compute commonology also finite dimensional compute commonology as a direct limit of Czech commonologies over finite over covers or it's our covers it's finitely many if it was not quasi-compact then to do this you would have to use infinitely many things and then if you write down the Czech complex there's some infinite products and they can become very big but if you use this information carefully okay so there's this thing that f might be unbounded but actually for the statement you can reduce to the case where f is bounded because you know by induction that canonical truncations preserve reflexivity by induction canonical truncations preserve reflexivity because the condition of being reflexive is a conditional commonology sheaves and so this means and each individual commonology group of the sky by some finite dimensionality only depends on some bounded piece of the complex so you can reduce to the case where this is actually just a sheaf in which case you are allowed to just use a direct limit of usual Czech but i'm saying it there is a result when the coherent top was the commonology is completely general by hypercovering if you want you can also do hyper you know i do have an analog of outings theorem on joints of an zel ring that allows you to do Czech so maybe yeah so maybe let's say hyper over it's all hyper covers but still will finally make terms in each degree and also you know that this sheaf f is not too big for example it has countable dimensional stocks and then you have to sit down and do it carefully but it implies that all the commonology groups they are at most countable so in particular you have to check that there are not too many etal things and so on but the critical part that ever makes everything work is a quasi-compacity here you have a co-finance a countable co-finance system yeah you have a countable co-final system of such things and then the second thing is you apply from grade wellity so because f is reflexive so in particular it's a dual of something else of some f prime and then this implies that actually the commonology of this guy with coefficients in this guy is a dual the r-home of the complex support commonology of this guy of the thing of which is a dual off into lambda and Ofer will again complain because I'm using a complex support commonology for something which is not representable the way to get around this issue is to not try to do this directly for the projection absolutely but for the map to the point mod k and then you use the description of sheaths on the point mod k so you can justify this but this means that actually the commonology is a dual space as lambda module the structure theorem for lambda modules right and then I mean if you have something the dual space which is countable then must be finite okay one can also do this argument and slight variance or one doesn't need this countability here and but argue slightly differently but then you need that everything is stable on the pullback to pull finite sets but anyway so that's the essential idea that on the one hand it's automatically dual and on the other hand by this quasi-compacity it's somewhat more of a direct limit nature and so playing these off against each other implies finiteness all right and so the other part is reflexivity so duality so again we start with our f which lives on this less mp part and we want to prove that the extension by zero so let's try to unravel what this means so what is the dual of j lower shriek well by the general formalism that's r j lower star of the dual and so and so we have an exact we have a triangle that j lower shriek of the dual of f that's r j lower star of the dual of f maps to um the stuff concentrated at the point for the push forward and so by the way this is a step will become clear in a second where we really already need to know that this kind of gadget here is finite to know that this extra this extra term that we get here is not too big and then we apply duality again so then first of all the order of the terms change and so okay so here in the middle we get the double dual f which is a dual what do we get on the right on the right we get the dual of this guy the dual again turns the j lower shriek into the r j lower star and then we get this off the double dual but that's just r j lower star of f and so then here's the duality commutes with the i lower star and so what we get here is the i lower star of the dual of this guy and we have our natural map here from j lower shriek okay so what is clear is that if i pull back this map under j then this is nice for example because then this term vanishes this term will be f again this map is a nice morphism this map is a nice morphism then this map is so it remains to check it on the complementary close so i upper stuff but if i pull this back under i then it's zero because i took the extension by zero so i need to prove that this is zero if i pull it back under i so in other words i need to prove that these two things become isomorphic after i pull back under i uh with a shift equivalently if i do this the i upper star then it's isomorphic to the dual of what i was getting for the dual sheave which is some kind of local from gradiality and let me actually go one step further and then we have to break and the proof of finiteness we figured out how to compute this procedure maybe instead we do it here and then if you do it there then the sections are given by these guys so going yet one step further you have to prove the following that for all k in gv of qp and open for p subgroup the sections of this guy so this is the left hand side it's isomorphic to the dual of the sections for the dual of pi upper star f maybe just destroy some new shifts so in words so in other words this is a ponker radiality on this deck which is this open part modular you also need that the map is the correct map and so uh let me say one last thing before the break um if you want to prove the statement for potentially unbounded f there's again a way to formally reduce to the case that f is concentrated in one degree so we can assume f is concentrated okay and so then i will explain how to prove this after the break no there's no it's not a bounded draft category sorry no no there's it's potentially unbounded i mean what you do in this case is that you see that well if you formulate the statement for the dual then both statements take directly filter direct limits to limits and so you can reduce to the thing where you bound it com logically bounded above but then by some finite dimensionality you can forget about stuff which is too far on the other side and anyway you can do this so break for 15 minutes so maybe i should have stressed that i mean there's one commutation of duality was j lower shriek which is a 10 transforms j lower shriek into j lower star but this other junction one might know it's actually failed so you it's not in general true that this r j lower star would be taken by duality j lower shriek that's what you want to prove and it's some of force for constructible sheeps you know it's true but only because you know by duality for constructible sheeps um okay so uh wait wait wait this time it's not the r here okay so actually something slightly funny happens here with this punker ideology namely it's a punker ideology whether pairing is an odd degree so let me actually tell you about an analog so it is that for punctured spectrum right the analog was a punctured spectrum exactly so uh so i assume you have an a2 over a k and you have the origin and uh let's say m is a strict generalization of a2 at x and then it has a closed stratum which is x and it has a punctured spectrum and sir and then well if you are get inspired by it's like a complex picture okay see then you would think that this is a very small neighborhood of this point and so it's like a three square and then there is a coordinate which gives you the distance to to x and so this has a three-dimensional punker ideology because for homology this factor doesn't matter so that's the kind of punker ideology which we want to prove so we have like the strict generalization of our space at this point and then there's somehow a homotopically trivial direction which gives you the distance to the origin and then you have some kind of Milner sphere around this point or whatever it's called um which is this compact manifold of one real dimension less and but now we do a trick and so let me explain the trick uh in this example first so assume you have a self map which is contracting towards zero uh for example on characteristic p uh you might take x y goes to x to the p and assume it's actually an automorphism so let me also assume that f is an automorphism uh so for example after passing to the perfection rectified which doesn't really matter as regards etalc homology so then what i can do is i can divide the punctured spectrum by f to the z where you would imagine that in the analogous situation what you're somehow doing is here is you're contracting uh in regards to the distance you're contracting to the origin so the idea is that this is something like this but then on this distance factor you're somehow folding this together with multiplicative period p so this would be in a three times as one and this satisfies some four dimensional punctured reality okay and um one can uh if one knows finiteness of chronology for m-circ one can deduce the three dimensional duality uh from the four dimensional duality in circumference f to the n times z where n is sufficiently large so the idea is that the chronology of this quotient is some of the group chronology of z acting on the chronology here but then if n is sufficiently divisible um this it's a chronology as finite this will actually act trivially on chronology and so the chronology of this quotient will actually just be direct sum of two copies of the chronology of m-circ and then you can deduce the desired geology here all right some hope for any fault and so in algebraic geometry it's a slightly tricky to make this work you can I think actually do something like this if you replace a strict generalization by this formal completion and then pass to attic spaces so then this m-circ is some attic space itself and you can make sense of this quotient and it will behave like a proper smooth object and so you can also make sense of these objects in our case so what you do is the following so in our situation fix some kind of up what just called up it's an element of gb of qp which is central that is some of x was positive slope on the important radical so the let me try to say an example what I mean so if g is g o2 and b corresponds to this bundle of 1 plus o then gb is this levy g o1 times g o1 sitting inside of the borrel and then I look at the element p comma 1 in this guy okay and so then I can take the quotient by the action of up so the idea is that up is such an operator as this f up there take some power of it is an analog of this operator f up there that is contracting mb tilde all towards the origin and but if you remove the origin then it's acting freely it can take the quotient so the action is free and now the map to the base is proper and smooth let me give you an example of what's happening here in this g o2 example still what was this mb cirq I said last time that this was the attic spectrum of this long series field modular pro for net group so this is a quasi compact space but the map to the point is not quasi compact because whenever I base change this to some non-archimedean field it becomes a punctured open unit disk modular pro for net group but in these coordinates the p corresponds to the map which takes t to t to the p and so if I take this guy module p to the z then it's this guy modular Frobenus to the z and then also modular profiler but just forget about this profiler group it's not the important part of what I'm trying to say so you get this quotient and so again there is this crucial difference of perspectives between looking at these just as a diamond or looking at the structure morphism if you look at this just as an object this is very bad it's not quasi separated just a sheaf on the perfecto it's basic characteristic p it's not quasi separated because if you take a quasi compact object over it like this attic spectrum and then takes a fiber product you get the many copies of it because you're dividing by the z actually but if you look at it relatively no it's still quasi compact but not quasi separated um but this map is wonderful it's quasi compact quasi separated it's even proper and smooth represented by it's relatively represented by diamonds by proper and smooth guys because whenever you base change the picture to an algebraically closed field what you get is a punctured open unit disk modular the action of Frobenus which is a wonderful object of the base change in some spa c I get punctured open unit disk modular Frobenus and so and this is a wonderfully nice object and so and because it is proper and smooth we can apply puncture duality on this space so if you then just use a formalism plus our assumption that f is reflexive we can we get this kind of even dimensional puncture duality on this stick and then get it right the argument on again to check some compatibility of maps um but so if n is sufficiently divisible uh the cosmology this quotient uh is the same as I mean uh u p to the n z x trivially on the cosmology of this guy uh by finiteness because if you have any automorphism of a finite object then some big power of it will be trivial and then once this happens uh this means that the cosmology by some uh who should say a spectral sequence the cosmology of this quotient is just two copies of the cosmology of you have got a spectral sequence but then you will need to know that it there is I mean it acts trivially some in the derived category okay I want that the action is trivially in the derived category and then you think it really splits as it acts and okay and then you need to check that the even dimensional puncture duality which you get on this space really comes from two odd dimensional puncture dualities relating this with the dual of this and the other way around so that's why I wanted to say about the proof of the main zero okay so let me make some concluding remarks about a few things that one can deduce from the formalism so this is somehow continuing the discussion uh of the first half of the first lecture so and for this I would like to pass to ally coefficients and so give me some kind of let me give you some kind of I don't want to claim that I can make sense of this in as good a way as would be desirable but you can make sense of some objects so you can just make sense of uh some dereflexive bungee with oe coefficients that's essentially by passing to the limit of the story modulo powers of l so if you would have done everything in an infinity category she could literally pass to the limit um there's a way to do it without infinity categories um then I can just formally set and so there's also no Thomas this is a canonical object that um where any possible definition would give you reasonable definition should give you the same answer in the end so for the next thing that's not true okay I define this with e coefficients to be just the idempotent completion nobody there is a problem in the limit process because then you go from oe modulo powers of l to one i don't want to do this just under the derived category yeah I don't take a limit of derived categories if I do a limit I do it of infinity categories no but I'm saying that that because of infinitely many comology groups and infinitely tau dimension when you pass from oe mod l to them to oe mod l to the smaller powers okay you don't necessarily go from a reflexive guy to a reflexive guy when you take mod when you pass from one ring to another because you have infinitely many possible comologies to given them and then you take tau's with a smaller ring and you can have many things on the left and on the right and then you will have many tau's that so you will get possibly will look okay so you can focus bounded maybe in a reflexive bounded let me do bounded okay and maybe I also want okay the idempotent completion of the single tensor with e and then I don't claim that this satisfies any kind of descent anymore or that you can check yeah I mean this is some kind of not good thing maybe but yeah let me do the idempotent completion because I might want to use some operators some excursion operators to cut to cut an object into pieces so I do want an idempotent completion but is it necessary is it you ask but yeah I think it's necessary I think because for DPC of nice no because schemes it's not necessary to it's already idempotent that is your any no but I would expect that you can have representations over oe of such periodic groups which are somehow which have some congruences but then generically that you compose and then if you want to somehow get the decomposition that you have generically if you want to get it integrally I don't think you can do it so I think you need an idempotent completion there um okay so I at least for the part of the drive categories I'm eventually interested in I'm confident that this is defined um and so let's now fix an L parameter fix same as simple parameter one can define a full subcategory which is somehow the five part what's this guy so there is also a question like in ellatic shift that you can have no for bernieus eigenvalues which are not ellatic units generally values I don't know if this occurs in this well I'm working over an algebraically closed field right now so right at the beginning I fix an algebraically closed base field and I'm working over the space field all the time but this w of q p so you have the uh yes so it might be well I can define this for anything it might be that it's just empty but I mean this real group it's not related to uh the fact that this very group appears here has nothing to do with this algebraically closed base field that I had so I'm asking if the for bernieus element is not kind of not ellatic unit so yeah so it's probably true that if I have a guy here whose image is not compact in some sense uh then what I'm just defining just now will be empty but I can just still define it okay lg is a Langland's Julian uh five part um by the condition that all excursion operators as I define them in the first lecture are given by phi or are determined by and so let me discuss what one what what what one would expect about this category uh so let me assume that this phi is actually a cuspital so assume that the group s phi which is a centralizer in the dual group of phi is finite this implicitly also says that g is well the central trawler says no split part okay so glm would be disallowed for g hat uh the dual group so lg is the semi direct product of g hat was available so in this case you would expect and these expectations are results that I would believe to be within reach okay um you would expect that all these reflexive guys in this component are concentrated on the semi-stable locus so actually um what this guy is this lump actual bar it's five part it should just be uh direct well it doesn't matter there's only finite number so under this assumption it's automatically true that there are only finitely many basic elements only finitely many connected components and direct sum overall basic b uh of the derived category of derived category with admissible homologies of this group jb of qp of these all these inner forms and then you take a phi part of that in the sense that you only allow those admissible representations all of those constituents have this fixed l-parameter phi where the l-parameter is as defined in the first lecture so this means that this thing which is a priority find in terms of some very complicated geometry boils down to some very concrete representation theory why would I expect that this can be proved so if you would have a sheath which has this l-parameter but lives on some other stratum then this other stratum is given by an inner the automorphism group there is an inner form of a levy and you would expect that if you have representation of an inner form of a levy that then the associated l-parameter should be induced which is wheeled out by this hospitality condition so you would only have to prove some compatibility with parabolic induction to get this statement so that's the first expectation you would have and the second expectation you would have is that uh all hack operators are t-exact for the standard t-structure that is not perverse but but well it doesn't really matter right because these are just just have points here so the perverse is the same as the usual t-structure ah but you are speaking about just well on this part and on this part some of the perverse and the usual t-structure are the same so um it's certainly not true that hack operators are except for any kind of t-structure on the fold arrived category you need to make some assumption and for these cuspial parameters it doesn't matter which one we take and so you would actually this way get an action um right now everything is a little undefined maybe let me try to explain this picture um so you actually get some kind of action uh we have here the algebraic representations of the outgroup times okay let me call c the category for phi to be the direct sum of all b and b of g basic so category of admissible ql bar representations of gb of qp takes a phi part so it's some kind of representation theory at the category attached to this cuspital parameter phi and so this would mean that whenever you have a representation of the outgroup you can act on this category and you get another object of this category but actually you get an object with a wave group action and the action is continuous okay uh and this action satisfies some conditions these are objects of c phi together with the action of w right so continuous means that it's like in like in the most representations that is right that every vector is an open stabilizer right every vector is an open stabilizer well after a while it might it might intertwine the allylic action the allylic part of the wave group with the allylic topology of this ql bar that will actually not happen and so um this gives you some purely category theoretic uh setup here where you have a category you have some category of representations of an algebraic group acting on it where after you act you get an extra action of some other group and satisfying some competibilities and you can write down all the structure there is and then there's the following theorem uh which seems to be well known to the experts in geometric Langlund so for example Gatesbury and Vincent LaForgue and so on they seem to know this um it seems to be well known but anyway uh somebody who actually wrote down a proof that I understood this you want to answer it um so yeah so assume that s phi is fine idea then there is actually an of the tensor category of representations of the centralized group s phi uh on this corresponding isotopic component which on under these assumptions uh under these expectations would actually be an action on this category uh c phi um such that uh the action up there actually this action is unique if you write down correctly all the competibilities it's given by the following procedure so if you have a representation of the l group so in particular you get a representation of s phi times the real group because you can restrict via phi to a representation of the real group but then this action of the real group commutes with s phi by the definition of s phi so you get a representation of the product group uh so which is the same thing as representations of s phi which are vqp a covariant and then if you have a representation of ql bar of s phi then you have this action so you get an endomorphism of this reflex subcategory so what do you mean by action of the tensile category just uh on the usual the rank category there is some higher no no on the usual the right category just on the category so this means there is some there are some opposites and some compatibility right so for every object you get an endomorphism and then you have two objects then the action of the tensor product is the composite of the actions and when you have three you have a query when you have three you have a compatibility but it's only on the drive category so it just stops at some point okay let me not try to remember at which point um okay so this means that you can describe the action of the l group actually in terms of the action of this s phi and the l parameter and so this actually if you do unravel what this means this is the form of the cotwitz conjecture so this action of the representations of the l group this encodes the cosmology of wrap up what zing spaces and you then take isotopic components under some guys in this file part and then this computes the output of this function in terms of uh the action of this category of representations of s phi which is doing some combinatorial thing like realizing some jucky length correspondence this takes one uh representation with l-point-meter phi and produces another representation so this gives you the representation theoretic part of the picture and then it also tells you how the real group acts namely well it's also given there's a form of cotwitz conjecture on the cosmology of basic wrap up zing space well this is the form of the cotwitz conjecture except that uh it doesn't really tell you what this action is so proving the cotwitz conjecture would then come down to actually understanding what this action is so let me say one last word about this and so there's a following conjecture which in this well which is essentially due to caleta except uh of course it didn't have this formalism to state it but caleta uh conjectured exactly how this category c phi or the irreducible objects of c phi look like um okay so let me assume for this a g-square z-split so then what caleta effectively conjectures is that this category c phi is equivalent to the category of representations of s phi where here's a trivial representation gets mapped to use a unique generic representation uh and fix the later caleta unique generic representation of g of qp which is the one which corresponds to a trivial element b and then in general he has a way to parameterize all the irreducible representations of all these what he calls extended pure inner forms all these jb of qp's and they should be given by certain representations of s phi which have a certain behavior um and well if you combine this of all possible different b's then they should be parameterized by all possible irreducible representations of the s phi and so he relates irreducible objects on both sides but actually this should be an equivalent of categories and and this should also be true uh what do you do at the gqp equal to what? g1 of qp so i mean if you take for b the trivial element so then uh comparatively with the irrepressible object so in other words there should be this unique generic representation and then this action of the representations of s phi would give you a whole function from the category of representations of s phi into here and this should be an equivalence so this isomorphism it depends on the choice of vitica data but some of the action of the representations of s phi on it they don't don't not depend on the choice of vitica data the unique generic representation here so some of the basis element here depends on the choice of vitica data but the kind of module structure over the representation of s phi do not depend on it so some base point three version of this conjecture some of that this category some of three of rank one over the representations of s phi okay and uh what Carl Leder really made much more precise is the he said he wrote down some explicit endoscopic character relations for this action and so so this is actually also something that seems to be in reach so there is work in progress of Carl Leder and Weinstein where they use the left shed trace formula and checks that it is compatible and check that the bungee picture is compatible with these endoscopic character relations so the way they phrase the argument is that they assume that Carl Leder's conjecture is known in his form that the irreducible objects up here in the expected way and satisfies the expected endoscopic character relations which Carl Leder has proved in a wide variety of cases and then they prove that what you see in the coromology of these basic robot thing spaces really is realizing this correspondence of representations and so let me just say that this generalizes uh work of fountains and straw for the limintate case of pharx for some unitary groups pharx and unitary groups i think u 1 n minus 1 right and also meeder had some results of gs for gsp4 i believe so there are fountains and straw proved by some purely local methods using the left shed trace formula and limintate or drain fuel tower as that in the coromology of these limintate spaces you realize the jucky langans correspondence and this is some large generalization of this and so my time is up so thank you very much very nice lectures are there any questions or comments i mean in this middle backboard of course there's also the action to deal with whereas in the lower blackboard there's no action to deal with well some as this serum tells you that the action of the very group is already as expected and everything comes down to determine and what happens on the for the periodic groups and for example for the limintate tower uh which you can't quite apply so if you pgl or something uh or prove a version of the serum which allows the center um the serum would already tell you that if you plug in some super customer representation of gln what you get is some representation of the division algebra tensor with the correct l parameter and then if you combine this or tensor with the l parameter that is defined by these methods um if you combine this with fountains and straw paper which tells you that you see the correct jucky langans correspondence basically see that you get the correct thing but actually the output of this is that actually what you see here in this picture is really the usual local langans correspondence using some global thing to determine this no this is purely local this argument but i'm using that uh there is a local characterization of local langans for gln that is what's realized in the limintate tower well i mean when yes but that's all that it is a local langans correspondence is proved by global methods


The Owen Patterson Affair has shone our Tory government in an awful light, but it's not only Tory MPs who have come off badly. The past 48 hours has also exposed some of the worst tendencies of parts of Britain's press. Most obviously this pertains to the most senior journalist in the country, the BBC political editor Laura Kuhnsberg. Kuhnsberg, as this story was breaking, as this row was erupting on Wednesday and before the enormous public outrage which forced Johnson to U-turn, tweeted the following. Senior MP says they've just been ordered by Whips to back the ledsome amendment. This is a proper Westminster village story, but it's really important if you care about how MPs' actions and behaviour is monitored. That is a really embarrassing. This is someone who is supposed to have a real sense of what is and what isn't a big political story. That's literally her job. She saw what has become the biggest political story of the year. Obviously there are bigger stories, COVID, climate, etc., but the biggest political row of the year. And she said, oh, it's a bit of a Westminster village story, this one. You're probably not going to be interested in this unless you're a real politics bod. 24 hours later, it's set the public conversation alight and the government have been forced into an embarrassing U-turn. Kuhnsberg did this morning return to Twitter somewhat with her tail between her legs. Let's go to this tweet. Blowback to yesterday is intense. One Tory insider says, people are going absolutely mad at what looks to many like a terrible calculation. Turns out there are lots of politics bods in the country if we are to accept Laura Kuhnsberg's previous analysis. A certain newspaper also didn't shower itself in glory. Spot the odd one out in this morning's front pages. So you can see here the Guardian went with PM accused of corruption as rules on sleaze torn up. The Times went with Tory's rebel over vote to block MP's suspension. And the Daily Mail went with shameless MPs sink back into sleaze. The Telegraph, though, let's look what their lead story was. NHS staff won't have to be jabbed this winter. So the paper, which most slavishly follows the lead from the Conservatives and, of course, who used to employ Boris Johnson, entirely ignoring the biggest political scandal of the year. Ash, Neva Kuhnsberg, nor the Telegraph have come off at all well in this story. Do you think they're going to be both feeling a little bit embarrassed that they sort of tied themselves to the idea that this wasn't a big story, that it might just blow over? I think if obsequious lobby journalists and pravda for the home counties, i.e. the Telegraph, were capable of feeling any sort of shame, their practices wouldn't be as egregiously servile in the face of power as they have been for quite some time now. This is by no means the most embarrassing thing they've ever done. It's just the most recent in a catalogue of failures to live up to the most basic purpose of journalism, which is hold the powerful to account without fear or favour. I think that the Laura Kuhnsberg tweet, the initial downplaying of the story of this is a proper Westminster village story. This is really revealing either of a deliberate attempt to manage the public reception of what is quite a scandalous story, or indicative of being so bad at reading the public mood and bad at taking the political temperature that she probably shouldn't have a job as senior as political editor of BBC News. It's got to be one or the other because you could put it another way, which is the expensive scandal, which was what? I guess around 2010, 2009, 2010. Stories can be as big or as significant as journalists want to make them by how much they pop on about them. Brexit was a matter of sovereignty and fishing rights. It became the defining political division of the last few decades. So I don't buy when journalists say, oh, well, this clearly has no cut through or this doesn't matter to anyone. Either it means that they are so caught up in the herd mentality of the Westminster lobby. You know, they're so aloof and so distant from what people think that they can't actually access it, or they're deliberately trying to make what could be quite a difficult story for the establishment, the government, or whoever it is, trying to make that go away. And I guess I would just wrap up by saying this because I was talking to somebody this morning about, is there any other profession on the planet as self-deluded as the journalist of the Westminster lobby? Because I do think that they truly believe that they are noble warriors in the quest for truth, crucial intermediaries between the public and the powerful, and they are indeed holding the powerful to account. Now, to anybody with half a molecule of critical capacity within their brain, you can see that's plainly not what's happening.  If your instinct is to downplay the significance of stories like an egregious breach of lobbying rules and the attempt to then rip up those rules, well, then clearly you're not being that good at holding power to account. If you are writing what is considered to be one of the most important political round-ups every morning and you are the Prime Minister's son's godfather, you're probably not the predatory fourth estate that you think you are. And there is this gap between the self-regard of much of the lobby and their ability to actually carry out that function, which I just find on a daily basis completely stunning.


Missing person cases rarely have a happy ending, but that isn't true in the case of Melissa Heismith, a woman who went missing back in 1971 and was finally located in 2022. Melissa was stolen from her home when she was only one year old, kidnapped by a woman who claimed to be her babysitter. The Heismith family had only recently moved into their new home in Texas when Melissa's mother placed an ad in the paper looking for a babysitter. Melissa's mother hired a woman who responded to the ad with the babysitter picking up the infant on the afternoon of August 23rd, but this was the last time Melissa was seen for decades. As it would turn out, Melissa was finally found after she'd taken part in a 23 and me test, having no idea that she'd been missing for all these years. When the results came in, the family reunited soon after. Follow for more true crime stories and be sure to check out my channel for full-length videos.


Currently Thursday, October 22, 22nd, 9 and 15 in the morning, cleaning out the garage, taking all the trash out because the trash is coming to pick it up and I've been delaying picking the garage up because we've been moving so truck is coming and so you get it.   Ethan! Hello? Buddy! Ethan decided to lock us out of our own house so that's cool. Now we're trying to get him to unlock the little button on the door knob so that's fun. He lost interest, I think we're done. He's gotta go watch the cartoons now. Ethan lost interest and now we're out here. Should we just go to Starbucks? Well, we finally got in. Don't ask how we got in, but we got in. Okay, yep, that's that's cool. Looks like we're getting a new door knob, Sarah. Yep. At some point in time, I'll give you a tour of the new house but right now I gotta clean that garage and it is 12.55 pm. I played Warzone, I played Fortnite, I finished about two videos for a client and now the family is having lunch. They're back, they're eating and I have to go clean the garage like an adult when it's 82 degrees and all I want to do is go on and run but I gotta go clean the garage. So remember when Ethan locked us out? These are the remains of our door knob which means I gotta call a door guy. Is it a door knob guy? I gotta call a door knob guy. This is honestly, this weather is incredible. Where I come from it snows. This time in fact they're just riddled with snow and I'm over here just moved. It's 80 degrees like I'm loving life right now. You hear that? Just hear the breeze, the trees that arrived. I don't want to clean this garage, I'm just stalling. Crap. Enjoy this time lapse of me cleaning the garage. My bags and I got more somewhere around here but I like gear, I like bags. Hour and 20 minutes. The garage is cleaned out. For the most part it's cleaned, it's organized. I've got a few things I gotta pack away but what I want to do is I'm gonna, I want to clean this entire floor because very soon I'm building the Johnny QR studio in here and I need it to be clean. That's what I'll be doing hopefully in the next video but right now it's 80 something degrees, super nice out. I'm gonna go on and run and it's gonna be so good and I'll see you guys after that. A few moments later. So I finished my workout, I went to the gym, really really good workout, pretty pumped in, I went to the sauna. Love the sauna. When I got back I brought in my paintings, I got about three full bins of stuff like this that I have not sold yet because I'm not put up on the website. So at some point maybe even tonight or tomorrow in the next few days I'm gonna be going on TikTok live and I'll be selling these pieces on my TikTok live and I'll send them to you guys after payment and so that's what I'll be doing. Stuff like this looks really cool. Okay then we got this one, it's covered in epoxy as you can see, really really cool and then also this one. Love the way this one turned out. So you know that's that's it for today and I'm really pumped to be setting up my like full-on Johnny Q Spinar studio in the garage. Obviously I'll bring you guys along for that as well but that's it and as always guys thanks so much for watching, thanks for joining and watching these daily vlogs. I will catch you in the next one. Peace.