OK, thank you very much for the invitation. So let me explain the kind of general structure of the talk. So the main purpose of the talk will be to describe certain conjecture sort of within local geometric langans. But the bulk and some sense maybe the biggest part of the talk will be reviewing some local geometric langans is. So I'll send and to discuss some applications maybe of this conjecture or corridors of this conjecture, applications and evidence. And I should say from the very beginning this conjecture, the conjecture, the way I'm going to describe it is essentially maybe slightly differently formulate. So it's essentially due to David Gallotto to Gallotto. So my kind of contribution to this is just being able to translate what he says into a slightly more mathematical language. So secretly, the talk is about computing or realizing explicitly as duality for certain specific boundary conditions or interfaces, boundary conditions in, let's say, 4D n equal 4 super young meals, topologically twisted. I should say, topologically twisted. But I'm going to only say these words once here. And I'm not going to mention them in the rest of the talk for two reasons. One is kind of lack of time. And the second is that definitely in this room, I'm not the right person to say these words in the first room. So in particular, I should say that the main conjecture, when we get to it, so I mean, David was able to produce it from sort of looking at this setup. And the way he was able to produce it is completely mysterious to me. So it requires some strength. I mean, the conjecture eventually will be completely mathematical. But it will be pretty surprising for experts. And he was able to use somehow some kind of string theory arguments to produce this conjecture. And I really sort of don't understand that at all. OK, now the actual plan of the talk is as follows. So we'll part 0, which will be extremely brief recollection of, so to say, usual local correspondence. Then we'll part 1, which will be sort of brief, but not extremely brief recollection, local, geometrical English correspondence. And so here, this will come into two disguises. So they'll be classical, both classical and quantum. And somehow sort of the emphasis will actually be on quantum. And then there will be main conjecture in its corollaries. And then some theorems. That's what I want to try to do. OK, now let me do part 0. So well, let's assume that we're given some field k, which is a local non-alchymidian field, for example, biotic numbers all around power series with k-efficient of finite field. And then usual local language correspondence deals with the following things. We'll also need some group G, which is a reductive group. And for the purposes of this talking, you can think about the group G ln, because eventually the main conjecture will be about G ln. And then the usual local angles deals with irreducible representations of the group G of k, well, in the appropriate categories. Some kind of smooth representation. And claims that this is closely related. Not quite, it's not quite a bijection, but somehow, well, let me make a dotted line here. Some kind of almost a bijection between these guys. And homomorphisms from the Galois group of the algebraic closure, actually separable closure of k, if it's positive characteristic, of k into G check, which is the language dual group. And again, for the purposes of this talk, you can think about G ln. And then the dual group is also G ln up to conjecture. And it is well known that there's a lot of problems with the exact formulation of this. So somehow, first of all, I have to be careful about what you mean by this. And also, even if you formulate this in some way, then somehow it's not quite a bijection. It's slightly bad, actually, for G ln or for general group, but still somehow, so we have to, I mean, there's some problems with this. And also, it's kind of not a very natural statement from this point of view. Because somehow here, you only talk about the reduced representations. And there's no way in this framework to formulate it for just all of us. For example, if you want to study the category of all representations, then somehow it's probably impossible to realize it on this side. And this problem is actually cured in geometric langlines. So now, local geometric langlines is about the falling. So here, I'm going to say what it does. And then it will involve some notions which I'll have to, well, if not defined, but then at least discuss what they're about. So main thing is that, first of all, we replace k by, say, Laurent-Pauwes series with coefficient complex numbers, so any algebraically closed field of characteristic zero. And then we look at the group G of k. And this is some infinite dimensional in-group scheme. And then, so the analog of this will be, so, well, here, I'm going to deal with an analog of all representations, not just necessarily of useful representations. And everything, the categorical level of everything will be one. So we should move one level up in the categorification of everything. So somehow here, we should look at categories. And then here, I'll also discuss in a moment what they mean by categories for the purposes of the store categories with, OK, let me write it strong, and also explain what strong means, G of k action. And this should be also closely related. Well, this is actually a two category. And these two categories should be more or less the same. And, well, there are some problems. Sorry, I want to say that what I'm doing now is kind of about classical local geometric langos. And then there was a quantum, and the quantum somehow some subtleties, which appear here will actually disappear. But the roughly structure is that this is more or less the same. And there's actually no correction on how to make it really precise. As categories, as the two category of categories leaving over, well, the stack of local systems. So I'm going to define all the players in the moment. So here, d star is a spec of k. This is the formal puncture disk. So let me talk about what all these words mean. So first of all, let me look at the left-hand side. So let me explain the meaning of this. So let me do it first, not for the group G of k, but just for, suppose that, let's say, a is some algebraic. Could you mention what's replaced left-hand side? If on the right-hand side you replace disk by cure, then what happens? No, that's a different story. I mean, you don't do that. So I don't think there is any sensible statement. So he's looking at surface operators. He's looking at surface operators. You're looking at surface operators. Absolutely, yes. So if I have algebraic group, well, there's a notion of, what, if you just have any group, there's a notion of group action in a category, which is pretty clear what it means. Now if you have an algebraic group, there's an action of algebraic group action in categories. So algebraic action, and so by a category, so let me just remark is that, well, I'm going to ignore this kind of problem, but basically category usually means some kind of derived category of something. So category, well, can mean, well, actually, for some application, it will be enough to work with the beginning categories, better to work with some kind of triangulated category, or if you want to be really careful in category, so you have to actually work with DG category. But let me, this is something I'm going to ignore completely in the stock, otherwise somehow I won't be able to get anywhere. And actually the main conjecture when I get it to, somehow, in the main conjecture, you can ignore this kind of category. The subtleties will even hold for some, because point AB line categories, but in principle should be thinking about categories, some kind of derived category of something. And so if you have an algebraic group, so there's an algebraic notion of algebraic group action in the category, which you can easily invent what it means. And so algebraic, so in fact, and one way to think about is that algebraic action of A on a category C, it's equivalent, this data is equivalent to the action of the monoidal category of quasi-K and sheaves on A, which is a monoidal aspect of convolution, so the monoidal structure has to do with the group law on C. And so in fact, DG category, if you want to work with co-complete DG categories, so DG categories which have arbitrary direct limits. And so, and well, this is what is actually called, I'm going to call weak action. And a strong action is the following. So intuitively strong action means it's a weak action such that it is infinitesimally trivialized. And one way to think about it is that strong action on C is the same as action on C. Of the monoidal category of D modules on C. Is it the same if we make kind of odd cotangent but we'll make DG group? Probably, yeah, probably. No, actually I'm not sure. Okay, I can't figure out on the spot, but somehow it means that, I mean, intuitively we should think about this one way. So what does mean in action? In action means that if you start with an object, you can make a family of objects over the group, which forms something like quasi-cagarenship. And a strong action means that this family will have a flat connection. That's, but it will be convenient for me to think about it in this way because, and maybe for future purposes, let me give you immediately some kind of generalization as construction. So suppose that you have now let A tilde be a central extension of A by, let's say, C star. So it fits into a short exact sequence, one C star A tilde A. And then for any complex number, there's a notion of strong A action or A tilde action depends on how we just noted. Of on, let's say, on level C. And this means that, so it's easy to define in this language. So somehow you can consider D modules on A tilde now, but now you can consider D modules with an index C. So it means C twisted D modules. So it means that D modules, which sort of have monodromy, you're given by this number C along this. So it's D models with on the space of A tilde, but which kind of monodromic with respect to a C star with monodromy given by this number. And that's again in monodrome category. And that means that this guy acts on C. And this notion sort of kind of doesn't change if you shift C by an integer. So this two category of categories with a strong action of level C is the same if you shift C by one. I have the same differential operations and C spirals of this line bump. It's, well, yeah, oh, you mean, what do you mean by that? Well, yeah, it's the same thing, but it's the same as to say D models on the total space, which are kind of monodromic with respect to C star with monodromic. But, yeah, it's the same as, it's the same as models over differential operators in the C power of this line bump. So there is such a notion. So it sort of depends only on image of C and C nodes. Okay, so that's, well, we don't even need, well, we'll need this guy in a moment when we discuss quantum local angles. But at least somehow now I can have told you what left-hand side is. So we have to apply to A being this group G of K, which is an infinite dimensional group in scheme. But it's fine system, it can be rigorously defined. Now about the right-hand side, so what I mean by this, so this local system, so by this kind of abusing of language, so local systems means the local system of the round sense. So this is the stack of, this is the stack of the round local system. The round local system, so it's a stack classifying, let me say, G check bundles with the connection, which is automatically flat. So it's not really an algebraic stack, but again, it's fine system. Now what do you mean by categories leaving over this? So let me just say that if S, let's say, as a stack, then there's a notion of category leaving over S. So kind of the most simple-minded example is if S happens to be an affine scheme, so if S is specter of some ring R, then life over, life of some category C over S is the same as sort of action of the ring R on C, so it means that it's the same as by definition as a map from R to the center of this category. This is the same as action of classical chemistry? Well, yeah, exactly, this was my next sentence, so that, but so, and that's actually definition in the general case. So in general, life over S is by definition same as action of the category of quasi-korean sheaves. And again, somehow, since we actually want to live in the world of DG categories, this quasi-korean sheaves should also be in the student sort of derived sense in a DG sense, but let me ignore all these problems. And so now, this is a monolithic category just with respect to ordinary tensor product. And so now, so this is what local language studies, so somehow this is almost true as stated, so in fact, there's some, I mean, but it's not completely true, it's stated, so there's actually, so there's a way to slightly change the right-hand side due to our ink and which actually, at least, can actually make something to actual equivalence of two categories, so it can actually formulate a precise mathematical statement. Now, I want to also discuss quantum, quantum local geometric languages, and so you know, is it visible if I write here or I should not write here? Because I want to keep this stuff on the board for some time. I'm sorry? So quantum, local geometric languages is the following, so this has to do with the fact that the group G of K comes with a natural central extension, so here you're gonna have to be a little bit careful about what the central extension is, prime first by, so in fact, if G is simple, so if G is simple, then the central extension, then there's kind of one parametric family of central extension, basically there's one canonical central extension, if G is not simple, if G is general reductive, then in fact, you have more parameters, but I'll kind of notationally pretend that there's just one parameter, so in fact, central extensions are primetized by integral even in varying forms on the Lie algebra, so maybe notationally I'll pretend that the group G is simple and then we have sort of G hat, which is a central extension of G of K, so again, there's not much choice here of G is simple, if G is not simple, then you have to be a little bit more careful, but let me sort of again not deal with this right now, and then quantum, local geometric languages says something which is kind of more symmetric with respect to G and G check, yeah, and sorry, before I say this, let me just introduce some notation here, so therefore we can talk about G of K actions on a category on level C, which is a complex number, which is just a special case of what I talked about for over there, and in fact, notation it will be convenient, so it's convenient, so we shift, convenient to shift C by H check, which is dual coaxial number, so we shift them by some integer, so which in fact doesn't do much, because of the marker over there, it doesn't change much, but somehow, well, you'll see that anyhow, so I want to change the just, when I say that level C, it means that the action is not by that number, by that number I shifted by the dual coaxial number, and you will see why it's important in a moment, well, although this is, well, approximate this is not very, very important, but just to be precise, let me do that, so then the statement is that the injector says that G of K actions, so let's say categories with G of K action of level C are really the same as categories with G check of K actions of level minus one over C, so, well, it's kind of an exercise to convince yourself why what's written over there is a special case of what's written here, namely, so if you want to take C to be equal to zero, then you get, I mean here, to get the left-hand side we had originally, it just should put C equal to zero, and then minus one over C becomes infinity, so what you have to convince yourself is that when you consider this notion of categorical action at level equal to infinity, then it's natural to think about it as just being a category living over the stack of local systems, that's, I'm not going to do that, I mean, it's kind of not a mathematical statement, it's a psychological statement in a sense, that somehow sort of you said, you can convince yourself that it's natural to think about this as a limit of this as the level goes to infinity. Okay, this is all kind of well-known stuff that I'm talking about, at least to some people, but so the main purpose of this talk will be to provide some examples of it. Yeah, and maybe let me just make a remark that if you define as things carefully, sort of, if you define as categories, DG, co-complete, DG categories with this section and so on, then at least if C is not a rational number, then it is expected, a statement that D is true as stated. Otherwise, you need corrections, and particularly in this kind of zero infinity case, you need, now maybe say that I mentioned this kind of boundary conditions and of course, I promise not to mention that anymore, let me mention one more time and that will really be the last time, that somehow if you can see that this four-dimensional and equal four-supremian meals, then it's known that it has a P1 worth family of topological twists, which are called Kapustin-Witton twists and if we can say the boundary conditions in this Kapustin-Witton twists, then boundary condition corresponding to, well, the twist given by number C is should be more or less the same as this category with GFK action of levels C. That's an expectation. In other words, one other way to say this is that if you have a four-dimensional topological field theory, then to a circle, it's supposed to associate two categories and the two categories associated to a circle should be this one. I have a question. Is this supposed to be kind of algebraic and C or transcendental, so in this complex numbers? No, algebraic, I mean, it's, I mean, you can replace complex numbers by any field of characteristic zero here. Is it known that the two monoidal categories are not equivalent? Does it just mean? Which two monoidal categories? Oh, you mean the corresponding category of D models? Well, it's probably not known that they're not, well, I don't know, but I don't think it's known they're not equivalent, but it's not known that they're equivalent, let me say it like this. Okay, now let me first do a couple of examples of how this works or how this is expected to work. And, well, this is kind of too abstract, but let me, I want to work out one example from which I can actually get some kind of equivalences of categories, which you can then test. And then I want to, so I want to do some, first of all, some kind of known examples and then I want to explain some examples which come from this Galliot's argument, which I kind of already completely knew. So, okay, so examples of how this works. Well, let's even forget about langons for a moment, let's talk about how in principle can you produce categories with action of A or maybe with action of A of sub-level. So, let me write here. So, how to produce categories with a strong A action or AC action, whereby AC action, I mean action of A on level C, just for general A. Well, there are gonna be two sources of examples and both will appear. So, source number one is the following. Suppose they have some X, which is some space and some variety or scheme or stack. So, well, let's say scheme with A action. Then we can consider the category of G modules on X and so this has an action of A in a natural way. That's one example. And if you want A sub C, then you need some additional structure in X. So, if X has a line bundle, which is kind of equivalent with respect to the corresponding central extension. So, if X is equipped with a line bundle, just a tilde equivalent, then we can consider DC modules on X. So, somehow, because we're in twisted G modules on X and here we have sort of, let me just, abusing notation, right? We have an action of A sub C. That's one source of, that's kind of a slightly sophisticated source of examples. Less sophisticated source of examples is the following. So, we can just consider, let's consider little a to be the lead algebra of A. And then we can just consider the category of modules over this lead algebra. So, then we can consider A modules has the category of A modules has a natural strong A action. And again, if you say have a central extension, let me consider the category of AC modules, which means that it's action, actions over corresponding central extension on which the central element X by C has a natural strong AC action. So, that's one source of examples. Well, it's two sources of examples. So, let me then do some specific, well, coming back to Langlands, let me give you a few examples of who goes to what or who goes to who. So, example one is the following. Example one will be still slightly abstract, but it will be important in order to, you will actually, this example will be important and to get some kind of concrete statements out of Langlands correspondence. So, example one, well, this is what I'm going to say is not a theorem, it's an expectation because this Langlands does not have a construction, it's a conjecture, it should exist some kind of correspondence like this. So, somehow we can only guess what kind of properties it should have. So, one thing is the following. Let's consider the fine-grass model of G. This is G of K mod G of O, where O is the field of Taylor series. Then it does have a natural line bundle, which is a covariant with respect to the exponential extension. So, we can consider the category of DC modules on the fine-grass model. And that guy is expected. So, let me denote this correspondence as by C going to C check. So, if I put check here, it should be actually a category of similar nature. So, I should write minus one over C here, modules on the fine-grass model of G check. I'll explain some importance of this result in a moment. So, again, you're welcome to do an exercise just to think what this means when C is equal to zero and this guy is equal to infinity. Then we get some kind of interesting statement. But, I'll ignore this. So, let me keep C in minus one over C finite. Then let me explain some interesting corollary of this. Corollary is the following that it says that if you consider the C, well, it's a category of G of K action. And inside G of K, we have the group G of O. And over G of O, the sample extension always splits. So, in this context, it makes sense to consider G of O invariant. So, this is the category of Geo-4 equivalent objects. At least when working in this DG setting, somehow this is all completely well-defined. And the corollary is that that guy should be equivalent to this guy. And so, why is it so? Well, if you believe in this thing as equivalence of two categories, then it should, well, it should match objects of this two categories, but it should also match morphisms. And so, one way to think about this, this is home over G of K from exactly from this guy to C. I should maybe put G of K, C here. And so, if you believe that this thing is an equivalence of two categories, then this statement immediately implies this. Although C and C check might be completely different categories, but if you take their Geo-4 equivalent objects, you should get the same thing. So therefore, any time we have a statement that one category is London's dual to some other category, we get automatically also some statement about two categories being equivalent, because if we take the Geo-4 equivalent objects, by the way, we can also replace Geo-4 by a Wahori group here, but I won't do it in the store. So for a Wahori, the same is expected, was also expected, but if we consider some smaller subgroups of Geo-4, then it will not be true. Only for Geo-4 and for Wahori, this is true. So in particular, it's actually nice to, you can actually do it for these guys themselves, then we get twisted Geo-4 equivalent objects, which are also Geo-4 equivalent on the left, both here and here. And we get actually some non-trivial statement, which is known, but with really non-trivial, in this case. Yes. I think something, the general discussion you said that C was different to an integer, the London's case, it's not- Well, it's actually, yeah, it's a good question, but see, you can actually ask this question here. So here, if you change C by an integer, you get, well, get just on the nose the same thing, but the effect of changing C by an integer on the number minus one over C is completely different. So therefore, we get some kind of, and this is absolutely non-trivial, so somehow it's very difficult to see it. So here, you get some, so in particular, this statement implies that, implies a lot of equivalences of these two categories, one sort of, you don't change the group, and they're absolutely non-trivial. I mean, knowing them is known, this is more or less the same, so it's, I mean, but you're right, you get some really non-trivial sort of equivalences just on one side of London's correspondence, and it's very difficult to see that, actually, whole. Question, if you want to have a quiz, you probably need some kind of bi-module, or two categories. Absolutely, yes, but, and the bi-module is just the following, it's like, I won't try it because I won't use it. You just take G-modules on G of K itself, and this has two G of K actions on the left and on the right, and then they just apply the slang ones to one of them. I mean, so it's, I mean, yeah, no, no, okay, sure, but, right, sorry. Do these two categories have any extra structure, like representations of a group can be tensored or something? Is there like a symmetric middle structure on them? Well, they do have, you know, you can't answer them over G of K for, well, no, you can, well, there's no issue of tensoring two categories of G of K, but I mean, I don't think this equivalent, I mean, I don't know any structures which should be respected by this equivalence, let me say it like that.  So now I want to do one example of this sort, and the main thing that I want to do eventually will be generalizing that example. So now it's natural to take C to be this G hat, well, G hat C modules. This has an action of G of K of level C, and you can ask what is its dual, and there's an answer. So, well, expected the answer. And again, so this expected answer can be tested because I said anytime you have a, anytime you say one category is dual to another, you also get some equivalence of categories like this. So, expected answer is the following. So I consider, so the only thing I always get confused about, yeah, so I can see that the group G check of K, and then I consider D modules, well, let me write it, and then I'll DC modules on that, sorry, the D one over C modules, and then I consider Wittaker of that. So what is Wittaker? Wittaker means the following. So if we inside this G check, we have the corresponding, which have some Borel subgroup, and some Borel subgroup, we have the corresponding important radical, and then we consider, then we, so Witt in general means considering N check of K comma Chi equivalent objects, where Chi is some, well, non-degenerate character of N check of K. So I won't go into what non-degenerate character means, but in some kind of generic character. So Chi is actually, well, it's a character in the sense that it's a homomorphism to the editing group. So, well, the central extension here, well, here what I mean by this is that I do this on the left, and so if you can adjust G modules on G check of K, it has an action on the left, it has G check action of level one over C, and it's also known that on the right, it has an action of G check of K of level minus one over C. So somehow the levels on the left and on the right are negative of each other. So here when I do this stuff on the left, when I apply this, this kind of equivalence on the left, then on the right, I still have an action of G check of K, which will be of level minus one over C, so at least the sense will make sense. And this is expected answer. And as a corollary, we get the following statement, which is actually theorem of Gaetz, Gore and Luger, that, so let me actually introduce some notation before, introduce notation here. So if one considers, so I want to consider the category of GC modules, and then I want to consider here geofold covariant objects, and in other words, it's modules over this, sorry, a fine algebra, which are integrable with respect to geofold, which this group acts, and this has a, this is sometimes called the Kashten-Lüste category of G check level C because it was started by Kashten-Lüste, and Kashten-Lüste actually, well, they introduce some tens of structure here, and actually it's a tens of category. So let me assume that for simplicity G is simple, and let's say simply laced, otherwise I have to be a little bit careful about how I formulate it, then Kashten-Lüste constructed an equivalence of this, well, for almost all C, so let me just say that it is equivalent to the category of finite dimensional representations of the corresponding quantum group, finite dimensional quantum group, not a fine work, U is e to the power two pi i over C, and here we have to exclude some C, so maybe we should write that C is not a positive or non-negative rational number for any C, which is not like this, this is going to work, and so the core I want to format, I want to take here this geofoic variant object, so here by definition I get the Kashten-Lüste category, sorry, this is modules. So I get this Kashten-Lüste, so this is called Kashten-Lüste equivalence, so if I take the Kashten-Lüste category of level C, then this thing should be the same as, well, and the equivalence is also, well, I mean, for this to work nicely, I have to make some assumptions on C, like best assumptions to assume this, but I think this equivalence I'm going to say right now is to work always, if you will format it precisely, so here I should write, sorry, I should write wheat, and so here I should write Dc, sorry, D minus D1 over C modules, and I should take geofoic variant objects here, but that's the same as to just consider G-monus on this affine-grasmina, which is geofk-mon-geofoic, so if you can see the Whitaker D-model on affine-grasmina, that's the same as Kashten-Lüste category for the dual group as if C is, if C and one over C, if the levels are matched in this way. So that's actually a theorem in this case. I guess, well, at least it's written maybe if C is not rational, for rational somehow there's some, it's more delicate, but I think it's proved that this one C is not rational. With C is rational, sorry, I mean it's proved I think in full generality, but written maybe only for irrational C. Okay, so, and again, thesis I have some, I think have some ways to explain why I should expect, why this is actually the right expected answer. And so in the remaining 12 minutes, I want to generalize this. So what I want is to slightly actually generalize this. And for this I need to slightly generalize actually this part of the board. So I mean I said that somehow here, one way to construct categories with this action of my group is just to consider the modules over the Lie algebra. Now I want to do something a little bit more general than that and again from, turns out that from the 4D Young Mills point of view, it's also natural to do that. Let me do the following. Let me assume, so assume that we're given a Lie super algebra. Let me call it, maybe it's a bad notation, but it wasn't come up with anything, with any other notation G prime. So G prime is G prime zero plus G prime one and such that G prime zero is my G. Then I can try to play the same game. I can try to consider this morning loop algebra. So I can consider G prime and I can consider it's central extension if it's sufficiently nice super algebra. And well, let me also put number C here. Well, you have to be a bit, again a little bit careful by, but by what the central extensions are parameterized. But again, let me ignore this, but you can consider just modules over this super algebra. And this category has an action of, well, let me write it G heads of sort of group G of K of level C. And so idea that if G prime is some nice super algebra, for example, if it's a simple super algebra in some sense, then this logless dual guy should be computable. Let me say it should be computable. There should be some explicit answer about what this thing is. And if one reads carefully the paper of Gaiot and Whitton from 10 years ago, then apparently there's actually an answer there for all, for at least for all classical super algebras. There's probably an answer, but today I'm going to concentrate on the simplest possible example. The simple possible example is the following and it's already pretty interesting. And again, for people working geometric long on some how it was completely unexpected that you could do what I'm going to say in a moment. So I want to take this G prime to be GLN slash M. So now what it means is that my group G is GLM cross GLM. It's also the same as G check in this case. And I want to compute, so I want to formulate an answer about what the longlist dual is. And this will imply some equivalence of categories as I explained by, if I take the corresponding geofoic variable. Before you have A, the algebra was simply dimensional plus Casmury. No, I mean, I said that, so here I started, suppose I have, I'm saying that here suppose that my fine dimensional algebra has some super extension, then I can consider the finization of that. So in particular, there will be a statement that the Casdan-Lousty category for the least superalgebra GLM M will have some kind of realization. So the question, so now, so I take this GLM M hat C modules, and I ask what is its langons dual when I dualize it with respect to both GLM and GLM. This is by the way also what Jesus would call interface between sort of young mills for GLM and young mills for GLM. I mean, that's an example of an interface. Okay, let me produce an answer. Now, and this answer is supposed to generalize this one. So this is the case, well, if G is GLM, then in particular, this is a special case corresponding to say M equal to zero. So, well, first of all, I'm going to assume that N is greater or equal than M. I mean, everything's symmetric with respect to N and M, so I can do that. And so somehow the easiest answer will be in the case when M is equal to N minus one for some reason. So, okay, so the structure and answer will be the following. It will be one type of answer when M is strictly less than N and slightly different answer when they're equal. So let me first do the case M less than N and it should generalize this, but I first do the case M equal to N minus one and it will not look like this at all. But then I will do the general M less than N case and it will be clear how it includes this one. So, okay, so, for example, M equal to N minus one. And let me do this for, so then the answer is the modules, well, of level minus one over C, sorry, minus, yeah, minus one over C of on just GLN, but consider as what? So somehow this has GLM of K, GLNK action on GLN of K. On the right, and on the left it also has GLN of K action. Well, I'm ignoring the levels now, but I'm kind of ignoring that. And I'm just, so I can naturally embed GLN minus one into GLN, let's choose this embedding in a natural way. And then on the left I can see there GLN minus one K action on the left. So on the left there's also full GLNK action, but I just ignore that and I only can see the action of that guy. So the claim is that then this two guys are supposed to be, well, sorry, I don't need minus here, sorry, that's just one over C. Because the minus will come because I can see the GLNK on the right. So that this is the langos duals in particular quarrel array, well, well, this is not a mathematical conjecture because we don't know what, we don't know a construction of langos correspondence, but the conjecture becomes completely mathematical. Is that the cash done loosing category of level C for GLM slash M is equivalent hat is equivalent to, well, I should take here objects, which are equivalent to respect to GLN of all on the right and GLN minus one of all on the left. So somehow one way to think about this is that, so let me write D minus one over C modules. And here I should take the affine Grassmannian of GLN. So this is thinking quotient by GLN of all on the right. And I should on the left also quotient by GLN minus one of all. So in other words, I can see the GLN minus one. So usually when people do geometric satake, you would consider just GLN of all equivalent objects here, but here you can see the equivalent aspect of smaller group and the claim is that, sorry, this is M equal to N minus one. This is N minus one. So this should be in equals like this. Okay, so I have three minutes left. So I probably won't be able to discuss any theorems, but let me still finish the formulation. So I should say that, again, the formulation of this is essentially due to Gallota. In fact, well, he didn't, of course, formatted this in this particular way, but somehow basically 99.9% of the formulation is due to him. So my only contribution is being able just to translate it into some kind of mathematical language. Okay, now let me do the general case. So let me do the general M less than M example. So M, sorry, less than M, general. Then the statement is that what you should do is the following. Then if you consider this GLM slash M hat C modules, and if you consider it's Langlund's dual with respect to GLM times GLM, that thing should be equivalent to the following. So now you should do the, you should, if you consider D modules on level one or C one, GLM, okay? And now I should consider it's with a reduction with respect to GLM minus M with respect to any important group in GLM minus M. So the two extremal cases, one is when this number is equal to one, then you have GL one here and the important group is trivial, so you don't do anything and that's what we did before. Another example is when you can see the M equal to zero, then I should consider the full Whitaker here and that's the theorem up there. So, well, no, I mean, that's the thing up there. So, but I mean, the theorem becomes a few. So in particular, GLM all times GLM all equivalent objects, the categories of GLM objects should be equivalent. There's also an interesting case when one of this number is equal to zero and the other is equal to infinity. Actually, when this number is equal to zero and this is equal to infinity, then it should also, this equivalence should also hold and now maybe let me conclude with a few things. So first of all, the equivalence up there, it's known that actually holds in a lot of abelian categories, don't need any derived categories and the same thing should be true here. One other thing is that let me also say what happens when N is equal to M. If N is equal to M, then we have here GLM, N have C modules and if I langons dualize it, then the answer is similar but slightly different. And the answer is that I should consider G1 over C modules on the following thing. I should consider GLM of K times K to the N. And so here this has two actions of GLM of K now. One is just say the left action just on GLM of K itself and the other is the following. So I have the right action on GLM of K and I also have the action of GLM of K and K to the N and I consider the diagonal one. So I can say these two actions and again if I take GLM of O times GLM of O equivalent objects, I get some equivalence of categories. And so maybe if I can have two more minutes, let me make a few remarks about this and formulate. Just tell you a few cases when this is known. So first of all, I should, one thing I want to say is that I never, until recently, kind of psychologically, I never admitted that the importance of, and then we acknowledge the importance of least super algebras, so to say. But that's kind of the first time in my life when something in least super algebras play some kind of a role. And now when I understood the formulation of the conjectures, then I started kind of researching both the literature and asking experts and turned out that this, for example, this casualistic category for least super algebras has never actually been studied. Although there are like a lot of people who start representations of least super algebras, but there's no, for example, analog of casualistic, well, not that there's no analog, there's no known analog of casualistic equivalence for this, I mean, nobody.  So I think that it might actually be sort of a good thesis problem to generalize casualistic equivalence to some kind of least super algebra like that. I mean, I think it should be relatively straightforward, but somehow nobody has ever done this. So, some remarks. The corresponding equivalence, so if I can see the g ln of all times g ln of all equivalent object, equivalence should hold for a billion categories. This is not something coming from a general, from any general expectations, but it's just effective. But I mean, if you work with examples, you see that that should be true. Also, in this case, this, again, I'm talking about g ln of all times g ln of all equivalent objects, somehow the corresponding categories, they're not just categories, but they can, what's called factorization categories. So also it should be equivalence of factorization categories. Okay, and the last thing I'm going to say is that there are a few known cases. The first known case, which is relatively easy, but not completely straightforward. So the first case is when n is equal to m is equal to one, so you can set gl11. It's not completely true in this case. So this is known except that I don't know how to prove at the moment that it's equivalence of factorization categories. So I can, the corresponding equivalence of categories I can check, but not with factorization structure. And by the way, I can also check the corresponding characteristic equivalence in this case. So the characteristic category for gl11, it's equivalent to representations of the corresponding quantum. We can define the corresponding quantum group and you can check that they're equivalent. So that's one thing that, the other thing is that when n is equal to two, m is equal to one, and c is equal to infinity, one over c is equal to zero. That's also, well, in this case, by the way, the super algebra degenerates to something much simpler, but still in this case, it's known and it's written in a recent paper of myself with Finkelberg, where it was studied for completely different reasons. Funny enough, it was also kind of an attempt to mathematically confirm certain conjecture of Gallota, but a completely different conjecture of Gallota. I don't see any relation between these two conjectures. So, but, and it's not, I mean, this is actually not very easy. I mean, it's not super difficult, but, and the other thing is that independently before, even these whole conjectures were formulated, Finkelberg, Ginsburg, and Trafken started the case, n is equal to n, and again, c is equal to infinity, one over c is equal to zero, and actually formulated the right conjecture in this way. So, I never actually formulated carefully what the conjecture says then in this limit, but that's easy to do. And so, they formatted the right conjecture, and so recently, just a few weeks ago, Trafken told me that he thinks he can prove this. So, probably, and I think that's the full list of known cases right now, at least known to me. And maybe I will conclude by saying that you indeed can actually, GLNM is just an example. So, just for any classical e-superalgebra, there's an analog of the statement, but that's just, that's just as simple as possible. Okay, thank you. What's the factorization category? Well, it's a form, it means that it's a following structure, it means that it's a category, it means that if you have any algebraic curve and any power of an algebraic curve, then you have a category leaving, sort of which leaves over any power of any algebraic curves, and with some compatibility between these. And so, the original category is the fiber, if you can see the just first part of the curve, then you can see the fiber of that over a point, then this is your original category. So, it's, if things are kind of finite enough, then that thing is the same as a tensor structure in the category. But, but, but. So, it's kind of first or actually, you have to put some kind of curve? Right, right. No, I'm saying that there should become some compatibility means that somehow if you have maps between curves, then somehow everything should be, yeah, sure.  And so, for example, this equivalence of Gaetz-Gurion-Lury, it's actually an equivalence of factorization. It's kind of trying to be in equivalence of tensor categories. But, I mean, can they actually are, I mean, cash-analytic, they actually define a tensor structure in this cash-analytic category. But the tensor structure is kind of transcendental because if I were to define this tensor structure, you have to at some point compute monodrome of certain connection. Sort of an algebraic world that doesn't exist. What does exist in an algebraic world is this factorization structure. And so, this, both of these, if you know what factorization structure is, then both of these categories have a factorization structure on the nose. And the claim is that it's an equivalence of factorization categories, which is actually a much stronger statement just in equivalence of categories. So, I mean, it's really kind of more difficult to prove this equivalence of factorization categories than that's an equivalence of categories. So, this equivalence is really very basic for local geometric language. So, Geyser calls fundamental local equivalence. So, he has a precise formulation of what local geometric languages should do, and it's based on that equivalence. So, you mentioned, sorry, maybe I'm just asking a physics question, but so you mentioned that this N and M are the Lanko gauge group. Yes. Between, I mean, two different gauge groups with interface. Yeah, but that's the same, I mean, it's kind of the same as the boundary condition for the product. Interface between two gauge groups is the same as a boundary condition for gauge theory when the gauge group is the product of the gauge group. So, yes, so, and so the point is that if you, just, I mean, where do these super-algebra appear from this point of view? So, somehow, if you want to leave where at the level being infinity, so you ask yourself, when can you construct naturally boundary conditions? So, somehow, the claim is that if you start with the representation of your group, of your gauge group, then you can construct a boundary condition at this topological, for this topologically-twisted theory at infinity. This you can do for any representation. Now, you can ask, what does it take to sort of extend this boundary condition to this P1 family of topological twists? Now, the claim is that the structure you need is the, so, if you start with G and the representation, then you can consider Li-algebra plus V. And so, again, this just gives you boundary condition to infinity, but if you want this boundary condition to leave everywhere, then you need to extend this to a Li-superalgebra structure on this sum. So, any time you have a Li-superalgebra structure on this, it defines the boundary condition for any point of P1. And so, this is just kind of the easiest Li-superalgebra you can consider. And it's only good to n, so it's just the product of the same theory, and then you just consider the boundary condition. No, no, it's the boundary condition for the product of GLN times GLN. So, it's an interface between GLN and GLN. And so, and actually for any physics argument, that's actually the most basic example. So, somehow, well, again, I won't be able to repeat this. So, somehow, this really requires some kind of string theory arguments, but no, I mean, the question is, where do we get this from? I mean, how do you know that the S-dual of this boundary condition is given by this? So, David has this kind of argument with, you know, brain construction of all of that, and so, for him, it's a kind of natural statement, but I don't see any kind of mathematical. I can form it mathematically what he says, but I can form it mathematically the eventual statement, but I cannot form it his arguments mathematically, even approximately.


I think that this is really important, particularly on the physical touch front, because there is a lot of content around out there, particularly like the manosphere or like for dating coaches that encourage men to sort of physically escalate by like touching on the shoulder or you know, things of that nature. It's good to be aware of this if you are wanting to or you are dating an autistic woman or just any autistic person in general, because that we might have quite a negative reaction to that and feel a bit unsafe, so just be careful around that stuff.


video I'm going to show you how to link your Instagram professional account or business account to your Facebook business page. So the first thing that I did was I logged into my Facebook page and I went to my Instagram page and logged in. So I'm logged in to the web browser with Instagram and Facebook. Next up in the upper right-hand corner I'm going to find my account icon which is usually a circle with your picture in it.  Next you're going to select the Facebook business page that you want connected to your Instagram business page and you're going to click on it and then now you're going to see your Facebook business page icon account up here. You're going to click on it one more time and then you're going to go down to settings and privacy. You're going to click on settings and privacy and then you're going to click on settings one more time. From here you're going to go over to the left-hand menu and you'll see a linked account with the little linked icons. You're going to click on that link and now let's say connect your Facebook page to a professional Instagram account. So we're going to click on connect account and sometimes also too you might want to make sure that you do have your Instagram setup as a professional account and the way that you do that is you go over here and down to the bottom where it says more and you're going to click on more and then from there you're going to click on settings. Alright and then down here it will say switch to a professional account if you do if you have it on a personal account but if you're already on a professional account it'll say switch back to a personal account. So if it says switch to a personal account that means that you're in a professional account so you're good to go. If it says switch to a professional account click on that and then follow the steps. So now we're going to go back to Facebook and we're going to click connect account. From here you're just going to click on connect. You're going to click on confirm and it looks like it's going to have me put in my login again and then you're going to put your password and you can put save login information if you like and then click on login from here and sometimes it will get stuck in here. I've had it happen many times got lucky right here gonna hit done and then now this is connected so you can see that it is connected now. If you want to double check you click on this icon again click on settings and privacy click on settings go over linked accounts and then here it is right here. So as you have access to manage Swan City agency Facebook so you also have access to manage the Swan City Instagram account.


What's up, guys? Mike Dakota here. Today we're going to go over the next problem called bit strings. This problem is significantly easier than the other two, but yeah, let's go over it. So you're given a number n, your task is to calculate the number of bit strings of length n. So if n is equal to 3, the correct number is 8 because the possible bit strings are 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1. Okay, this problem is very easy. So there's two things you need to know. Two things you need to know. Okay, so let's say we have, I don't know, length n is equal to 3, right? So here in this case, n is equal to 3. So there's three places. And remember, bit strings, you could have possibility of 1 or 0, right? So there's either 1 or there's 0. So the two possibilities you could have, the total possibilities is 2 then, right? So then we would just have 2 here. There's two ways to put it here. And it's the same thing. There's two ways to put it on the second bit also, right? And then there's another two ways to put on the last bit, right? So n is equal to 3. So our total answer in this end would be 2 times 2, which is 4 times 2, which would be 8. And that's the answer, right? That's the answer. So yeah, n is equal to 3. There's three places. And for bit strings, you could either have a 1 or a 0. So there's two choices for each of them, right? Two choices for each of them. Okay, so essentially is if you have, if you generalize this for like n equal to like 9 or 10 or whatever, the answer would just be 2 to the n, right? 2 to the n. It would be your answer. Okay, so the hard part is this problem is that they want you to print the result modulus 10 to the 9th plus 7. Okay, so the hard part is that this number will get overflow. And you want this number to mod 10 to the 9th plus 7. Okay, so you want to print this to the number to mod 10 to the 9th plus 7. So this is the answer, 2 to the n mod 10 to the 9th plus 7. You're going to get an overflow sooner or later. So basically what they want you to do is that you want to use this property of modulus. So if I explain what they mean is that, so let's say I have like x is a number and y is a number, right? Both of these are numbers. So if I take 3, I don't know, x is a number, y is a number. If I take x times y, and I do a modulus mod by like z or something, right? Remember, mod is finding the remainder, finding the remainder of the number, right? This is equivalent to doing x mod z times y mod z. Okay, and then you could, yeah, this times that, and then you could mod again mod z, right? This is a property you could learn for modulus. x times y mod z is equivalent to x mod z times y mod z mod z. Okay, so, yeah, to do this problem, once you know this property, you know 2 to the nth is equal to 2 to the n mod by whatever this number 10 to the 9 plus 7, right? So you know this is going to overflow sooner or later. So what you're going to do is you're going to actually use a for loop and you're just going to separate this into like individual 2 mod 10 to the 9th plus 7, right? You're going to do this and you do times 2 mod 10 to the 9th plus 7 and then you keep on going so on and so forth until you get to n, right? Do this n times and then the end you could just mod by 10 to the 9th plus 7 and that'll be your answer. That's essentially the gist of it, right? You're going to do this n times so then you would get, in the answer, you would get the same answer. So, yeah, the code is very, very easy, super easy. Read in the number n and then let's create a variable. Let's call it x. I use long, long because I don't want to, ll means long, long, okay? So long, long x is equal to 1. I'm going to loop from 0 to n and every time I'm just going to multiply it by 2 and I'm going to mod by this number mod. I create a variable called mod. It's a constant. It just represents 10 to the 9th and I just wrote it out 10 to 9 plus 7. So, 10 with 0, 9s and then 1, 7. That's a variable mod and I just use that and every time in this for loop iterates I take x and I equal to x mod this mod and in the end I just print out x. So, yeah, this problem is very easy. There's nothing else to it. Just make sure that you understand the mod property and then this is super easy. Rate, comp, subscribe. I'll check you guys later. Peace.


Welcome to this episode of Talk of the Town. I'm James Milan, and today I am talking to Laura Kiesel. Laura is a resident here in Arlington for nearly 10 years now and an independent advocate and journalist. And one of her focuses of concentration for sure is the area of housing. And Laura, first of all, let me thank you for being here. I appreciate you taking the time. Yeah, thank you so much for having me. It's great to be here. Great, and we do want to talk about housing, the issues around housing and affordable housing in particular, generally, but we want to concentrate mostly on how things are playing out here in Arlington. And I know that you are, it's a funny word to use, I was going to say, you're in the privileged position in a sense. You're in the special position, I would say, to speak to this of somebody who is in an affordable housing situation yourself, right? Yes, I live at Capital Square Apartments, which is one of the main properties for the Housing Corporation of Arlington. And I've lived here for going on seven years next month. Okay, and so I think before we came on air, you and I were talking about the fact that it is rare indeed for warrant articles such as the two that you are part of the sponsorship team for, and that we will discuss in further detail as our conversation moves forward, but for warrant articles around affordable housing to be written by somebody who's actually in your situation is a rare thing. I think you haven't heard of anybody doing this yet, right? Yeah, I think this is the only warrant, or we have two warrants that is authored by renters entirely by renters and I think it's the only one, and I think it's the only one we've had at all that has been authored in part by an affordable housing tenant. So I'm going to ask you to kind of set the parameters of the conversation for us here, but just, you know, leaning on your expertise in this area and having you talk about what the situation is with affordable housing in this area, you know, in the Boston area and then kind of start to zero in on what the story is here in Arlington. So the Boston metro area, it is one of the most expensive rental markets in the country. I believe it may be at this point it's been vacillating back and forth between the third most expensive and the fifth most expensive I think as of a year or two ago it was the third most expensive rental market in in the country. So, and there's also other parts of Boston metro that are particularly notorious in 2017 I believe it was there was a study by rent cafe that found that the Boston metro area was the single most expensive and cost prohibitive market for single so unmarried women, due to a combination of the cost and wage gap issue being larger here than the other major cities. And we have one of the largest, I think it might be the largest wealth, racial wealth gaps in the country there was a really explosive report that came I think it was from the Federal Reserve Bank of Boston in four or five years that show that the median net wealth for black and African American households was $8 so not $8,800 but $8 whereas the median wealth for white households was basically almost a quarter of a million dollars so $248,000. So we have a lot of disparities by, by race and gender to that really exacerbate that housing problems here. Yeah, and as you said it is, I think everybody is aware of a couple of basic kind of facts or premises that we can proceed forward on and one of those is yes, this is an incredibly expensive place to rent or try and own a home. And then Arlington itself is, you know, introduce us to what the situation is here in Arlington clearly it mirrors, it mirrors the situation in Greater Boston area to a large degree. Yeah, so the Arlington is reflective of to an extent the problems with the larger metro area but in some ways and this is why we felt the need to do these two specific warrant articles. Arlington is behind a lot of our municipal neighbors in affordable housing so the state mandate is that each municipality should strive to have at least 10% of their housing stock be affordable housing, and Arlington is barely over half of that threshold we're at 5.7%. And we're behind all of the municipalities we border with the exception of Winchester, as well as a lot of other ones that kind of compete with us with population and demographics. So, even though we can say well, you know it's a problem with the whole state or the whole city like we, we can't just fall back on that we have more responsibilities to we need accountability to do more for affordable housing because even by the standards of the Boston metro area we're falling behind and we're not doing enough to provide genuine affordable housing for those those who need it. I was surprised by this because I knew it was 5.7%, but when we were doing our research to compile an FAQ. I was surprised that in the past, almost 20 years so 18 years. We've only increased our affordable housing stock by 0.1%. So it's only gone up from 5.6 to 5.7 since between 2001 and 2018. We really need to catch up and do do a lot better than that. Yeah, I have to say, I'm glad that you brought that last statistic up because I was going to if you didn't in that I was struck really really struck by that. And in some of the materials you shared with me before, before our chat today to see that, and especially that compared to the surrounding communities which you said already, we are lagging here in Arlington but the degree to which we're lagging when you put that point 1% to those figures that you had cited from other communities, Cambridge at 14%, Belmont event, I shouldn't say even but Belmont at 6.5%, etc, and other surrounding communities also really for want of a better term, I'll just say it kind of putting us to shame in that in that sense, especially because I'm sure it's not that Arlington was that much closer to affordable housing goals and aspirations beforehand that clearly is not the case. So there is work to be done in town and I do know from speaking to residents and town officials over a number of years now that this rhetorically, at the very least, this seems to be a very high priority for a lot of people here in town. So let's talk a little bit about about what to do about it then assuming people recognize there's a problem. People want to find a solution. How do the warrant articles that you are that you are sponsoring or co sponsoring with with others. How are they meant to address that issue in a concrete way. So, the first warrant article which we just had our hearing last week with the redevelopment board, they vote April 5. It would basically increase the inclusionary zoning requirements for private developments right now the inclusionary zoning is that one out of six new units or 15% in private developments have to be, quote unquote affordable. And we originally proposed it as a range between one out of four or a quarter or one out of three and 33% but because we felt that maybe a third might. Well one hit a lot of resistance but to maybe it might not be as feasible because of a lot of issues with economics we settled for one out of a quarter so if this past it would be that new developments. One out of every four units have to be affordable. Now, well, I think the minimum buildings right now as it's written would still start at six build six units or more so in other words, as it gets bigger but we're actually thinking or I know some other citizens have talked about adding an addendum or a friendly amendment I think it's called to make it start at one at four units so if someone has a four unit building that they build that one of those units has to be affordable I got the spreadsheet from Aaron's work of I think, you know, different buildings from the past 15 years different developments and I think several there were several four building developments where I was thinking wow, you know, if we had higher inclusionary you know one of the each of those several buildings would have been affordable because I know some people have concerns that with one out of six well we're not we're just not getting enough affordable stock in. And also that you know it's easier to evade if people want to do just for if you know developer builds a four or five unit, then there's no affordable housing in in those units and at the same time the market rate when you're just building. There's no new high end developments a lot of times that just escalates the rent prices in the area if there's no affordable housing to kind of offset those costs and stabilize it. So, I think one way to do it is to increase the affordable housing minimums that we need for this town and I know that that was something that was actually recommended. In 2016 with the housing production plan was one way to start actually increasing the affordable housing would be to increase our inclusionary zoning amendments, but it hasn't been done yet. Yeah, so I just just to be crystal clear because you just gave a lot of information. And the, what you're seeking is to ensure that we that we have zoning such that any developer or anybody who's going to be making a building that has six units or more for now. That a quarter of those 25% rather than the current 15% or a sixth of them would have to be again affordable housing. Yes. So as you mentioned, from on the behalf of some other citizens is to actually lower that threshold from a six unit building to a four unit building, which again would would kind of include or encompass a wider, a wider array of potential new buildings. Yeah, I think as we have it right now it will still be an improvement since a lot of buildings are a lot more than four or six but I think if that threshold gets lower it will be more meaningful for if and when there are more of those four building for unit buildings. Did you, does it make sense to you right now to mention the other warrant article as well or should we table that for a little bit. I mean I can mention it really quickly. The thing the other part of this is that a lot of times even affordable housing isn't very affordable to low income people. I'm proud that Arlington at the end of fall for fall town meeting past an affordable housing trust fund. And I think that that also because some of the argument or pushback I've gotten about increasing the inclusionary zoning is that it's a disincentive for developers because it's it's not just creative affordable housing and they're not recouping their costs, but there've been a lot of changes, better in federal policy with the low income housing tax credit as well as statewide to increase the value of those credits quite a bit and also I think we're doing things locally here to help fund more affordable housing one of those things is the trust fund. However, I was disheartened that I think it was at the 11th hour and there were two competing amendments one would have earmarked the funds of this trust fund to go toward households making 60% or lower area median income, which is what actually HCA is standard is 60% am I or lower. And then there was another one that would allow households making up to 100% area median income to qualify for these funds and the 100% am I got passed. I was concerned about that so to put that in real numbers 100% area median income. I think that is 80. I think that's $86,000 $8586,000 for a single family household for a two family household that is $93,000 and for once you start getting into three family household that's like $107,000. So you're basically allowing these funds people who are middle, middle class or even upper middle class or borderline upper middle class to qualify for these funds. And let me, sorry. Sorry to interrupt you were saying one family and two family and three family do you mean like one person household or. Yeah, so it's if you're a single family household just you a single person not married no dependents no it's I think it's $85 or $86,000 my notes are in here but I that it's around there.  It's $93,000 and then when you start getting into a three family households so say a married couple with a child. It's $107,000. Those are the incomes of that level, and to just plainly say what my concern is. So I talked about the median wealth gap between black and white households, the area median income in the Boston area for black and African American households is $47,000. The area median income for Hispanic households, I think it's $42,000 the area median income for white households is $86,000 so about, or $85,000 so about the same that this level is and whether or not this becomes a proxy of discrimination it can exclude black and Hispanic households from here. And I know that we've had a lot of talks in this town discussions about how single family zoning has been a proxy even if it was not. It didn't name it as such but it was a proxy to exclude black and Latinx households there's been a lot of discussion in housing justice circles here in the Boston metro area primarily led by people color about how these very high area median income determinants for affordable housing has become another proxy to exclude black and brown people from neighborhoods so whether it was intending to or not my concern is that very high am I is exclusionary to people of color and also, that's very high people who are on section eight vouchers I'm on a section eight voucher. Again, section eight vouchers are disproportionately held by people with disabilities, and again black and brown people, or seniors. This is too high for people with vouchers to use to access affordable housing in towns like Arlington. And so I wanted to. All right, just to, to again just just for in the interest of keeping things as crystal clear as possible. Section eight vouchers are for a certain amount of money, and clearly if an affordable housing unit in one town versus another is a lot more expensive that that doesn't those those vouchers don't increase. It's just that you have to supply the difference between the voucher and the rental. And that, and that's a, that's obviously a big problem with higher rents. Yes. And there have been studies that show a lot of section eight voucher holders particularly section eight voucher holders of color are being hyper concentrated in extreme poverty neighborhoods and aren't having access to higher income neighborhoods like Arlington. So our, our warrants in that respect is seeking to earmark the majority of those funds we have a proposed range 75 to 85% for people making at or under 60% am I just to try to ensure that most people who are the most needy can actually have funds without that affordable housing trust fund. You know I, I know that some concern was well what for home buying programs you might want to hire and am I so that's why I think this is a compromise so let's have a little bit put aside for the higher incomes for, you know, home buying opportunities or whatever but I really, I would like to see the bulk of those funds go where I think they were initially intended to go and especially during the coven pandemic where people are being pushed out into the streets or in Congress settings like shelters and institutions where because of the virus that is could be a death sentence and has been a death sentence for many people. So can I just ask you about about this because I'm sure folks are, are, you know, generally following what we're saying but is the warrant article then seeking to set as a, as a stipulation that whatever the range is that you've determined 75 let's say 75 to 85% of the funds in the in the affordable housing fund would go to people whose am I or just immediate income is 60% or less. So those truly in need compared to those from the 60 to 100% above.  What is it that the warrant article is seeking. So you know how sometimes what warrants they can solidify before town meetings so we haven't had our hearing with the select board yet to get their input. I mean I hope for a stipulation it might be a guidance it's a little bit of an experiment when I spoke to Doug Heim and I asked about doing this he said, you know, we lawyer by the way. We had town council so you know he mentioned that since this has passed but we haven't established all the parameters yet it's a little experimental to be trying to do this and do an arrange. But we could try so we're trying and at the very least we're hoping to direct a lot of attention and a conversation for how these am eyes can be exclusionary but at the most we are hoping for either a stipulation or a guidance about where the priority of these funds and the bulk of these funds should go. Okay, that's great. Thank you for kind of setting these things up as clearly as you have because in a lot of ways these things are complicated and they involve various calculations and various acronyms, which are always difficult to sift through a lot of the questions for people even even with the best of intentions, and speaking of the best of intentions. I wanted to ask you about the fact that here in Arlington, I think, many of us, most of us feel that way about ourselves that we have the best of intentions in terms of our black and brown neighbors in terms of our seniors in terms of people with you know section eight and affordable housing. I do honestly think I don't think that that people are diluting themselves necessarily, but I do believe that that's part of our of the way that we see ourselves. I think we also want to see that reflected in town officials those making putting policy into place and making decisions that affect all of us. I do have a couple of good points about to make about the fact that good intentions might be genuine and still lead to some, you know, unintended and unwanted consequences. So speak to that please. Yeah, I mean so I'm a disability advocate I'm a disabled person and I mentioned this I held a housing equity panel last month and I mentioned that we have the saying that I really like try to abide by it's called nothing about us without us because again, no matter how well intentioned if anyone who's working on a policy or program if they are not getting significant input from the actual populations that are most impacted there's going to be blind spots. And whether again, whether they're well intentioned, they're going to miss stuff and it's going to make the policy. To some extent, inequitable. So it's always important to really try to talk to the people most impacted and and listen and take, take credence from what they're what they're asking for and implement that the and I. And that's what I, why I also wanted to design this warrant in mind with my own experiences about am I. So I live in the housing corporation of Arlington, and when, and a few years ago I was, I first I started out market rate housing here in Arlington and I was within two years priced out it was going up hundreds of dollars in a couple years and at the 11th hour I got into HCA housing. Which I was, you know, happy to stay in Arlington I wanted to stay here. But then I had health issues and my due to to those health issues I had to scale back my work significantly and so over an eight month period my income was halved but a luxury condo went in next before the mass ab project happened and my rent started going up higher than my income, and very fortunate for me I had savings a lot of people don't that I could skim but it still was not going to be sustainable long term I eventually was going to run out of savings if something didn't change but what surprised me is when I spoke to town officials when I spoke to people on HCA is on board and I explained that I was being priced out several people were very surprised they thought that because I lived in affordable housing my rent was in cut you know capped at a certain percentage of my rent so that I could not be priced out and I was like no that's not how it works you know if you have a voucher. You know the voucher will absorb up to a certain you know they pay three quarters of your rent and you pay up to a third of your income but I'm not I wasn't on a voucher at the time. And I was like so there is no cap and since the AMI is set at a rate equivalent to what the area is if the area gets very expensive, then even the AMI you know 60% AMI can be a lot. So people were didn't know that and you know I, I'm on Medicaid because of my disability and just to also put it in context you know a lot of low income people are Medicaid when I spoke to a case worker at Medicaid and I told them what my rent was at the time. She paused and said wait I thought you lived in affordable housing and I said yeah I know I do and she's like well that that's affordable housing nowadays that's how much it costs and I was like yeah that's how much it costs in the Boston metro area and in Arlington. So I think a lot of people just think oh affordable housing, you know people, you just go in and you're not paying much and so affordable housing people can afford it. Yes. And so I think that that sometimes I've talked to people in the redevelopment for a lot of people don't understand these factors and if they're creating policies. I'm worried that some of the effects again whether intended or not is going to be that it's not going to be as genuinely affordable as they think it's going to be or protective of the lower income populations that need it. Makes a lot of sense. I appreciate very much you being here today and I want, we're running out of time but I wanted to invite you with the two or three minutes we have left. If there is anything of import that we have not failed that we have not yet mentioned and you'd like to. I invite you to do so. Just to elaborate again. The, so people on disability I wanted to mention to people on social security income only make $7 or $800 a month and that's it.  I was like because my rent is literally hundreds of dollars more than what the payment is for SSI and people don't understand that so if you're on social security income, you cannot find a single housing market like market rate apartment that you can afford on an SSI check it has to be affordable housing and even then you usually need a voucher in addition to that. So, especially since we have coven coven has created an additional population of disabled people who can't work and now might temporarily or maybe permanently need to be on disability. So there's going to be people who need, who need that housing. So I want people to understand that I think also there's a lot of misunderstanding I, I know that it wasn't meant to be a harmful statement but the redevelopment board made a comment in response to affordable housing a few weeks back saying well we should look at workforce housing and the way it was presented seemed to suggest that it was mutually exclusive that people don't who are in affordable housing don't work. Many of us do. I think studies show that 40% of people who live in affordable housing work who are working age, especially my next door neighbor is a health care worker she's a medical assistant who works for private doctor practice right here in Arlington. We have daycare workers who work for people here in Arlington. So I think I want people to know that a lot of us work. Those of us who don't work due to disability or age still have a right to housing here and we still also contribute a lot economically to our towns and that's another thing to I think there was a comment about growing our economy or contributing toward progress, saying why we should invest more in market rate versus affordable housing and that suggests to that, you know, sometimes people who live in affordable housing or drains on the economy. And there have been studies that show that people who live in affordable housing they're more housing stable than other renters. So, compared to homeowners in the rental market affordable housing people actually stay in their communities longer than market rate renters. And it's unlikely to be investing our dollars in the immediate community because we're not going to Europe we're not going to California we're spending all our recreational dollars here. So I also think the town needs to also consider us an asset and I think that will be more reflective than in the policies and willingness to invest in increases in affordable housing to understand that it does pay off in many ways. Thank you so much. Yeah, thank you for your activism for your advocacy your energy and and for shedding light on issues that are of real importance and often misunderstood ways that you've described here in town. I'm coming with Laura Kiesel, who is, as you can tell, an ardent advocate for affordable housing for those with disabilities for our brown and black neighbors, and and others. And it's been a really worthwhile conversation. Thank you Laura. Thank you. This is Milan, and this is talk of the town. Thanks for joining us. We'll see you next time.


Wiki. Build knowledge together. If you've ever gone to Wikipedia to look something up, you might have noticed that you can actually edit the page. You can add information to it, because this is a Wiki which is a collaboratively created and edited document. Moodle has a Wiki too, which you could set up for your students, for example to do a group project doing some research, or to write a combined essay. To add a Wiki, we go to our course page and we turn the editing on, either by clicking the button top right, or by scrolling down to the administration block and clicking the link, turn editing on. Then in the section where we want our Wiki, we click add an activity or resource. This brings up the activity chooser. Scrolling down we'll see Wiki in the activity list. And if we click once, we get some information giving useful teaching ideas about a Wiki. And then we can click the add button at the bottom to add it. The name is important because it's what the students will see on the page to access it. And then you can give a description of what you want them to do with the Wiki, and if you tick the box display description on the course page, they'll see the description there. Normally a Wiki would be group work, collaborative. But you can have an individual Wiki by changing the drop down menu here, and each student would have their own copy, for instance as a personal notebook. But we're going to keep it as collaborative, however. So the next thing to do is to choose the first page name, and this must be carefully chosen because it cannot then be altered. There are other settings, but for now we just go to save and display and get our Wiki started. You see now that we need to press a button to create this page. If you don't understand any of the other formats, leave it as default, it's absolutely fine. And we now have our first page of the Wiki. To set up links to make up the pages, you put double square brackets around the name you want to give the page. So here are a couple of examples, and this is how your learners would also make new pages on the Wiki. When we scroll down and save it, we see that we have links. These aren't yet pages until we actually click on them and create those pages, in the same way that we made the first page. So if we click on the reading link, we then prompted to create that page, and we can start adding information to it. This is how your students would then add text and information to a new page. Now if we scroll down and save that page, we can take a look at some of the other links in a Wiki. If we look at the tabs at the top, map takes us to the pages which have been created and we can access them from there. Currently we only have two pages. History is a useful way of the other learners and the teacher seeing who has changed what on the Wiki, and in case of any problems, that's a handy place to go to and to revert changes. And comments enables you to see and add comments to the Wiki.


A cell cannot be closed until it's opened, okay? Hey guys, what's going on? It's Andy. In this video, I want to talk to you about the four things that are going to teach you how to open up the cell. Now, by the way, if you want to go to the next level, you need to understand this. A cell cannot be closed until it's opened, okay? And a lot of people don't know how to open up a cell to actually be closed. They don't create interest. When they go to ask for the customer or the client to say yes to them or their product, whatever they have, they don't get the answer they want because they never open the cell. So these four things are very important, okay? So how to open up a cell. Number one, you got to get their attention. You got to get their attention, okay? So how do you get people's attention? Number one, be a freaking professional and do a great job.  And by the way, you're going to want to know what I know because when you're around me, you're going to see that I'm an intelligent person, but also I really, truly care, okay? And by the way, I'm not on my phone. I'm not messing around. I'm not distracted. I'm not BSing with some buddies. I'm all into you.  So number one, get their attention, okay? Be good at getting people's attention quick. That's one of the first ways to open up a cell. Number two, establish common ground. Listen, I'm going to tell you this, all right? This word right here, rapport, okay? Number one, the second you fall out of rapport, you never close a deal again in your life, okay? That customer is dead and they're cold when they fall out of rapport. So I want to talk to you how to build rapport, okay? Because rapport right here, once you fall out of rapport, you're toast. You're done. Hey guys, what's going on? It's Andy. A lot of you leave comments telling me that you need help. Do me a favor. I'm going to tell you the best way to get a hold of me. Shoot me a text message right now,<PHONE_NUMBER>.<PHONE_NUMBER>. I'll help you with whatever you need.  Let's get back to the video. So the main thing that you want to do is establish common ground and build rapport with your client. So look for a shirt they're wearing. Look for a ball cap. You know, look for a bumper sticker on a light. I don't care, license plate, bracket. Find something that you can spark something with, okay? But building rapport is huge. By the way, if you want to build great rapport, people ask questions, ask great questions, get great answers. So four ways to open up a cell. Number one, get attention. Number two, establish common ground. You and them got to speak with familiarity and feel like you've known each other your entire life. Build that rapport. So number three, really important, how to discover the need, even when they don't realize they have one. Look, a lot of people don't even realize that they need your product as bad as they need it. It's your job to ask great questions, discover the need so that they know that they need you. Number four, make sure that you add value to their lives. This is really important. As you're looking at all these, when you're opening up a cell, as you're building common ground, you're getting their attention, you're discovering a need that maybe they didn't realize they had or make them feel even how important this is, that they didn't realize it was even that important, you're going to show them that you're going to add value to their life or this product's going to add value to their life. And if you do that, guess what that's going to do? That's going to open up the cell, they're going to want to know more from you. And now you get the opportunity to do your job. So there's a deal, it's called hook story close. Let me explain this to you, okay? So this right here, this is the hook. Okay, it's like when you're fishing, you feel something taking the bait, boom, you hit the hook. Once you set the hook, you get a chance to tell the story, which is about what you're going to do after you open the cell and then you get to close it. Hook story close, this is the hook. This is how to open a cell every single time and you'll crush it, learn these four things. Hey guys, I just want to tell you the true one percenters you made it till the end of the video, do me a favor, share it with the friend that wants to go to another level, make sure you like the video, comment below so I know who you are, set your notifications and then subscribe to the channel. We got daily sales training videos dropping. I'll see you soon.