 has given several different contributions to Schiff's theory. So today is going to talk about the Schiff's on T-topology. First of all, thank you very much for the invitation. And OK, I will talk about Schiff's on T-topology. I will start with a brief introduction, and then I will recall some basic notion that Bolivia also recalls in lecture about Schiff's on topological space. And then I will restrict Schiff's on T-topology. The idea is that in this setting, we can consider more objects. And we can study from really more objects and work with them. And after that, I will talk about a couple of applications on algebraic analysis. So I will see how, thanks to these new objects, we can construct enlarging a little bit of topology. We can deal with, we can construct functorially some analytical objects, which are very important. And then also, I will talk about some application to differential operation operators. So I will say that I will explain how enlarging a little bit the category of analytical object, we can distinguish a different kind of differential operators. It cannot be distinguished with a classical Schiff theory. OK, as I said, a Schiff theory is used in several mathematical contexts and has an application in various fields of mathematics. In the same context as semi-algebraic geometry, sub-anithic geometry, and a minimal geometry. And each of these theories has been very useful in various fields of mathematics, but each of these theories has some elements which was missing in the other ones. For example, in sub-analytic Schiff theory, we didn't have the notion of fibers or stalking fibers. In a minimal geometry, we didn't know about the operation or didn't know about how to construct acyclic objects. And the theory was to find something to generalize these various constructions. And so a media, which was introduced by Kashiwara and Shapira, but I think it is known by other mathematicians as well, was to consider less open and less coverings. So a very easy notion of site and extend the classical machinery of Schiff's to this context. And in this way, we can give a first, a fine description of the various constructions, so the sub-analytic, semi-algebraic, and the minimal one, using the notion of t-topology. So I recall briefly what is a Schiff. Again, I'm interested in Schiff's of vector spaces. So we consider a topological space in the field. A per-Schiff is a correspondence, is the data, corresponded from the open subset to the category of vector spaces, to each open associated vector spaces, and to each characteristic distribution, to each inclusion, we associate a morphism, which is the restriction morphism, which is compatible with the inclusions of open subsets. And that means that it is a contravariant function from the category of open subsets over topological space, where the arrows are the inclusions, to the category of vector spaces. So given this contravariant function, we can define Schiff, which is a per-Schiff, satisfying some gluing condition, which is represented by this exact sequence. That means that if we have a section of an open subset, which is zero on a covering, then this section is zero. And if we have a family of sections on a covering of an open subset, which glue on the intersection, then we can find a global section extending all of them. The classical example are continuous functions, because all we know that if a continuous function is zero on a covering, then it is zero, and then we have a gluing condition on coverings. So these correspond to define this basic example of a Schiff on a topological space. But there are objects which are interesting, but do not define Schiff's on a usual topology. And this example is this one, for example, bounded continuous functions. For example, on the real line, we can find coverings, infinite coverings, and bounded functions, which do to a function which is not bounded. For example, we can consider the covering of R, even by the open subset minus nn. Of course, each continuous function is bounded of minus nn, but it is not a bounded, not every continuous function is bounded on the real line. For example, the x to the x square. So its correspondence is not a Schiff. If we consider the usual topology, we have to change something. And on topological space, we have the notion of Stokes, which is represented by this inductive event, the inductive limit, the inductive limit on the section. The stoke on the point is given by an inductive limit or the section on the neighbors of the point. And the element of the stoke are equivalence classes. So an element of the stoke is represented by a section on a neighbor of the point. And two sections are equivalent at the level of stokes if they coincide on a neighbor of the point x. Stokes are important because thanks to them, we can check if two shifts are isomorphic. We can check if sequences are exact. And we do everything on Stokes. And the definition of a topological, a topological space, if we look at it, only depends on the notion of open subset and the notion of covariance. And we can slightly generalize this notion by choosing only a sub-family of open subset T and for each element of T, a sub-family of covariance satisfying some suitable hypothesis. And in this way, we can define a site, a very easy type, a very easy example of site because we just take less open subset and less covenants. Arrow size are always the inclusions, so we only have one arrow between two objects of T. And in this case, we can define pre-shifts as a contravariant factor from this new family T to the category of vector spaces. And we say that a pre-shift is a shift. If it satisfies the gluing condition, only on this restricted family of open subsets and covenants. So we still have the exact sequence of the gluing condition, but it is restricted to a smaller family of open subsets and a smaller family of covenants. Why we choose less open subsets and less covenants? Because we can consider more objects For example, we have seen the example of bounded continuous functions. And if we take topological space, we consider all the open subsets, but now we consider only finite covenants or better covenants, which admit the finite refinement in order to attain the site. And we consider the correspondence we have seen before from U to continuous bounded function of U on U. In this case, if we consider finite covenants, we can glue the bounded function to a continuous function, but we can also take the max of the bounds. So we obtain a bounded function with glue all the continuous function on the finite covenants. So if we consider less covenants, we have one more element which becomes a shift. In this case, a continuous bounded function becomes a shift. So with this idea in mind, we try to extend the notion of shifts or subanalytic or minimal semi-algebraic and other kind of sites. So we consider a topological space. And on this space, we consider a family of open subsets which is stable under union and intersections, such that a difference of two elements has a finite number of connected components and such that it is a basis for the topology of the topological space X. And starting from this, we defined the site XT where the elements are just the elements of this family of open subsets. And the covenants are the topological covenants admitting a finite refinement. So I wrote finite covenants in this sense. So it is important to have finite covenants. Examples of this family T are, for example, the family of semi-linear subsets in RN, so the open subsets which are defined by linear equations. The family of semi-algebraic subsets of RN, the subsets which are locally defined by algebraic inequalities. And the family of relatively compact subanalytic subsets of a real analytic manifold, which is the subanalytic site XSA. And this example I will use later for the applications. And more generally, if we have a minimal structure, the family of open definable subsets of this minimal structure. All these sets are stable under union intersection and have this nice property of having a finite number of connected components. So if we fix this family T of this property, there is a nice category inside the category of classical shifts, which is the category of coherent objects, which is the category of shifts, such that there is an aprimorphism from a finite sum of objects of T and such that for each morphism, if we find a sum of constant shifts on T, the kernel also satisfies this property one. So there are shifts which have a finite resolution on this constant object U. This is a full balance of the category of the category of shifts on X. And if we consider the natural functor of sites, which is obtained by the inclusion of the family T and the family of the open subset of X, we have a functor of direct and the inverse image from the category of classical shifts, topological shifts to the category of shifts on the site XT. We have direct and inverse images related to them. And the direct image is also a fully faithful inclusion, exact inclusion of the category of coherent shifts in the category of shifts on the site XT. So we can really see this category of coherent shifts as a subcategory of the category of shifts on XT. And we have an additional description of this category of shifts on XT because it is equivalent to the category of coherent shifts on T. And that means that an element of shifts, the category of shifts on XT, can be seen as an element of an ind object, as an ind coherent object. So a limit of a fantasy from a coherent to a set. And this category is also isomorphic to the category of left k-additive contravariant functions from the coherent shifts to the category of vector spaces. So a shift on XT can be seen as a contravariant function from coherent to mod k. Thanks to this, we also can construct in a very easy way the shifts on XT. So if we have a pre-shift, so if we have a correspondence from the formula contravariant function from the family T of open subsets to the category of vector spaces, it is easy to check if it is a shift or not. We assume that the input set is sent to zero. And we only have to check gluing conditions on couple of objects of this family T. So we have to check that this sequence of gluing conditions is exact only on couples of open subsets. If the second is exact, we can say that f is a shift on XT. For example, on bounding function, we just have to check that two bounding functions glue on a couple of open subsets, which is quite straightforward because we take the max of the two bounds. So in the case of shifts of T topology, we don't have the notion of thoughts because we are in a grotting topologies. So we cannot see if two, for example, if two shifts are isomorphic, only checking fibers on the points of the topological space X because even if this family T for my basis, so if this limit is actively the limit on the neighbor of the point X, the stocks on the points of the topological space are not enough to check if two shifts are isomorphic or not. Because, for example, if we consider continuous and bounded continuous function, we cannot distinguish continuous functions only checking or on stocks because on stocks any continuous function is also bounded. We consider the limit of neighbors on a point because any neighbor contains a component. So if one wants to consider only stocks on classical points, we lose information about the shift. We need more points. And these small points, we take these small points considering the spectral topology associated to this family T. So we consider the prime filters of elements of T. So filters such that the union, if the union of two objects is inside, one of the two is the filter. And if U is an element of T, we consider the U tilde of the filter such that U belongs to that. So in some sense are the filters containing, the alpha contained in U. And a neighbor of a prime filter is, this sense is U tilde with the U tilde element of T contained in alpha. And we call the X tilde the associated topological space. And in this space X tilde, the open subsets U tilde are quasi-compact. So they admit every covering, admit the finite sub-cover. I give an example because when X is equal to R and T is the family of finite units of open intervals, we can classify the points of R tilde. We have five kinds of points. The first one is the filter of U containing X. And this corresponds to the real points of the topological space somehow. Then we have the points which are represented more or less by left and right neighborhoods of the point X. So the point X plus and X minus. And we have to add the points at infinity. So if you want left and right neighborhoods of the infinites. So you see that we have much more points in the X tilde than in R, 30 line than in R. And thanks to these new points, we can distinguish effectively continuous function from continuous bounded function. Because for example, we can consider the continuous function one over X. Then it is not clearly, it is not bounded on each open subset of the kind of zero epsilon. That is a continuous function on zero epsilon. So it is not bounded on each neighbor of the point zero plus. So it is not X minus one. It does not belong to the stokes of bounded function on the point zero plus. But it is, it will also the stokes of continuous function on zero plus. So on the point zero plus continuous functions and bounded functions are different. But they coincide on each point of the kind one. So on each, if you want classical point of R. And if you want other example, for example, that function f of X equal to X, it is a continuous function, but it is not bounded at plus and minus infinity. Stokes at plus and minus infinity are different. But we have much more. One can prove much more. And sorry, X is a topological space, not the Riana 19 manifold. The category of shifts on the site XT and the category on topological space X tilde are equivalent. The category of shifts are the same. Level of topos, they are the same thing. So if we consider the category of shifts on XT and we want to work on stokes, we have to consider the topological space X tilde. And this is useful because sometimes to check something or to make some operation, we really want to consider fibers, stokes or fibers. And we have a good topological space making the job. Another important construction we can do on this site XT is the subcategory of T-flabby object. T-flabby objects are shifts such that for each couple UV, which U containing V, the restriction morphism is subjective. And this is very useful when we want to find the resolutions of funtors because this family is injective with respect to the functor. On F with F coherent. And this family includes all the section on opens. And with respect to direct images as well. Of course, once you have defined a morphism of topological spaces from X to Y and which is compatible with the family PX and TY on the topological space. So these T-flabby shifts are enough to derive and to find a resolution which are a C-click which are injected with respect to HOM and to the functor F-star associated to a continuous function respecting family's T to topological spaces. Okay, so thanks to this construction we can have a global approach generalizing subanalytics, semi-algebraic and and all these construction are useful in all these settings. But now I want to focus on the subanalytics side and talk to you about some constructions we can make on this setting. Thank you. Which have been part of my research. So we consider the subanalytics side as I told you before open subsets are relatively compact open subanalytics subsets and coverings are covering or relatively compact subanalytics open subsets admitting a finite sub cover. Okay, so we have the idea is that we can consider more object as the basic example of a bounded function in this setting we can consider for example, temper functions. So function which have a polynomial growth at the boundary of an open subsets subset infinity functions. Again, this is not a shift with the usual topology as in the case of bounded functions. So you see that when you want to consider condition at the boundary of open subsets functional spaces we depend from conditioning the boundary of an open subset they usually not define a shift if you consider the usual topology because we lose this condition at the boundary. But again, in this case I give you an example of this factor. For example, if we consider the real line we consider e to the x one over x and this function is a temper it is even bounded on each open subset un of the x between one of n and one with n natural but if you consider the interval zero one which is the union of this un the function e to the one over x is still well defined it's still a continuous function it's not tempered because it at around zero near zero it grows so fast so the growth is not polynomial anymore so it does not define a shift if we consider the r with the usual topology but it is possible to show that if we have two open subset which are subanalytic and we consider the intersection union and intersection we consider couple of open subanalytic subsets we have the exact sequence corresponding to the gluing condition so we have gluing condition on couples open subanalytic subsets and as we have seen in the general case if we check that a pre-shift is a shift on couples open subset of t in this case relatively compact open subanalytic subsets we obtain a shift so temper functional shifts are a shift on the subanalytic site and moreover in this setting we can also define the six grottendig operation we have all tensile product direct image and inverse image which are always well defined on a site but for a subanalytic shift we also have a proper direct image and one corridor and now let's talk about funtorial constructions so we see how using this funtorial operation we can construct analytical objects which are important we consider a real analytical manifold dimension and a family of submanifers and we suppose that they intersect transversely as a basic example we consider c2 and zi equal to 0 i equal to 1 or 2 and we need to construct the notion of normal deformation when we have an analytical manifold we can construct an analytical manifold x tilde m1 and an application from this x tilde m1 to from x times r satisfying its property d minus 1 over x minus s1 m1 is isomorphic to x minus s1 m1 times r minus 0 d minus 1 of c is isomorphic to x for its different from 0 and d minus 1 of 0 is isomorphic to the normal bundle of m1 we consider the restriction now of m1 to d1 different from 0 and we define m tilde 2 as the inverse image of m1 and we perform again the normal deformation of x tilde m1 with respect to this m tilde 2 we obtain another map and we define recursively the multi-normal deformation along the family i of submanifers as the x tilde m which is one with respect to m tilde we consider the 0 section of this deformation the inclusion s little s the set omega given by t1 to tl greater than 0 and we get this commutative diagram we have the inclusion of the 0 section and the set omega and we have the projection from x tilde 2x starting from this diagram we can construct the functor of multi-specialization which send a shift to an element of the 0 section of the normal deformation we take a shift we take an image by p we cut on the sets of parameters greater than 0 and then we restrict to the 0 section and I show you an example on dimension 2 to understand what is this specialization which corresponds to the setting considered by Majima when he studied asymptotic functions x tilde m1 is given by c2 with an extra variable and the projection is the multiplication of this new variable t1 on the first variable then we perform again the deformation with respect to m2 which is given by the coordinate zeta 1 0 t1 locally and we can construct x tilde adding an extra variable and the projection is the multiplication of the two variables t1 t2 z2 in this case the 0 section is equal to the product of the normal bundle over 0 on the normal bundles on m1 and m2 this is the construction we can perform on c2 and now we consider a shift of witness infinity functions which are functions which are bounded with the bounded derivatives on a relatively compact subanalytic open subset this is again a subanalytic shift on x we check a couple of subanalytic subsets of course we still maintain the bound on the function and on all the derivatives and then we consider the shift of witness holomorphic functions holomorphic functions which are bounded with bounded derivatives from these shifts we can construct funturally the shift of multi asymptotically holomorphic function multi asymptotically developable holomorphic functions applying to the shift of witness function the function of specialization specializing the shift of witness functions again when it is equal to c2 if we consider the witness holomorphic function c2 we can apply to this shift the functor of specialization and if we check on fibers it is the limit of the fibers of witness holomorphic function of the product of sectors containing a direction points of the product of this of the product of two normal bundles are direction when outside the origin and this shift this funtorial construction correspond to an analytical object which is very well known which are magemas asymptotics which are constructed starting from an asymptotic expansion which is given by this family of some sort of formal power series with holomorphic coefficients which is an approximate function and we say that a holomorphic function is only asymptotic redevelopable if and only if for any poly sector properly contained in a poly sector S we have this estimate the difference from S between F and the approximate fashion is founded by the power of Z1 and Z2 this is for each a sub sector containing this we have this classical analytical definition of asymptotic expansion and it is possible to check that a function is asymptotically redevelopable if and only if on each poly sector a each poly sector properly contained on a poly sector it is written so it is bounded with bounded derivatives and that means that the chief of asymptotically redevelopable functions can be constructed by specializing the chief of written holomorphic functions so if we know written holomorphic functions we can talk about asymptotic asymptotics on sectors and starting from written holomorphic function we can consider other subanalytic shifts which are the shifts of a flat written holomorphic functions and written holomorphic functions on Z is the the union of of the submanifers and we can prove functorially that we have this exact sequence of specializations and this is a this corresponds to the boring late exact sequences and we can probably we can prove it this sequence functorially by studying the of flat functions and written functions this is the exact sequence because when we consider the case of asymptotics we can check that asymptotics are the specialization of written functions the flat asymptotics are the specialization of written function flats on the axis and the third term corresponds to the consistent family of coefficients of magima and the fact that the sequence is exact corresponds to the classical boring rate theorem which says that if we have a family of coefficients we can find the function which is asymptotic to the family so we can obtain a general boring rate exact sequence for multi-syntotic developable function just for fixing the notation the family of consistent family is is given by in situ the family of F1, F2 and F12 such that F1 and F2 are function on the first and the second variable which are asymptotic to this F12 which are just coefficients and as a last application of some analytic shifts we can want to see how adding new objects on the category on adding new analytical objects we can distinguish differential operators so we consider the shift of things of differential operators locally a section is differential operator with holomorphic coefficients and holomorphic functions are a shift of the model so this is d x on holomorphic functions by derivation and multiplication of coefficients and we can define given the model on the ring of differential operator the complex which is the complex we obtain applying p to gamma and this complex have a homology and degree 0 and 1 and the degree 0 corresponds to the homogeneous solution of the differential operator and each one is the co-kernel so we have the solution of ps equal to g if you want and we see that two operators are equivalents if kernels and co-kernels are isomorphic so that means sol fp1 and sol fp2 are quasi isomorphic an easy example if we consider two operators zdz minus alpha and zdz minus alpha plus 1 if alpha is different from one we can easily verify that the solution of these two shifts are isomorphic because we can the solution of the first one is z to the power of alpha and to the second is z to the power of alpha plus 1 we won't go on to the other side or deriving and then dividing by alpha plus 1 so these two complexes are isomorphic and the operators are equivalent but for example if we consider these two operators zdz plus 1 z squared z plus 1 they have z minus 1 and x z minus 1 as a homogeneous solution so if we consider meromorphic functions and enable of 0 then the h0 of the solution of zz plus 1 is c times z minus 1 but for the second operation the h0 view is 0 because the exponential is not meromorph these two operators are not these two operators are not equivalent even if in this case the holomorphic solution are because x of z minus 1 is holomorphic in general if we define the definition of a regular operator a regular depending on the order of the terms and the derivatives of this fundamental theorem saying that if p and q are two operators which are regular as 0 in the neighbor of the region they are equivalent if and only if the holomorphic solution are quasi isomorphic so holomorphic solutions are enough to show if two differential operators are equivalent or not if they are regular so these of course have a huge generalization in the Riemann-Illberg correspondence by Kashiwara and Nebuchut what about if an operator is not regular in this case subanalytic shields can be helpful we consider the subanalytic shift of temporal holomorphic functions I said before that temporal holomorphic is infinity functions are a subanalytic shift now we consider holomorphic functions and they have a structure of a you have to add this lower restrict in order to allow multiplication with the coefficient it does not create any problem and we consider two irregular operators so they do not satisfy the if we check with the definition I gave before they do not satisfy the definition we consider z squared z plus one and z cubed z plus two the solution are the exponential x prime of z minus one and x to the x minus two and this is not really a theorem it is easy to check that there are open subanalytic sets containing the origin of the boundary such that hx a x of z to the minus one is a temper x of z to the minus two is not temper for example in the left half plane so in particular that means that the h zero of the temper solutions of z squared plus one and z cubed z plus two are different because the first one is given by x of z to the minus one times a constant and the other is zero because on this open subset it exists z cubed z plus two is not temper anymore so we can distinguish these two irregular differential operators thanks to temper the holomorphic functions and this is not possible with the holomorphic the two exponentials are well defined in the set where in a set not containing the origin and this can be generalized because there are some particular differential ordinary differential equation of degree one which can play important role in classification of differential equations which are the one which have as a solution the exponential of p z where z where p is a polynomial z minus one z to the power of one c of z different from zero all these operators are irregular and for all these operators the holomorphic solutions are not enough to distinguish them so we cannot distinguish them using the sheaf of holomorphic functions but this is a theorem of morando if these two polynomials are not multiple by a constant so for large family of polynomials the holomorphic functions are enough to distinguish the operators so we can find open subsets such that the solution on one exponential is tempered and the other not so thanks to temper the holomorphic solution so going to the subanalytic site we can find an object which now can be represented as a sheaf which are tempered functions and thanks to this object we can distinguish irregular differential operators we cannot distinguish with the holomorphic solution but we can distinguish a very big family of irregular differential operators and I want to add that this problem was completely solved with a very huge technique which is why Daniel first have to add one variables and then go to the category of enhanced sheaves this is a very technical process but you can solve completely the problem of distinguishing irregular differential operators and irregular differential operators and now just final slides about because I told about these topologies generalize a minimal sheaf theory as well and when we consider minimal sites we can perform all the construction but also we can develop the formalism of six grottendick operation thanks to the fact that we can work on fiber and for this a for various kind of minimal structure so we can define the six grottendick operations and we can define the six grottendick operation and thanks to the grottendick operation we can develop a very important formulas and computations so as I derive a projection formula the derived batch change formula coonet formula and develop the computation of these homologies and in this case it's different from the case of subanalytic shears where we could really work on the site and then compare with the classical sheaf theory where a lot of this formula were state in this case we work very work on fibers we really work on the associated topological space we work on fibers and the fact that topological space are good because they are normal we can perform on these constructions and ok thank you very much for your attention your attention thank you so much