 So I want to thank the organizers for inviting me to give these talks. It's a great honor to speak here and So I want to talk about something that I call perfectoid spaces and Let me first give a sort of some general introduction to What perfectoid spaces are so the general aim somehow to get a comparison between objects a Mixed characteristic zero P was objects in Equal characteristic P and somehow the basic theorem somehow is so the idea is to get some more in geometric and nice version of the theorem of Fountain and Winter-Berger Which says that the absolute gara groups the absolute gara groups of Qp to which you for example there are many other choices you join all P par roots of P So mixed characteristic field and of the long series field Over Fp are isomorphic and in some sense even canonically isomorphic and So one sees some of that after joining lots of P par roots to a mixed characteristic thing You get something that in some ways looks like an eco characteristic thing and so we generalize this to the following so we want to have some kind of geometric objects over this Mixed characteristic field and some kind of geometric objects over this eco characteristic field and compare them and so Let me just as notation put k this field and K flat This field Well, some slight variations of this or someone and then It becomes true that Certain category of so-called perfectoid spaces over k is canonically equivalent to the category of the same kind of spaces over This eco characteristic field. So I denote the flank function by X maps to X flat and so let me state some general properties of this equivalence To give you some idea of what's going on. So first of all perfectoid spaces are locally ring topological spaces and So in particular one can talk about the underlying topological space and in fact that's just the same on both sides and Even better not only the topological spaces are the same but even the etal topoi are canonically equivalent So somehow if you just take X to be the point Spec K or something then this will exactly re-prove somehow the serum of Fontaine-Vincent-Berget In fact, we will need this as an ingredient, but somehow This serves as a generalization somehow to the relative setting of the serum of Fontaine-Vincent-Berget Some other another property is that this isomorphism there between the categories is somehow of a deeply analytic nature So in some sense there are some limit procedures involved in this isomorphism And so we have to leave the world of algebraic geometry and instead we have to use some language of Richard analytic geometry and There's a reason that they are called perfectoid namely the objects are Perfect in a certain sense So meaning that if we are in characteristics P and you could characteristic P Then all the rings that occur are perfect and Mixed characteristic We have a similar condition So guaranteeing some of that there are lots of p-power roots So one thing that we see from this is already that the rings are that we will deal with are highly non-Esserian So perfect rings are basically never an Esserian and so it's maybe technically not so easy, but Okay And I want to add some words about how perfectoid spaces are related to varieties. So for example in characteristic P We have a function from varieties over K to perfectoid spaces over K so okay, flatten and So that's the notice by X map maps to X perf and in fact It's clear that this functor Factors over the category of Richard analytic varieties. So RIT's over Okay flat so X mapping to X rig Which then maps to X perf somehow first you associate the identification to X and then you make the structure perfect basically and So These two are sufficiently close to another for example, it's true that the atal topos of this associated perfectoid space is just the same as the atal topos of The associated which an analytic variety and in particular the atal comority of the two objects will agree But there's no such function characteristic zero So someone characteristic zero. There's no canonical way to Join P power roots in general Nonetheless, there are some favorable situations where one can still Give nice constructions of perfectoid spaces also in characteristic zero. So let me give some examples of perfecto its places in characteristic zero Which also somehow gives a flavor of what such a space looks like so one thing one could do is one takes The a1 over K and say it takes the associated widget analytic variety Well, I will take some slightly different thing but think of it as the rigid analytic variety and then It takes the inverse limit over the piece palm at the T is the coordinate on a1 So of course for a1. There's some of canonical way to join lots of people roots. That's what we do So more generally for any torque variety we can do the same We can take the inverse limit Or the map phi which is on monoids given by multiplication by P of the Widget analytic variety associated to K and then in this Say first example One can even describe the tilt so I didn't say this so this Isomorphism between the equal and mixed characteristic Series is what I call tilting. So in this case the tilt of this inverse limit It's just the same thing constructed in characteristic P But there are some other cases where one more or less has a canonical way to join something like people our roots and say A of a case and be a variety Then you can look at the inverse limit over multiplication by P Of the associated which is an analytic variety and again, this is a perfectoid space and At least if a has ordinary reduction one can describe tilt again as the inverse limit over multiplication by P over some Right other variety a prime over this other field where a prime over K flat you can describe it via ser Tate canonical parameters Described or ser Tate canonical Parameters Okay, and there even more examples so you can take over the ring of integers and K. You can take some pity with a big group Somehow from here on it's all in progress and I haven't Checked in which generality exactly I can say what which things but For ordinary reduction somehow It's the easiest For good ordinary reduction. Yes. Yeah, sorry. Yeah, so if X is a pity with a big group then one kind of again take It's generic fiber where I say it's a form of the pity visible groups is supposed to be something like the open unit ball And then again, you can take the inverse limit of a multiplication by P and then this also related to some so-called banach con mess spaces occurring in p at a coach theory and Also the modular spaces some of the beam varieties of pity with the big groups have this nice property. So if you have The shimura variety depending on a level then you can Take the inverse limit of all levels at P of the corresponding widget and only stick variety or You can take Xk K running through compact open subgroups of some group g of Qp a rough-porting space and Can look at the inverse limit over K of X K So this means that in this book I mean it's been a problem for a long time if you have say the lumen T tower that you would like to look at it at the Infinite level and now it turns out that at the infinite level this clearly becomes a space namely a perfectoid space and so for example one sees that One can take the inverse limit of the dreamfell tower and this really knows Canonically isomorphic as a perfectoid space to the inverse limit of the lumen T tower So there's also Jared Weinstein working on this now. So as perfectoid spaces Okay So that's some of the general introduction So these perfectoid spaces are really some are rigid analytic spaces, but of an infinite type So they're not the Syrian not anything and in particular for example. I don't think that One can describe perfectoid spaces as a category of certain form of models modulo as some So sort of admissible blow-ups because these flattening techniques seem to require them to see in this hypothesis So we have to Look for some other approach to which is analytic geometry and the one that I choose is a Huber theory of attic spaces and in the first lecture I will basically now explain what an attic spaces and compares the theory to other Existing series and rigid analytic geometry so Throughout let me fix so the definitions of holding a non-archimedian field K is a topological field whose topology is induced by a Non-trivial valuation of rank one so throughout let me fix a non-archimedian field K and There's an extent essentially unique norm map from K to the positive wheels and Let us fix one which is not really necessary for who bus theory, but for comparing them with other series with my maybe helpful Okay, so some of some basic idea of Widget analytic geometry Is somehow it should be an analog of the theory of complex analytic spaces over C and so there should be a functor Well, I mean attic spaces now There should be a functor from the varieties over this field K to Now the category of attic spaces Over K sending X to XR and Somehow on this attic space it should now make sense to talk about the subspace where function is somehow bounded by one or something so and sort of The following should make sense some of four for all functions on your space Something like this should make sense so in particular One basically sees that any point has to give rise to some sort of valuation map. So so, okay, so there are now three approaches which Have this requirement. So some of this approach of rigid analytics geometry Take you take X rig, which is a set of closed points of X so somehow if X so if X in X is a closed point and Somehow there's a and say this is just a fine Then we get so this corresponds to some X corresponds to some and so Maximum ideal so we get a map to arm or damn which is some finite field extension of K and The norm on K extends to any finite extension. So we Get a natural map to the positive fields and this gives the valuation corresponding to this closed point Yes, I wanted to say this but I forgot. Yeah, I assume throughout that all my rings are complete Some of the whole theory will not change when one replaces A ring by its completion, so I take the freedom to assume that they are complete Okay, then there's Birkowitz theory where one considers certain norm maps from During our to the positive fields So extending the valuation which you already have on K So this gives you more points and I will later an example describe these points but There is some of something still more general that one can do When the laws basically all possibly higher-rank valuations, right? I will explain the comparison to other theories later, but Again in the situation that I will perfect your spaces. I don't know whether these comparisons still work so Let me start by defining what Hubert calls the valuation. So let R be some ring then a valuation on R Is a map Which I mean Hubert writes It's just usually be called a norm, but I stick with who was terminology of calling it a valuation So it's a map from R to gamma union zero Where gamma is some totally ordered to be in group so for example, it could be the positive fields and Satisfying the following requirements So The way you have zero zero the value of one is one the value of X plus y is at most the maximum of the values of X and Y and So this is that's multiplicative. So the absolute value of So some I think he calls it valuation because one One is used to talking about higher-rank valuations, but one is not used to talk about higher-rank norms, but Okay, so And Now far as the topological ring as it will be in the cases that we are interested in Then this map is called continues it for all gamma and gamma the set of all X and R such that the absolute value is less than gamma This is open right necessary. There must not necessarily the value group, but I Will now define a notion of equivalence which somehow does not see the value group. So So in this context that we make several definitions so we have lead gamma absolute value subset of gamma he's a subgroup generated by all Absolute value of X X and R such that they are not zero and What else do I need? An absolute value has its support which is a the set of all X and R which have absolute value zero this is an prime ideal and The last thing I need to say somehow any valuation factors as You first project to the quotient by the support and then You can even take As this now integrals the quotient field of this and then there's a Norm not to gamma you mean zero from there and Let me call this K then the norm gives some a rise to this whether you field K, but it additionally gives rise to the valuation suffering of K, which is a set of all X and K such that the absolute value is at most one so this subset of K is a valuation suffering and Now we say that to valuations Norm and norm prime are equivalent if the following condition equivalent conditions are satisfied So one way to say it is that There is an isomorphism between these subgroups Such that The obvious diagram commutes so we have our mapping to this is anything But there are other ways of saying the same thing so The obvious diagram commutes so we have our mapping to this is anything But there are other ways of saying the same thing so It's also equivalent to saying that the support is the same and The valuation the ranks that they define are the same Which then of course live in the same portion field third condition is that the absolute value for A and B and R Come on the absolute value A is Bigger than B, but only if the same is true for the other valuation Okay, so and Some of the topological spaces that will underlie these locally ring topological spaces are now certain equivalence classes of continuous valuations on these rings and And who might define such spaces and very great generality, but We will not need these utmost generality and I will specialize them into the case that is of interest to us now That's what he calls the Tate K algebra This is a topological K algebra R and one has to put some assumption on how this topology looks like and The following assumption is put by humor. So he assumes that that's their existing suffering Are not in our such that a times are not was a Invertible element of K such that this forms a basis of open neighborhoods of zero. So for example, you can take our The ring of convergent power series and several variables so Meaning the set of those sums A I 1 up to I so a G to the I I bigger than zero using multi index notation Such that the coefficients go to zero And then you can take are not inside it just as the suffering of Those power series with integral coefficients and I forgot to introduce it cannot Is a set of all elements which have absolute mayor at most one? no, it's no and Second is that an affinity K algebra is a pair so somehow Spaces and who were set up are not associated to a single ring, but to a pair of rings and So this is a pair our R plus where our state and Our plus In subset of our up or not which is to find to be a set of power-bounded elements in our Is an open and integrally closed suffering but in fact in all cases of relevance and widget analytic geometry it will be just The ring of power-bounded elements It's not of great importance and The last part is that Such a thing is called this is of topologically finite type. So I just write TFT If R is a quotient of Such a thing for some N so that's just a kind of algebra that appear in classical rigid analytics geometry and also R plus is really just And again one need not put the assumption that ours complete But it does not change the theory of on dust so we assume From now on that our plus is complete okay, and Now we can define the topological space associated to such a thing. So Be a phenol K algebra No, no, no it does not imply it's a it's a condition I've put this condition So some of there is a general definition of what a topologically finite type algebra is and in this case some of it boils down to this condition If you're K algebra Okay, so then we define space called the attic spectrum of our R plus and Defined to be the set of all Continuous variations There's a property that on these integral elements. They are at most one and you take these up to equivalents Okay, so this is some set and We want to end out with the topology. It's a topology can be defined in the following way. So First for any x and x we write f max to the absolute value of f at x for the corresponding valuation and So this apology spaces of open neighborhoods Given by the rational subsets and they are defined as An open subset depending on certain functions f1 fn of mg and It's a set of all those x and x such that for all i The absolute value of fi of x That most the absolute value of g of x where The f's are certain elements of R that generate R and g is just any element So one if one is used to Berkowitz a series and one might wonder that one puts a non-strict inequality here And this is supposed to define an open subset, but I will give a justification for using a non-strict inequality here later so Okay, and In fact It turns out that This topological space Is somehow very scheme-like in a sense That's an ideal. Yes. Well, you can just always add it at the g to the f's because this condition for g is always satisfied Okay, so this is a somewhat scheme-like topological space So it really feels like doing algebraic geometry if one is working with them and For this let me recall the following definition A topological space x is called spectral It's the following equivalent conditions are satisfied the first is that X is topologically isomorphic to the spec term of some ring Okay, so it is very scheme-like The second condition is something which does not somehow uses This strange definition it can be written as a fine as an inverse limit of finite not spaces. It's still maybe not very concrete And there's another way of saying it is that X is quasi-compact I would just write Qc and has a basis of quasi-compact open subsets basis for topology of quasi-compact open subsets Which is stable under intersections and every irreducible subset has a unique generic point Every irreducible closed subset is unique Generic so and for example one can take the rational subsets In definition three Okay Well, somehow there is a paper where this equivalence is proved and this proceed paper proceeds in the following way first of all for any finite not space it constructs some example and It does so in a functorial way and then you can just pass to the inverse limit that There's I just don't think there is any relevance of this ring, but Well, there's no there's no ring no no but somehow This implies that X is in particular quasi-compact quasi-separated meaning That's the intersection of To quasi-compact opens is again quasi-compact open. It's again quasi-compact And it's t naught so it has some nice properties Some of which will be used later and in particular There's also a good theory of constructible subsets some of this topological space Okay So more over Somehow this space is somehow not empty It could be that there are no valuations at all and then it would be ought be very interesting, but So it is large in some ways so for example if X is zero then R is already the zero ring Secondly if F is an element Such that the absolute value of f of X is non-zero for all X and X Then F is invertible and certainly if F Is an R such that it's absolute values at most one everywhere then F isn't the suffering of positive elements and Okay, now we have associated to some such a finial k-algebra some topological space and now we want to endow it with some structure sheath and so Now we want to define the structure sheath and the definition is as follows so it's some more enough to define the structure sheath on rational subsets so and Let us choose are not an R as above so somehow such that it somehow defines the topology on R and Then we consider the following algebra we so both of these are somehow sub-algebras of R where you invert G and we equip it with a topology making Bases of neighborhoods of zero and then we let or X of you or X plus of you is a completion and One may wonder about the notation as the definition obviously involves some choices but we have the following proposition is that first of this is a As defined it's a an affinity k-algebra again and secondly It has the following universal property For any complete we have always assumed complete, but here it really becomes important For any complete affinuate k-algebra SS plus With map from our plus to SS plus such that the corresponding map on Spaces Factors over you then there exists a unique map from O X of you O X plus of you to SS plus some are making The obvious diagram commute and hence it really does only depend on you No, I don't yet, but it is true. No, I Mean this is universal property and the statement only involves the subset you and not nothing else. Oh Yeah, you're probably right. Yeah, and sorry, okay and so and now for any you and X open define or X of you to be The inverse limit of all W contained in you, which are rational of O X of W and Same for X plus and then there is a basic problem that It's not in general known That this O X is actually a sheaf. So there's only a pre-sheaf here But there's a foreign serum of fuba and it's probably should not be true in general But there's a following serum saying that if Any such ring of convergent power series on an open on a ball somewhere over your R is in a Syrian fall in then O X is a sheaf. So some of the basic problem. So this is satisfied If ours of topologically finite type The basic problem is somehow that Completion is not so well behaved when you are in a non-Syrian situation And later we will somehow be trying to associate spaces to such so-called perfecto at algebra which do not satisfy this Serum, but we will still be able to prove that this O X is a sheaf Topologically ring wave achieve So somehow the maps are strict or so some of it up here Which rings I'm not sure If it's topologically finite type then it's of course just a usual case. So our deals are close. Yeah, yes Okay, let me Say a few words before making a short break. So About these pre-sheafs O X and O X plus in general. So We're back in the summer Even if O X is not sheafs the following state true for any X and X the variation Extends to O X X the local ring and O X X is the local ring and O X X plus is a set of those F's and O X X Was absolute value at most one so it's a valuation subring No Somehow then you have the residue field K of X of this local ring and You have the image of O X X plus plus and Then this is a valuation suffering of course so somehow at each point of this attic space you get a residue field and in this residue field you also get a corresponding valuation suffering and In fact is through that for all you and X open This sheaf or X plus can be reconstructed from O X somehow by Saying that it's those functions such that for all X and U The absolute value is at most one and so in particular before any X we get some O X a certain pre-sheaf O X and certain maps The absolute value at X for any X and X so certain triple and then definition is that if O X is a sheaf Then this is called an affinity at X space and Okay, let me make a break here for say 15 minutes Let me continue. So what we have done up till now is we have started with some Kind of topological algebra and we have associated some kind of local ring spaces to them And now we want to glue them and for this we consider as a category of triples As above so we have some topological space X some O X and some valuations for any X and X so where X O X is a locally ring topological space, but in fact O X is a sheaf of topological algebra's and F maps the absolute value of F at X is a valuation on the local ring and it's even continuous variation and They form a category in a natural way and then we define an attic space is such a triple That is locally on X affinity is such a triple. So finally we have defined one and attic spaces and So somehow one has a category of such complete Let me stress this affinity K-algebra's and in there one has the ones Where our X is a sheaf these now are now equivalent to the affinity at X spaces over K and Of course, they are then a subcategory of the category of attic spaces of okay Okay, and now let me draw a picture in a simple example so we take the ring R to be just convergent power series on the disk in one variable and As the integral suffering really just the power series with integral coefficients and then the picture looks as follows We have some points then we have some point here then we have lots of arcs as in the Birkwich picture summer then it can also end somewhere in the middle and Around this point, let me zoom in so we have this point and we have some arcs going out and Additionally, we have one point here for each arc going out of this thing and the same thing happens summer at each other branching point so Let me enumerate all of these points so I take the same enumeration as Birkwich uses except that there will be some additional ones at the end so first of all so So I assume that K is complete and algebraic enclosed for simplicity so First we can take a point of absolute value at most one then We have some apps that sense the power series to the evaluation at X Which is then some element in K bar and then you can compose with the norm Sorry These are some of the points which I draw on the outside here So then there are the points of type 2 which is some for example such a branching point or Anything else which lies on this ray so Let me do both of them together. So again you fix some point in K bar of absolute value at most one but additionally you choose some real number between zero and one and then you can Produce what is often called the Gauss norm. So you send The power series which now you rewrite centered around Centered around X and you sense this to the supremum of the absolute value of a n times r to the n so and It turns out that this can also be written as a supremum overall Why such that the absolute value the distance to Exes at most are of the absolute value of f at y so somehow here f gets mapped to the absolute value of f at X So in particular this depends only on the disc around X of radius r So this means that for any disc inside this picture We get a new point and somehow any point X is connected with a ray Which is the interval of length one to one point which is some of the same for all points because this disc of radius one will always be the same and Somehow for any point you get such a branch which goes from this point to there But they start to merge at some points depending on the distance then there are some rather strange points If you have a decreasing sequence of discs Such that the intersection of all the is empty such things can exist for example for CP and Then we get a point F maps to the infirmum over all I of the supremum over all X and di the absolute value of f at X so somehow Going the other way some are from The interior now to the outside it may happen that somehow You get stuck in the middle here somewhere But the most interesting points now are those Which they have type 5 which are the ones that are Around this point there for example and they we get them in the following way we take the following Totally order to be in group. We take the positive fields But we introduce the second factor which I call gamma to the z. So as a group it's just a copy of the integer of the integers and Would require that This new variable gamma less strictly between any real number less than one and one so far Then we can send such a power series to the maximum Of absolute value of a n times gamma to the n and again this defines a point and Somehow this is sort of of the same sort as in two or three it but except that we are now taking this Parameter our infinitesimally close to one. So this means that Somehow on this line here it lies infinitesimally close to this point here. So that's what I wanted to draw here and So we have a similar picture around any point of type two so I should say what type two means mm-hmm to corresponds to the case that R is in the value group of K and case three corresponds to the case that this is not the case and Then one easily checks that somehow a branching occurs at some point on such a ray if and only if it's of type two And at such branching points one gets for any ray going out one gets an additional point of type five Okay, so it seems that there are lots of points and maybe One is no clue of what they're good for but I will give now Several other ways in which this topological space naturally appears Somehow in nature so First let me give the theorem that compares widget analytic varieties to attic spaces and it's the following So there is a fully faithful function from the category of widget analytic varieties over K to the category of attic space over K so X maps to X add and It maps the widget analytic variety associated to some usual Tate algebra to this attic thing so for any R not of topologically finite type okay, and One may wonder what the image is and of course one has to put Finiteness assumption on on those attic spaces so One would expect that the image of this should be exactly in the attic spaces locally of finite type over K So meaning that it's locally Spa are are not Where are not this topologically finite type That this should be some of the image And obviously the image is contained in it, but in general there are some Strange ways to glue attic spaces which are not present in the widget analytic world But as soon as you assume that your space is quasi-separated you get an equivalence so But basically everything is quasi-separated so it's not such a huge restriction and So now let X be a quasi-separated widget analytic variety Then the following nice statements are true So we're genetic varieties carry a natural growth in deep topology of so-called admissible covers and admissible open subsets So and it's a nice thing about a spaces is that you get a very convenient reinterpretation of what an admissible open is It turns out that the set of quasi-admissible opens in X is Just the same as the set of all quasi-compact opens in the associated attic space Sending any You summer to the intersection of you with X and it's a commerce function. I denote by UMAP so you till that So I recall that summer any classical point gives you an attic points on this way You can consider the classical points as a subset of the attic space and so This intersection makes sense and what's even better is that Certain UI and X cover from it admissible cover if and only if certain The associated subsets of X of the attic space cover X at just in the usual sense so in other words the language of admissible opens and admissible coverings is just Reinterpreted just as opens and covers in the attic world. So very nice and so it follows that sheeps over the growth in the topos of X are just the same as sheeps over the underlying topological space of the attic space and In fact this already determines X at uniquely if you make a slight technical assumption that it should be sober So there is somehow Some out of the theory of admissible subsets. It's even possible to directly cook out cook up this attic space And let me also give an example of how you see this not admissible in the example above so a typical example of a non-admissible cover would be to cover some of This disc which we consider so basically the subset of all X Which have absolute value at most one so you can cover this by the set of all X which have absolute value one or Plus a union over all reals less than one of the set with the absolute values at most are But this should not form an admissible cover because left-hand side should be connected and so what happens in this picture here is that Somehow the subset where it's the absolute value is equal to one is somehow this thing and Then the other thing gives you increasing balls here. And if you look at this now at this point Then you will get larger and larger subsets here. You will get this is subset But this one point is not in there Precisely because some of the absolute value of X which is this gamma is strictly between One and zero for all are less than one It's a tree or it's not the total what what is the topology finally well summer Well, I mean it's generated by these rational subsets But for example, if you look at this subset here, then this is topologically the same as a spectrum of The a1 over the residue field copper. So copper is the residue field of okay So meaning that this is somehow a generic point and it specializes to all the other points which are somehow from the close points of an fn line and so For example, if you just take out this one point, this will somehow be open at that part Okay, and there is a second way to construct this unaligned topological space of an attic space, which is to use the following serum of Renault is that If you are only interested in quasi compact and quasi separated which is analytic varieties. Oh, okay Then each of them admits a formal model may saying a quasi compact admissible form a scheme Over some of the ring of integers, which is cannot But of course such a form model is not unique and any tool somehow You have to localize by the category of admissible blow-ups Meaning blow-ups that only change the special fiber in a sense So let me you know this function by X maps to x rig and Using the equivalence for rigid analytic varieties That's all the same as a quasi compact attic space locally of our finite type over K so it gives you also an attic space and Then there exists a continuous specialization map from the unaligned topological space of the attic space to X and So it's continuous not anti continuous as in Berger which is setting and If you take X any given quasi compact quasi separated attic space locally a finite type Okay Then it admits formal models and so the following does make sense you can look at the Inverse limit of all formal models of the underlying space of X So the specialization maps give you a map to this inverse limit and it turns out it is just in homeomorphism So how does this look like in the example? so you can one formal model would be just to take the spiff of This thing and then the special fiber It's just the a1 over the residue field and How does the specialization map look like so you have This generic point and then these strange point around and then some other bunch of stuff And this just maps to the a1 over K, which is again this point by contracting all rays Okay, and Hence you see that Such a subset for example should be closed and If you think about it and this precisely tells you that you should put non strict inequalities to define Open subsets and strict inequalities to define closed subsets and Also summer what happens in this procedure some of you have one formal model, which is the a1 and Then you're allowed to do some blow-ups and so this introduces some p1 here and then some additional p1 and so on and so this builds up to three summer and they introduce many more p1's and summer the strict transform of any such p1 summer Survives up to the inverse limit and it gives you precisely some of these p1's which you have around any such point of type 2 so some of the points of type 2 are precisely the ones that specialized to a generic point and some form a model and then somehow get always mapped to this generic point and Yes, okay Why the dual graphs there's no dual things Yeah, but I mean you can really stop a lot really says what it happens without sort of talking about you or stuff and so on And somehow Yeah, some of the attic Language is compatible with the formal models in a very nice way For example, this Specialization is continuous But even more who defines attic spaces the much more general setup which would also include formal models for Widget spaces and so and then passing to the generic fiber as in the sense of Renault really is passing to the generic fiber somehow so okay, and So finally I want to say a few words about the comparison with Burkowitz spaces and for this I need the following definition an attic space X is called out th u t Ta ut If it's quasi separated and for any quasi compact open You and X also the Closure is quasi compact. So that's some technical condition, but It turned out that I mean most most spaces that you will encounter in nature are taught but There are now some examples which occur in nature where you have really non-toward attic spaces and Then we have the following serum is that there is an equivalence of categories between the category of Hallstorff strictly care analytic Burkowitz spaces with the category of Taute Richard analytic work taught here Richard analytic right is for example Okay, or then what is some of the same as? Taute attic spaces locally a finite type So sending some of a Burkowitz space that we denoted by X work to the attic space so okay, and Then there is a total logical map from the underlying topological space of the Burkowitz space to the attic space But this is not continuous So somehow the image is just the subset of rank one valuations But there is a better map. So there's a retraction from the attic space to the Burkowitz space Which identifies the Burkowitz space with the maximum house of quotient of X so continues no Identifying maximum Hallstorff quotient Of X art And so How does this look like in the example Okay, so let me draw it again. So So you have this in the attic world you have some of this and then the whole tree and again some strange points here and As a Burkowitz looks basically the same except that it doesn't have these strange points So you just have some of this tree Nice that it is and Okay, and So this Contracts any point of type 2 with all points of type 5 around Okay so I Mean and then it has to summer because this Burkowitz spaces Hallstorff and some of these points of type 5 lines in topological closure of this point so any map to a house of topological space has to contract them and That's precisely what it does So in a sense one can somehow reconstruct these this Burkowitz space from the attic space Okay, and so the last thing for today is to explain somehow the fibers of this of this map to the Burkowitz space more or less so or in other words giving a different interpretation to the points of an attic space and For this let me assume that access a phenoid and in fact we can go back to the Complete generality not even assuming that the x is a sheep or no, so necessarily a sheep okay, so so for any point x and x We have associated somehow a pair consisting of the residue field of x and a valuation suffering and This gives rise to the following definition that an affine rate field so someone classical algebraic geometry points are more less maps to fields and so in the attic Geometry the point will more or less be a map to an affine rate field so an affine rate field is a pair kk plus where case a non-archimedean field and and in k plus in case an open valuation suffering and Again, we will often is only work in the complete case, but These are not in general complete so somehow. No, I mean, yeah, I should say the power bond elements Okay, so then we have the following Proposition That our plus be an affine rate k algebra and then the points of this attic space arm by Jackson with maps are our plus to kk plus where so where kk plus is an affine rate Field complete affine rate field and I mean that I mean you can always make some of the field larger and to exclude this case we sort of assume that The quotient quotient field of the image of r and k Stints and so somehow in Berkowitz's theory Point is simply given by a map to a complete non-archimedean field and here we sort of additionally have to remember this valuation suffering in there and so Somehow The fibers of this map to the Berkowitz space are precisely somehow parametrized by the valuation sufferings inside of this corresponding non-archimedean field and Using this description of points. It's also easy to say when one point specializes to another point in this Edit space. So let me recall the definition of specialization of points say x specializes to Y which is written x somewhat greater than y if y is in the topological closure of x and then we have the following proposition that If x y are in The edict spectrum of r r plus then corresponding somehow to maps k k plus or k prime k prime plus then x specializes to y if and only if first of all they give rise to the same non-archimedean fields or some of You have such an isomorphism here commuting with a maps from r in some of the valuation were suffering corresponding to x is more general in the sense which means that It's larger So so this way And so we see that In particular any non-archimedean field admits the set of power-bounded elements in k as Variation suffering and this is the one that corresponds to the unique rank one valuation unique rank one valuation and one sees that summer funny case as the unique generic point so Which specializes to? any other point for the same x therefore the same field k and for any x and x corresponding to a rank n valuation the set of all Why such that why specializes to x? It's the wrong way, but okay, and so this forms a chain of Lengths precisely in so in the example summer there was This generic point and the special points around it this one was a rank one valuation And there's only one point above it in me itself So the chain is of lengths one and for the other point summer There's precisely this one point which specializes to it and you get a chain of lengths to which is also rank of this valuation so that summer Well as a general picture about attic spaces and that's it for today