Good afternoon, so welcome to all of you. Thank you for coming. We have one of the special events of the year today. We have this ceremony for the Ramanujan Prize, as you know, it's the prize in honor of Ramanujan, who is one of the top mathematicians in history, very young, so it's given to a young mathematician from developing countries. And we are happy to have today here, the first secretary of the Indian Embassy to give the award, the prize. And I have to also mention that today with all, we have the new students who are just arriving for the new, for the year, for the diploma course. So we also want to welcome them. I hope that this will be the first nice experience at ICTP. And by saying that, I will let the work to Fernando Villegas, who is the director of the mathematics session, to introduce the prize and the awardee. Thank you, Fernando. So as Fernando said, the Ramanujan Prize is awarded yearly to a mathematician from developing countries under the age of 45, is funded by the Department of Science and Technology of India, and is awarded jointly by DST, the Department of Science and Technology, the International Mathematical Union and the ICTP.  In the early 1900s, while he was working at the Port Trust office in what was then called Madras in India in an accounting job, wrote letters to English mathematicians with his discoveries. And only G.H. Hardy, which at the time was one of the most well-known mathematicians in England, paid any attention to these letters. Initially, he thought that they were possibly a fraud, and he eventually came to recognize the Ramanujan's originality and wrote in amazement, the theorems defeated me completely. I had never seen anything like this before. These theorems must be true because if they were not true, no one would have the imagination to invent them. Our 2015 winner of the Ramanujan Prize is Amalindo Krishna. Amalindo was born in India, did his PhD at the Tata Institute of Fundamental Research in Mumbai, India, where he's now in the faculty. After his PhD, he went to the United States and was a Hendrick assistant professor at CLA from 2001, 2004. He was a member of the Institute for Advanced Study in Princeton, 2004, 2005, and went back to India to the Tata Institute. He has received a BM Birla Science Prize in Mathematics in 2009, the Swarna Yanti Prize in Math from the Government of India in 2011. The prize is given in recognition of Krishna's outstanding contributions in the area of algebraic K theory, algebraic cycles and the theory of motives. In his work, Krishna has shown an impressive command of a very technical subject, applying the modern theories of algebraic K theory and Wojcicki's theory of motives to study concrete problems. As one of the US CLA students said in a review of his teaching, Professor Krishna is one hell of a math professor. I think this is something we all in this profession aspire to. And so, without any further ado, I welcome the first secretary, Aaman Verma, to give the prize to our winners this year. And now Aamalindu will give the Ramanujan lecture whose title is algebraic cycles and cohomology theories. Okay, so good afternoon to all of you. Before I go into more serious thing of talking mathematics, I have to thank various people. So please bear with me for some time. So, I had heard of this Ramanujan prize from ICTP before, but I could never think that I was kind of good enough to get any of these prizes. So, I was kind of basically pretended to be oblivious of this prize and kept working. So, until I got an email from Unraveled Director of ICTP Professor, Kivu Adho Fernando, I was kind of very surprised to see this email. So, it took me some time to believe it. Then I asked one of the senior faculty member in Tata Institute, is that true that? And he said, yeah, maybe possibly true. So, then I got another email from Professor Villegas. So, I thought, oh, maybe this is indeed true. So, and then when I read the letter from Professor Kivu Adho Fernando, and then later there was a mention of the list of members in the selection committee to choose this award. And when I saw the names of the mathematicians in that list, the kind of eminence those mathematicians had achieved and getting a recognition from those people was really humbling for me. So, I would like to thank each and every member of the selection committee for this Ramanujan Prize, including Professor Fernando Villegas for recognizing my mathematical piece of work. I would also like to thank ICTP, IMU and DST for organizing this prize. I would like to thank each and every individual or organization who are which directly or indirectly played any role in giving this prize. I would like to thank every individual present here today to make this event memorable for me. And finally, I would like to thank my family, especially my wife and daughter, who just see our side every day makes my day. So, thank you all of you once again. And now let me begin my lecture today. Okay, so this lecture is about algebraic cycles and cohomology theories. So, cohomology theories in mathematics is a very old thing. Everybody knows about cohomology theories, but algebraic cycles are not as old as cohomology theories. So, what is the motivation and background of this problem? Well, so let K be a field and let X be a smooth variety. So, for people who are not from algebraic geometry, variety means basically you have some field and you have basically some collection of polynomials and you look at the set of points where those polynomials simultaneously vanish. So, a smoothness means basically just like not any kind of variety, but it's kind of like in topology what you call a manifold. So, it's a kind of algebraic and a lot of smooth manifolds. So, recall that associated to any such variety X, there are various known cohomology theories, namely algebraic K theory, which is one of the most complicated ones among all this. Similar cohomology, which is the most well known among all. And then something called Alleric cohomology discovered by Glutendic. And then there is PID cohomology and of course, well known Dirac cohomology. Also recall that except K theory, each of these forms, what is called a well cohomology theory, if X is projective. So, well cohomology theory is a cohomology theory, which was invented by Glutendic because she wanted to prove what is called now the well conjectures. So, recall that what is an algebraic cycle? Algebraic cycle on a variety is a linear combination. Here linear combination means the integral linear combinations of closed sub varieties. One applies various equivalence relations on these linear combinations to define cohomology theories using algebraic cycles. So, a homological way of saying that a variety is determined by all of its points of various dimension is that a variety is basically determined by its algebraic cycles. If you know all the algebraic cycles, essentially you know the variety itself in a kind of loose way. Now the question we are interested in can be roughly praised as follows. Given a cohomology theory at star of X as above, is it possible to describe it purely in terms of algebraic cycles? So, because you know you want the cohomology theory to give you information about the variety. So, if I can detect this cohomology theory by the algebraic cycles, that will help me to understand the variety itself using this cohomology theories. Because then it will say that this cohomology theory themselves are some kind of cycles. So, this question may look a little vague at the moment, but let's try to make some sense. Well, there could be two possible meanings to asking if this H star X can be described by algebraic cycles. So, the first many of the problem could be the following. Let us assume that X is a smooth, projective variety. Then we know that every well cohomology theory and also algebraic theory received cycle class maps from the chore groups. These chore groups are basically algebraic cycles on X. Model of some kind of equivalence relation where the equivalence relation is generated by a family of cycles which are parameterized by rational points on the projective lines. Now, there is a cycle class map from the chore group of co-dimension I cycles to H2I of X where this is any of those cohomology theory. Now, the question is how much of H star of X can be described by cycles via this map? Well, it is immediately clear that this is a weak question because if the way it was the cycle class map was defined, odd degree co-homology classes can never be detected by algebraic cycles. Even in even degrees, only a small part can possibly be covered by the algebraic cycles. And the well-known Haas and Tate conjectures give precise formulation of this measurement namely how much of these co-homologies, even degree co-homology classes be algebraic. Furthermore, Haas and Tate conjectures are so hard that we do not know if it is a good idea to pursue this approach of understanding the co-homology via algebraic cycle. So, let us turn to another way of describing this co-homology by cycles. So, now the question is can we completely describe the sieves defining this co-homology by algebraic cycle? Notice that all the co-homology theory which we described, they can be described as sieve co-homology on the varieties. So, basically you can define a sieve and then you can take its co-homology and that will give you so the which co-homology you take depending on that you have to choose appropriate topology on the variety. Okay, so the more precisely the question is is there a Jarevski sieve? So, now I take the Jarevski the usable topology algebraic topology on schemes. Jarevski sieve of chain complexes of algebraic cycles which I call here TCHX on X such that if I take these sieves of cycles and take its co-homology then I completely get my given co-homology. So, that's the question that I have a co-homology theory of a variety and I want to understand this in terms of cycles in this sense. Namely I can take some collection of cycles make a sieve out of it and then I take the co-homology and that will give me the co-homology theory itself. So, it looks kind of a very surprising thing but let's see how it works. So, the aim of this talk is to address this question for the Diram and the crystalline co-homology. We shall concentrate on the crystalline co-homology because its original construction is very complicated. So, before we state our results let's give a brief introduction to the original and other known constructions of crystalline co-homology. So, crystalline co-homology was a well co-homology which was invented by Gruthendig and described in a complete way by Barthello. And this was the idea behind the genesis of crystalline co-homology was you needed some kind of co-homology theory where you can prove some kind of well conjectures and Gruthendig et al. co-homology had this problem that it worked only when you take L which is prime to the characteristic. So, the et al. co-homology could not detect the periodic phenomenon. So, you needed some kind of periodic co-homology and which is now called the crystalline co-homology. So, let K be a perfect field of characteristic P positive and let WK denote the ring of P typical bit vectors. So, for any integer n you have let W and K denote the ring of P typical bit vectors of length n. Recall that so, what is this ring of bit vectors? So, well before without getting into the technical definition basically it is a mixed characteristic discrete valuation ring with the field of fraction K and it is a studio field K. So, if you have a perfect field K of positive characteristic you can always construct a discrete valuation ring which residue field is exactly your given field K and its fraction field contains the field of rational numbers. So, this helps in many of the lifting problems. For example, if I take K to be the finite field Fp then this ring of bit vectors is exactly the ring of periodic integers. Recall that this ring WK is equipped with two distinguished operator called Provenius F and Versaibung B which satisfy certain properties. So, F is a ring endomorphism while V is just an additive group homomorphism for WK. It is also known that W and K and you have this because a discrete valuation ring. So, when you take W and K you have this principle ideal canonical principle ideal that I call P. So, it has a canonically defined what is called the divided power structure gamma such that this becomes a divided power ring. So, I will not go into the divided power what divided power ring is. Basically the idea is that in any ring you want to sort of exponentiate certain kind of elements. In characteristic 0 you can exponentiate a function but characteristic P you cannot because you cannot divide by P. So, you kind of artificially put some a structure where you can do this thing. And this kind of artificial structure is called the divided power ring which allow you to exponentiate certain elements in the ring. So, actually this construction of ring of bit vectors can be done for any K algebra R it need not be just the field K itself. So, the same a story goes in particular if you have X a smooth scheme of finite type of what this field K which is a perfect field of characteristic P you can use the functoriality of this assignment R going to ring of bit vector W R. You can make W O X as a sieve of rings and similarly W and O X. So, in fact, one can prove that this locally ring the space X comma W and O X is a scheme of finite type over W and K. Furthermore, it is projective over W and K if X was projective over the field K. Now, setting Sn equal to a spectrum of W and K and Xn equal to the spectrum of W and O X. Remember this W and O X is a sieve of rings now. So, this Xn need not be a fine a scheme it depends on what your X was. So, we see that Xn mapping to Sn is a morphology of finite type which a scheme theory closed fiber is your original X. Well, okay. Now, what you construct something called the crystalline site of X. So, the crystalline site Chris N of X is a category with a growthandic topology whose objects as a growthandic topology was again defined invented by growthandic in order to define his etal topology etal topology. Because etal topology is not given by open subsets of a scheme. So, you need some kind of outer kind of topology on your scheme and that not well known as growthandic topology.  And remember that since our characteristic is P it is essentially force that the ideal of this close immersion U inside T is nilpotent. So, in particular the topological space of U and the topological space of T are homeomorphic. So, and then you have this idea of morphisms of so, morphism of these two triples is a morphism of PDA schemes over W and K so, that when I restrict this map to W U prime this is a usual open old open immersion of U prime into U. The crystalline topology is generated by the family of morphism like that. So, that if you are a stick to T alpha of map into T this is a usual jaroski open cover of your T. Now, category of sieves of sets on this growthandic site with the canonical topology is called the crystalline topos of X. This is all in the language of growthandic. Now, there is a morphism remember on your original X that is a jaroski site where the open subsets are usual open subsets. And you have this crystalline sites and you have a natural map. And so, such that if you take the jaroski sieve on X then you can take its upper star and you can also take its lower star. So, such a more basically you can think of as a map of topological spaces in a very like broad categorical ways. So, basically it tells you that this UX is a continuous map of topological categories. Okay, in other words, yeah. So, now, so let us define what is the crystalline cohomology of X? So, the crystalline cohomology of X is the cohomology in the crystalline site of the sieve OX which is the structure sieve which is given by for any triple T u delta you evaluate it just take the ring of functions on the space T that I call the structure sieve of X in the crystalline topology. So, you take this sieve and take its sieve cohomology in the crystalline site that is called the crystalline cohomology of X. And remember that this N was any positive integers and WN plus one maps to WN. So, I can take the inverse limit of this cohomology and that I write it as X over W which means that this is a crystalline cohomology of X where X was any smooth projective variety over a perfect field of characteristic. So, that's the definition of crystalline cohomology. Well, now you see the way the crystalline cohomology is defined is fairly complicated. How do I sort of compute it? It's like you define some some some some authentic site and then you have some sieve and how do I compute this cohomology? So, a priori it looks very complicated. So, Bloch gave a different description of this crystalline cohomology. So, Bloch's idea was the following. He says that well you replace this crystalline site by the usual Jaroski site of X. You don't discover any new site. Only thing you have to do is that you replace this crystalline sieve by some kind of Jaroski sieve which are given by the sieves of algebraic K theory of infinitesimal thickening of Jaroski open subset of X. So, basically he replaced this OXX by some kind of Jaroski sieve of K theory. Well, the K theory themselves are very complicated, but nonetheless to Bloch K theory were much more well known than this crystalline site. So, it was more familiar to people.  You take your ring and take the truncated polynomial ring and it has a miller K theory and you map it to the K theory of your ring and the kernel is basically that is the sieve which Bloch's takes in this complex. And then here is the theorem of Bloch. He says that let's assume that your P is at least three and also the dimension of X is a strictly less than P. This is a very, very strong condition which Bloch had to put. Then he says that if X is a smooth and protective over your K, then crystalline comology is nothing but the Jaroski comology of his complex of K theory sieves. So, this was his theorem. It was a remarkable theorem at that time because he basically says that crystalline comology basically is Jaroski comology of some much simpler kind of sieves. Well, it's not easy to compute these sieves but nevertheless it's kind of there are tools available to handle them. Now, after Bloch there was another approach of illusory to define, to understand this crystalline comology. So, before gothendic Seier had suggested that some higher version of this comology of sieve of ring of it vectors should give rise to a PID comology theory. Seier did not know this what is called higher version of this ring of it vectors. Higher in the sense of you take the ring R itself and then on the other hand you have this ring of killer differentials. So, higher in that sense that is you replace your ring by some kind of differential forms. So, but Seier did not know what exactly this higher thing is. So, Deline outlined this higher version of W and OX in the form of what is now known as the P typical Dirambit complex. Deline outline of this W and omega X was completed by illusory in this famous paper. And here is a theorem of illusory. Let X be a smooth scheme over K. Then there is a canonical map from this sieve, the crystalline sieve to the Dirambit complex. And it is a basically isomorphism in the derived category. So, it gives you isomorphism of the comology theory. So, in particular the old crystalline comology is again a jarristic comology of some sieves of differential forms. They are not exactly sieves of differential forms, but some kind of like infinitesimal techniques of these differential forms. So, to SP which I call the Dirambit complex. So, you have you take the Diramb forms and then you put in the Witt vectors in that then you can I say it is a Dirambit forms. So, notice that unlike block illusory they had no condition on dimension X. So, there was there was this improvement over the Bloch's theorem. There is no condition. P could be 2 or X could have any dimension. So, in conclusion we have the following diagram. We have this original crystalline comology. On the bottom we have this illusory's Diramb whatever Dirambit comology. And on the top right you have this block comology, a comology of sieves of p-typical curves on K3. That is what he said. So, you have this diagram. Now, there is a question mark here. So, the question is that can you construct something over here such that this whole diagram commutes. So, that blocks construction and illusory's construction can be basically unified. So, our result can now be stated as follows. This is the result theorem with my collaborator, Genian Park. So, I still have to assume that the characteristic is not equal to 2 for some reason at the moment. But no assumption on the dimension of X. So, let X be a smooth scheme over K. Then there is a pre-seep of chain complexes of algebraic cycles. If you take their simplifications, then there is a map from the illusory's Dirambit complex to this cycle complex. And there is a map from the cycle complex to block complex such that the first map is a derived isomorphism. And the second map is an isomorphism if X is projective and dimension of X is strictly less than p, which was the condition in block. So, in particular it tells you that there is a complex constructed purely out of algebraic cycles which recovers illusory's theorems and also recovers block's theorem. So, basically yeah. So, basically you can unify this blocks construction and illusory's constructions and original construction of crystalline cohomology by algebraic cycles. So, what are these algebraic cycles? We now describe our cycles which give rise to this complex. In order to construct this, we do not need any perfected assumption now on K except that we only assume that so K has characteristic different from 2. So, now we have any field K of characteristic not equal to 2 to define the cycle complexes. Only to prove the theorem you need the assumption that characteristic is positive. So, for a smooth defined scheme X the complex is made up from what is called the additive higher chore groups of X. So, this is now a new theory which basically gives rise to this connection between crystalline cohomology and Dirac cohomology and blocks cohomology. So, what is the motivation behind this additive higher chore groups? Well, the motivation behind the discovery of additive higher chore groups was the belief that the relative algebraic theory of nilpotent ideals in a ring and the modules of differential forms on a smooth scheme should be motivate in whatever sense. So, remember that when Spencer Block constructed his theory of higher chore groups there was a problem with that higher chore groups in the sense that it worked well only for a smooth varieties. If you have a smooth variety then Block proved that his higher chore groups rationally is same as higher K theory. However, this Block's theorem miserably fails if the variety is not a smooth. Then we have no theory of higher chore groups of motive cohomology which can describe your algebraic K theory. And this idea behind the discovery of additive higher chore groups was something like this to construct a motive cohomology theory which could describe K theory of any scheme whatsoever. So, for a field K, the additive higher chore groups of zero cycles which I write at TCH n this one where this one is basically something like this Wn in Dirambir complex. So, now this is there you can think of n equal to 1. So, it was invented by Block and NO in this paper and what they showed is the following. They showed that if I take a field K and take the the chore group additive chore group of zero cycles then it is essentially isomorphic to the module of absolute killer differentials on that field. So, this is a first indication that this has something to do with the Dirambir complex. Then ruling generalized Block's inner theorem where he replaced this one by any m and this m is or is basically this the length of the bit vectors in the Dirambir complex. So, this m is corresponding to that. So, he showed that there is a natural isomorphism from this the field other the chore group of zero cycles to the Dirambir complex which was defined by Hessel-Herrlton-Madsen. You should notice that this Dirambir complex is not quite same as basically this one is not quite same as the p typical one where I just wrote w. So, this is slightly is a more general general form of Dirambir complex was defined by Hessel-Herrlton-Madsen and the p typical one is a basically is a direct amount of this. So, now here is the construction of additive cycle complex for any. So, remember that the Block NO or ruling only define zero additive zero cycles that also only for field. So, at that time there was no definition of additive higher chore groups for any scheme or not only zero cycle, but any cycle of any dimension. So, this was done by part and independently by Krishna and Levine in 2008. It was shown by Krishna Levine that this there is a functor from x going to this additive higher chore groups which is a kind of non-homotopy invariant Borel Moore homology theory on the category of quasi-projective schemes working. So, it is a theory which is not homotopy invariant in the sense that if you take if you take a vector bundle over the scheme then this theory becomes different unlike the usual co-homology theory like singular co-homology and so on. And that is kind of expected because algebraic k theory of singular schemes is not homotopy invariant. They also showed that additive higher chore groups have a canonical structure or module over the motor vehicle homology ring or the blocks higher chore groups. So, these additive higher chore groups are modules over the blocks higher chore groups which are already known before. So, here is the definition which is kind of technical. So, let me just skip this. So, the basic idea of this additive higher chore groups is that you define exactly like the cycle complex of block but what do you do is that you you put some kind of modulus conditions for every cycle. And this modulus is basically this described this length of the bit vectors which you are going to connect these two. So, there is some modulus conditions. This modulus condition is basically you know like you when you define your simplex then you take the Euclidean space and take set of all vector with sum is equal to 1. So, I can by sort of like suitable normalization I can take this term is some all such points where sum is equal to any non-zero integers. And then when I take the limit when the non-zero integer becomes sort of like infinity and I write like 1 over lambda then it becomes sort of 0. So, basically this shows some kind of like what happens to these cycles at infinity. So, this is some kind of condition and that is what gives rise to some additive higher chore groups. So, here is some definitions. So, you know basically there are some cycles with some modulus conditions. Now, let me give you an example of how this additive higher chore group work. So, let R be an essentially of finite type of smooth K algebra. Then I can define a map from the ring R to C H 1 R 1 comma 1. So, here this that C H upper 1 means the co-dimension one cycle in a spectrum of R cross A 1. That is what meaning of that is this. So, take a spec R cross A 1 and take co-dimension one cycle over there with the modulus is 1. Then I want to define a map and the map is very easy. You take any element A in R and take the cycle which is the zero locus of the function 1 minus A times T in A 1 over R. And you can see that this in A 1 R modulus condition is equivalent to saying that your cycle does not touch T equal to 0. And that is clear because T equal to 0 and 1 minus A T equal to 0 have no common solutions. So, it is a well defined cycle. And then you can define. So, this defines and then you can prove that this defines actually a group homomorphism from R to this cycle group. And here is some sketch of the proof. So, it defines a ring a group homomorphism and it turns out that this map is actually an isomorphism if R is a unique factorization domain. So, this gives you another evidence that this cycle group unlike block cycle group they are more sort of additive than multiplicative. In the block language if I remove this T then I will get something called R star over there. But here by just by putting that T here I replace R star by R. So, this becomes this one yeah. So, that is the basic difference between block higher chore groups and additive higher chore group. If you take the block higher chore group you will get R star, but you will get R. More generally for any semi local ring R you can define map from the module of theory of differentials to co-dimension n cycle complex in a spectrum of R cross A, A, N. And they define they are defined in this form. You can check that it is a well defined cycle. And remember that if R is a semi local ring containing a say infinite field then any element in ring R can be retained as sum of two units. And that is why this differential form is generated by the logarithmic forms. So, the proposition is you have a map from the scalar differential to additive chore groups. And in fact it turns out that this is also an isomorphism if R is a regular semi local key algebra. So, now what are the main results? The example given above is a prototype of the more general result which was motivated by the following slogans. The slogan is the differential forms as well as the operations on them lemony differential waste product etcetera on a smooth schemes are all motivate. It is kind of surprising thing if you if you try to think about it you know algebraic cycles and k theory they have no like additive thing there are no differential forms over there. And you are saying that even the modules basically the quasi coherent sieves can be described in terms of cycles that is what it is saying. Because this differential forms are kind of quasi coherent sieves. So, the credit for this slogan goes to a special block whose various work indicate that he must have already imagined this long time ago. And his work with Elena on additive dialogue which gave birth to additive cycles or a step towards achieving this vision. In order to state the local forms of main visual we need to recall the following concept of bit complexes over a ring and this concept is due to illusory in the p typical case and Hesselrod and Madsen in the general case. So, here is a definition of restricted bit complex which I will skip some technical definition. And the p typical dirambit complex of the lean and illusory was generalized by Hesselrod and Madsen for any truncation set S. In particular if I take S equal to 1 to up to m we get what is called the big dirambit complex W m omega r. Now, if you take S equal to 1 p up to p to the n minus 1 where p is a fixed prime then you exactly get by illusory's p typical dirambit complex. So, one of the result of Hesselrod and Madsen is the following. They say that the big dirambit complex W m omega r is a universal restricted bit complex over r. It is a universal object in the category of bit complexes. In other words, given any restricted bit complex e dot m over r, here dot means they are basically differential graded algebras. There is a unique morphazine from that big dirambit complex to e dot m. Now, we said W dot r could take the reverse limit about all m and there is a simple W is that we have p typical one if r is a zp algebra. Another useful fact about dirambit complex, the following if r is a zp algebra then there is a p typicalization functor. If you take the big dirambit complex you can apply this functor to get the usual p typical complex of illusory. And if r contains q then this dirambit complex is nothing but some direct sum of usual diramb complex. So, basically this dirambit thing is a new thing in characteristic p only in characteristic 0 it is nothing but the old dirambit complex. We can now state the main result which generalizes the previous example. So, you look at you write this TCH rm as the sum of all erative chore groups of r and here is a theorem. Let K be a field of characteristic not equal to 2.  Then the erative higher chore groups of r form the universal restricted bit complex over r. So, this is a universal restricted bit complex. In other word by universality it tells you that this must be isomorphic to the big dirambit complex. So, it tells you that the dirambit complex of illusies or the big dirambit complex of Heselon and Madsen are nothing but erative higher chore groups. They can be described completely by algebraic cycles. So, notice that the part of the assertion of this theorem is that this erative higher chore groups is are equipped with many a structure. It must have a differential map because it's supposed to be differential graded algebra. Must have waste product must have provenious must have the veritable operators and so on. And now how do I give all this structure is given by these maps. So, since TCH r is a restricted bit complex they can apply the p-typicalization function to get p-typical erative higher chore groups. So now, so we have seen that for a local ring or semi-local ring a dirambit complex is same as erative higher chore groups. Now I want to simplify these to make a see part of erative higher chore groups. So, how do I do that? So, this is achieved by using the following theorem by Wataru Kai in 2015. He proved that erative higher chore groups on the category of a smooth affine schemes over k is a pre-seep in the sense that it's a contra variant functor if you take a map from a ring a to b then erative higher chore group of a maps to erative higher chore groups of b. So, it's a pre-seep on a smooth affine schemes. Now I can use some standard construction to make it a pre-seep on all a schemes by some sort of like limit cold limit construction. So, here is how you do it. So, basically the idea is that if you have a nice kind of pre-seep on the category of a smooth affine schemes then you can extend this pre-seep in a canonical way to a pre-seep on all a schemes that's what this this this method does. So, and you do the same thing to define this to define this dirambit complex the seeps of dirambit complex on all a schemes. And it turns out that this is same as if you take w i omega n x that the same as the Ulysses dirambit complex. Now as a corollary of that previous theorem if it follows that if x is a smooth scheme then there is an isomorphism of chain complex of Jarecki seeps where this side is the dirambit complex modded out by p to the i and this side is also the the seeps of additive higher jaw groups modded out by p to the i. So, the so the basically the dirambit seeps are same as the seeps of additive higher jaw groups. Now it follows from a theorem of Ilyssi that if you take w omega this one and this maps to this w i omega x and Ilyssi says that this is this is not an isomorphism but it's a quasi isomorphism. Combining these results with above corollary and the main result of Ilyssi we conclude the following. Let k be a perfect field of characteristic p at least three let x be a smooth scheme then there is a canonical isomorphism where this side is the old crystalline co homology and this side is the co homology of seeps coming from additive higher jaw groups. This is just additive higher jaw groups made into the pre seeps on the Jarecki site of x. So, so here I related crystalline co homology with I related additive higher jaw groups with crystalline co homology and which is okay. So, now how do I relate additive higher jaw groups with the blocks construction? To relate additive higher jaw groups with blocks construction you need to construct a cycle class map here from the additive higher jaw groups remember this was the seep taken by block the k theory seeps. I have to connect them I have to construct some kind of cycle class map from here to here. How do I do this? And this is a requires a very hard theorem. And here this is a theorem which we will use is a theorem again with part and this says the following let R be as above then this additive higher jaw groups are generated by some very a special kind of cycle. This I call a moving lemma for additive higher jaw groups for semi local range. So, you can basically by move your given cycle in such a way that they are for very nice they are a very nice form basically they are of this form. So, any cycle it can be generated like this but once you have a cycle like that it's very easy to define the cycle class map. You just take any cycle Z which is of the form given by a b 1 b 2 b n minus 1 you simply take the Milner symbol here. So, here this curly bracket is missing here this should be a curly bracket here. So, this is a Milner symbol in R prime and then you because R prime to R is a finite map I can take the norm map on Milner k theory or which is called the push forward map on k theory and I get back to the element in element in this ring this group. So, this is the way I define and it's and then you saw that this actually kills the equivalence relation over here. And so that it actually defines a map and this you achieve by using some results of illusory again. So, this gives you a map again from additive higher jaw groups to the block constructions. So, if you simplify you get the maps of sieves and the combining our main results with some results of illusory obtained let k be a perfect fill with the same condition then and the extra condition which was given put by block the dimension of X is bounded by T then this additive higher additive higher jaw groups are same as the blocks construction. So, as a combination of all these things what we have achieved is the following diagram. Remember this was our crystalline cohomology this was the blocks cohomology this was illusory cohomology and here you have this higher jaw additive higher jaw groups and it tells you what we have proven that this is an isomorphism this was illusory isomorphism this was blocks isomorphism when X has bounded dimension and we prove that this is also an isomorphism X has bounded dimension. So, this is the main result which we stated in the beginning and this is how this is using the cycles this is how you achieve it. So, if the characteristic of K is 0 then basically the all the result tells you that if X is a smooth affine scheme of finite type or a field of characteristic 0 then you look at the dirame cohomology of X it tells you that this dirame cohomology here the usual dirame cohomology of X is same as the cohomology of cycle complex cohomology of sieves of additive higher jaw groups that are given like this it is a stresky sieve which is stock at any point if the additive jaw group of this local rate. In particular the absolute dirame cohomology can be described by algebraic cycles. So, now some applications of this theorem apart from connecting this those pictures. So, when Heselholt and Madsen define this this dirame bit complex there is a question by Heselholt that suppose R to R prime is a finite map of noetherian regular K algebras can you define a nicely behave trace maps on these dirame bit complexes. A trace map for killer differential was constructed by Kuhn's as a consequence of existence of remember that for additive higher jaw groups it was a Borel Moore theories it has a push forward map for finite map. So, I have a push forward map on the additive higher jaw groups and I have proven that these guys are same additive higher jaw groups using that now we can indeed construct this trace map. So, this solves a problem raised by Heselholt. Another consequence of so, this application was how to use additive higher jaw groups to conclude something about the known dirame bit things. Now, if you turn the picture around suppose you know something about the dirame bit thing can you construct prove something about additive higher jaw groups and this is what it says. For dirame bit complex the Gersten conjecture was proven by M. Gross and now using this isomorphism we can prove the Gersten conjecture for additive higher jaw groups as well. Now, some new questions well the isomorphism between dirame bit complex and additive higher jaw groups provides a new perspective of looking at the crystalline co homology. One knows that the crystalline co homology is not well behaved if the scheme is not proper and not a smooth also. And a substitute of was invented by Barthelot this is called the rigid co homology. If X is not a smooth and proper the crystalline co homology can behave very badly it could be like huge group. So, which is not allowed for a well co homology theory. So, now the question is there well now you have this something coming from algebraic cycles. The question is can one use this additive higher jaw groups to give a simplified or an entirely new construction of rigid co homology. Again the rigid co homology is something the construction is quite complicated. So, the question was that what does this does this theory of additive higher jaw groups allow you to say something about rigid co homology. Now, another question is the following the additive higher jaw groups also makes sense if you replace there was some A1 factored in additive higher jaw groups which I did not show you the full definition. Now, you replace A1 by P1 again you can make sense of some additive higher jaw groups where I put this C for this is compactified. And this says that the theorem again with Park is let X be a smooth scheme over a filled up characteristic not equal to 2 then there is a split sort exact sequence. So, look at here this is blocks higher jaw groups this is additive higher jaw groups which I have been talking about and this is something new which we can construct just by replacing this A1 by P1 in the definition of additive higher jaw groups. So, this theorem tells you that there is a this tells you there is a theory of jaw groups which unifies blocks higher jaw groups and additive higher jaw groups. So, now look at this diagram. So, remember you have this miller K theory for ring here you have the dirambit complex for the ring. Now, I remember this exact sequence here this this is also an isomorphism which was proven by Elvaz Vincent and Murad Eshta and this we have proven now. So, now question is if there are some kind of algebraically defined object here which unifies the miller K theory and the dirambit forms. Remember that this miller K theory is multiplicative theory and this is an additive theory. So, the question can be in a loose way in other word does there exist an algebraic object which semi-simple part is the miller K theory and the unipotent part is the dirambit complex. So, this lower diagram exact sequence tells you there should be something here which can be completely purely described in a algebraic way. So, I stop it here. Thank you. Thank you very much. Are there any questions for Amalindu? Anton. So, you have aomorphism between the big dirambit complex and the complex constructed from additive cycles, additive Chao groups. But in the characteristic zero, you stated the result only relating cohomology theories. Is there a more precise result relating the diramb complex in characteristic zero and some kind of additive Chao cycles complex? Yes, that works for characteristics zero as well. You have this theory of additive higher Chao groups and if you make it a seep and take its cohomology that is same as the diramb cohomology in characteristic zero. So, is the story become more simple in that case and more... It depends from what perspective you are looking at. So, for people like us who want to understand things like cycle class maps and so on because and you want to understand the variety using diramb cohomology so to speak. But then it tells you that diramb cohomology basically can be constructed from cycle on X itself. So, sometimes it helps you understand the variety without going into diramb cohomology. So, that's the kind of perspective we choose. So, because the cycles are more intrinsic to X than the diramb cohomology. Any further questions? Chao groups are defined and very important also for singular varieties. Now, when you have singularities, things become much harder and many things don't work and so on. But is there any hope that some of these theories which are so deep and so profound can be adapted to varieties with, say, mild singularities? I think it's a very good question and this is answering your question is the part of the goal of this. So, additive higher chao groups is directly related with the supposedly chao groups of singular varieties which I did not, I kind of, I just kept quite about it. But the whole idea of additive higher chao groups is to describe a cohomology theory or some kind of chao groups for variety with any kind of singularities. And you cannot right now see how additive higher chao groups describe the chao groups of singularities but there is a direct connection. So, our hope is that this theory of additive higher chao groups can be suitably adapted to define a cohomology theory for algebraic variety with singularity, a cohomology theory which can allow it to describe its k theory. That's indeed true. Any other questions? Yes, the field, you did not never assume field algebraically closed. No. Not needed in this? No, it's absolutely not needed. It could be any field as long as its characteristic is not equal to two. We have some problems in characteristic two because to prove like this, the additive chao groups form a differential graded algebra and so on. Sometimes you have to divide by two and that creates problem. Except for this mild problem, the theory goes through everywhere. So, the problem is there that to prove that additive chao groups are differential graded algebra, I have to construct a differential. And to do that, and then I have to prove that the delta square is zero. And to do that, I have somehow divide by two. And that so far, I don't know how to avoid this. So, that's the only place where you use characteristic not equal to two except that you do not have any assumption on the field. Yes. That's right. Yes. No, you don't need perfectness also. So, basically, there are a lot of tools where you can reduce the problem to perfect guess.  So, in the final situation, you do not need perfectness. I have a question. So, you started by saying that crystalline comology was very difficult to compute in the original form. To what extent this new perspective changes that? Well, so one of our motivation for proving this was that it was very, it was not clear that dirambid complex, things like that can be motivated because they are differential forms. And then we had some correspondence with the Spencer block and then basically he was very excited about this and we thought that first let's prove this to say that crystalline comology can be obtained by algebraic cycles. And this is a very recent result and we could prove some applications of this. Now coming back to your question that how much this is going to help understand the crystalline comology, I think our result is too new to say more about this. It will take some time for us to think about it more. Maybe eventually in a year or so we might see some more about these things. So, a lot of work for everybody else as well. Yes. Any further comment or question? In the topological K theory. So, you had the same kind of homomorphism you defined in the beginning of your slides like churn corrector from the topological K group to the diramb comological. Yes. So, the use started by asking question means how can one describe this comology grouping tons of algebraic cycles like this. Can we ask same type of question in the topological K theory also? Well, you know like in topology. So, basically what I have been doing is what I did is I, when you say that when I constructed the map from the RT higher jaw group to K theory this map was the universe to supposedly churn corrector map from the algebraic K theory to some kind of comology theory. We proved that RT higher jaw group is a comology theory and now what is left to be done is to construct some kind of churn corrector from the relative K theory to this RT higher jaw group. We do not have the churn corrector right now but what we did right now is we proved we defined the universe to that because we often know that churn corrector particularly with tensor with Q in isomorphism. So, if you are in a characteristic zero it is known that RT higher jaw groups are automatically rational vector spaces. So, you do not have a tensor with Q. So, then you expect that the RT higher jaw groups are isomorphic to relative K theory via the churn corrector. So, what I did here one consequence was that I constructed the universe of that churn corrector map still we do not have the formal churn corrector from the relative K theory to RT higher jaw groups and remember the topological setup this does not make sense absolutely because there is no question no concept infinitesimal thickening of topological spaces because all topological spaces you take a scheme X and take is reduced topologically they are same. So, topology this question does not make sense at all. This is a kind of nilpotent phenomena which we are talking about which is purely algebraic. In topology this has no role to play because the spaces whether X or X reduce they are topologically same spaces. Any other question? Let's thank Amalindu one more time.


Good morning. My name is IT2, Villalobos. I'm an ACFL here at the Naval War College. Today we're going to be doing a workout that you can end the week on. We're going to be mixing upper body and lower body workouts as well as cardio workouts to work that cardio respiratory system. Let's go ahead and get started. For the first part of this workout, we're going to do a workout that involves upper body and lower body. We're going to be super setting these workouts, which means no rest, which is involving our cardio respiratory system. We're also tricking up the body from not just working upper body at once, but working the whole body completely. So we're going to start with regular push-ups followed by 30 seconds of squats. So I'll begin the demonstration of the push-ups. We're not going to do a numbered amount of push-ups. Your body doesn't know reps. It only knows resistance. So try to do these till failure, till your body gives out. I'll go ahead and demonstrate. Go ahead and get in the push-up position. Remember, hands out. What you want to do, you want to keep your body tight to pack up in the air. Keep everything firm. Make sure your core is firm. And when you go down, remember, do not flare out your elbows like this. You're only putting pressure on your joints and on your shoulders and taking it away from the rest of your workout. Remember, go all the way down 90 degrees. Come back up just like that. So what you're going to do, you're going to do that as many times as you can till failure, till your body gives out. You're going to come back up and you're going to go right into 30 seconds of squats. So I'll do the squat demonstration. What a lot of people do incorrectly is they just go straight into the squat. And what you're going to do is you're going to put a lot of emphasis on your knees and that's why a lot of people have knee pain. So go ahead and start with the glutes already flaring out like you're going into stance, like if you're in football.   What you're going to do is you're going to come straight down. Just like this, come back up.  We're going to do that for 30 seconds straight, no rest.  The whole point of this, we're going to do this three sets till failure on push-ups and the 30 seconds for squats. You're really working your sagittal and frontal planks as well as working your cardio respiratory system, which builds muscular endurance and that's the first part of the workout. So for the next part of this workout, we're going to do another form of squats as well as we're going to do various forms of push-ups. So if you want tips and tricks while you're doing the squats, you can use your household items kind of like a chair or a sofa, just to make sure you're really getting the most out of the contraction of the squat going all the way down. The more you go up, the more you're emphasizing on all those body parts on your legs. So we're going to do some explosive squats, which is going to help us with velocity, speed, and power. So you're going to go get down in the squat just like this. We're going to do four sets of this. You're going to do this for 10 reps. So you're going to go down nice and slow on the squat. And then when you come back up, you're going to explode up like that, just like that. You're going to do the jump in motion. This is going to get your heart rate going, but you're also with going down in the squat and then exploding. You're really working that cardio respiratory system as well as your muscular endurance being built overall. So we're going to do 10 of those. Right after you do 10 of those, we're going to go ahead. We're going to do some planks. You're going to go down into the plank motion. Remember, you want to make sure that your arms are under your chest.  This is the best way to do this. This is the best form to hold the position a lot longer. Make sure you keep your core tight. You're going to hold that for 30 seconds. You're going to hold it just like this for 30 seconds.   You're going to do that for four sets. You're going to do this simultaneously without rest. The whole point is you're also getting your cardio in besides working your legs and your upper body. And that is the second portion of the workout. This will be the final portion of our workout where we're going to be focusing on a lot more cardiovascular workouts. A lot of people are upset right now that the local gym is closed, your base gym is closed. Man, can I use my stair climber? Can I use the treadmill? Can I use the elliptical? Well, you still have the whole area in your house and you can do a lot with your body to get a really good cardio workout. So for the cardio portion, you might want to use your phone, timer, microwave, clock, whatever you want. You're going to time yourself while you're doing these exercises. So what you're going to do first is going to do a minute straight of four-count jumping jacks. So I'll demonstrate and then come to the ready. One, two, three, one. That's one. You're going to do that for a minute straight. After you're done doing that for a minute straight, you're going to go into jump ropes. So if you don't have a jump rope at your house, that's fine. You don't need one. Pretend you have one. Still do the motion. You're going to do that for 30 seconds straight, just like this. Even do everything with the arm movements, everything the same. 30 seconds straight and then after that, you're going to go back into the jumping jacks a minute four-count jumping jacks. You really want to try to do this with no rest. You're going to do this three times. So you're going to do a minute of four-count jumping jacks followed by 30 seconds of jump rope. If you really want to challenge yourself on this portion, what you want to do is you want to do it faster. So instead of just doing it just like this, you can always go faster. You don't have to do the four-count count to yourself. The whole point is that you don't stop for that time. As soon as that minute's over, go straight into jump roping. You can even mix it up. Do what you want. Be creative. Go as fast as you can. The whole point is to not stop. This is going to be building our endurance and you should feel a sweat coming on. That'll be the final portion of the cardiovascular workout. So now that we've completed this workout for today, we've worked out both parts of our body. Upper body, lower body, as well as got in a little portion of a cardio workout. Any workout you see on any of these videos, you can mix and match. You don't always have to do the same thing. The whole point is that we don't want to get comfortable with what our bodies are doing because we'll get into a state of plateauing. So mixing up is the best key for the body to get used to something different. And that's how you get stronger. That's how you get faster. And over time, that's how your body grows. We adapt. We overcome. We might not have the gym right now. Everything might be closed, but that doesn't mean we can use stuff around us like items in our house, our living room, our garage. We can always have a good workout no matter where we're at. My name is Zaiti Chuvialovos, and thank you for joining me on this workout.


Well, today we have tools that we never could have imagined even 50 years ago. We have the ability to see the molecular structures down to the atomic level. We can make images of individual atoms. We can feed a DNA molecule through a little hole and read out the sequence of code that's on the DNA molecule one mark at a time. We have the most astonishing tools that we now have and we're just beginning to know how to use them. We can now buy a DNA analyzer for $1,000 that can do your genome. So this is pretty spectacular. So there's almost no limit to what you could imagine will now be possible. And it's all happened very recently. So it's a very, very fertile place to be, to be in the process of creating our future for the whole human race and for the whole planet Earth. So to me that's very exciting. There's a whole range of things in physics we're learning about too. Not long ago we learned that the universe has dark matter and dark energy in it, which we can't see. We can measure their presence. There's a hope that we'll be able to learn what they are over the next few decades. So huge numbers of mysteries there. In astronomy, where I work, we're learning how the expanding universe has produced stars and galaxies, how we come to be. The history of the atoms that are in our bodies has been worked out to some extent. We know that it's a pretty dynamic process, including supernova explosions that produce the chemical elements, gas that goes back out into the intergalactic space and falls back in, reassembles itself under gravitational forces to produce stars and planets. Then the solar system like ours is pretty young relative to the whole universe. We have some evidence for the history of the solar system, including bombardment of the Earth by asteroids and comets for hundreds of millions of years, and then a quiet time following that where life could possibly occur. The geologists are now working out how that goes and enables life to be here. We even have signs that the first fossils of life occur only shortly after the end of the bombardment by the comets and asteroids. So we're beginning to learn our own history through the physical sciences and the biological sciences and beginning to be able to do the most spectacular things with that knowledge.


And one of the things that when you think about a wall, one of the things that happens when you build a wall is that you trap people inside the United States. So before there was a wall, right now there's a fence across the border and there's a lot of enforcement across the border. So it's very hard to cross the border. It's expensive. It's difficult. It's a risky proposition. Used to be much easier. When it used to be easier, people used to come into the United States, work here illegally, and then return. Or they used to come for the season and go back. Or they come for a few years and go back. But if you build a wall, and this has already been shown to be the case by building fences like in San Diego and places like that, then they come and they stay because they can't go back. It's too expensive, too difficult, too risky. So you're actually increasing the number of illegal, so-called illegal immigrants in the United States by building a wall. Now, I don't have to tell you, walls can be tunneled under. Look at what Hezbollah and Hamas have been doing. And you say, oh, what's the motivation? The motivation is the amount of money that'll charge the people that are smuggling in. Oh, we can catch the tunnels. Oh my God, really difficult. Ask the Israelis how difficult it's been to get those tunnels and to find them very difficult. And Israel has a much bigger incentive than our border patrol does. Very difficult to do. She's not gonna get the tunnels. They can go, they can, you know, the fence, I can't remember the number, but the border patrol spends hundreds of millions of dollars on fixing the fences. But Trump wants to build a wall, a 30 to 50 foot wall so they can't really just cut the fence like they do in normative cases. They'd have to bring ladders and climb over it. How would they get down, somebody once asked? Let me think, oh, wait a minute. Oh, a rope. How about a rope? They could get down using a rope. Okay, so it slows them down a little bit. True, and then you could also dig tunnels underneath. Slows them down. Maybe you'd cut a little bit off of the number of immigrants. Maybe they would find other paths by boat off the coast. We'd have to beef up the coast guard. People will come. And this wall, 30 to 50 feet high. How much does it cost? Tens and tens of billions of dollars. I mean, the five billion dollars that Trump is asking for now is just for a little section. Just for a tiny little section. How long will it take to build this wall? 20, 30 years. Because it'll have to go through courts. The environmentalists will sue over environmental restrictions. Private property owners will sue over eminent domain. States will sue because some of it's state land. There's no way to do this fast and quick and efficient. They will go over the wall. They will go under the wall. They will go around the wall. Right now they're going around because there's 700 miles of wall right now. How well has that worked? Chinese try to wall, right? What happened to the Mongols? Did they get stopped? So there is no justification for building a wall other than Trump promised it. It's one of the dumbest policies ever invented. Somebody asked, what's the other side of the argument for a wall? There is none. The only other side of the argument is it becomes more expensive to get here. That's not a good thing. It's not a good thing. I like immigrants. I want more people, not less. I want us to fix our legal immigration system so that there is no incentive to be an illegal immigrant. Fix the legal immigration system. Allow more immigrants to come in illegally and you will stop, dramatically stop, dramatically stop the amount of illegal immigrants. Somebody says a solid wall hinders preparedness by security forces. This is true because you can't see what's on the other side. If you build it, they can't come. No, if you build it, they can come and they will come and they do come. Walls don't stop motivated people. They don't stop them. It's all about politics. It's all about rhetoric. It's all about emotion. There's no facts around the wall and there are no facts about the disastrous consequence of illegal immigration. There are no facts about the negative impacts of legal immigration, of any immigration. It's just emotion. It's just scare tactics. People who are scared are easy to control. People who are scared are easy to manipulate. The first thing every authoritarian on all of human history does is scare his people because then they can be controlled. There is no security issue from illegal immigration. Again, all the data shows that immigrants are less likely, particularly the legal ones, less likely to be criminals than native-born Americans. So those are my views on immigration, right? I think it's great and I think we should encourage it and I think we should have more of it. And I think to do that, we need to fix our legal system. We need to get rid of illegal immigration by making it legal to immigrate to the United States. There is no issue of immigration. There is no concern of immigration. There's no problem of immigration into the United States, granted. There was a problem in Europe with Islamic immigration. Other countries might have immigration problems. The United States does not have one. Indeed, illegal immigration into the United States has been a decline since 2007. The number of illegal immigrants today in the US is lower than it was in 2007. Net illegal immigration is negative. More people leave than come. One other statistic about the wall, 58% of all illegal immigrants in the United States who come illegally, who have come over the last 10 years into the United States, came by illegal because they overstayed their visas, not because they crossed the border. A wall will not protect us from people overstaying their visas. Won't stop people from overstaying their visas. Currently, does the border need better security? Action Jackson asked. I don't think so. I don't see what the threat is. I mean, I'm willing to change my mind if somebody can show me that there's a legitimate Islamic terrorist threat that could manifest crossing the border, either from Canada or from Mexico.  Haven't seen evidence that such a threat exists. There is Islamic terrorist activity in Latin America, so you have to watch it. And at some point, you might need increased security across the southern border because of that threat. Wouldn't legalizing drugs solve the crime problem? Absolutely. Legalizing drugs is one of the best policies you could have in terms of crime, but also in terms of the bad elements that involved in immigration, the more criminal elements, the gang elements that come in with the illegal immigrants are motivated to come in because of the drug trade. So if you get rid of the drug trade, absolutely. Much of the crime problem is solved. Okay, Taylor asked, I work with people on a daily basis around the world and my company would move them here as quickly as they could if allowed. They have to lobby every year for visas right now. It's sad. Absolutely, it's sad. People talk about jobs leaving to China. Why not bring the jobs here? By being the people here. The problem is we don't have enough, for example, engineers in the United States. We don't have enough certain types of skilled laborer in the United States. Bring them here. And they would. Companies would do that. And what problems would these people create? Name a problem these people would create. And again, don't give them citizenship. They can't get citizenship in their lifetime. The kids can, but they can't. Fine. Would you allow free flow of Germans into US after World War II? After World War II, yeah. After World War II I would have. Not during World War II. Once the war is won, yes. Japanese, Germans. And I don't think there were many restrictions in Europeans. There are many restrictions in Europeans coming to the US after World War II. It was restrictions from everywhere else in the world after World War II that held up. So again, during a war you can have all kinds of restrictions on immigrants. But once the war is over, it is over. All right.


Those are medieval robots. On January 14th, VHS tapes containing footage from the haunted mansion were found in the library room, located in the ballroom of the mansion. It was reported that the footage was recorded during the first occasion of the mansion during 1982. One tape was Incident 82, and the other was Incident 92. The other tape was yet to be found. The first tape was reported to be recorded by Anthony Farnsville during a walkthrough of the mansion, where said mansion reportedly experienced a blackout due to heavy thunderstorms. It was said that during the period of 240 p.m. to 410 p.m., four of the ten guests mysteriously vanished. Surprisingly, audio of the footage was leaked but the audio was very grainy, so people kept digging further. It wasn't until March 20th, 2014 when the entire footage was recovered and entirely repaired with some help of repairing the tapes themselves. Due to mold and mildew in the tape, some parts of the tape were cut halfway or all of the tape had to be cut. So, some things may be abrupt. The tape started with a group of people entering the elevator and the famous dialogue from the hallway starts. You remember the one, right? Welcome, foolish mortals, to the haunted mansion. I am your host, your ghost host. Time misstep all the way in, please, and make room for everyone. There's no turning back now. Our tour begins here in this gallery, where you see paintings of some of our guests as they appeared in their corruptible mortal state. As the dialogue continued as normal, and when the ghost host said, Of course, there's always my way. Thunder was heard followed by a crash, causing the mansion to shake. Everyone was heard panicking and screaming before the tape cuts immediately to the ride continuing through the hallway, with thunder shaking the ground and causing the ride to go uneasy, along with the music getting slightly more and more distorted with each second. Butful dialogue of the narrator was heard with some notable rippling sounds. The camera cuts to a bird's-eye view of the graveyard. Three shadowy figures were shown. Now, people may confuse the three shadowy figures as the ice angels from the holiday variant, but the variant didn't open until 2001. As the camera followed one of the doom bodies, something happened. The audio stopped. The music stopped. The atmosphere stopped. Instead, a near-rapingly loud evil cattle was heard, followed by thunderous thunder clap.  It was reported that the projector faces suddenly caught on fire. One of the workers had to use a fire extinguisher to stop the fire. If that wasn't bad enough, four of the people were reported missing after a check-up from the workers revealed an empty doom body when there were four people in the doom body before. The only reported thing from the mansion was a transcript of a report dating August 30, 1982. Status, doom bodies are stationary. All guests on the ride. Four missing. Cause of death? None. Left? Maybe. At 7.50 pm, guests of the mansion entered and things seemed fine until the graveyard scene where four individuals vanished. To this day, no one has surfaced at the moment. On December 30, 1987, the first tape was given by one of the workers of the haunted mansion, Anthony Gulliver, who reportedly buried the tape under one of the tombstones in the cemetery. Along with a few secret Disney memorabilia, the tape along with the second tape was buried itself. It wasn't until January 14, 1996, the second tape wasn't covered after a freak accident involving one of the pieces of the ride breaking down and smashing into a tombstone. The tape was heavily damaged but the reel itself was recovered by two unknown individuals who were revealed as Walt Disney Imagineer workers. The tape wasn't recovered until February 20, 1998. It was fully recovered by archivists and workers at Disney who said themselves that people needed to see what really happened at the haunted mansion. Update. It turns out that the two WDI workers were named Jonathan Price and Nathan Blakefield, and the archivists were from a secret Disney group called the Disney Recover Archivists Society, also known as the Drass for short. The Drass were best known for recovering old Disney reels and known for recovering some of the test footages for movies during the period of time. As of right now, the Drass had split in 2004.


So ladies and gentlemen as I'm the last speaker before lunch I'm going to ask you to please stand up and stretch your legs for a moment. Everyone if you could stand up Thank you, and I'm going to ask you to also stretch your arms and make this gesture This gesture is a sign for equals Okay equals is the global partnership to bridge the digital gender gap and we make this sign Because we all need to work together to bridge the digital gender gap Okay, you can put your arms down and sit back down. I thank you Excellencies ambassadors distinguished guests Ladies and gentlemen and most importantly the girls a Very warm welcome to all of you And we'd like to thank you so much for joining today's girls in ICT celebration This is my first girls in ICT Celebration since I took office as director of the telecommunications Development Bureau of the ITU in January It's also the first time that I have personally celebrated girls in ICT day here in Africa On the UN official calendar girls in ICT day always falls on the fourth Thursday in April And I can't imagine a better place to be on this fourth Thursday in April than here in Africa here at the African Union Commission and here in Ethiopia What energy what? enthusiasm and what entrepreneurial spirits This continent stands poise on the brink of a new wave of social and economic development and information and communication technologies ICTs are going to be the engine of that growth Thanks to ICTs for the first time ever we truly have new ways to tackle old problems ICTs have the power to overcome the tyranny of distance and bring services to communities That have been for far too long marginalized Marginalized from higher education Marginalized from routine health care From access to libraries and other sources of information From financial services from commercial opportunities Marginalized from the basic foundations for economic growth that other parts of the world have long enjoyed and Have taken for granted for far too long I believe that we have reached one of those inflection points One that will overturn our old ideas of what might be possible of what might be achievable the internet of things artificial intelligence big data and Innovative new online platforms have changed the game and What we've seen to date is just the tip of the iceberg Africa's digital transformation will open up vast new opportunities for a continent with so many untapped assets and Africa's young people They will be the ones that will be the major beneficiary and here in Ethiopia As you know 60 percent of the population is aged under 25 50 percent under 18 and That huge youth Demographic is something that African nations can harness Because we all know that young people are natural users of technology My 17 year old daughter is here in this room Anna Now myself as an ICT professional for over 25 years, I think I'm sort of tech savvy But Anna can do things much more quickly Five times faster than I can when it comes to technology She hasn't had any special ICT training Although I forced her To do some coding weekends But it just comes naturally to young people They understand technology and they get it in ways That older folks like myself just don't and That's why girls and ICT day is so important To the 250 girls here in the audience today.  It's about showing you that studying technology Will be a huge asset to you regardless of what field you choose to pursue In these days, it's hard to think of an area where ICTs don't play a role Journalists use ICTs to instantly amass global news to write their stories Art historians are using ICTs to analyze lost painting techniques to identify unknown artists and to date ancient works of art Lawyers use ICTs to help them go through decades of legal precedents Archaeologists use ICTs to find new treasures Doctors use ICTs for diagnosis and care of previously untreatable conditions Physicists are even pushing the boundaries of human knowledge by using ICTs to photograph unexplored phenomena. I don't know how many of you saw This photo which made very big news in the scientific community just three weeks ago It's the first ever image of a black hole in outer space Until now Nobody had ever seen a black hole. It existed only as a scientific theory This photo has made history as one of the great astronomical breakthroughs of the 21st century and Here's the scientist who wrote the algorithm that enabled this photo to be taken Are you surprised? Yes, well, perhaps you shouldn't be Because women have actually been instrumental in many of the great scientific breakthroughs of our day Women were actually the first programmers of the first IBM computer Margaret Hamilton wrote the code that helped guide the Apollo 11 moon rocket Nicola Pello wrote the code That helped to turn the internet into the worldwide web as we know it Today's girls girls and ICT day program is about showing you that tech can be fun It's about showing you that tech can be incredibly creative It can pretty much be Anything that you want it to be because you understand it much better than we do And that means you can use your imaginations to harness tech to do whatever you want it to do Technology can help solve some of the most chronic Greatest challenges that we have on the development front and later today You're gonna be hearing from some Ethiopian graduates who are gonna tell us about how they are harnessing Technology to help accelerate the achievement of the United Nations 17 sustainable development goals Ladies and gentlemen girls distinguished guests and friends I'd like to leave you with this thought Right now in Africa only 18 percent I repeat 18 percent of women have access to the internet Just imagine Imagine the development potential if those millions of unconnected women were online just Imagine the transformational impact on the education of Africa's youth on Family health care and on the economic opportunities that a new generation of female digital entrepreneurs could tap into We don't have to imagine it we can make it happen you can make it happen I Believe that technology is at the very heart of the African Renaissance So I encourage all of you girls here today all of you to play your part To play your part in building the networks and services that will transform Africa The future is really yours. I thank you


I'm Gabrielle and today I'm here at Oxmer Toyota. You guys, I have a rare vehicle to go over. This is the 2022 Toyota Corolla, the SE, and it's the Apex. The exterior color is in the cement. It's going to have the black sand pearl roof. Now with the Apex edition, you'll have that sport tune suspension.  You'll have 18 inch black alloy wheels. There is the black bronze side rocker panels right there underneath the doors.   So extremely sporty looking exterior. Now moving into the interior, you're going to have the moon stone. There is going to be an eight inch touchscreen. You'll have six speakers. There's going to be Android Auto and Apple CarPlay. And lastly, you'll have the paddle shifters and the sport drive mode. All right you all. So again this is the Apex in the cement exterior color. Absolutely beautiful. Now if you have any questions, if you would like to stop in, please let us know. All of our contact info is right here. We look forward to hearing from you and have a wonderful day. Bye.