 So, first, I'd like to thank the organizers for this nice conference, and I hope I'll be fast so everybody can go for lunch. But so I'll talk, now you're going to change to classical long-hanging interactions again. So I'll talk about topological approach to micro-economical thermodynamics and phase transition on interacting classical speeds. So I don't have a hobing hood in my place, but I would want to use 10 seconds to advertise so the university is coming from. And the first plot I want to show is the temperature plot, so it has function of time, that it's between 22 and 34 along the year. And I want to say that because we now have six positions open, so then maybe someone is interested in doing nice physics and being in a nice place as well. So the thing that I'm talking about today, it was simply submitted to archives like two weeks ago and has to do with this work, but it's now under evaluation. But I want to give a very short introduction. So this work has to do with previous works from like Lapu, Cassetti, Mikhail, Kastan and other researchers that are now here as well. But I would say that the first work on this topic comes from 1997 when like starting the dynamics of some classical systems. So it was conjectured that maybe non-ality cities in like critical points related to phase transitions were due to proper change in the topology of configuration many folds, like in that potential many folds. So then there was a quite exciting period because there were many works on this feature and there were many examples on that. But for now, for today, I will talk about only for the Hamiltonian infield model. I think maybe one in three talks are about this model today. So basically, so then I want to remember this work as well. So basically for that model, it was shown that a topological change in that potential many folds like associated to the Hamiltonian were like the kind of the origin of phase transition in that model. And at that time, there was a so-called topological hypothesis that maybe a phase transition would directly connected to a topology change in the configuration space. So and then like most of the work, it's based at least the technicalities is based in this work 2003 from Lapu and Petin and Cohen. And I want to remark as well that in 2004 there was like a theorem that stated like this theorem is not for long-range models, but it's a necessity theorem which claims that under certain conditions you may have a necessity condition that a phase transition is related to a topology change in the configuration space. So that is a really short review of it. And just the thing that I want to focus today is the topological invariant because that was the first among many topological invariants that was the topological invariant that people focus in. So basically, if you remember like basic high school, you see that in like convex polyadure, you know that like vertex minus edges plus phase is a constant equals to true for like convex polyadure and then you can generalize this definition for any dimensions and then you can compute this for your like Hamiltonian like a potential surface and stuff like that. And in that work that I just mentioned, so that was the first time like at least I know that was introduced in this context. It was shown that there was a huge topological change in at the critical energy level like at the phase transition for the Hamiltonian mean field model. So I can give more details or you can go to the reference, but basically most of the critical points are very, very close to the phase transition. There is a huge like attachment of critical points in that level. And then we can, so we found a one-one relation between the critical energy and topological change in the critical point, but then maybe we can have a question because so if you assume that the topological change in a potential manifold is associated to a phase transition and also think about connection with other ensembles or with thermodynamics because when you are in micro canonical ensembles, a simple derivative of our entropy gives us the temperature. So then I will focus today in how we could, if you assume that you can study like systems that way, how we could find a critical temperature using these methods. So basically the first thing we would guess, okay, if you think that this quantity represents some sort of topological entropy or Euler entropy, we would like to try to make kind of the derivative of this quantity and then we would try to find temperature with that. So then if you try to do that, we are not so happy with these results. So this is just the derivative, like if you try numerical derivative of this curve because this quantity is an integral number. So we cannot define properly how to do this numerical derivative because of these fluctuations in this quantity. So then I would like to use all the tools from applied topology to construct a new or not appropriate definition for this derivative. And then we may connect this derivative with the phase transition and the critical point and things like that. So just to make clear how we like this kind of concept emerges from this historical review. At some point in 2003, it was conjectured also that at least around, so I just took this sentence from that paper in 2003. So at least around the transition, the entropy should behave like similarly to the Euler characteries in the sense that the entropy could be split in two parts, one analytical part and one non-analytical part. And so the non-analytics could be found through the Euler characteries or through this topological invariance. And then that motivates us to search for this kind of topological derivative. And then later we had new results, we include lots of dynamics in this approach. So there is an outstanding result from Michael Casten in 2008 that he basically, and he's collaborated to develop the necessity condition so we could verify that there is a phase transition after computing some geometrical quantities, like I call now density of Jacobian's critical points. So I want you to remember that curve for the infinite range xy model because so basically if you find a divergence in this quantity at the critical point, so this is a necessity condition to find a phase transition but not sufficient. So I want you to remember this result because since this is a not sufficient condition we may can ask you what are the conditions of when this divergence happens, not only at the phase transition. So and then until 2008 I would recommend a review in these three, kind of three papers like by Lap with collaborators and Casten in the book by Petini. But so it's not always a bed of rows so because that was a kind of a trial to do a general theory that could describe topology in terms, so statistical mechanics in terms of topology of configuration space and things like that. But then after this many years of research it was found a contract sampler of this theorem that I just showed to you two minutes ago and basically for the 5.4 model it was shown that this hypothesis does not work well so and then made this made the error to be a bit calm so that because there was not this like search for universality was not at all a question anymore but so I'm aware that there is a new work on archives that may want to circumvent this topic but I don't have technical like I cannot judge the technicality of this work but I'm aware of this like kind of extension of the topological approach so that may fix or extend this theorem or not but this is still an open question but anyway so that's how science evolves and we cannot do a universal theory but I would like to discuss so if you assume that this approach works what can we do what can we find so could we find also temperature like like we do with micro-conon ensemble and find a connection with like a non-conon ensemble things like that so then that's what I want to address so there was also some not contra-examples but model season that the necessity condition that I just show you also don't work that that one about the paper in 2008 it's like just to see the next example that it's on an audience as well like the paper by Tasizio 2011 also shows that this dynamic approach for like to study the stationary points of energy landscape also don't does not work at all like has a sufficient condition so just a necessity criteria but so that's how science evolves so from now after this introduction I will make strong assumptions that I could be comfortable that I could do something with topology okay so then that's why I want like a warning so we are we are assuming now that like the distance between critical points approach zero so that in the thermodynamic limit the other interval of energy that you are interested in in study like is dense and basically technical it means that I we are restricting our hypothesis in that this topological approach works and then we are going to put together these ideas with some modern ideas for applied topology so I would recommend this book to see the idea that we are going to discuss now and basically I want to answer in this context two questions so it's possible to derive thermodynamic quantities by using information by the only topological invariants like in particular the only characteristics or since the only characteristics is a discrete quantity how could you perform this discrete derivation with respect the only characteristics so is it possible like so at least do the best of my knowledge how could you define a topological derivative in the context of most theory and this approach then we have some hints for an applied topology usually physicists are more like fasting using maths to do science but in that moment like I think engineers are doing faster than us but basically because like in 88 in 8 there was an interesting paper that you could actually do like integral calculus using this topological invariants so we could do integral calculus based on all the characters this is at least for physicists for me it was a bit a true advance in terms of technicalities but if you go to this book in applied topology you can understand how it this definition works and basically these two guys from Bell Labs they developed a tool to do enumeration and do target like in networks like kind of an application for robotics using this kind of integral so then that's why there was these two nice papers very recently using this method so in basically if you come across to this idea of integration you work with statistical physics okay if I can integrate I can define a volume I can define a proper entropy and I can see what happens at least like that's what they make very naively just let's see what happens if you use this measure and we can maybe find some information about this topic so basically then we define using that like analogy with the papers that I showed I showed you that we can have all the integration which means that basically instead of integrating like summing up like a normal Riemann sum for every level that you integrate we compute the weights according to to the topological invariant like the MOSI index so that if you integrate a constant function we have the the Euler characters and then we can do an analogy so this like between Boltzmann and Euler like entropy so that I don't want to create an any new entropy and actually that was proposed by Clape 10 years a few years ago but just I want to compute what's the topological contribution for for the entropy and how we can find this critical temperature so and then you can define average and everything so basically the only question is how could you define a proper derivative and then I can ask to Euler and they basically there is interesting results called MOSBOT theorem or non-critical net theorem basically if you have many critical levels if you if you want to see which topological change happens in your manifold if you observe like an energy interval that is less is lower than two critical levels we don't see anything it's not see any topological change if you if you look for a huge interval you see too much so basically we choose exactly the distance between two critical points to compute this derivative but that is a bit tricky because you should change your like delta E with like time or with energy because you always have to track distance between two critical levels so then if you do this kind of derivative you find a smooth curve instead of that one I showed you before so basically then we want to apply these ideas to this do the Hamiltonian field model and I want to show that we could find actually analogous results using only topological arguments to this to this model at least so then then we'd solve it this model with in a field so I am not aware about proper solution in a field in equilibrium because that most focus in this model is on dynamics and stuff like that so then we we may we use this normal set of point method but we include a field in the system and basically what you have to do it's just like to track all the instabilities and stuff like that like in this set of point solution and then we can find a proper classical mean field diagram for for us I spin system in the presence of a field that has like this kind of magnetization or like a phase transition for zero field and then we can find like canonical diagrams and micro canonical diagrams and also you can find instabilities and so I can find find this spin of the outcores like that that are related to instabilities so I want to remark here that the instabilities are in the interval zero and one for field because that will be useful to some topological interpretation and then we can find proper entropy so the continuous lines are like stable entropy the dashed lines are like unstable entropy is associated to this one of our loops that I just show you and then we start to try to connect this with the topology of the configuration space the first idea okay I can try to plot a small configuration space only with two spins then we can see that through this diagram that basically all the instabilities that we draw for a large system also holds for this tool the like systems in the sense that the instabilities I show you evolves with the field and then with the field is which the the maximum value to the stability we don't have any stability anymore like you can see that this maximum approaching and matching for few this equal to one and here you have the entropy and things like that so then we do the same analogy to compute and do the same topological approach so in the sense that we compute the index and we found the same behave we found force for this interval of fields between zero and one we found kind of stability lines that has to do with the van der Waal loops as well so we can complete and find a curve that is analogous in the shape for the for the entropy so that's what I want to show you now basically and then if you go back to that result I show you from Kassner we can find also relation in this divergence in that quantity that not only diverged for for the phase transition diverse for the whole interval that we find van der Waal loops and then we can characterize like topologically this van der Waal loops as well so then we can conclude that the singularities in this Jacobian dance that I just mentioned corresponds to van der Waal loops and then remarkably like this may has to do with the Augustus shape talk that he gave like few days ago because we basically the defining like an ansatz another measuring and he mentioned that can be several like equivalence between measures we basically found that at least for this model I don't we don't know what happens for the other models but if you compute magnetization use the order average and then both magnetization we found same magnetization and if you try to do analytically you can also find critical temperature only using these these definitions so basically that was the the main result that we found it's very technical but I tried to summarize we can also compare this with all the proposals like terms of energy energy landscapes and past works that was done like in this paper in 2003 by Lapu and Petin and Cohen and basically if you do this if you try to understand using that quantity you may have the nearly same results but when you try to find this loop of this kind of topological contribution for the entropy the correct slopes come from the other characteristics it's they differ really slightly but when you try to find this loop it it's it's like like that so the this loop that gives you the critical temperature for phase transitions is like embedded in the other characteristics at the critical point and basically for that model we also have like results for short range model in the x5 1d model actually but CCA by lack of time and the focus on these conferences on long-range interaction so we I want to stop here and conclude that we propose a topological approach that may allow established connections between thermodynamics and topology of configuration space and we found evidence that the information embedded in the other characteristics at least in the more in this model suffice to describe the magnetization and and the canonical microconon pardon sorry and at the critical temperature and I would like to say that so we started the other classical season and related context like the contents of energy landscape and things like that are very very visible so that's that was what we did