 Thank you. And also thanks to the organizers for inviting me to speak. So of course the theory of quantum groups and its applications needs no introduction, but maybe the theory of quantum symmetric pairs does. So for now you can think a quantum symmetric pair is like a quantum subgroup. And in the past decade it's become clear that these quantum subgroups have analogues of all the things you know and love about quantum group theory. They appear in categorification, they admit a canonical basis. There's a sure while dualities. And today I'd like to tell you about their applications in low dimensional topology. So this is by means of equivariant factorization homology. And so before I'm getting to applications I'll have to explain what all these words mean. So this is more or less the setup for the talk. So let's dive into it. So what is the geometric context that we're talking about? It's a tangent orbital and so what I mean is a global quotient orbital. So we have a smooth manifold M. There's a finite group acting on it. This action is not necessarily free. So this could have singular points in its quotient. And then so we're specifically talking about framed quotients. So what does it mean for M to be framed? Well we know what that means. It just means we've fixed the trivialization of the tangent bundle. So now you might wonder what is a framing of a global quotient. And so if you think about the orbital in terms of the equivariant geometry of M, that should be an equivariant trivialization of the tangent bundle of M. So equivariant with respect to what? We need to fix some representation of the group gamma such that we can trivialize the tangent bundle equivariantly with respect to that given representation. And that's what I'll call a row-framed global quotient. Now what are examples of these? So let's look at a sort of small group example. So we're taking Z mod 2Z, which I denote by Z2. So for example you might take the 2-disc. You rotate by 180 degrees. That gets a fixed point at the origin. And now with the normal blackboard framing you see that this framing is inverted by the Z2 action. So the sine representation is here, the framing representation. Or similarly you might take 2-disc and we have the free quotient but we've framed them with the opposite framing so that also here we get the sine representation. And finally you might take a torus and rotate it 180 degrees like this. So you see 4 fixed points appearing and this again is a sine-framed orbital surface. So these of course fit into a category of framed global quotients and the type of maps that we're considering are framed embeddings but these are framed only up to a prescribed homotopy. So you don't need to preserve the framing on the nose. And in fact there's a whole space of these framed embeddings so it's rather a topological category or an infinity category if you prefer. Now to obtain invariants we're going to think about these orbitals that's glued out of simple pieces. And so if you think about manifolds there's of course only one local piece that one needs namely the disc Rn. But for orbitals locally if you're around some points they might look very differently and this is completely determined by the isotropy group once we have a fixed sort of representation that controls the global geometry. So what are these local pieces? That's a full subcategory that we're going to be looking at of discs and these discs all look as follows. We take some subgroup of the finite group which receives a faithful action on the row and then we can consider this quotient by which I mean we take gamma and Rn and we mod out the diagonal i action. So this is just the amount of cosets of i and gamma copies of Rn and it naturally has one of these row frame structures that we talked about. And so we're going to be thinking about global quotients that's glued from these and we have a symmetric monoidal structure on both these categories given by this joint union. Okay so now this is sort of a natural generalization of the setup of factorization homology as introduced by Ayala Francis so we can define our invariance in the same way. Namely we fix an algebraic gadget called a disc algebra which just means a symmetric monoidal function from our category of discs into our favorite target symmetric monoidal category and then we integrate that's algebraic gadget over the orbit folds by means of a left con extension construction. So if you don't know what left con extensions are let's just not worry about the details but let me tell you some properties of what this construction has. So first by definition it's a functor so this thing is functorial with respect to the framed embeddings which were amorphisms but also the higher morphisms like isotopes of embeddings, isotopes between isotopes and so forth and it's a it's monoidal so taking the disjoint union of two orbit folds the invariance assigned is naturally the tensor product of the separate invariance. And now finally the key property that actually allows us to make computations is so-called tensor excision meaning if we have say some some global quotient that we divide up into two pieces so we've got a piece M plus here which is a gamma invariant piece we have a piece M1 and they intersect over a color gluing as such which is a product of n cross r and what's important here is that this r direction is completely received a trivial action from gamma so gamma is not doing anything on the r direction on this thing. Now what that means for us is that the invariant assigned to this color gluing is a natural E1 structure coming from the r direction so there's an associative algebra structure on this piece of the invariant and it's in fact acting on the invariance left and right just given by embedding that color gluing piece into either the left or the right chunk of this cut-up manifold and so we can compute the invariant as this relative tensor product construction maybe important to note that this is in fact a defining property so if you have some abstract function out of this category of orbitals into a nice enough symmetric monotone category and it satisfies excision then in fact it must be given by a factorization homology construction. So that's the abstract setup and in fact we don't need to think about frames you could do other structure groups but now let's just move to a specific example the one where these quantum symmetric pairs show up okay so we're going to move to two dimensions back to the Z2 group with the sine representation and so first we need to think about what's the kind of algebraic data that we need to fix in order to get invariance so the question is what does it mean to have a symmetric monoidal function out of this category of disks? Well there's only two types of orifold disks in this case we have three quotients and we have the singular quotients at least for this given representation now D and D star which are the names for these orifolds we'll get assigned something and so we're going to be taking our invariance with values in k linear categories so that means one of these functors assigns a category A to D and a category M to D star and the question is what are the structures categories need to have in order to define one of these Z2 disk algebras okay and so the monoidal structure I've made a comment there it's kind of basically what you would imagine it to be it's like a k linear tensor part of k linear categories okay so just let's just look at some embeddings and some isotopes to see what kind of structures appear so one thing you could do well we can take this free quotient and then flip the two disks okay that's an equivariant embedding from this thing to itself and so it's a morphism in our category so that should get assigned a functor which goes from A to A we could also do something else we might take two copies of this D and embed it into D again which results in a functor from the tensor part of A with itself to A okay and then I invite you to think of that as a tensor product functor or we might take a copy of D and a copy of D star and embed that into a bigger copy of D star and what results is a functor from M tensor A to M so these are some of the embeddings but of course there's also higher morphisms so there's also certain isotopes we might look at so for example we might compose the flip embedding twice but that's just a normal embedding the identity embedding so in particular there's an isotope between this composition and the identity embedding given by the trivial isotope and what results then is a natural isomorphism between the square of this functor phi and the identity functor on A otherwise we might take this embedding for the tensor product we can rotate these two disks and what results is a braiding natural isomorphism between the tensor product and the opposite tensor product so this of course reminds us very much of an E2 algebra and now what's interesting here is when we rotate these actions sort of embeddings around the singular points what we notice is that once we end this rotation which is like a half twist the colors red and blue are interchanged which for us was exactly that embedding corresponding to phi so this doesn't get back to the action tensor but actually we're twisting the A component by this functor phi now of course there's a lot more data you might draw like wild embeddings and worry what is corresponding to this but it turns out that all the data you need can be expressed in a small finite list of functors and natural isomorphisms which have summarized here as follows so you well so I've shown a coherence theorem that says if you want to have a z2 disk algebra in this two category of categories actually this is the data that you need you need a braid monodic category with an anti-involution and you need a module category which has what I call a z2 cylinder braiding I'll leave the coherences out at the moment but I'll draw some pictures later so you see what kind of axioms this cylinder braiding should satisfy but then of course the question is where do I get examples of such and this is where the symmetric pairs enter so let me remind you so what is a symmetric pair well I have some semi-simple Li-algebra G with an involution, a Li-algebra involution on it and I'm looking at the sub-Li-algebra fixed points so this is what is called a symmetric pair I've given you some examples and classically these were studied as the infinitesimal data corresponding to a symmetric space so the symmetric space here so think of G theta as the complexified Li-algebra K the complexified Li-algebra of G now people were interested in studying quantum symmetric spaces but they didn't know how to define those and so a natural idea was to try and quantize this infinitesimal data because of course the quantum group was well known so then the question becomes how does one quantize that sub-Li-algebra and interestingly it turns out it doesn't quantize to a sub-hope-algebra of the quantum group rather it's slightly more subtle the thing is you can indeed quantize these which is done in the work of these various people but the sub-algebra B that quantizes U of G theta is actually a co-ideal sub-algebra we'll get back to the relevance of this later now as I promised quantum symmetric pairs should have all the cool stuff quantum groups also have so what do they have that is particularly nice well the quantum group of course has the universal R matrix there is an analog for quantum symmetric pairs called the universal K matrix so what is this so this is due to Balagovic and Kolb and they say every quantum symmetric pair is quasi-triangular by which they mean there is a universal K matrix which solves in every B module this particular equation called the reflection equation and so here I've drawn a topological picture which explains what that equation is saying and so this is the equation corresponding to the braid group of a cylinder and so this K as you can see is appearing here literally as a K in the picture which explains its name and well and so that's more or less all there is to say but there is a subtlety so in fact if you study what the quantum symmetric pairs solve it's not quite this equation rather it's the following equation so there's twists appearing in this reflection equation and this is the actual equation being solved by the quantum symmetric pairs so here phi is some involution on the quantum group and somehow the R matrix is being twisted throughout this equation so if you look at this picture it's not quite clear how to interpret this and in fact I'm proposing we should rather look at some other picture so this was a picture of the braid group of the cylinder which we can think of as the fundamental group of points in a plane but rather I want to propose we need to look at the fundamental group of points in the orbit fold plane so think about this D star then we can draw the following picture so this is a picture of the configuration space of two points in D star this singular orbit fold where this black line is now representing the singularity rather than just drawing the points themselves I've also drawn their Z2 orbit in a different color so a configuration of points in the orbit fold is now a Z2 orbit of points within the orbit fold and what we can see now is that the R matrix is just a double R matrix happening on the points and their Z2 orbits but the K matrix is exchanging the blue braid for its red Z2 orbit braid and so if you think about coloring these pictures by representations in the quantum group we would color the blue braids with normal representations but their Z2 orbits are colored by the twisted representations by phi and so now if we do K2 and then we apply the R matrix we see we need to twist one leg of the R matrix by phi and similarly there twist this one by phi twice and here once which is exactly capturing this twisted reflection equation okay so this is not just some funny way to reinterpret that equation this is actually a profound connection between quantum symmetric pairs and orbit folds namely let us move back to the observables that we discussed before so on the one hand we have the quantum symmetric pair with the various sort of structures that are naturally present and on the other hand we have the local observables for these Z2 disk algorithms as we discussed before so now if we imagine we take the quantum groups category of modules and denoted A and we take the category of modules for the symmetric pair and denoted M then we see that we perfectly match up the structures present between the local observables on the one hand and those present in a quantum symmetric pair on the other so what does that mean? That means that if we take the following notation that the pair rep QG and rep QK is exactly the type of data of a Z2 disk algebra in categories in particular this means we obtain invariance of two dimensional orbit fold surfaces by integrating this Z2 disk algebra and so the question is what do we get? So let's go back to my favorite example of an orbit fold surface we want to understand what is the categorical invariant associated to this orbit fold or rather maybe let's not do this orbit fold but the one I actually computed namely we remove two points on the end and there so we're only left with two fixed points and think about this orbit fold quotient so we are puncturing the torus twice and considering the orbit fold like so now what is the neat thing? Now I can take two copies of D which was this free quotient here and there and I can embed them together with this punctured torus into itself just by drawing back that torus a bit okay? Well by functoriality if I have an embedding as such on the assigned invariance I get a functor as such where D is now my shorthand for this invariance that we are trying to understand So what does this mean that we have a functor as such? Well D is appearing as a module category Now module categories if you're lucky are much easier to understand because there's a lot of structure present in fact we'll make use of the following slogan module categories should be categories of modules so we are supposed to understand what is the algebra object for which this category is appearing as a category of modules now rather than running you through the computation let me rather just explain to you what the algebra is okay so the particular algebra that we're interested in is the algebra of quantum differential operators so let me recall what that actually means so think of differential operators on a group G what should that be? Well we have functions on the group we have sort of this enveloping algebra of vector fields on our group and there's a cross relation because if you want to move the vector fields past functions they're going to take derivative you could write that as a certain specific cross tensor product of UG and OG which is actually a very general hope algebra construction known as the Heisenberg algebra in particular this just generalizes to the quantum setting by replacing UG by UQG and OG by OQG the dual quantum group okay so that's what I mean by this algebra of quantum differential operators so this isn't just an algebra in vector spaces rather it appears naturally inside a certain category so the group G has a natural right action on itself and a natural left action and so this algebra of quantum differential operators is equivariantly quantizing the differential operators with these two actions okay so where should this DQG live it naturally lives in rep QG tensor rep QG which is encoding this left and right action that is naturally present well if DQG is an algebra object in there then in particular we can look at the category of modules for this algebra object internal to rep QG tensor rep QG now this allows us to formulate our results namely if you want to compute the categorical invariance of this punctured torus we get the category of modules for DQG but not in rep QG tensor rep QG rather in rep QK tensor rep QK now recall rep QG acts on rep QK in particular if you have an algebra object inside rep QG it makes sense to look at modules for the algebra object inside rep QK so in particular or similarly if you have an algebra object in the tensor part of rep QG tensor rep QG then this expression makes sense as well but so if we try to sort of unwrap what does that mean well we're looking at modules of this algebra DQG which are equivariant for a left and right K action so those are K cross K equivariant D modules on the group G or if you like those are exactly the quantum D modules double quotient K mod G mod K okay so this is the main result let me maybe say something more because I see I still have some time left so this is interesting and it's quite nice now what can you do with this so this is some category of modules that appears from this equivariant factorization homology construction but that means that this category in particular has lots of extra structure because it is coming from factorization homology so let me maybe try to draw a picture to see what kind of things we can do oh and meanwhile if there's any questions about anything this would be a good time to ask where is the relation double point algebra that's an excellent question let me answer it immediately okay so what can we do so we take our punctured torus now let us take so that's T prime and let us take some copies of D maybe let's take one of those and let's take two copies of D star these clearly embed into T prime just by mapping in these disks in various places so the result is a factor from rep q g n times rep qk two times into this category of D qg modules okay now this factor if we start moving around these disks corresponding to the quantum group these can move around freely so there are two isotopes between this function itself where we've moved around these points now if we just shrink these disks to points that's just the configuration space of points moving around in that orbital torus and so by functorality of the invariant this function naturally receives an action of the orbital group of this torus now in particular one thing the two is just choose things to plug in here so you might choose V being the vector representation of the quantum group for each of these entries and take two characters here of rep qk and then choose a D module here to home width so we're going to home the image of this function landing here this is an interesting D module which means we're going to be looking at the following product and copies of this vector representation a character new mapping into M and maybe we're going to take dual here so that we can move this to the other side because that's actually how it appears in the literature in any case this vector space is receiving an action from bnt mod z2 and for this particular choice of representations this descends to the type c check cn so this was done by David Jordan and Xiao Guan Ma in a paper 10 years ago and now this construction is naturally reappeared in a more conceptual way ok let me end there if you take other surfaces have you computed what you get? so yeah that's a great question so what I've said so far is of course specific to framed things and there's not a large supply of two dimensional framed orbital surfaces so one thing that you would have to do now is say ok the extra structure you need to go to an oriented version of this TQFT and so for the quantum group it's quite clear there's some ribbon element for this quantum symmetric pair actually that's still to be worked out so that's work in progress now once we've pinned down that structure which must be there then we can start computing for general surfaces what the invariants are but if you look at sort of the work that David and his collaborators did some of the key ingredients for building all surfaces are DQG together with the annulus computation those two are done so I think that gives us a lot of information about what to expect in two dimensions is there a reason you'd expect only the only sort of interesting group to choose in order to take all the folding of the spectator to be Zmod 2? what would you expect Zmod 2? would you take much more expectations? that's a great question well so for example Zmod 3 I think would be actually very interesting so there's not yet a theory of quantum symmetric pairs for order 3 in volution of the quantum group but I and others expect there will be examples if you look at D4 and the triality this is hinting us that there will be an analog if you look away from like more general things say you do D8 somehow the like if I draw a picture of the type of strata that appear you would get something like so maybe so what kind of the shame is is that if I take an embedding of like the free quotient with all these copies there's no way to move so the embedding cannot move through these I guess singular strata and so somehow these isolated points they have very kind of rich braided structures and for something like D4 that will not be present so I guess in that sense for me because I care about braid groups these more like Zmods and Zs are the most natural place to look for invariance what are the parameters of the DAF here that's a great question let's discuss that in the break when I pull up the paper I don't remember exactly so there's so in choosing the symmetric pair there's a choice of a parameter here so there's one parameter in choosing these characters there's two parameters there's the parameter Q of the quantum group there's actually even more for these but it's already 1, 2, 3, 4, 5 so in fact the characters come with two parameters so it gives seven parameters that are somehow interdependent and recovers five parameters of the DAHA I'll give you the specifics later but so what's interesting here is that we see well there's more room to glue in so we could have glued in more symmetric pairs which is really unobvious from the previous construction of Ma and Jordan and so we're hoping that if you do this maybe one can recover all the parameters of the DAHA but that's unclear at this point