 Thank you very much Sergei. So I would like to thank the organizers for inviting me to give a lecture course in this school. So let me, so first of all, let me briefly review the plan of my lecture. So in the first two lectures, I will review the notion of WRT environment and also some general notions of TQTs and how WRT environment, briefly mentioned how WRT environment can be understood as a TQT. And so the last two lectures will be more about more recent developments. So in this first lecture, I will explain what kind of analytical continuity assignments here, which convinced us kind of a mathematical approach to try to formalize the past integral, and which is mostly was developed by Kantsevich and Witten. And in the last lecture, I will mention the so-called BPS environments of three manifolds, which has close intimate relation with WRT environments, but they have nice properties that they expected to be some integrality properties, so they expected to have an accurate verification. Okay, so let me first start with some general motivation behind the story. So first of all, there is a close, very close relation between links and three manifolds, which is given in terms of surgery, which I will explain in detail about what this procedure is. And so in particular, this leads to also close relation between the environments of links and three manifolds. So let me give you some table, for example, of environments of links and the three manifolds counterparts. So the left-hand side will be environments of links, and the right-hand side will be environments of three manifolds. So first, you probably already seen the environment which is Alexander polynomial, and the counterpart of the corresponding three manifolds environment is known as drive torsion. So in particular, if three manifolds obtained by surgery, one can express the drive torsion using Alexander polynomial. And then there is, well, to be precise, it's a colorate, colorate Jones polynomial, which you also already see in J. Krasmusen lectures, and the corresponding environment of three manifolds is W-R-T invariant. So to be precise, this is an environment constructed for quantum SL2. And actually, in this kind of more general setting, the drive torsion can be understood as a W-R-T invariant for quantum SL1 slash 1 supergroup. And then we know that there is, so there is a categorization of Alexander polynomial which is not for homology, which was already also mentioned, and the categorization of Jones polynomial is Havana of homology, or more generally, Havana of Razanskiy. And so here, we actually know in the case of not-for-homology, we know that there is a counterpart environment for three manifolds, which is, so, just for example, this was also already mentioned, this is called Isis Abigritan, or equivalently Higart for homology, three manifolds. But here, it's not the big question what actually we have here. And so part of this DPS invariance is aimed, kind of very smart, I aim to fill this gap. And also, Havana can actually also continue table in this direction. So here, there was links, three manifolds, and then also, actually, a related to homonorrhance of four manifolds. So in particular, if you understand this kind of homology, homological invariance as a vector space assigned by Tqft functor to a four-dimensional Tqft functor, to a three manifold, then there should be some corresponding numerical invariance of four manifolds. And here, is Isis Abigritan invariance. And again, it's not clear what should be a counterpart in the lower line. Okay, so this is just general motivation, and now I will proceed to more concrete things. But are there any questions about this part? Okay, so now I want to describe in some detail what is a dense surgery. So this can be the operation which I take a framed link and from framed link in, so suppose in S3, but this can actually be done in any S3 manifold. I construct oriented closed S3 manifold. And so usually we want to consider early links up to ambient isotope and S3 manifolds. Up to, well, in S3 manifolds, if we're in S3 dimension, we can consider either up to homomorphism or diffomorphism in S3 dimensions, it doesn't actually matter. So let me remind you what is a framed link and also fix the notation, fix kind of conventions we shall be using. So framing, what is framing? Let me take, so consider, so let me, so L, I will denote by L some link, some framed link, and by the dense surgery S3 of L, the result of the dense surgery. So let's, so suppose this link has all disjointed components for framing, so framing on a link component Li, so which is a embedded circle is a choice of a section in a normal bundle. So this is one way to understand this. So if I suppose I have some node in a S3, and essentially what I want to choose, I want to, so choice of non-varnishing. It's important that it's non-varnishing. So at each point I want to choose a vector, orthogonal to a node, to this node, which is a component, a certain component of a link. And so they are equivalent, they're kind of, and we usually want to consider this up to a kind of, up to a homotop, so up to any continuous deformations. And, but there are other ways to kind of specify framing more combinatorial. So in particular, so framings on a node in S3, they want one, there's a canonical one-to-one correspondence with integers. So in general, if we consider framing, so framing can be also understood as a trivialization. So as usual framing, it's a trivialization of the normal bundle, so the completarization is given by just the second vector, which gives me, so the second vector in the basis of trivialization is, I just pick a orthogonal to both this red vector and the link component. So how does this one-to-one correspondence work? So I take, so this number, so this map is given by realized, so the result of this map, so it's some n is, can be calculated as a linking number of, on li, and li prime, and li prime, where li prime is a push-off of li towards framing. So what does this mean? So this was my link component li, and so I can see the small shift of this link component towards the framing. And then I, and so then I have a link between li and li prime, can calculate this linking number. And it's well-defined, but it's only well-defined when the link component is in S3. Yeah, well, I can understand either S3 or even R3, it doesn't really matter. So in other words, one can understand so framing, framed, so a framed node, I can understand as embedded ribbon. Sometimes people can see the, just can understand this as a ribbon. So the ribbon is formed by again a small, you can understand a small ribbon, which is one of the boundaries li and the other boundaries li prime. And another remark is that one can, so in a node diagram, one can specify framing by what is called blackboard framing. The blackboard framing means that I can understand essentially the, well, node kind of line as almost in a single plane and just a little bit with resolved crossings. And then the framing everywhere is goes, is parallel to the plane or the blackboard. And so in particular, if I, for example, if I draw a node, something like this and assume this has blackboard framing, I can accurately understand as just, I can alternatively specify this by just drawing a node and fixing this integer n, which also specify my framing. In this case, it'll be plus one, okay? So in particular, so when we consider, so if you want to consider framed links up to ambitizotope, there is no longer, there will be no longer the first red and white stream. So this thing doesn't work. So instead, we only allow to do a modified red and white stream. If you have something like this, then this we can replace by just a stri-strand, okay? Questions? Well, it depends if you have, so if you have, for example, orientation on the node, then there is a canonical choice. Well, if you, but in principle, well, there is another choice, but it's always up to homeotope is the same because we always can rotate, at any point we can rotate it by 180 degrees. Any other questions? So to summarize, to specify framing, either we just, for each link component, we specify an integer number or in terms of plane diagram, we draw it in a certain way so that we don't, we are not allowed to do this first random master move. And essentially, yeah. So more or less, this integer number is convinced how many times this framing vector winds along the link component once we go around. Once we go around. So now I want to describe a denser, so L, so it will be L component framed link. So the poses look something like this. In this case, there are two components, L1, L2. And so by integers, the I are specifying my framings. And so the dense surgery operation which produces a rented closed C manifold from this data can be understood in, so it's easy to understand this in two steps. So first steps, step is cut out a tabular neighborhood which I would denote by NL of L. So let me, so this means that for each link component, I remove a small solid torus which surrounds it and some particular boundary of this tabular neighborhood is homomorphic to a copy to the joint union of L copies of two torus. And so let me specify copy of two torus by index I. And then we can choose, so in the first homology of the torus, there are the bases, the particular bases generated by classes of two curves which are usually called meridian and longitude. So what is the meridian? The meridian is any curve which is contractible and to specify another cycle, I need to use a framing. So another cycle will be the longitude can be understood as a push-off of this link component towards the boundary of the tabular neighborhood in the direction specified by the framing. I'm doing this, okay, let me try to look something like this. And so more formally, for each at each point, I can consider a ray in the direction specified by the framing vector and the intersection of this ray with the boundary of the type of frame. Boundary of tabular neighborhood, which is the T2, is my cycle LI. And so actually I also want to specify, so essentially I actually want to specify orientation, but the final result of the surgery won't actually depend on the choice of orientation. So let me fix orientation, some orientation on each link component and this orientation will determine the orientation of my longitude and meridian. Again, from the definition of the framing, it follows that so the, for example, the linking number between the longitude and the link component itself is A2. And so the second step of the dense surgery, so we want to glue, so I take a disjoint union of L copies of solid torus and glue it back in using the following prescription. So here I can, so I have a standard, I can choose a standard longitude and meridian. So LI will be just S1 times any point, so it's the boundary, so on the boundary. So LI, the longitude will be just S1 times any point of the boundary here of the disk and the meridian will be any point on the circle times, sorry, in this great point, the boundary and the meridian is the boundary for disk times point. So let me draw, so suppose I have this copy of solid torus S1, so D2 times S1 with some standard. So I want here, so this will be denoted as primes. So this will be MI prime and this will be LI prime. So the gluing should be used using the homomorphism phi from one boundary to another boundary such that it's pushed forward on homology axis follows. So MI is sent, so MI prime is sent into LI and LI prime is sent to minus. So essentially what happens is when we glue back in the longitude and meridian, they swap around. So L2 is identified with M2 prime and M2 is identified with minus, sorry, yeah, minus L2 prime in this particular example. And so just, of course, this, and the easy statement, of course, is that if I choose any other homomorphism such that it's pushed forward on homology is like this, this gives me the result will be homomorphic, three-manifold. So finally, so one can write more explicitly the result of the dense surgery on this frame link is a union of the complement of the tabular neighborhood with the joint union of L-copies of oscillators. And the union, the gluing is done using homomorphism phi which satisfies this condition. Questions? Yes? Well, I mean, of course, there are certain, there are certain freedom which you can change a bit more phi, yes, so that the result will be the same. In particular, of course, I can do a homomorphism on these oscillators which will change my longitude but it's not gonna change meridian, yes. Is this, does this answer any other question? So what is the, let me give you some examples. So if I make a dense surgery on a node with framing P, again with specified frame by some integer, the result is a lens space LP1. So my definition of lens space is that there's a quotient of S3 mod Dp, namely, so there are different quotient. So this quotient is realized as follows. So if I understand the sphere by the following equation on two complex variables, Z1 and Z2, then my Krullens relation is that where Z1, Z2 is identified with a pair where I rotate one of the variables by e to the 2 pi divided by P and the other variable by e to the minus 2 pi i. And so another example, so consider...