 I will talk English. So first, thanks the organizers for allowing me to present here. I will mainly talk about something related to the optimal code embedding problem, which is based on the joint work with Xiao Lu Tan and Yisar Tuzi. The code embedding problem aims to present some private distribution on real line as the law of a Brownian motion stopped at some stopping time. So that's to say, given some Brownian motion defined on some private space and a centered private matter, mu, the problem consists in finding a stopping time tau such that the stopped process by tau is uniformly incurable, and the random variable beta has a given law mu. Such a stopping time is usually called an embedding or a solution to the problem. Roughly speaking, there are two formulations of the problem in the existing literature. In the first one, we seek the stopping time with respect to the Brownian filtration. I mean, the filtration f is a natural filtration generated by the Brownian motion. In the second one, we introduce some additional randomness, and we consider a larger filtration for the stopping times. This problem is initialed by scrollhold and the various solutions are achieved afterwards. A very overstick idea is following. They compare a realization of a Brownian path to another well-controlled process, usually depending on the Brownian motion, and use later to decide when to stop the former. So if we take this well-controlled process phi to be t, a deterministic process, we get root solution. If we take the running maximum, we get azimah york solution, and if we take the local time, we get valois solution. So more interestingly, several solutions enjoy optimality to be defined in the following. We take the root solution, for example. So this root solution is constructed through some broad site called a barrier. A barrier is a broad site in R2, such that it is closed in right. So let me see for any element of this site. We take any element that is right, that lies right to this given element, so it belongs again to this site. So such a broad site is called a barrier. So root constructed this solution by constructing some well-defined barrier this way as the first heating time. And it is shown by root. Root solution maximizes this optimization problem, defined this way. So the root embedding is the maximizer for this problem among all the embeddings to the given probability mu. However, it is natural to ask if we consider some non-Markovian function phi, I mean it is part-dependent of this S4, could we start systematically this optimization problem. And we wonder why there's the optimization problem given under the different formulations, means the strong formulation and the weak formulations are equivalent in general. And moreover why we wonder why they were made constructed for a general functional phi as the optimizer. If we not, can we derive some characterization for this optimizer? Okay, the main result of this talk is two-fold. First, we give some weak formulation of the optimal scrolled embedding problem and prove the corresponding duality. And the second, secondly, we derive some principle which is called multi-state principle allowing to characterize the optimizer via its geometric support. Okay, let's go. To formulate this problem, we introduce some space continuous fractions starting at zero and we define the enlarged space omega bar by the product of omega and r plus. So, given this space omega bar, we define the canonical element bt defined this way. So, then we introduce some filtration. The objective of to introduce this filtration is to ensure that the random element t is a stopping time with respect to this filtration. So, roughly speaking, we construct some private space and then we will represent the embeddings and the stopping times using the probability measures on this space. Okay, let p bar be the space of probability measures on this product space omega bar, such that b is a Brony motion, it's an f bar Brony motion and the measure and the process stopped by t is uniformly integrable. So, for any given set of the probability measure mu, we consider the sub-site p bar of mu as the site of embeddings relative to mu. Okay, as for the optimization problem, we consider this problem for a restricted class of fractions which are called non-anticipative. That means this function node depends the path up to t, up to theta. So, for such a non-anticipative function phi, the optimal scrolled embedding problem is defined as the maximum of this expectation over all probability measures p bar in the site p bar of mu. Okay, this is the primary problem. So, we wonder why this optimizer exists and how to characterize this optimizer. So, to do this, we first introduce a dual formulation and then establish the required duality. So, let lambda be the space of continuous fractions defined on r with linear growth and let fb be the natural filtration of the Brony motion b and p0 be the linear measure. So, the dual problem is defined as following. Let h be the collection of some suitable process such that we may define this stock cut ceiling to grow. So, we require some technical conditions for this h such that it is I'm sure to be a super martingale. So, define d by the couple lambda h such that this inequality helps p0 almost really for all omega in the space omega. And then the dual problem is defined as the minimum of the integral of lambda with respect to mu for all lambda h belonging to this site d. So, roughly speaking, for the primary problem, there are two kinds of constraints. First, the bt is required to have the given law mu and second as b is Brony motion and it's a martingale. So, we finalize respectively these two kind of constraints by introducing a stock I think row a super martingale and some some function depending on omega t. So, assume that this fractional is the upper sneak continues with respect to the first component with respect to the second component theta. And the result is that there exists an optimizer piece bar star and moreover this charity house. Okay, what is the main issue to have this charity? With the help of this charity, we may revisit the monstrosity principle which is introduced by Becker-Bock, Hausmann and Cox in 2014. And we provide a new proof using this charity and the optional cross section theorem. Moreover, with every continuous martingale is a time-changed Brony motion. We may use this charity result to study a class of continuous time martingale optimal transport problems to derive the so-called Carnowitz charity. Okay, to describe the monstrosity principle, we need the definition of the so-called stop-go pair. So, what is the stop-go pair? It's a couple of elements in omega bar such that the two paths determine at the same level. It's called omega theta is equal to omega prime to the prime. And moreover, this property is satisfied for this pair. I mean, if for any new path which is not reduced to be zero, it has some strictly positive length, so for any path e omega bar, if we paste it to f here, f represents omega prime and j represents omega. So, if we paste this path along f, it is always optimal to paste it to j. So, that means according to our optimal criterion, it is always optimal to paste any path to omega bar prime than omega bar. Okay, then for every sub-site of omega bar, we define another site gamma-info such that each element could be extended strictly in gamma. Okay, with all the notations above, we may provide the main result. It's called monstrosity principle. So, if there exists some optimizer and there is no joint gap, then there exists when we find some broad-site gamma such that the optimizer is supported on gamma. And moreover, the intersection of the site of SG is a collection of all sub-go pairs with this product is empty. So, it is easy to interpret this equality. So, lose this thinking. If not, if we may find some element in this product of these two sites such that it belongs to the site of sub-go pairs. So, imagine we may cut the every path along this element belonging to gamma-info and paste it to the gamma. So, we may again optimize to improve the value of the expectation. So, this contradicts to the optimality. So, with the result we have obtained, we may prove this result in another method of completely different nature. Okay. So, let's go back to the Ruth's solution and see how to interpret its construction by this principle. So, if we take gamma as the support of the optimizer p bar star and then we define the we may easily compute by the concavity of the factor five for the form of the site of sub-go pairs. It is easily to deduce that SG is right in its way. And by the monocyst principle, this intersection is empty. So, we may deduce easily that the barrier must be written in this form. So, this can interpret the Ruth's solution. And as far as I know, other embeddings that enjoy the optimality in the literature can be interpreted by this unifying principle. Okay. To finish, I would like to make some comments. First, because I claim that there exists another site SG star that depends on the optimizer, such that this inclusion holds. Let's say we find some bigger site SG star than SG such that this intersection is empty. And moreover, we may extend our analysis to the multiple marginal case. Let's say with a slightly modified formulation, we may consider a multiple scrollhold embedding problem for a vector of marginals. Obviously, this vector is required to satisfy some condition, which is called a peacock process useful for complex questions in French. So, for such a given vector of marginals, we may study equally the problem in the same way and establish the similar results. And the last comment I would like to make is thank you very much. So, some of these stopping times can be easily calculated or approximated like the original scorehold one or the Dubens one or SMI or at least what I remember from the original root construction. It was based on a fixed point argument which was not really constructive. But maybe with these optimality considerations, you're able to compute it. So, given a random variable x, suppose you want to compute what is the barrier with some approximation epsilon. Do you know how to do that? Actually, that depends on the very particular structure of this function phi. For example, if phi is equal to x square or the square root of x, so the explicit construction of such a barrier exists. But in general, yes, it cannot be very explicitly constructed if we consider just consider strictly concave function phi. But afterwards, it is this problem started by the potential theory and extended to the multiple marginal case. So, in the recent paper of Cox and Obloy and 2D, where the optimal embedding is constructed in a similar way but using more technicals. So, I just want to emphasize that using this unifying principle, we cannot compute explicitly for all the existing embeddings, but we may find how it can be constructed. For example, for the Adi Major, so we know there is some increasing function and for the running maximum that the optimal embedding is defined as the heating time such that Bt is equal to this one. So, generally for some very specific example, the phi can be calculated. So, I mean it can be calculated for all examples, but using this criteria, we cannot recover the explicit form of this function phi, but we know there must exist some increasing function.