 Okay, thank you very much for this invitation has been so long that I've been at a conference and now this is a possibility to say the things that I really want to say. This will be a kind of a survey talk. So I don't want to talk so much about proofs and maybe not even so much about precise statements of everything but more about the big picture and how I see it. The first one to tell you about period numbers, what they are, how they can be defined. And then turn to the period conjecture and then finally speculate a bit on the connection. So that's the hominimality part of the title. So let's talk about periods. Periods are numbers that you can write as values of integrals. So you're integrating omega over delta. We're both omega and delta of some kind of algebraic or arithmetic nature. I'll be more precise later. If you don't put any conditions, of course, you can write any complex number like this, but we are putting conditions. There are several equivalent definitions around and with several, I mean like a dozen. Some of them are just variants, slight variations of things, but then there are also really big different types of definitions of what a period number could be. They appear in many different settings. So one setting where they appear is in quantum field theory, is Feynman integrals. So this explains the interest of people with a background in mathematical and physics. So this talk won't mention mathematical physics at all. That's not where I come from, but it explains the interest of other people. Let's do examples first. The very first example that everyone knows is Cauchy's integral. We're integrating dz over z, viewed as a complex variable here, and I'm integrating over the unit circle. And then, of course, the answer is 2 pi i. So 2 pi i is the first example of a period number. You can keep the same differential, dz over z, but integrate not around a closed loop, but along some path from 1 to alpha. And then the answer is log alpha. Maybe traditionally, you would say alpha is a positive real number, and then this is just the ordinary logarithm. So values of logarithms in algebraic points would be periods as well. Another old example is the case of Euler integrals. So the differential form is of dx over y, where y is the square root of some polynomial of degree 3. And I want this polynomial of degree 3 to have coefficients in the algebraic numbers. So for example, x times x minus 1 times x minus lambda. And as a path of integration, taking a straight path from 0 to 1. So Euler was studying these things. Of course, a much better way to look at the same integral is to say we are on an elliptic curve. This is an elliptic curve. And then dx over y is the invariant differential on the elliptic curve. And this path from 0 to 1, it lifts to the elliptic curve. And if you work it out, it goes around half. It's not a loop. It's only half of a loop. But then if you go back the other way along the other branch, then you get a closed loop. And the value of the integral will double. So basically, this Euler integral is a period of an elliptic curve. And that's where the whole story gets its name. So when you're integrating forms in the plane and then you get periodicities and the periods, they are the periods. There are also higher dimensional examples of differential form dx wedge dy over 1 minus x times y. And delta is a triangle. So like this. And I hope x is less than y. Yes. So it's the area under the triangle. And the value of this integral is zeta of 2, classical identity. The same trick also works for all natural numbers. If you evaluate this particular integral over the simplex in higher dimensions, then you get zeta of n. So values of the Riemann zeta function. And I hope that you all agree that these are all really interesting numbers. Periods are a really rich source of interesting numbers. Because there are many more non-periods than periods. But the non-periods usually are boring. They're just something. But these numbers that you write down in a special way, coming from arithmetic data, they are interesting. So let me come to a proper definition. And I'm calling this the elementary definition. Depending on where you come from, you would think that this is a difficult definition or an elementary one. So this is the notion of a naive period. So a naive period is a complex number of the form integral omega over domain of integration g, where g is contained in R to the n. It's compact. There's some orientation because we want to integrate less dimension d. But the really important condition on g is that it's a semi-algebraic subset of R to the n. And it's q-semi-algebraic. So this means the subset is defined by polynomial equations with coefficients in the rationals. And you're also allowed to use the less or equal sign in your definition. So that's a semi-algebraic. So all the integrals that I gave, they were written down with semi-algebraic sets. And omega is algebraic. You can even take omega as a rational differential form with q-bar coefficients, regular on g. So there's no problem with convergence of anything. Degree d, it's closed. So the important condition for omega, whoops, no, I want this one, it's an algebraic differential form defined over q-bar. So both omega and g have these algebraic properties. This definition, this precise version of the definition was written down by Friedrich. This version of the definition shows up in traditional transcendence theory. So going back to people like Lindemann and Schneider, Baker, of course, they didn't have the notion of a period or this definition. But these were the numbers, the type of numbers that we're looking at. It's very similar to a definition that you can find in a very influential paper by Konsevich and Zagje on period numbers. It's not precisely taken from there, but very similar in spirit and in fact, equivalent to that definition. So let me give another example of a naive period. When you take a semi-algebraic subset of r to the n of full dimension, n-dimensional semi-algebraic subset of dimension n, and then as differential form, the volume form dT1 wedge dT2 up to wedge dTn. So t1 up to tn are the variables on rn. And then you integrate, of course, what you get is the volume. So volumes of semi-algebraic sets, they are prototypes of naive periods. This looks a bit special, but in fact, it's not special. In fact, a complex number is a naive period if and only if both the volumes are as close to periods as they can possibly be. If you have a volume, you always get a real number and it's always positive. So if you look at complex numbers whose real and imaginary part are up to science volumes, that's the best you can hope for. And that's actually what's true. So yeah, this is the theorem. You get a comparison result like this. So why should this be true? Well, integration, if you're integrating a function, that's the volume under the graph, right? That's the key part of the argument, and then you have to work. So this was pretty elementary, I think. I had this notion of a semi-algebraic set in there, which I didn't really define. There's a completely different approach via a homology notion of a homological period. So for this approach, I'm fixing an embedding of q bar into the complex numbers. To be honest, I did this before. When I was doing these integrals, I was also fixing an embedding of q bar into the complex numbers, and we are using algebraic varieties over q bar. So x is an algebraic variety over q bar and y is a closed sub-variety. And when I say homology or homology, then what I'm thinking of is singular homology. So I'm looking at singular homology of the pair x, y, and elements in there are represented by chains. So by q-linear combinations of simplices, and the singular simplex by definition is a map from the standard simplex, standard i-dimensional simplex like triangles into the analytic space defined by the algebraic variety. So every, when you have a set of polynomial equations, it gives us a complex manifold or complex analytic space. And you can ask this map sigma j to be continuous, or you can ask it to be c infinity, you get the same theory. And the y comes in by the boundary condition. So these are maps to x or the analytic space defined by x and the boundary of this chain is supported on y. So that's relative homology. There's a second homology theory that comes into play, that's the Ramco homology. Ramco homology was introduced, algebraic Ramco homology was introduced by Kotendijk. It's defined in the smooth case, it's defined as hyperco homology of the complex of algebraic differential forms. So the nice thing about Ramco homology is that it's defined in purely algebraic terms. It only uses the algebraic variety. Singular homology is defined using analysis and topology of the analytic space, but now the Ramco homology is completely algebraic. So in particular, these Ramco homology groups, if the variety is defined of a K, then the Ramco homology will give you K vector spaces. I'm working over Q bar, so these will be Q bar vector spaces. And then Ramco homology and Singular homology are linked via the period pairing. So every class in Ramco homology is represented by some differential form and every Singular homology class is represented by some cycle. And then we can integrate. If you want to do this really precisely, I think it's customary just to just to write integral Sigma Omega integrated differential form over a singular chain. And what this means is you pull back the differential form to the standard simplex and then you integrate on the standard simplex. This is the pairing. And what Grutendick showed is that this pairing is well-defined. So you have different differential forms giving rise to the same homology class. You have different chains giving rise to the same homology class, but by Soek's theorem this is well-defined. And that's the more important part of the statement. It's a perfect pairing. If you rewrite this from a pairing into a duality, you can also see this as an isomorphism between the Ramco homology and Singular homology. So the dual of homology is co homology and having a pairing is the same as having a map to the dual. And then this becomes an isomorphism after extending scalars to the complex numbers. Now we have the period pairing and it even induces this period isomorphism. But now we have the same vector space of the complex numbers and it has a Q structure coming from Singular homology and a Q bar structure coming from Duramco homology. And they interact in a non-trivial way. The definition is a comological period is a complex number that you get in the image of this period pairing for varying varieties X and sub varieties Y and also the index I, the homological degree is allowed to vary. So you get a set of comological periods. This definition really goes back to Gordendijk, but then it was Yvonne Dray who really worked it out and explained it to everyone. But you also find the same point of view in papers of delinear and in conceivage and in many other people. So now I've given you two very different definitions while they're not so different. In both cases, you're integrating differential forms or something coming from differential forms over something algebraic. Sarah pointed out that these are the same numbers. Well, he did this in the case of elliptic curves and elliptic intervals. So in the very first paper when Gordendijk defined algebraic diromcomology and he defined the period isomorphism, he also had these numbers coming up in the period matrix of comology of elliptic curves. And then by, it's a remark in Gordendijk's paper, Sarah told him that these are the same numbers that people in transcendence theory are looking at. These are elliptic periods. And this really extends to the full picture. So these two sets of period numbers are the same. You can pick your favorite definition and you always get the same set of complex numbers. It's not only a set, it's a Q bar algebra, if you think a little bit about it. You'll get the same Q bar algebra of period numbers. So I don't really want to talk about proofs but I can at least give you a hint. So the main ingredient into the proof is the existence of semi-algebraic and triangulations. When you have a semi-algebraic set and every algebra, the complex analytic space attached to an algebraic variety that can be viewed as a semi-algebraic set. So every semi-algebraic set has a triangulation. So it decomposition into simplices and all these simplices are also semi-algebraic. And then you can use the triangulation to describe homology. And you can also, if you have a semi-algebraic G in the first definition, there was a semi-algebraic set G. So you decompose this into simplices and then these simplices give you a chain in homology. So you get from semi-algebraic sets to homology classes and you can also go the other way. Every homology class can be described via any triangulation and then you have a description via semi-algebraic sets. So this is the main ingredient, the statement from semi-algebraic geometry. But even after this main idea, there are lots of technical problems. So one technical problem is that, yeah, you really want to reduce to the affine case. And this uses noise and basic grammar, if you've heard about this. And then also you need to reduce to the smooth case because there's nice connection between algebraic varieties and differential forms that only holds in the smooth case. I was cheating you. I gave you the definition of algebraic deramco homology in the smooth case. But then when I wrote X and Y, there was no condition whatsoever. X and Y can have singularities that don't need to have to be proper or anything. So I really want to have all X and all Y. There is a definition of deramco homology in this general case. But if you want to do the comparison then you have to reduce to the smooth case. So this relies on the argument of appearing in a paper of Belkada and Brosnan. So it goes on for quite a number of pages to get on this comparison done. So actually I've given you three characterizations of periods now. One via semi-algebraic geometry, then the one that went even more to the elementary side, just volumes of semi-algebraic sets. And then on the other hand, the characterization via homology. They form a countable subset of the complex numbers. So you can see this either via the volume description or via the homological description. In both cases, there's only a countable amount of data and then the numbers in the image that's on the only countable set. Okay, so this was about, yeah. This was about definitions. So maybe now would be a good point if you have questions so far. No, okay. So let's talk about the period conjecture a bit. So this is really the question. Now we have all these interesting numbers, these period numbers. Question is what are the relations between them? That's really a question of transcendence theory. For a complex number to be transcendental means to be algebraically independent of one. Or linearly independent of, you bar linearly independent of one. So they are these different points of view. But transcendence theory, first asking whether a number is transcendental and then later on asking what are the relations between these numbers? This really is transcendence theory. So you want to talk a bit about the history of this question. It really goes back a long time all the way back to Lindemann in 1882, in Freiburg proved transcendence of Pi. You also proved transcendence of numbers like log two or more generally log of alpha when alpha is algebraic. And then the next big stack was in 1934, when Gelfand and Schneider proved, for example, that log two and log three are cuba linearly independent. So it was two proofs coming at the same time. So more generally log alpha and log beta are cuba linearly independent. If alpha and beta are algebraic numbers that are not in a multiplicative relation. So if you look at log of alpha and log of alpha squared then there is a relation but that's the only type of relation that you can have. Siegel started the transcendence theory of elliptic periods and then really this was fully developed by Schneider asking about relations between linear relations between elliptic periods. And then a really big theorem of Baker in 1966 who was considering a whole space of values of log. So if not one or two values of the log function but a whole bunch of them and then you determine what the rank is. Okay, I don't want to formulate this precisely. And this goes on and on. And I'm probably not the best person to tell you about results and transcendence theory. So let me stop here. The big conjecture, period conjecture says that the only relations between period numbers in any of the senses that I described are the obvious relations. This raises the question what the obvious relations are but it's really meant to be obvious. Obvious as in as obvious as it can be. So in the case of log, yeah, I told you log alpha and log of alpha square is a relation, it's that kind of obvious. The first formulation of this conjecture is due to Grotendieg, but he didn't really publish it. So there was a hint in this DRAMCOMOLGY paper but then the one who formulated it for us was Yves-André. You also find it in the work of Deligne. So this version of the conjecture is based on algebraic geometry and relations that come from algebraic cycles and more generally on motives. So the category of pure motives, this was the first case that we're thinking of, category of pure motives is fully described in terms of algebraic cycles. And then whenever you have a map between motives, then these motives have periods, again the same period numbers, and then all kind of funclarity between motives. You can formulate this as a statement on funclarity of motives or you can formulate it as a statement on every map on a homology induced by algebraic cycles. This is the kind of obvious relations that we're thinking of. And then if you replace pure motives by mixed motives, that's like the Grotendieg formulation of the period conjecture. The only relations between motives are the only relations between period numbers are the ones coming from funclarity for motives. And if you specialize to one motive or one algebraic variety, if you would like, then you get a finitely generated algebra of periods of that particular variety. And this point of view gives you a formula for the transcendence degree of that algebra. So the prediction is that the transcendence degree should be equal to the dimension of the motivical algebra. So that's one point of view. Uses relies on homology and on motives. But then Concierge formulated a version of the period conjecture that a priori has a very different flavor. This point of view was really developed by Norrie. There's work of Ayup, who actually proved it in the function field case and is also spelled out in my book with Stefan Müller-Stach. So this version of the period conjecture has more a calculus flavor. If you think of the definition of periods as integrals over semi-argebraic sets, then the obtuse relations are the ones coming from the transformation formula. That gives you a formula relation between your map between semi-argebraic sets and you pull back the differential form. So you get the transformation formula that gives you a relation between period numbers. And the other type of relation that you get is from Stokes's theorem. Whether you integrate d omega or some domain or omega over the boundary, it gives you the same period number. This point of view involves a formula for the cubar dimension of the space of periods attached to some, I don't want to say motive, but it's motive. So if you look at the cubar dimension of the space of periods generated by HI of some algebraic variety, that's a finite dimensional space and the conceived version of the period conjecture makes a prediction for the dimension of this space. But again, these two versions, these two point of view are essentially equivalent. It's a very small difference that you don't have to worry about if you don't know the details and then you would know it anyway. So the conceived point of view and the broadened point of view for the period conjecture, they are equivalent. You can think of motives or you can think of transformation rules for integrals of semi-argebraic sets. And this gives you the same type of obvious relations. So I'm attributing this theorem to Norrie because really the way to prove this is to introduce a theory of motifs. That's what he did. He developed a theory of mixed motifs. And then using this theory of mixed motifs, the Grotendig version in terms of motifs is equivalent to the explicit one written down by Konsevich. So I've been using the word motive a lot. So let me say a few more words about this. When I'm, a category of motifs is always an abelian category. So it's a category where you can compute. You have exact sequences, you can take kernels and co kernels, you can compute. And it should be, and it's an abelian category such that I'm comology factors. So we have, we have algebraic geometry. We have the category of algebraic varieties. And we have Q vector spaces. And then we have comology. So this could be singular comology or the RAM comology, comology series. And category of mixed motifs sits in between. So it has the, has the flavor of a category of Q vector spaces in being an abelian so that you can compute. It's something linear. But it's still as close to algebraic geometry as it can be. Well, that's the category of motifs. So in the category of mixed motifs, you have all the morphisms coming from algebraic geometry. Whenever there's a morphism between algebraic varieties or between pairs of algebraic varieties, then there's also a morphism in the category of mixed motifs. That's what this frontier from algebraic varieties to mixed motifs says. And you have long exact sequences. Whenever you expect a long exact sequence in comology, that all really comes from a long exact sequence in the category of motifs. And mixed motifs are universal with this property. So as close to, to algebraic geometry as they can be. And contrary to rumors, this theory exists. There's a well-defined category of mixed motifs defined by a binary that has all these properties and that can be used to give a precise formulation of the period conjecture. And then you can use this in a Gordendick style formulation of the conjecture where you talk about factoriality of motifs or you can use it to translate to the conceivage version. Okay, it's a big conjecture. Yeah, going back to Gordendick. What's the evidence? Because we have all the classical results from transcendence theory. So all the, I gave you these examples that these classical numbers like log of two and two pi i, they are elliptic periods. They're all examples of period numbers. And all the transcendence results from classical transcendence theory can be read as confirmation of the period conjecture. The conceivage version of the period conjecture holds for periods coming from algebraic curves. So that's the one-dimensional case. Or if you like, it's the period conjecture for one motifs holds. This was worked out by Gisbert Wistals and myself. And the really big input into this is the analytic subgroups theorem of Gisbert proved quite a while ago. Yeah, so you have to be careful. I'm not saying that the period conjecture holds for curves or periods of curves or periods of one motifs. So this is the conceivage relation. It makes a prediction about the Q bar vector space. It's not a prediction, it's a theorem. For the Q bar relations between periods in degree one, one-dimensional integrals. And all the relations that we get are precisely the ones coming from algebraic geometry. This is different from the Grottendijk version of the period conjecture for one motifs where you would look at the algebra generated by these numbers and make a prediction on the transcendence degree. But we haven't said anything about that. We say something about the Q bar dimension. So the two conjectures are equivalent if you do them for all objects. But if you restrict to a subset, or in this case, one-dimensional objects, there are two different term statements. So the Grottendijk version, we really don't know anything. What we know is MCM elliptic curves. So that's not all elliptic curves or all curves. It's elliptic curves with complex multiplication. Now this was established by Shudnowsky in 1978. He proved that the transcendence degree is at least two. And then in this case, that's what you expect and it gives you a sharp bond. So this is what we know about the period conjecture as I stated it or as I hinted, it's really next to nothing compared to the width of the problem that's next to nothing. Generalizations have been established. So there's the independent proofs, Ayub and Nori in the function field case. So then all the relations between periods of this about transcendence theory of functions now, the other ones come and predicted by the period conjecture. So can also read this as a confirmation. To give you a feeling on what we don't know, I told you that the values of the Riemann-Zieter function in natural numbers are periods. And this period conjecture implies that the values at odd integers should all be algebraically independent. But even integers you have the powers of pi. So we understand that. But if you look at the odd values, that they should be algebraically independent. And we really don't know this. We know that Zeta three is irrational. You don't even know that a single one of them is transcendental, let alone that they're algebraically independent. So this is, we have now a very nice framework to talk about these conjectures, but we don't have a lot of confirmation. We have the one dimensional case. Oh, that's already like good. We have a question in the chat by Dinesh Thakur. Maybe you just unmute and ask away. So maybe the mic's not working. The question was, what does essentially stand for in the Nori theorem? So Kort Konsevich defines a space of formal periods by explicit generators and relations. And then this space of, so basically expressions in terms of H, I, X, Y, the coromology and then an omega and a sigma. And then the relations are the ones coming from boundary maps in coromology and factoriality for pairs of algebraic varieties. So yes, this explicit algebra. And then the period conjecture says that the evaluation map from formal periods to period numbers should be injective. And the Grotten-Dick version would only say something about the, so the essential part is that if you have the Grotten-Dick version of the period conjecture about the transcendence degree, then you also need to know that this space of periods is connected. So we know it's a smooth from absolute theory. It follows that it's a smooth space, spec of the period algebra is smooth. It's a torsor under the Motivi-Galwa group. And if in addition, this torsor is connected, then the Grotten-Dick conjecture implies that also the evaluation map is injective. Yeah, sorry. So it's this connectedness. And I think some people include this integral in the Grotten-Dick conjecture then that would be completely equivalent. Oh, and I didn't localize. I didn't mention that you have to invert the Tate motif. Nevermind, but that doesn't make a difference. Let's come to the speculation part. There's a question that I want to raise is this relation of the theory of periods to semi-algebraic geometry. Is this more than a curiosity? Yeah, we have these very different flavors of definitions of periods. All the structural insights come from this point of view of comology and description via Tanaka group. So the algebraic geometry point of view is very good for in a comological point of view is very good to give a conceptual interpretation. But on the other hand, we have this completely different description via semi-algebraic geometry which also has its own set of obvious invirations. Is this just an accident or is this something important? That's really a question. And I would like to argue that maybe it's important. Maybe it's something that we should explore. So what's the evidence? Why we should explore this? The evidence comes in the shape of a generalization of the whole story. Comes via the theorem of exponential periods. So this goes also back to the same paper of Conceivage and Zaghe. They also consider exponential periods. So they are integrals. Again, you're still integrating. But now under the integral, there's this extra factor e to the minus minus f. If you put f equal to zero, then you get back ordinary periods, but now you allow something more. And as with classical periods, there are several very different points of view. You can look at this from the point of view of comology. And in this case, this was really the original the way they were introduced. Classical periods, Durham comology, that's the theory of period isomorphism, the Riemann-Hilbert correspondence for regular connections. So you have a vector bundle with a connection on it. This gives rise to the Durham complex. And if this connection has only regular singular connections, then you're in Hodg's theory and the period isomorphism. Locke and Denau wanted to study irregular connections and go towards a Hodg's theory for irregular or something like Hodg's theory for irregular connections. So this was their paper with the starting point and then the theory was really developed fully in a series of papers of him. So there is a homology theory that you can use instead of singular homology and to use the Riemann comology of now an irregular connection on a complex of differential forms. And the differential is twisted by some exponential factor. So the F comes in in the differential of the Durham complex. So, and then there's again a period isomorphism. And once you have a period isomorphism, you can define periods as the elements, the numbers in the image of the period pairing. So that's the first point of view for exponential periods. There is a theory of motifs underlying the whole story that was developed by Fressan and Jossen. And then the picture that we get from the motific point of view is really as nice as in the classical case. So there's a nice theory of motifs and all periods coming from comology and also the periods coming from motifs. And then you have a Tanaka group and then you can do all these things. And then in this case, the elementary point came last and that's in a paper of Johan Kuhmelin, Philip Harbecker and myself, where we also give an elementary definition or what I call more elementary definition of these exponential periods. And as in the classical case, all these different points of view are actually equivalent. So let me give you a definition, proper definition of the exponential periods. Yes, the definition that we came up with in this paper. So yeah. There is a question, please, by Hershey Kicilevsky. Will you please send Newton and ask away? So he asks if zeros of zeta functions are periods. Zeroes, I don't know, I don't think they are. I don't think they are, but I don't know. And Andrew Obos also asked, what about the real parts of the zeros? I don't know that either. So good question, thank you. Yeah, good question. No, we have this definition of a naive exponential period, which looks very much like the definition of an exponential of a naive period that I gave in the classical case. So again, we have a semi-adrobike subset of C to the end of dimension D and there's some orientation. In the classical case, we were asking G to be compact. Now it's only a closed subset. So it's not necessarily compact. Omega is S before. It's an algebraic D form on affine space, a rational differential form. And F, F is a rational function on affine space. So again, algebraic function. And we ask it to be irregular and proper. So there's a second compactness condition here. We ask F to be proper on G and one more condition. The image, if you apply this algebraic function F to G, then the image is contained in a strip. So this is the complex plane. And then F of G is contained in a strip like this. So the real part is bounded below and the imaginary part is bounded. If you have these conditions, then this E to the minus F, the real part will go to infinity and you'll get exponential decay. So this condition makes sure that you get absolute convergence. So this is a well-defined number. So these are the naive exponential periods and we get the same kind of comparison result. So they, yeah. This gives the same set of numbers as the one coming from comparison of rapid decay homology with twisted theorem homology. So this is a good definition because it captures the information from homology. And as in the classical case, we find these numbers as volumes. So if a complex number is an naive exponential period, then it's real and an imaginary part are up to sign as much as you can hope. The volume, not of a semi-algebraic set, but of a definable set in a certain O minimal structure. So that's a finally where the O minimality of my title comes in. So note, in the classical case, this was an equivalence. We don't get an equivalence in this case. We just get an implication in one direction and we do not expect an equivalence. We expect the left hand side to be strictly contained in the right hand side. This is usually really hard to prove. So you would need to prove that, oh, sorry. You'd need to prove that e to the e is not a naive exponential period. And it's always next to impossible to show that something is not a period. So we don't know this. Nevertheless, there is this relation. So these naive exponential periods, they are volumes in this O minimal structure. O minimality comes in. Let me tell you just a little bit about O minimality. So what does it mean for a set to be definable in this structure that I can't unpronounce? It means that the set can be written in terms of Boolean operations. So you allowed finite unions, finite section, taking complements, products, that type of operation. You're also allowed to take images under a projection of two coordinates. And then the interesting part, you can write down the basic building box for the definition, polynomials again, with Q coefficients. But now in addition to semi algebraic geometry, you're also allowed to use the real exponential function and you're allowed to use the sine function restricted to some compact interval. For example, sine function restricted to the interval zero and one. So an example of such a set definable in this structure is the set of pairs of real numbers where y is at most e to the x and x square is bigger than two. So x square bigger than two, that's algebraic. But you're also allowed to use the exponential function in your definition. So this is between minus square root of two and m square root of two. And then you have the graph of the exponential function and y has to be equal to something like this. So that's sets of this shape. And the big theorem is that, not our theorem, but the big theorem that we are using is that this structure where I'm allowing these operations is O minimal. So this means that the subsets of the real numbers that are definable, they are just finite unions of intervals. Well, and possibly infinite things, like- Well, they could be, the intervals could be infinite. Yes, when I'm saying interpret, yeah, I'm thinking of half open intervals like this. So going off to infinity, that's also allowed. Yes, but only finitely many of these things. You're also allowed to use single points. That's also fine. So this is due to Wirki and then van den Riesmiller. So you have to combine two big theorems. So let me comment for those people who don't know about O-minimality. The complex exponential function is not definable in any O minimal structure. That's because the complex exponential function is periodic. If you take the preimage of zero, wait, the image of the exponential function. But I'm, yeah, okay. That's a stupid point you to take, just any other point. And the preimage is a discrete set, e to the two pi i is, no, it's pejori. And an infinite sequence of points is not definable. Subsets of the reals have to be finite. Finite unions of intervals or points. So complex exponential function is not allowed. But if you're bounding the imaginary part, then that's fine. We have the signed function on a bounded interval. So if you bound the imaginary part of the argument, then that's definable. And that's how you get to the things that we have in the definition of a naive period. This definition of a naive period, I was bounding the imaginary part. That's precisely in order to make this true. So this condition on F of G can be used in order to ensure an absolute convergence. But it's also used in order to get this relation to this O minimal structure. And O minimal theory, the geometry that you get is very similar to semi-algebraic geometry. It's a very nice type of geometry. And it's a direct generalization. So another example of an O minimal structure is the structure where you only allow polynomials, not no complicated monomorphic functions. And then you get back semi-algebraic geometry. So let's look a bit wider. What's the relation between O minimality and the period's conjecture? Well, there certainly is a relation between O minimality and transcendence theory. So I'm viewing the Andre-Aud conjecture as a higher dimensional transcendence theory. And there's the theorem of Pilar Wilkie and the whole strategy to prove the Andre-Aud conjecture that has been applied to prove or reproved the Andre-Aud conjecture in some cases. So there is a relation just roughly, there is a relation between O minimality and transcendence theory. There's also a relation between O minimality and Hodge theory. I didn't stress this so much, but of course, the period isomorphism is the starting point of the theory of Hodge structures and the Hodge conjecture. And there's a very close relation between the Hodge conjecture and the period conjecture. To some extent, they are very similar. And there has been an application of O minimality to Hodge theory. So consequences of the Hodge conjecture have been proved using O minimality. These are papers by Bebagar, Wim Berg, Winger, Zimmermann in different combinations of these people. And now we have this small relation between periods of irregular connection, exponential periods, and also O minimality. So I can just ask again, is the period conjecture really a conjecture about algebraic geometry or is it a conjecture about O minimality? Maybe we should try to use the tools of O minimality to approach the period conjecture. Thank you very much.