 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. This is probably a well-known story to all of you, but it's worth repeating, I think, Srinivasa Ramanujan, who is the mathematician whose name, the prize, is given. 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. So, growthandic topology with objects at triple T U delta where U is a jaroski open subset of your given a scheme X and T is a scheme of finite type over W and K such that U sitting inside T is a closed immersion. 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. So, more precisely Bloch constructed a chain complex of Jaroski sieves like this where the sieves are basically given by, you see that it's basically some kind of miller K theory. 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. Let r be an essentially a finite type regular semi local K algebra. 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. First you prove perfect and then you use the lot of tools and then you sort of, you build it up and then you eventually remove this perfectness condition. 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.