 Thank you for the invitation and obviously I would rather be in France and close to Paris for this occasion. But yeah, let's keep fingers crossed that next year we'll be under better science. So today I want to present some joint work with Gleb Koschaboy on the maximal green sequences. So first I want to talk a little bit about cluster algebras since I don't want to assume that everyone is familiar with them. And then introduce maximal green sequences and then give some combinatorial constructions for certain maximal green sequences on certain cluster algebras. And so let's see. First we start out with a quiver queue, finite quiver and we don't allow loops or two cycles. So here we have such an example. I mean, we can see there can be cycles but just two cycles are forbidden and one cycles. And due to this setup there is associated a cluster algebra due to the work of Fomen and Silevinsky. And we denote this cluster algebra by aq. Okay, so this cluster algebra lifts inside the function field generated by the cluster, initial cluster variables and these are just some internals associated to the vertices. So in this quiver up there we have four vertices and to each vertex I associate a variable. And in the field of rational functions of these four variables there will be our cluster algebra living. And to obtain it we start out with the polynomial rings of these four variables. And then there is an inductive procedure which takes as an input this quiver and our initial set of cluster variables and produces a new set of cluster variables and a new quiver. And this procedure goes on and on and well whatever we obtain after doing this infinitely many times possibly is our cluster algebra. And this procedure is called mutation at the vertices of q. So for each vertex we can do your mutation and obtain a new set of cluster variables which we have to add to our algebra. So essentially the datum is we have a quiver q and our set of initial cluster variables and we do mutation at the vertex k to obtain a new set, a new quiver and a new set of variables. And we have to do this infinitely many times to obtain our cluster algebra. So how does this look in more detail? This procedure of obtaining of mutation, of obtaining the new datum consists of four steps. So one is we take our cluster variables and we change them but in fact we only change the one where we mutate. So if you mutate at vertex k we change the cluster variable ak and it's going to be replaced by one over ak times the sum over two terms and one term is the cluster variables attached to the incoming errors and the other the product and the other is summoned is the product over the cluster variables of the outgoing errors. So this is what we do to the variables and then we have to do also something to the quiver to keep track of things and first we need to add transits through the vertex k then we reverse all errors at k and finally we remove the two cycles if any of them occurred. So let's see how this looks in our example. So our quiver we had x4, x3, x2, x1. So we replace x3 by x3 prime which is x1 which is the incoming error plus the product of the two outgoing this is x2 times x4 divided by x3. And then we have to change the quiver so there are transits from x1 over x3 to x2 and to x4. So I get two transits from x1 to x2 and from x1 to x4. Then I have to remove to reverse the errors at k. Sorry you have variables for a now a is a x. Oh yeah that's probably because I drew them by hand. And you know there are a and x cluster varieties so it's really central. Yeah absolutely it's kind of unfortunate. So yeah this should all be a's. Yeah as pointed out there's a dual procedure on the x side of this but this looks a little bit different. I mean the quiver stuff is the same but on the variables there's the dual thing happening. Alright thank you. Okay so this is this mutation. Let's do another example to see what kind of stuff we get from this. So the easiest I guess non-trivial example is the a2 quiver and we have two variables x and y so we can mutate that both of our vertices. Let's first start our off our algebra comes with x and y already. Then we mutate the first vertex the error there is no transits of course we can just turn around the error there will not be two seconds so we see the error will always flip. I forgot to say if you do this process twice at the same vertex then you go back so you won't get anything new. So in this example we have two vertices so we can only mutate one two one two one two we cannot do one one I mean then we go back and don't get something new so and if you compute then the x variable goes to one plus y over x and then we mutate at two and we obtain this term down here and well it's the rational function one plus x plus y divided by x y and we go on mutating at one then our first variable changes again and we obtain one plus x divided by y before we had one plus y divided by x we mutated two and well we see that the second variable becomes x again which is already a little suspicious but then we go on and mutate at one and we see the first variable becomes y so in conclusion we are done because we already have symmetric expressions up here so this will be our cluster algebra in this situation. Usually though this procedure will not will not stop so it will be infinite in fact due to Fomenko Lewinsky there's a theorem that there are only finitely many cluster variables if and only if the cluster algebra comes from a Dünken quiver of type ADE so so most of the time it will not stop. There is however second notion of finiteness due to Keller and also appearing in the work of Gerjoto Moore and Neitzke implicitly called Maxime Green sequences and this is a more more broad notion of finiteness and this is what I want to discuss today so for this we start with a quiver q and as to say to the quiver q a framed and a q co-framed quiver so what is this? Well let's just do it on an example so if we take a three quiver we obtain the framed quiver by adding copies for each vertex and then errors from the corresponding vertex downstairs to the new copy and the co-framed is dual I mean the dual construction so let's say if this is v1 v2 v3 we call the downstairs vertices v1 bar v2 bar v3 bar all right and they are a little bit special so they are frozen which means we are not allowed to mutate them they are basically there to record some stuff for us so this is the framed and co-framed quiver we just add vertices and errors downstairs or upstairs now we call in a vertex of our quiver green if there is no incoming error from a frozen vertex yeah I mean this is not I mean not just for the framed and co-framed quiver but if I have any setup where I have a set of frozen vertices and a set of not frozen vertices I can make this definition and say in vertex screen if all errors which connect this vertex to frozen vertices are outgoing and the same I can define red is if they're only incoming vertices from frozen so let's nevertheless look at the example we had of the framed quiver and here we see of course each vertex has exactly one error to a frozen the downstairs error and it's always outgoing so all three vertices so all vertices are green so basically framing is like making everything like defining everything to be green like as an initial setup but now we cannot imitate this quiver not that the frozen but at the mutable vertices for example at the vertex v1 and what will happen while we get a transit here from v2 to v1 bar and the error from v1 to v1 bar will turn around so that will make v1 red and while the other two vertices will stay green but we see that the situation is a little bit more complicated than in framed or co-framed so yeah however the definition still makes sense and in fact there is a theorem due to Dirksen Weimar and Silevinsky which says that if you start with a framed quiver so not just any ice quiver but with a framed quiver and mutated by a sequence s so s is just the finite sequence of mutation steps so finite sequence of vertices then after doing this every vertex will have precisely one of the two colors red or green yeah so like in this example we have red green green but this is completely general that they are partitioned always all right so this is the setup and now Kehler defined the following so we look at again at a sequence of of mutation steps or a sequence of vertices of our quiver and this sequence we call green if all of the vertices are green yeah I mean of course not in the initial quiver but when when they are due to be mutated so basically we only mutate at green vertices so after I mutate the first i minus one I want to again mutate at a green one and this is also the term of reddening which is like making red so this means that after I finished with the sequence everything is red and then there is maximal green which means it's green and redding so um so I cannot extend it the green anymore because everything is red already okay so um let's make an example let's look at a two again so we get our framing we have two green vertices and let's mutate at the second vertex first the arrow here will turn around and we see there's no transit so just the arrow from the second to the frozen second will turn around and will turn this arrow red and now we can mutate to the first one and similarly with just the first arrow will turn around and we already have a maximal green sequence consisting of two mutation steps here so that's nice we could however also have started with the first vertex and mutate this one um and in this situation we get the transit so here's a little bit more complicated the ice quiver we obtain and we see what happens when we mutate the second quiver now the second vertex now we have a transit here from the first to the second to the first frozen and this will give us a vertex from the first to the first frozen but this will create a two cycle which we have to cancel so um so we will obtain this quiver here and um we see we mutated the second vertex which becomes red but here the first vertex became green again so it's not always just that this one vertex flips and then we mutate the first again and now we have a red sequence and a red ink sequence or a maximal green sequence which was three lengths three here um all right and um something we see here is that uh these two quivers are isomorphic as ice quivers um well okay I mean they're not that big so might be a coincidence but this is a general theorem due to Prusler, DuPont and Perotin which says um whenever I have a maximal green sequence S then there exists a unique isomorphism between well I mean the red ink I created and the initial quiver when I co-frame it so that's nice what does it give us these green sequences so um they have nice consequences so if a quiver Q has a maximal green sequence then um under some my technical assumption um the for contour of conjecture holds for um for the cluster algebra AQ um which says that there is a nice canonical basis a parameterized by uh by um the mirror dual cluster tropical points of the mirror dual um cluster algebra um and this is um due to a cross-hacking KL and conceivage so this is very nice um there are other consequences so um the refined Donas and Thomas and invariants due to a conceivage Stoiblmann they can be computed as a product over certain quantum dialogues um where the um factors are indexed by um the mutation steps in the maximal green sequence so this um is due to Keller and also um pops up in work of Gaiotto Moore and night skill so in all in particular also we obtain quantum dialogue algorithm identities from this so for example if we look at the a2 quiver we obtain um the identity uh e x e y is e y e q to the minus one half x y e x where e is um this uh quantum deal algorithm so and of course for for other quivers um yeah kind of complicated identities arise these ways you should add x and y q commute yeah come again you should add that x and y q commuting variables not yeah x and y yeah yeah and i think it's all think it's one need not not not necessary maximal green sequence just redening sequence yeah yeah yeah that's that's that's true yeah um i yeah do you know if there's like any theoretical um advantage only uh inherited by maximum green sequences or is i mean besides that they are easier to find somehow no no i prefer i prefer redening sequences yeah normally yeah but okay but also to construct them or for theoretical purposes i don't know yeah okay yeah i mean um exactly so usually the properties are are inherited by by redening sequences and not maximal green sequences as far as i understand um but the maximum green sequences there are somehow in some situations more canonical so which makes them easier to find maybe okay so a similar one can formulate this that uh for physicists that the bps specter can be computed with these maximum green sequences and i guess that's the point of view of gaiotto moore night square so class algebra's where the quiver possesses maximum green sequences have very nice properties which we want to have so the question is which quivers possess maximum green sequences and i mean like some examples i want to list is an acyclic quiver has a maximum green sequence um there are a lot of examples in in physics so for example gaiotto moore night square and other works where where um maximum green sequences are implicitly at least um um constructed um there is also work of mage where he constructs a maximum green sequences for um the double prouhat style associated to s l um for e for the identity and um the longest word so and this this has a cluster structure due to Bernstein form in sylvinsky and cluster three so one question is what about other types here uh than type a and what about um other double prouhat cells um yeah what's the picture here so and um one way to do this is to look at this triangle product q r of two quivers oops q and r i kind of like to call them a and d for us cyclic and dunken but um it's also confusing a and d so let's call them q and r um and this is defined by um keller this triangle product and basically you take the product of the quivers and you triangulate it in the sense that you uh like everything which comes not triangulated due to the product is going to be triangulated um so let's do an example d five um triangle product a four so first we take the product of the two quivers and um now we have to triangulate it in the sense that if when I go blue and red then there's a diagonal always like if the quiver before the red quiver is not triangulated then I don't change that okay so this is the the box product triangle product and one question which arises is if we can combine maximal green sequences for the individual quiver to obtain a maximal green sequence for for this product and um the answer here is yes if s is a source sequence and t is the sink sequence now so they are the maximal green sequences by assumption s is the maximal green sequence of our first quiver and t um the maximal green sequence of our second quiver and if you further more assume that s is a source sequence so it only um like the points where we mutate are always sources at the point um when we mutate them and t is the sink sequence then we can combine them to obtain a maximal green sequence for this triangle product q r so this I might I want to um demonstrate on an example so there is this nice um uh applet written by Bernhard Keller where you can um work with um mutations and and and quivers of of cluster algebras um so once we open the page it already starts out with this quiver and um what can we do we can um add some nodes we can add arrows and um then if you click on an on a vertex here then um the mutation has been performed and also we can uh freeze nodes and um this way we can con control for um green and red names sequences and these things so let's see here in our example um we have here the fork lever and we want here um a sink sequence so let's see uh maybe you should check for green so we can also do an at framing which adds like the framing all at once we don't have to do this individually all right and we can also hide them so now um here we want a sink sequence let's see if that works all right sink sink sink sink sink so that's okay and here we want a source sequence but that always works for our cyclic good so now how to combine them for for the product for the triangle product well we start with the um with the um the sink sequence here and do one first one whole round of the um source sequence here and then we move on and you see what happened downstairs now happens upstairs um in the product so this is all very um well behaved in this situation so we stay always within this um realm of yeah okay this was yeah there probably is sometimes it gets some numbers it confuses them yeah anyway but um let me see I somewhere prepared this so I guess this example is um d4 times d4 which is a little bit harder to um to see um and let me see if we can do it here so I already implemented it and now I just forward and um work through the sequences so this was one time the um sink sequence in in the base and now we do it again second time and we start again with our sink at seven all right not done yet so here we have this um reddening property so we found a maximal green sequence here and um so this works in general if you have these um sink sequence downstairs sink maximal green sequence and source uh maximal green sequence on on this side okay good so let me go back to the slides so now where we have established this the question is um how to find these source and sink uh maximal green sequences or when do we have them and um well one thing is the source maximal green sequence um is equivalent to the quiver being a cyclic and about the sink maximal green sequence well we know that if the quiver is um duncan of type ade then there exists such as sink maximal green sequence due to result of pristler du pont perotin um and we conjecture that this is equivalent I don't know if uh someone in the audience has an opinion about this so um so this would mean that um this this construction works um for exactly for products of acyclic quivers with triangle products of acyclic quiver with duncan quivers so what's the upshot for um our initial question well um for these cluster structures introduced by um Bernstein, Forman and Silevinsky on the double prouette cells we can actually um obtain um the the quivers um as full subquivers of box products of um the q is a duncan of the of the same type as g and r is just an a to the n so this means that um for these um cluster algebras um we obtain maximal green sequence with this construction um we furthermore obtain optimization sequences for the frozen vertices of of these um double prouette cells and um so this is basically the same technique um um and um thus we can uh the the work of crosshacking Klan Konsevich produces actually um the desired basis here okay so let me show this optimization sequence property so so here we have uh d four times a four and um well we freeze the ends of the um duncan diagram yeah so this that's why they are blue they're frozen um and this is because um yeah this comes naturally from from the cluster structure of of these double prouette cells so it's something one one has to deal with an optimization sequences mean like for example the vertex 10 here has an ingoing error from 11 and outgoing to two and optimized is if there's only ingoing error so um um so it's like yeah i mean it's like similar to these to the screen but it's on the on the on the um for the frozen vertices so the colors here we can ignore this is just because um we we add yeah because of these frozen so but it's they don't have meaning for us now so what we do here is um if we have um the situation that the outgoing error is on the level we can just mutate on the level line and one can show that this is an optimization sequence so for example if i look here at the 16 i can mutate 15 and 14 and then i have only an ingoing error and this is optimized so this works fine but um if i want to optimize the left hand side i the same principle wouldn't work so um what we do here is um we apply similar techniques as for the green sequences but for maximum red sequences for this quiver and for the mutable for the mutual sub quiver here so i start here with the um sync and mutate here and this looks a little bit messy on the right but that's fine because we we did that already with the right i mean we already found optimization sequences so we basically can cut them off and forget them we don't need them anymore as long as the left hand side stays nice it is just because it was not a sink in the in the full quiver the mess on the right all right so now if you forget the right hand side we see that that this the rest of the quiver is is um well isomorphic to what we had before so we can we know we already done when we now go level lines so here then i have only in going errors so we get these um optimization sequences so in some sense it's it's a little bit of a similar game these optimization sequences and maximal green sequences but in the optimization sequence we don't start with the framing like there are several errors this somehow is related to um well i mean this is like the non-competitivity of of matrix multiplication in a sense okay so this is this and last thing i have to um i want to say is um so the sync source type of maximal green sequences they uh can be defined more generally so this was not just for triangle products so for example i mean there are several versions to formulate this but if you take gamma is an acyclic graph and are just for simplicity is now a n then um we can do the following how much time do i have yeah 15 15 yeah well i won't need 15 anyway so so this is very simple acyclic graph and as a base we take a n here and now we just write some copies of um these some some parts of this a n quiver here and um we um draw inclined inclined um errors which just alternate between between the um the level lines here but not in the in the product way they can skip something and be a little bit more irregular and here we can also produce a sync source a maximal green sequence because the graph is acyclic so essentially we need that the projection onto the graph is acyclic in each step if you define it the right way but if the graph already is acyclic then this will always be an acyclic quiver that's one thing we can do another generalization is we can also allow for example multiple inclined lines here but they have to be compatible so they have to induce the same orientations on this graph gamma in in this in this projection way i ended it so um so just two examples so if you want a sequence for the circle um we can write it as like this where we have here a whatever and here just the one point and this is our inclined thing and this produces us um here a maximal green sequence um or we can here multiple inclined errors so if there are five errors here five here five here so let me get the pointer five here five here five here then there are three then there are two um and here there's one and one so they are a little bit um messy but um for example we write them as there's a blue one then there's a double red one which stops here and so on and to this blue red and um and yellow we associate now the the um the sequences with which they produce like zero stands for the bottom line and one for the top line and if I look at the blue one it produces zero one um while the yellow one produces zero one one zero zero one zero and well the green line and the red line as um indicated here and if you write them underneath we see that they they they agree in the initial pieces like in general if you have more level lines they have to be um shuffled and accordingly um but since they agree here um this works and we obtain um a maximal green sequence here and it's given by zero one one zero zero one zero this means we first go level line zero all the way then we go one then we go again one then we go zero then we go zero then we go one then we go zero so let me um also this also display this yeah this is the quiver we just looked at and um I first go in the bottom line then I had a one so I go in the top line then I have again a one I go on the top line zero in the bottom line zero uh one zero and the last was three um yeah so I mean the the way they are constructed is that that the bottom quiver never gets disturbed by the top quiver and the other way around right I mean if you have this property then you can always combine the sequences for the sequence of the bigger quiver and it's also be compatible with the with the um how it moves the framing because um yeah the level lines don't interact in this way okay so I have one more sequence so thank you for the attention thank you worker um do you have any question from worker yeah I have just very small questions this triangle product which you defined yeah um it's for quivers and quivers are in fact we understand this is quiver's potentials and generic potential yeah uh does it's is it some notion of product for quivers with non-generic potential well I don't know yeah because yeah I don't know I didn't think about this because in our applications right we take like acyclic and duncan to obtain something for these double brouhats cells and there we don't arise at this problem right oh yeah yeah you don't have potential from the very big yeah yeah the potential comes up in the triangle but not in the in the factors yeah but yeah I don't know what to do if the what happens with the potential I mean yeah and also by the way if you get this product for acyclic quivers which you get triangles so you get some generic potential but can you exhibit kind of explicit example of this generic potential because it's it's kind of a bit elusive object you should remove infinitely many hyper surfaces in some parameter space what do you mean by we get the generic potential no no uh you see that if you have a quiver in general then you consider all possible cyclic quivers which could be started from cubic quarts and so on and make all possible linear infinite linear combination get infinite dimensional general vector space and then in this vector space infinite dimensional space you get countably many algebraic hyper surfaces when you get troubles after some finely many mutations but in the complement the complement will be like very irrational yeah okay but and for this there's a product you get a lot of yeah you get infinitely many cyclic quarts so so exhibit just one concrete generic potentials it's not obvious problem instance yeah you say it's not not obvious how to write down a generic potential yes yes yeah because we only make choices you can make wrong choices yeah I always thought it I always assumed it's um for this kind of quivers like triangle products it's not not a problem but I don't know yeah I never yeah yeah yeah yeah for example here it's a naive story you get all small triangles yet just all small triangles is weight one that's it yeah but maybe it's the right the right choice yeah but maybe also for the more general situation it could be a question yeah yeah okay okay so another question Maxim so they say theorem so if we have quiver with which possesses maximal green sequence then the Jacobian algebra is finitely dimensional ah yes yes in this case problem is no problem is no maybe for this maximal green sequence you you get kind of unique yeah so maybe from this we can extract some canonical potential yeah in fact I'm not sure you see that for for any quiver you get certain field of maybe of transit this degree which could be finite infinite this number of parameters of all but potentials modulate gauge transformation so it gets this huge field of unknown transcendence degree and yeah this was class p of quivers when you get point you get generically only like field will be rational numbers but for such things I don't know yeah and in principle when you make his mudding this reading sequence you get automorphism of this huge field so you don't have really canonical potential but what's what's your point of view is it like that the the point is to get to have a nice potential and just to work with it or is it the interesting thing also to study how the potential yeah yeah no no no the right questions maybe you can see there's a this field of rational functions and coefficients of this potentials just rational functions and infinitely many variables and consider invariants under gauge under gauge transformation for this pass out completed pass algebra get certain huge field which is preserved under mutations and then if you make this maximal reading sequence you get automorphism of this field kind of like rational automorphism of some variety maybe infinite dimensional and the canonical potential will be fixed point of this rational morphism whatever it means but then does the canonical potential does it have meaning or is it just to work with I don't know no no you see that for surfaces from people who consider this you partied graph on surfaces there is a kind of canonical collabial theory category so there is a canonical gauge equivalence class of potentials which is preserved mutations but can you provide us with reference or maybe just I don't know yeah but no but this is perspective from 4k categories for some non-compact collabial three-fold gives you a kind of implicit definition what is the potential I don't I'm not I'm not sure that it was really calculated okay so then we ask you okay okay thank you so generic potential is somehow still more interesting to consider than exceptional cases it's it's not clear yeah even for surfaces what is the dimension of space of potentials this transit this degree is not clear yeah so but what could be an interpretation to a study like potentials which are not generic no you see these mutations make sense for this collabial sphere collection of spherical objects in collabial three categories just the troubles that you not not always you can do mutations yeah sometimes you can you should stop it belongs to different perspective how we can use this existence of a rediamen for a maximum green sequence to construct potential it's I don't know it's it's not it's not it's kind of different story yeah I don't know just will give us some equation like for this die logarithm identities no okay so then no no no it's strictly speaking this potentials modified die logarithm identities because uh when you have potential then you consider this shift of vanishing cycles and trace of rabbinus and stick things like this to get identities for something and only for generic potential you you're sure that you're doing with die logarithms otherwise you're doing with something else but still get some identities but the kind of will be different identities this we have kind of special potential not generic one do you have some examples not really but yeah but for example I remember we're gonna have three vertices and three arrows going from first to two to three or maybe two goes from first to two to three to two to one the something which is under mutation goes to itself yeah there is hidden I think elliptic curve and it depends on elliptic curve if it's it's complex multiplication one can using this method we can variance can produce something slightly different yeah yeah it's awesome it's awesome little ambiguity so where's the elliptic curve uh just a second elliptic curve maybe I'm wrong but there was actually some one parameter here and in this case when you consider dt invariance it's determined up to some series in some center and this personal algebra and it was uh I know I think it was elliptic curve when you consider space of uh representation of a query get as a parameter spaces will be some elliptic curves motives of elliptic curves appearing in the middle how much rising but what I mean representation of yeah because I even consider vanishing cycles and fix you can see the model space of objects in three color categories then uh it's some dimension vectors you get uh non-trivial varieties which are critical points of potential since it will be and the commode will become is if I'm not mistaken commode of some elliptic curves depending on uh how exactly choose the potential for the square yeah so it yeah so it gets some non-rational varieties it's end of the day it's a model space of representations definitely okay yeah I mean as a time yeah okay thank you very much everybody thank you very much for the nice talk