 My talk will have maybe three parts. The setting, I mean the first part will be what it is by I divide setting. Then I will discuss some maybe results in functional transcendence. I will close up some questions and results on algorithmic aspect. So it will be more, it will be more a description of some results, some methods. I would have very few time for proof, I'm sorry about this, but I try to give a general picture. So if I start with by algebraic setting, I'm interested in the situation where you have a transcendental map between algebraic objects. And I will give three examples and I will work with only two. So I should say that I will just look at the geometric side of this and not the number theoretic side. So the first example is just the exponential map. The second that I will more discuss is the uniformization map of an obedient variety. And also I will discuss the situation where I have an Hermitian symmetric space, maybe realized as a bounded symmetric domain in some C-victor space, if you want. And I look at the map from D to S, where S is a Shimura variety. And the situation, the Hermitian symmetric space. And gamma is an arithmetic lattice. So it's not completely clear what I mean by a map between algebraic objects. The first thing is that S, for an obedient variety, there are some hypotheses to be algebraic, but I make this assumption. For S, in fact, it's always true, but I would just say that S is quasi-protective. So D is not at all an algebraic variety, but at least D is a real semi-algebraic and complex variety, which is a kind of lot of structure. And so maybe this is a torus. I will mainly discuss the Hermitian case and the Shimura case, but we could make this old description for a torus like this. Okay, so from the geometric point of view of bi-algebraic setting, your interesting objects are coming by a bi-algebraic object. So a sub-variety of say, okay, maybe V, or say T, A, or S as previously, is bi-algebraic. If a component of pi inverse of V is algebraic. So I have to explain algebraic in the case of Shimura varieties, so maybe I give a definition. Some W inside D is said irreducible algebraic, if it's an annihilating component of W tilde intersected with D, with D tilde, algebraic in this olomorphic tangent space, C power N. So in this situation W is a real semi-algebraic. Now you're talking about the case of D, not the case of C tilde. No, it's all right. Yeah, I take the intersection of some algebraic sub-variety of the olomorphic tangent space with D. And when I take an irreducible component, and I say that this is an irreducible component. So components mean reducible components? Irreducible components. What do you mean by components? Yeah, in the analytic setting, there is a notion of components. And in this situation, you have only found actually many components. And the components are semi-algebraic and complex. So this depends on the embedding of D in some... So D, okay, I said here that it's an emission-symmetric space, but I said realize that it's not in all this terms a bounded symmetric domain. No, but you have different realizations to the related biolomorphic maps and not the semi-algebraic maps. Okay, so I have different realizations. A lot of what I will say will not depend on realizations, but for this talk, I stay with this particular realization of the bounded symmetric domain. So this notion does not depend on the realization? I mean, the notion of being bi-algebraic doesn't depend on the realization. There are a lot of things, but the main theorem will not depend on the realization. But I don't want to discuss it. But if you perturb the realizations by a semi-alomorphic way, there is no reason that something algebraic would remain algebraic. You can always... An isomorphism, okay, but still it's true. So if you have an isomorphism of realization, automatically it has to respect the D of R action. And then all this biolomorphism are in fact semi-algebraic and complex analytic. And you can verify that all the notions are in fact independent of the biomorphic class of... So, okay, but I don't want to discuss realizations. I just start with the bounded realizations. Okay, so example. An algebraic superiority of A is bi-algebraic. It's the same thing as saying that B is P plus B for P point and D-elevation superiority. It's the same thing as saying that B is a totally geophysic variety. And we give a name for this. We say that B is weekly special. So when we are in this station, we call this weekly specials of varieties. And for... For chimura varieties, V inside S is bi-algebraic. And only if V is totally geophysic. This means that either V is sub-chimura variety or there exists a sub-chimura variety of the form S prime is S1 plus S2. S2 as a product of chimura variety and V is S1 plus the point of S2. And by definition, we call this weekly special. Why do you make two names for this kind of definition? Why two names? Three names. Why you don't work with just bi-algebraic? By definition, so this is just bi-algebraic. Weekly special is one of the three things. I don't prefer one or... An emotional context. An emotional context. You have to prove it. Sub-chimura definition. Maybe this is the natural definition, but I'm not sure. It depends what you want to do. Okay, so now I want to speak about transcendence results and explain what I mean by flows or somehow. So, the static point. Okay, so I should say that this part is due to Moudin and maybe you're heard by yourself. And talking with your desic is related to some Riemannian structure or you don't care which? No, I care which. I don't explain more. So the first result, the setting I want to describe is the following. You look at the case where you have a uniformizing map of an Abelian variety and I start with W in CN, an irreducible Ademite variety and then the result is that if I look at the Zeinski closure of Pi of W, then this is really special. So you can check that it's more precise than this. So this result somehow was proved a long time ago by Axe and a new proof using hominimality was given by Zanier and Pilar and it was important for a new proof of learning method which could be extended to the case of Fimura varieties. And we finally proved an hyperbolic version of this statement with Fimura and FF. So the statement is the same, and W inside the irreducible Ademite then the Zeinski closure of W is really special. So I don't want to comment too much on this. The proof uses hominimality theory in a very serious sense and some hyperbolic geometry inputs and it was important for the results on the health projector. So when you are in this situation, I stated in this way on purpose, you have two natural questions. So what about for interesting subset of CN or D? So mainly to give examples, I could look at F from C to CN, a neuromorphic map and W is F of C or the value is F of C intersected with D or for example a real analytic sovereignty. So we say something. And the second question which is even more natural for me, if you start with W, the interesting subset of CN or D, what can be said? The usual topology. So I should say that this is a certain new question in transformable functional and this is more a question in algorithm theory or topology because the Zeinski closure is a very crude definition and here you really want to see where your image of F of W goes and it's more... So when you are used to this kind of question, you know that generally you cannot prove anything about topology without using measures so you have to reformulate the question in terms of measure. So I explain this. So in the Abelian case, I can use the invariant when one forms on CN and in the Shimura case, I have a Bergman metric, it's also a G of R invariant one-one form, a positive one-one form on D and when you have a complex analytic sovereignty of CN or D, we can define for all R, positive, a measure on a W, R or S, such that for all F, let's say it may be continuous. Okay, I say one of the three. The way you compute this measure, you take the integral on the ball for the local metric with the W times omega at the power R, R is the dimension, the complex dimension of W and you divide by the volume. And now the question becomes be easier or we can limit U, W, R as R goes to infinity. Okay, so first I want to describe some functional transcendence result with natural order. The following theorem by Blokosia is a quite old statement and I can formulate it in this way. So it concerns a variety. We do take the limit after projection because it's easy to define a measure on CN and then you want to project it. And I take... Yeah, it's really a... Yeah, it's a gamma invariant. Okay, Blokosia, you start with another variety and a nonomorphic map, you look at W is F of C and the result is that if you take the Zariski closure of pi of W, it's beneficial. And the proof is using Nevenina theory. So one statement I want to do is a hyperboleic version. I don't... it's so good to do this. Okay. Hyperboleic Blokosia. So with FIF. So it concerns a Chimora variety. So you look always to this situation and you start also with a nonomorphic map from C to CN and W, you take F of C intersected with D. And for what I know I have to assume that S is projective. And then the conclusion is that the conclusion is a little bit... a component of the Zariski closure of pi of W is with this position. So in terms of method the main part of the proof is a... you have to assume something. Okay. I assume it's not empty if you want, but if not... You mean that every component is a little... Any component? Any component. Every or any component. But you can also take components because W is not connected. You can also take components of W. I mean W is not connected and you could have infinitely many components. Absolutely. The Zariski closure, you have infinitely many components. Each of these is with this position. And the proof is a minimal theory. And so it's somehow the same method as our proof of the hyperbolic ax in the matter, but we use also the result a lot. We have a short... I mean we use some vanilla theory but for a very small part. Okay. So... Okay. Maybe the last thing I want to say on functional transcendence it's a very simple result also. I did this one. And I would like to say that three days ago I received a very simpler proof and started to go. So it's really... And so the result is the following. Okay, it's really... So you start with an analytic variety definable some full minimal structure and unbounded. Then if you are in a rebellion case and are... the result is the second time. I mean all method is using all minimal theory and the P-W-T counting theorem. But they have been able to give an argument without even the P-W-T counting theorem. But okay. So maybe it will be important for generalization. I don't know yet. And... Okay. So... The W is not... You start with just an analytic... Yeah. Does it include some... No. Component. The same one. Sorry. Any component. So there's a kind of generalization of the experiment. Yes. So it's definable including all the break. Yeah. It's definable. Because you don't say anything about the song. Oh, I mean you work for example in Anex for any minimal structure. The only thing you need to know is some minimal structure containing Anex. Maybe I'm not... Does this also include this part? So I'm not able to do it. So I have to say that I'm not able to prove this statement using the minimal theory. Only. And I'm not able to say anything about this in the Shimoakis. Okay. So... The more on the algorithm theory side are... So algorithm... Okay. So... Just to start I think I will give several examples of what we are aiming to do and then a more general theorem. So... And the example should explain the kind of picture you have to understand. So all the examples will be a billion surface and the first thing you can show could be a consequence of the statement if you start with a curve inside C2 say the curve is, for example, hyperactive. Okay. Then the kind of result you can prove is that for all ladies I mean I'm not sure I'm using this but such as 8 is a million dollar idea. If you look at the usual topology you find that pi of C is everywhere dense and that in terms of measure this tends to the high measure under the invariant. And this is independent of it's true for all of the invariant. This is the risky closure? No, this is the usual closure. So it is not true when it is completely closed possibly but only when it is not. It's a statement for hyperactive curve in C2. I will give a general description but in this situation you always find that so I give a second example maybe explaining more what kind of trouble you can have. So now I start with the question of square root of minus one i plus z of square root of minus one i plus z of square root of minus one i. So my identity is in fact the product of C2. And then you can prove the closure is a 0 cross E and in terms of measure mu C r goes to one half and if okay to continue on this example if a dual lattice gamma hat has an element of the form a zero a or b zero then phi of C bar is level in by T goes to mu A and okay so this is the generic situation and okay so just to give a last example to see the kind of trouble so it's even simpler somehow what a gamma hat so it's a dual lattice so it's the set of element in C power two such that the scallop product the European scallop product with any gamma is easy. C2 of a gamma I take a very explicit so I assume that I have two basis vector E1 E2 and my lattice will be minus 5 for example E2 and I take for W just a complex line generated by E1 plus E2 and then phi of W bar is the real toss dimension 3 and the one when you look at V is our E1 plus E2 square root of minus 1 E1 square root of minus 5 E2 and this torus is V divided by gamma intersected with V which is a lattice in V and this is the torus so it means that you cannot expect for this kind of project to avoid real objects real and antique objects okay so okay so now I can state the general result I have in mind this is the result produced example so really what do you mean the gamma hat which gamma gamma is the lattice you are talking about what do you should take any lattice in c square oh you mean you are talking about the situation arbitrary arbitrary situation so I just just to say that the result is I will state is quite explicit you can compute things so is that doesn't mean so you mean A is not I so you need to product you look at first no you can have some situation where you can have some situation where you have only you have some element of the form A0 in the lattice but no element you can change the structure I think there is some spring to gamma Z1 but Z1 is a square bracket maybe you should multiply Z by by 5 not 4 okay so it's Z of Z times square root of minus 1 yeah you are right Z times square root of minus 1 Z times square root of minus 1 oh you could just yeah I could okay so it's a product of two political so the general statement is referring that A is Cn divided by gamma let C be a curve an algebraic curve in C power n so let C1, C4 be the set of branches C all points of infinity for all one R we can define there exists a translate T alpha which is P alpha plus T prime alpha a real sub torus T prime alpha I want to say so I can give a definition a bit later so I call this the asymptotic non-forte torus associated to C alpha so it's something you can really compute in fact such that the first thing pi of C a bar it's just the image union of all this real asymptotic tori Cn of 2 UCR we click on this to a linear combination a positive linear combination of the measure the R measure on this tori I should say that in the general situation all the T alpha equals 8 you should say how this is related to Casp the closure of the germ will be I can so what you say is that when you have a branch at infinity you fix a point at infinity in your branch and there is the smallest real sub vector space associated to your curve and which intersect as a lattice and this is essentially this torus sorry what is branch of C you take you define C star closure C star finite set of points and when you look locally near a point at infinity and you look at the connected components once you throw the point at infinity and you call this branch so what is the relationship between the translate torus and the branch it all intersects ok so the statement is offering that C alpha a real branch a real branch you do some partial some pi then a smallest real so the pi is the same so the pi alpha is the pi no you cannot you can have several branch through a point at infinity you have this point at infinity alpha is a translate by some point yeah ok stop the same ok sorry ok so there is the smallest real nothing sub space q alpha plus w alpha such that w alpha intersect with gamma is a lattice in w alpha and such that t alpha is asymptotic then what does it mean exactly so it means that the distance then to 0 when you go to the point at infinity you take a point on the curve on the branch you take a point at infinity and you you say that the distance the usual distance close to 0 and we say then we say that t alpha which is pi of q alpha thus is the asymptotic associated sphere when you use the one appearing here and you can compute it in terms of the duality so very precisely so my time is almost over so I wanted to say that my main interest was in trying to do this kind of thing in the hyperbolic case this said the only thing I was able to do using algorithmic theory it's the case when I start with some totally geodesics of the IT of D I take the image and then the usual topological closure I can describe them in terms of group theory so this is the only thing I can do but I'm for example completely unable to do if I look at say the delta cross delta in C cross C I look at the map and I take any curve intersecting I don't know any example this could be reasonable and you want to understand pi of pi of C intersected delta cross delta and this seems to be quite difficult I mean the the proof of all this is really using harmonic analysis and oscillatory integral and you have almost no chance to do it by harmonic analysis here so you have to expect some more things like atmospheric theory and I'm not able to do it at the moment I cannot but so there are a lot of questions that I'm not able to solve in this setting so I guess we first ask some questions questions from Sanya Sanya but there are no questions from Sanya no questions from Sanya so we will ask people in Tokyo what is the sum of questions what is this asymptotic amount for torus sorry what is asymptotic amount for torus asymptotic amount for torus so so I should say that the month of torus is the first natural thing you can define is you look at the smallest affine subspace containing the curve of C the linear part intersects the lattice and this gives you a first upper bound for the topological curve and this I call it the month of torus and generally in the generic situation this upper bound is a but when you push more analysis you have to look at the point by branch at infinity and for each point at infinity and through any branches through a point at infinity you can define the asymptotic month of torus in this way and this is the one who is governing the you said that if there is some notion it's out of the system what is it so the notion is that you have a curve inside cn or anything inside cn so assume 0 is in c for something what I want to say then there exists the smallest real subspace so you consider this a real subspace of r2n such that the intersection bkma is a lettuce and c is included in this that's really the notion of an unfortunate and this you call it an unfortunate torus if 0 is not in c then I'm going to bring back to 0 and it's it's the first I mean when you look at this it's trivial that pi of c bar is contained in this torus the more delicate analysis is why it's not equal in the general situation it's equal again yeah that's a question in Tokyo so in this one dimensional case there are several real toruses so in high dimension are you expecting something similar or it can be more complicated I think that this result should be true in any dimension so the fact that we have some real torus and so if w is any of the y sub iq I expect pi of w bar I should say that I didn't expect that you could deduce it from a result on curve and the idea is that more if you have an edge of iq by the higher dimension you could expect to have a generic curve as the same topology as the edge of iq but I didn't write a proof but I think this should be true for any other dimension generally this kind of question or more related to curve I think w is a question about curve so I don't know but I think it's not very difficult for this question thank you are there further questions thank you Dr. from Tokyo so some more questions when you really study topological pleasure the way you think about this is really you take the intersection between both this bar and you see how it's moving with bar it's really what is the flow in homogeneous spacing I think it's really what you're doing to see the behavior of pi the intersection of w with p and this is really emotional flow I really did this because the first statement we had about hyperbolic axin demand it was not stated in this way and it was more directly related to what we wanted to do on the earth but it's really unequivocal and in this way you see that it's really a question of flow okay more questions no okay so let's talk to our speaker again