 Okay. Well, thank you very much. I'm honored and thrilled to be here. So the first thing to say, I guess, is happy birthday, Dirk. I guess is very belated at this point if I'm not mistaken at some time in July. So Dirk was a very important kind of mathematical influence on me about 15 years ago when I arrived in Boston, where Dirk was spending half the year at the time. I started talking to him and learning about his work on the half algebras of renormalization and thinking about how these structures could be kind of lifted to a categorical level. So in fact, today's talk, which is sort of based on recent work is very much, at least in my mind, related to that set of ideas. And over the years I've had the pleasure of visiting him in Paris and then in Berlin and I have great memories of those visits. So thanks for all the inspiration, Dirk. All right, so let me, let me get started. So the main idea of this talk is that one can interpret things like the half algebra of rooted trees or the half algebra Feynman graphs, as well as a number of other combinatorial half algebras as hall algebras and what's a half algebra well a hall algebra is an algebra whose structure coefficients count extensions or short exact sequences in a category. And so, so I will talk about these types of categories, and I will, the ultimate goal is to describe the construction which, which attaches to a project of TORG variety, a half algebra. And this will arise as applying the whole algebra construction to some some category of coherent sheaves in the, in sort of a somewhat strange setting. So, my plan is to is to first talk a little bit about hall algebras in the kind of traditional setting they they are a tool that is important in representation theory. And so for people who study quiver representations and and quantum groups and things like this. But the, the setting in which we will ultimately apply them is actually somewhat different, which are sort of categories that are combinatorially defined which are non additive, but, but nevertheless one can do the same thing. And then I'll, I'll discuss some some elements of sort of the algebraic geometry of monoid schemes and, and the kind of combinatorics we see in that setting. Okay, so what's the, what's the traditional setting of of hall algebras. It has been sort of, they have been used in, in representation theory since at least the 80s. You start with an abelian category which has some strong finiteness conditions so it's this is called being finitary. And it means that the the set of the set of morphisms or the set of maps between any two objects as well as the, the set of extensions between any two objects, or finite as sets. And this, this is pretty, this is not so easy to, to achieve somehow because, you know, many, if the, if the categories linear over over some field that field better be finite or else a finite dimensional vector space is not going to be a finite set. So the main sources of, of examples of these types of categories are quiver representation so quiver is a directed graph and representation is, well we attach vector spaces and linear maps along the edges. So the category of quiver representations over a finite field has, has this property, and another geometric source of examples is the, as the category of coherent sheaves when, when x is a projective variety or a finite field. Okay, so these are the kind of two main examples one one can have. So what is a whole algebra well as a, as a vector space, it's just functions on isomorphism classes in the category so we just take this, this is the most naive version of all algebra I should say there are more sophisticated versions but just take functions on isomorphism that have finite support. So they're non zero and only finitely many isomorphism classes, and you can equip these with a type of convolution products so the convolution evaluated on the isomorphism class of an object M is obtained by summing over all sub objects and of M and evaluating F on the quotient and G on the on the sub sub object. If you squint your eyes and replace the summation sign with an integral here and think of M mod n as x minus y and, you know, and as why then you can see why this is called the convolution is reminiscent of convolution of functions and harmonic analysis. And a basis for these finitely supported functions is given by, by functions on just individual isomorphism classes delta functions. And so if you, if you convolve to delta functions, which you, what you see happens is you, you get you get a summation, or where the structure coefficients. And that's the following, or the structure coefficients correspond to the following numbers so you're counting the sub objects of K, which are isomorphic to N, and such that K mod the sub object is isomorphic to M. So, up to some automorphism groups, which you're counting are short exact sequences of this form so n goes to K goes to M. Okay. So that's what I meant when I said that's the structure coefficients of hall algebras count short exact sequences. And, and of course this implies that the structure coefficients are non negative also right since they count things. So to, I'm going to try to sort of connect this stuff with with sort of quantum groups and when if you've seen quantum groups you know they're a bunch of powers of cues floating around. And to, to get those, we have to take a slight twist of the multiplication. So you introduce something called the multiplicative Euler form and you. So, so here is a formula so of course the for this formula to make sense you really need a category where, you know, X are are non zero only in in finitely many degrees so something of finite homological dimension. And you can think of this as as basically Q to the power of the ordinary Euler Euler form. And so, so we take our old multiplication except we multiply through by this, what will ultimately basically be be a power of Q that depends on Eminem. Okay. And then the theorem of wringle and green is that is that if you have a finitary billing category, then these these these algebras either the one with the Euler form or without it are our associate of algebras. Okay, so you get a you get an associate of right. So if you want to study co algebra structures. This becomes a little bit more subtle so you, if, if a happens to be be hereditary so that the, the, the global dimension of a is less than or equal to one, then you can, you can equip this with a with a co product and an antipode. And get a get a half algebra in some cases this this won't, this will be something like a topological half algebra because the, the co product might not might land in the completion, for instance, but roughly speaking you get a get a half algebra but only in this in this nice case where the global dimension is less than or equal to one. And now, the claim is that that if you, if you see if you look at what you get, and as these hall algebras you get sort of interesting quantum group type things. So, as I said the kind of two main examples of categories were that are that are finitary and a billion being quivers and coherent sheaves on some on some projective variety. So, so here's the, the first theorem tells us what happens when, when we look at quiver representations. If I take a quiver I can view the underlying graph undirected graph as a dinking diagram for for some for some cuts moody algebra, and the theorem states that that this that this hall algebra that we computed by counting short exact quivers contains sort of a positive half of the quantum group corresponding to this to this cuts moody algebra so roughly speaking, just like ordinary. Let's say semi simple because moody algebras these quantum groups have kind of triangular decompositions, and roughly speaking this plus means that we're looking at the kind of upper triangular part here. And this map is, is, is an isomorphism in in sort of finite type so when the, when the, when the dinking diagram is of type edd, then this is actually an isomorphism and there is then kind of a procedure by using using the the Drinfield double, which you could then use to to recover the entire quantum group so I guess what I want to emphasize here is that if we hadn't learned about quantum groups from from Drinfield and Jimbo we could have discovered them in principle by using this this whole algebra construction. And another thing to kind of point out is that this, the, the order of the field, which is q right this q is a prime power, the square root of this thing appears as the, is the deformation or the quantization parameter here. Okay, so, so somehow prime powers have something to do with quantization and and I'm not sure if anyone really understands why that is. And so, so this was, this is a theorem that somehow tells us what happens with quivers and as a just a sampling I'll mention this theorem of component of and castle Bowman which tells you what happens if you take the category of coherent sheaves on just P one or you know simplest projective variety you could have over a finite field. And then also you see a quantum group type object you have. But now it's a it's a quantum affine algebra so so roughly speaking this is this is what happens if you if you were to to quantize the loop algebra vessel to rather than just vessel to itself. So, so in both of these cases we get some some interesting kind of quantum group. And there's been lots of work on this by, by, by many authors so just work of bourbon Schiffman, Eric Vasero, Capron of, and, and many, many other people. And so, so as you as you increase the so in the case of coherent sheaves. If you increase the complexity of the of the variety, this thing gets complicated fast so already already for elliptic curves it's it's it's not it's not easy to see what what happens and you get interesting algebras like the sort of double affine heck algebra show up. Sort of more generally, this the study of hall algebras of curves over finite fields is sort of related to to the theory of automorphic forms on function fields so the the action of the hall algebra on itself corresponds here to to the action of sort of geometric heck operators. So, already for hygienists it's sort of, aside from kind of abstract results it's not easy to see what happens concretely and when when x is a variety of dimension greater than one than basically it's, you know, very little was known and understood about what these what these hall algebras look like. And, and also these these categories of coherent sheaves no longer have global dimension one or less so the sort of co algebra structure I mean you can write down a ring, but you know whether this is a or not becomes a much harder question. And, and so the stock is ultimately aiming to, to look at this last case of sort of higher dimensional varieties but in a certain kind of combinatorial limit, a type of classical limit. Okay, so, going back to to the formula for for the multiplication in the whole algebra. If we look at this. There's nothing about this formula which somehow explicitly requires the category to be a billion or added. Okay, so, so, even though historically, people are looking at quiver representations or coherent sheaves. There's this this formula makes sense and serve a more general context. And, and that's the context that will be interested in. When, when I started thinking about this. There wasn't really necessarily a very good framework for thinking about these non additive examples but since there's been very nice work of Dickerhoff and Capron of. They define a class of categories. They are called proto exact categories. Okay, and these are. These are somehow. These are generalizations of Quillen exact categories but which are allowed to be non additive, and which are somehow tailored to, to hall algebras. And, you know, there's a very nice formalism that that that that was developed by them and and sort of related to kind of higher categorical aspects of the story also. And one thing to note is that that any, any of these sort of proto exact categories has a has an associated algebra key theory. Okay, so you can, you can define a key theory of sort of higher key groups of anything that is, that is proto exact. The show is that if, if you have a proto exact category, which is, again, finitary so that the morphisms between any pair of objects and the extensions of any two objects are finite as sets. And then you can write down an associative algebra using the formula that we saw before. So everything works, kind of as, as expected. Alright, so. So now there are lots of examples of these proto exact categories that are combinatorial and non added. And, and so, let me just mention a few. The simplest, the simplest examples that is not additive is perhaps is the category of pointed sets. Okay, so. And sort of related examples are, if you take a monoid and the category of modules over the monoid whereby module I mean appointed set with with the action of the of the monoid so things that are by people in semi group theory or called acts, sometimes. Other examples are, you know, you can take a quiver and instead of putting vector spaces on each vertex, you can put pointed sets. And again, things can be made to work more or less as before. Other other structures in combinatorics such as pointed matroids. And then the examples that come up in renormalization, things like rooted rooted trees and forests, as well as Feynman graphs are examples of these, these proto exact categories, which means as I mentioned already that that therefore they have associated algebra k theory so which which we know is interesting because I mean I'll get to this in a second but these these hierarchy groups are somehow known to be very interesting. But the thing that will be of the main interest in this talk is this last example which is the category of coherent sheaves on something called the monoid scheme so I will this this monoid schemes are basically sort of versions of algebraic geometry and in a non additive context and sort of correspond to sort of combinatorial units of ordinary ordinary schemes, if you will, but they are they are a good source of these sort of non additive categories where hall algebras still make sense. So, my goal is basically to do the following so since we know that we know that hall algebras give interesting quantum groups. And, and so, so in particular if you have a higher dimensional variety something like a surface and beyond, you would expect to, if by looking at this, this hall algebra of coherent sheaves over FQ you would hope to get some sort of interesting, maybe quantum group like object but this seems very hard. So, so let's try to do something simpler. And let's try to compute as classical limit so I want to take the limit as the deformation parameter. This Q goes to one and see what happens to to this whole algebra. Okay, so it should, it should become somehow more commutative in this limit. And hope to to use this information about this classical limits to understand something about the, the original structure of the thing that we were really after which is which is the kind of quantum object. And this somehow ties in with this, this kind of philosophy of doing things over the field of one element and this, this stuff comes up when you're, when you look at limits of calculations over finite fields as, as Q goes to one. So this is, this is an old story and, and you know, kind of a set of, I would claim it's more of a set of sort of interesting analogies and ideas rather than than maybe a sort of a complete theory at the stage but, but the ideas are kind of neat I think so let me give a couple of examples here. So for instance, let's consider the enumerative combinatorial problem of counting subspaces of n dimensional vector space over FQ over finite fields so in other words I want to count the number of points of this across over FQ. So if you do this, it's an elementary exercise to see that that this is given by a rational function which is called the rational function in Q which is called the Q binomial coefficient and choose K. And if you take this rational function and it has a has a well defined limit as Q goes to one, in which case it's this that limit is just the ordinary binomial coefficient. So, this, this sort of leads to this idea that the, that the limit as Q goes to one of the category of vector spaces over FQ is something like the category of sets or maybe better pointed sets. Okay, because you want some some something corresponding to zero in your, in your quote unquote vector space. So, again, this leads to this idea that a pointed set is a vector space over F4. Let me just mention another sort of classical observation here this is due to teats, is that if you take, if you take a simple algebraic group over FQ and you, you count so you count the number of points of this group over FQ and you, if you take the limit suitably normalized here as Q goes to one, what you will find is that you get the order of the vowel group of G. And so, so this again sort of led to this notion that maybe, you know, if we if we had a good theory of algebraic groups over F1 then then the so F1 points of an algebraic group should be the vowel group. And to some degree, this has actually been been made precise in, you know, the work of, of Lorscheid and and Konkensani and others. So the, I'm not going to get into this very much but the you know the basics of the sort of dictionary is is the following so the category of vector spaces over F1 should be something like pointed sets. So an algebra over over F1 should be a monoid. So, all these, all the structures over F1 are somehow non additive. So, so, so you lose addition there. So, as you go from vector spaces to sets you, you lose addition as you go from algebus to Monoids again you lose addition. So the notion of the module is becomes a pointed set with an action of a monoid. And so it's not surprising that the notion of kind of a scheme or algebraic variety over F1 should also be something built out of Monoids. So, let me, let me say a few words about what when monoid schemes are. So, I have this one additional slide here. So, you know, in, at least in my mind, these various sort of proto exact or proto abelian categories that are non additive are somehow, you know, can be in some cases at least sort of thought of as as limits. And it goes to one of, of, of sort of additive things. So, so these, these are somehow the, the analogs of, of abelian categories, abelian or exact categories over F1 things like matroids and graphs and things like this. And in this combinatorial setting. So I mentioned already that this, if you're working over FQ the the issue of the existence of a co algebra structure becomes kind of subtle. You need some conditions to to define a co algebra structure, but if you're working with these combinatorially defined categories you can do something very simple minded so our whole algebra is functions on isomorphism classes. And most of these combinatorial gadgets have some, have a kind of a co co product in that category which more or less amounts to disjoint union or wedge some which is kind of the pointed version of this joint union right so we have two Feynman graphs we can take their disjoint union. If we have two rooted trees we can take their disjoint union and get a get a forest, and, and so on with with combinatorial objects, the co products or somehow more or less disjoint unions and so. So you can define a co algebra here co algebra structure which is sends a function to the function evaluated on this disjoint union or wedge cell. This turns out to actually be compatible with the product for in these combinatorial categories so you, you get something which is manifestly a co commutative hall algebra or a co commutative by algebra. And it's, you can it's also easy to see that there is a natural natural grading by by kind of a positive cone inside the growth in the group, and this thing is connected so so you always get something which is, which is kind of graded connected co commutative half algebra and so, at this point we can apply the Milner Moore theorem it'll tell us that what we have is an enveloping algebra. Okay. So, and this, this, this Lee algebra, this this Holly algebra is just corresponds to indie composable objects so so things like, you know graphs that are connected or trees that are. Or I mean, things that are honestly brooded trees and not not for us and so on. I should say here that in connecting the sort of hall of a story with the usual story of the half algebra of graphs or trees, what we're getting here is the dual. Rather than, then, so we're getting an algebra which is non commutative but co commutative as opposed to how, maybe most of the time the these these half algebras are viewed. So, so let me. So let's, let's, let's see how this kind of this classical classical limit idea works out so we want to we want to use this kind of story about hall algebras together with this sort of f one philosophy to to compute some classical limits so Well, let me just give, give a couple of examples so if you, if you take the, if you look at the category of modules over the sort of the, the free monoid and one generator. This is the monoid, which is just powers of tea. Then then this than the, the whole algebra of this category that you get is is is basically a dual of Dirks half algebra of rooted trees. And this is because to give them to equip a set with an action of this of this monoid is basically to draw a directed graph. Which tells you how T acts and then you can see that the type of graphs that can rise or are either rooted trees or sort of cycles with with rooted trees attached. And if you take, if you take the, if you take a quiver and you look at at the representation of that quiver and pointed sets which somehow you should view as the kind of kugos to one limit of the category of ordinary quiver representations. What do we get. Well, so the naive guess would be that I mean the sort of quote unquote classical limit of what we had before which was the positive half of the quantum group should be just the sort of the positive part of this, of this cosmode algebra. What you actually get is, in general you get this module a certain ideal, which I'm not going to describe here but this somehow reflects kind of a non flatness of this this kind of q goes to one limit so something non trivial happens this q goes to one. Something which is kind of maybe smaller than expected. However, if the if the quivers of type a. Everything works nicely so you get exactly the, the enveloping algebra of upper triangular matrices. So as we know in mathematics, everything works nicely in type a and then then it doesn't work as nicely in other cases. And so the, these, these hall algebras applied to other to other kind of combinatorial categories recover sort of other types of. Well, for for Feynman graphs we get the dual of Dirich's algebra of graphs and for for matroids you get the dual of Schmitz matroid minor half algebra, and, and so on. And as I, as I mentioned these these categories have have have have an associated algebra k theory and just to indicate that this k theory is interesting. In the simplest case if we take vector spaces over f one which is pointed sets, and these k groups correspond to the stable homotopy groups of spheres. Even in the. And this is somehow the simplest case so so in several other cases like for instance these, the case of matroids, you can show that this, that this k theory here is at least as big as homotopy groups of spheres in the sense that there's, there's the, the, the k theory of f one vector spaces sits inside of these things, quite often. So these are these k groups are are are interesting and and somehow hard to to compute. Okay, so, so let me, let me finally talk about these. These trying to do algebraic geometry in this kind of non additive setting, which will ultimately lead us to tutorial varieties, or, or rather to their kind of monoid versions. We know that that you know an ordinary scheme and algebraic geometry is obtained by gluing spectra of rings. And you can do the same thing with with spectra of Monoids. So if we have commuter to Monoids you can define prime ideals, you can, you can define as a risky topology and, and, and do the same exact same thing you do for rings. And, and what you do, what you obtain when you glue these things together is, is a space which is a, it's a minuital space so it's a topological space with a sheaf of Monoids on it. And this is, this is what a monoid scheme is it's the exact same story based on commutative Monoids rather than commutative rings. Just to just to give you a sort of a flavor for for this so so what are what are some of these kind of simplest schemes. The thing to kind of notice here is that these schemes have very few points. So, if we take the affine line which is which normally a spec of polynomials. So instead, in the Monoids version we would look at spec of a monoid on one generator and this thing has only has two points. So it has, it has exactly one prime ideal the one generated by T and it has the kind of generic point corresponding to zero. So, so the kind of monoidal version of affine and space is you take the free commutative monoid on n generators. And you look at prime ideals in that and in effect, these correspond to subsets of these variables exactly the kind of coordinate subspaces. So part of the reason why there are very few points is is so so as I will maybe say this monoid schemes are closely tied to kind of Torah geometry and and their points in this kind of monoid sense correspond to to Taurus and sub schemes in the in the sort of ordinary story of Torah variety. So there are much fewer of these sort of Taurus equivariant things than there are of points in general, of course. And so here I also give an example, well you can take two copies of the affine line to glue to to get to get P one so P one now. It has three points it has zero infinity and a generic point. Okay, so, so how is, how is our monoid schemes related to to kind of Torah varieties well. It's determined by fans, and a fan gives you a monoid scheme. So, so a fan in in the kind of Torah consensus is a is is a is a collection of cones. Satisfying some some properties let me just give an example. Yes. So here's a here's a fan it's basically a way of dividing our two into into three chambers in a nice way. And the way that so these three colored regions here these are the three cones. Each cone. So if you look at the, at the kind of lattice points that live within each of these regions you get a you get a finitely generated semi group and or monoid. And, and these, this picture kind of tells you how these monoids are glued together. And this data can be assembled to sort of gluing data for for a scheme for for for, you know, for monoid scheme. And if you linearize this object, meaning instead of taking the monoid you take the monoid algebra, you would the monoid ring you would get a Torah variety in the, in the kind of ordinary sense. Okay, so in the three cones in that picture and give us give us these three monoid so so Sigma naught is is the is the is the first quadrant here and and the the the semi group of lattice points here is generated by the standard basis vectors you want to need to. And so this corresponds to the variable sex one x two. And these, these other cones give us give us different, different semi groups. Okay, so let's, let me discuss what's what coherence sheaves on on this object kind of look like so the story is again parallel with the ordinary story so if you have a, a module over over ring that module gives you a coherent chief on the, or quasi coherent chief on the on the scheme corresponding to that ring and the same thing here happens if you start with a module for the monoid, then using standard constructions you get a quasi coherent chief on on that on that monoid scheme that I described. Of course now this, these categories are not additive so you can't an ordinary algebraic geometry coherent sheaves form an abelian category on a monoid scheme they can't because it's not additive, but they do. They do form sort of proto exact categories in this in this Dicker Hoff Capron of sense. So that means that we have a chance to to talk about hall algebras of these coherent sheaves. Now, you kind of run into a certain problem which is to to to get hall algebras we need finitary conditions we need to objects to only have finally many extensions between them so only finally many short exact sequences between two, two objects. She fails in this in this monoid context, even when the the kind of the fan underlying the monoid scheme corresponds to something which is projected. And that doesn't happen in the kind of ordinary setting. When we, when we do ordinary algebraic geometry we have results of Sarah and and the tell us that X in that case just finite but but things can go wrong in this in this sort of non additive world. So that's a problem, but what we can do is to pass to sort of a smaller category of sheaves. So, so we define a class of sheaves called T sheaves. And what are T sheaves well, I'll show you a bunch of pictures pretty soon but these are basically these are sheaves which, which locally admit a grading by the by the semi group. So we've seen that our our our monoid scheme is somehow is covered by by by subsets corresponding to cones, each cone corresponds to a semi group. And, and so locally we want the sheaf to sort of admit a grading by this, this a signal. So the second condition, which actually this, the second condition is something. Something that comes up a lot in this sort of non additive world which is you want, you want to type of cancelativity and in these sheaves to to hold. And this actually fixes the finitarity issue. So, you get a, well, everything, what once you impose these conditions everything works. So how does this connect to to kind of combinatorics and other things well. The basic theorem is that this these T sheaves on affine space just corresponds to skew shapes and dimensional skew shapes and in the kind of ordinary combinatorial sense. Let me, let me give an example here in two dimensions but this works in any dimension. So, so here's a, here's a skew young, kind of young, young diagram skew partition and how can I think of this as a as a module for a monoid. Well, so in two dimensions I have the for for kind of the affine plane. So we have the monoid on on generators x one and x two. So how does this monoid act on this on this set. Well, x one moves one box to the right until we fall off the diagram and then things go to zero and next to just moves up. So any, any diagram like this can be thought of as a module. Over this over this monoid and therefore it can be thought of as a coherent sheaf. Okay. And so, I can make the remark here is that this, this, this is a picture of in this kind of monoid setting of a torsion sheaf supported on on the sort of third formal neighborhood of the origin, because three is the smallest power of all that kills all, all elements here so if you, if you, if you take any positive path of length, you know, at least three you're going to fall off this diagram and that's the smallest number that works. Okay, so here's some, here's some other pictures of sheaves. So these diagrams can be infinite. So, so here, I'm thinking of a diagram that continues off to infinity and the y direction and the x direction. And so this was correspond to a coherent sheaf on supported on the union of the x and y axes. This is a diagram which is just kind of, you know, has something missing in the lower left hand corner but is infinite beyond what I've drawn and this is a picture of a torsion free sheaf. So here's a picture in three dimensions. Okay, so, so this would be a picture of a, of a sheaf on a three supported on the union of the, of the three coordinate axes. And, and in general, if I have a monoid scheme, I'm going to glue these skew shapes together to get something global. Okay, so, so these T sheaves are just sort of objects that are that are glued together from from skew partitions. The problem is that, that this, that this category of these T sheaves is, is, is nice it's finitary is proto exact, and, and so you can define a whole algebra, and by all the kind of abstract nonsense I've said earlier. That's going to be an enveloping algebra. So, so in other words, for each torque variety we get a le algebra, and you can ask well, what, what did easily algebras look like so. Let me, let me just here give a, give an example of kind of how you would compute a le bracket here between. So here I've chosen sort of two torsion sheaves these are these are finite diagrams that they create this is in two dimensions so these are two torsion sheaves and in two dimensions. And so I want to look at all extensions between these two diagrams. And so this, this amounts to all ways of the product in the whole algebra amounts to always of stacking one diagram on top of another. And, and then anti symmetrizing this operation. So, so s times T is, is the sum of these three terms here, always of kind of sticking T to the right and up of s, and then we anti symmetrized so now we're sticking the diagram corresponding to s, up into the right of T. Okay, so, so would you would you get in as a sort of a byproduct of this is that for instance that the, you know, that skew shapes and dimensions have a le algebra structure and this under this bracket and this in fact these these the structure constants for this the algebra always plus or minus one and zero. Okay, I'm starting to basically run out of time here, but we, we are able in certain examples to actually compute these, these le algebras attached to Torah varieties, and you see stuff that looks like, you know, loop certain certain pieces of loop algebras like one, you get a certain piece of loops to GL two, which is, you can be thought of sort of a classical limit of kind of Kupronov's theorem that I mentioned earlier, where he was working over FQ. And, for instance, and if you take sheaves supported on the second formal neighborhood of the origin and a two so point sheaves. Again, you can look at the kind of hall algebra of this category. And in this case you get, you get a certain sub algebra of positive loops to to GL two. And there are other examples so so we're able to do, you know, P two. Well, we're able to exhibit a basis for P two and and and in some cases, when we truncate this le algebra we're able to sort of identify it with things that that are known but in general the structure of these things, which again are believed to be classical limits of some type of quantum group are still pretty mysterious. All right, I'm going to stop here so thank you very much. All right, let's thank Matt. Are there questions. Yes, I would like to ask a question on this black dots on the shapes you want this image twice this black dots to get to get a lead bracket. So, no no it was at the end. And. Oh, okay. After that. Yes. Yes. So, do you do you have some properties for this black dot for example is it a pre list or whatever. So I, I don't think in general, I mean it's certainly for for for general proto exact category I don't. I don't expect this to be freely because I mean this is more of a kind of intuition that I have more than a theorem but what freely is about insertion and sort of in a single spot. So, if you have these skew shapes the interact when you, you're not really inserting in a single kind of place like on a tree, and that ruins that freely property and so. So, so this came up I thought about trying to do sort of an insertion elimination type thing for these for these skew shapes and then things kind of fall apart precisely because when you stick them together they kind of that the interaction is non local in some some. Okay. Thank you. I had a quick question. You mentioned that the K theory of these, these categories. And in the very specific example of rooted trees, which have been studied in great detail what is then what are the, what is the K theory for those. It's So, I think we're free. So, I'm embarrassed to say I don't. It's something that again contains the stable homotopy groups of the of the sphere spectrum I think you can kind of, you can write it as a sort of more or less a sort of a smash product of that with something else but I don't, I'm going to get it wrong if I say it now but I can, I can look it up I think and tell you. I mean, the fact that the trees are like a co free kind of hope father but I mean is that, I mean I would have expected something trivial or something very generic for, for these kind of catering questions. What one wouldn't expect pointed sets to give you the stable homotopy groups of the sphere spectrum right. Yeah. So, so yes I agree that it's. You ask what happens for grass. Thank you. Well, I don't know but that that's I think that would be an interesting question. Yeah. Thanks. I think I have a question but cluster algebras are very popular in physics at the moment and they have people who say that cluster algebras are all algebras of quiver labs or something like that. Do you have a comment. No, I mean, so I've also heard this, I've heard people trying to kind of relate this. I mean, I think they arrive, they arise in a slightly different ways if you start mutating a quiver but but I don't understand the precise relationship. The whole story. Yeah. Thanks. Any other questions. I always find it very amusing that this F one philosophy it's almost the opposite of the way you do q counting where you would often start with a binomial identity and generalize it to the q form and here you're doing the opposite where you're taking something that makes sense at the q level. And going the other way. Anyway, if there are no other questions, then let's thank Matt again. Thank you.