 So I thought I would start first by some background from physics, since many of the math grad students might not be familiar with the physics motivation for this. So in physics, we have, well, we have first of all classical field theory. So yes, in physics you usually start by studying motion of particles, but then, for example, Maxwell's theory of electromagnetism is not in terms of particles, but in terms of fields. So something that takes a value at every point in your space. So basically what happens is you have a manifold, the general setup is you have a manifold and you have some fields on M. And this, well, mathematically, these are just sections of a sheaf, but what can they be? So typically they're vector fields or connections or maybe maps from M to a fixed manifold, but there's some assignments that, yes, assign something at every point on the manifold. And then depending on your physical system, you study the time evolution of the fields. So roughly you're working on a spacetime, which is M times, well, it's basically M times zero, one. And another thing that usually happens is that the evolution depends on a metric. So M comes with a Riemannian metric. I mean, physics usually depends on distances. Great, so that's all I'm gonna say about this. Now, later, physicists developed quantum field theory. Okay, so what is quantum field theory? I won't be able to tell you now, but it's a field theory that also incorporates quantum effects, meaning that the energy is quantized, it's N times, it's some integer times quantity, the Planck constant. And then when you set, when you let H bar go to zero, you get the classical limit. And then it also should incorporate special relativity, meaning Lorentz invariance. Okay, so physicists developed theories that involve this, all these things. And we will actually have an introductory electron quantum field theory. We added it to the schedule on Friday by Max Zimut. And yes, he will tell you more about this if you're not familiar with it. Let me just say that this was a very powerful idea in physics, so it explained kind of three of the fundamental forces of nature. It explains, well, electromagnetism. It explained the weak forces and the strong forces in particle physics. Great, so this formed the standard model of particle physics. The one force that is not, that doesn't quite fit into the picture is gravity. Well, physicists have tried to incorporate gravity into a QFT and then they developed various other theories like string theory. And in the process, they started studying QFTs on their own. So there are some interesting QFTs that you can kind of write as toy models for nature and you can see what, yes, what could be like just theoretically what QFTs are interesting. And in the process, that's how they discovered that some QFTs are just topological. And this led to the, I mean, they do not depend on the metric. So this led to the concept of topological quantum field theory. So TQFT, and some people call this just topological field theory. In fact, so this is a mathematical concept. It was introduced by Artea in 1988. And I think Jake Rasmussen talked about TQFTs last week and you're gonna see them again in the other lecture series as well. There's nothing quantum in the definition. I think they were called TQFTs because they came from quantum field theories but the definition doesn't refer to quantum. So some people just call them topological field theories. All right, so let me remind you the definition. So instead of working on a cylinder, now you work on any co-boardism like manifold with boundary, maybe from Y0 to Y1. So this is a manifold W. So the dimension of W is D plus one and the dimension of YI is D. And then you wanna have some spaces associated to the D-dimensional manifolds and some maps associated to the co-boardisms. Okay, so yes, and they should satisfy some properties. They should compose well. Typically, these things are vector spaces or modules. More generally, you can talk about, let's see, a C-valued D plus one dimensional TQFT where C is a symmetric monoidal category. So a symmetric monoidal category is, well, it's a category where you have a monoid. Okay, so I'm not gonna give the precise definitions but you have basically a tensor product of objects and of morphisms and satisfy some properties. And also it's symmetric in the sense that you have isomorphisms between A tensor B and B tensor A. Okay, so the examples to keep in mind are just vector spaces over the field K or modules over ring R. And yes, another important category, symmetric monoidal category is co-boardisms in dimension D plus one where the objects are closed, maybe yes, closed oriented, let's say D manifolds and the tensor product is the disjoint union and yes, and the morphisms are co-boardisms. So in this language, a TQFT is a symmetric monoidal functor from C to, sorry, from co-boardisms to C. So it's exactly something like this and but it should preserve the monoidal structure. So if you have disjoint union then you should get the tensor product of disease and so on and some properties with regard to the morphisms as well. Okay, so I think Sorin will talk more about the exact definition. Let me say one thing. So in a TQFT you can also consider D plus one dimensional manifolds closed and then you can think of them as co-boardisms from the empty set to the empty set. So let's say that our category is vector spaces over a field. Then this gives you a map from K to K, Z of X and that's just an element in K. So basically to close D plus one dimensional manifolds you associate elements in your base field or base ring. And yes, what else? I mean, this is the cleanest definitions but in practice you have some variations of the notion of TQFT. So you have to be flexible with respect to the definition. So for example, you can talk about manifolds equipped with various things like base points and paths. So this is kind of what happens in Hegat flow homology. You have to, yes, you have to give a base point here and then maybe a path and that's when you can associate a map. You can also sometimes they should be equipped with spin C structures. So yeah, so it's not exactly from the co-boardism category but from a variation of that like manifolds equipped with something or a very interesting case is equipped with embedding. Embeddings may be in some fixed order manifold, let's say RD. So this is the most important case is that of knots inside R3. So knots are just one dimensional manifolds but they come with embeddings and then you can talk about TQFTs for knots where you not only have the co-boardism but everything is embedded in R3 and then times 01 and you associate maps, you associate let's say vector spaces to the knots and then maps to co-boardisms between the knots that are embedded in the cylinder. Okay, so now let me give some simple examples of TQFTs. Okay, let me start with the simplest case which is one plus one dimensional TQFTs. Ah yes, and these are the same as what's called Frobenius Algebras. They're basically determined by that, by an algebraic structure. So why is that? Well, there's only one connected closed one manifold namely the circle. So Z of S1 is some, let's say some vector space A and then you have maps. So you have a multiplication corresponding to this co-boardism. So this is mu from A tensor A to A. So that's why this is an algebra but you have some extra structures. So oh yes, you also have a unit corresponding to this. So this is a map from K to A which gives you a unit element in the algebra. Maybe we can call it eta. You also have a co-unit, so a map epsilon from A to K and then you have a co-multiplication given by this, yes, given by this kind of co-boardism. So going from here to here. So like A goes to A tensor A, let's call it delta. Okay, and they have to satisfy some properties and I'm not gonna say what these are but basically every co-boardism between one manifolds can be decomposed into pairs of pants and then these caps and cups. Okay, and they have to satisfy some properties so if you write down what it is this gives you the concept of a Frobenius algebra. All right, so actually what's an example of a Frobenius algebra? Here's a Frobenius algebra. H star of S2 which is F of, well let's say K of X so over a field X over X squared, X squared equals zero. X is the generator in degree two and then I can write down what these maps are so the multiplication is just the polynomial multiplication. Epsilon of one equals zero. The unit is the unit epsilon of X equals one. Delta of one is one tensor X plus X tensor one and delta of X is X tensor X. So you might recognize this from Jake's lectures when he constructed Havana homology he exactly used this, well this is kind of for the R naught and basically for every edge map when you have a gluing or a split and these are the maps that determine that. So some Frobenius, so this Frobenius algebra was used by Kovanov to basically gives Kovanov homology which ends up being some, well some TQFT for knots. So this gives a one plus one dimensional TQFT but it also gives one for knots in a more complicated way. So Jake explained how to do this. Yeah, so I'll start writing numbers on the board to make sure I cover all the lectures so this is lecture number two comes in play. Great, okay and yes, Kovanov Lee homology is also from a slightly different Frobenius algebra. All right, let me talk now, okay so another example it's invertible TQFTs. So this means that, so this is a particular case of TQFTs where the objects and morphisms are invertible under the tensor product. So okay, so another way to think about it is TQFTs form a symmetric monoidal category. You can tensor two of them together and get another one and there's a unit TQFT which just gives the vector space K to everything and the identity map to every cohortism and these are the ones that are invertible. But basically what this means is that kind of all the vector spaces has to be just K so yeah, so they're much simpler than the general case but they're still interesting. So for example in one plus one dimensions you get Frobenius algebras but now A has to be K and then you have, well you have this co-multiplication co-multiplication which goes from K to K times K which is K so basically delta is given by some it turns out to be, it has to be invertible so it's a unit in K and then epsilon is I think it's mu inverse. So they're basically determined by a unit in the field. So yeah, that's a C star if K is C. And okay, so these are easier but that means that we can actually study them and yes, in general it's hard to classify topological quantum field theories, they're very complicated especially when you start adding decorations and embeddings but invertible TQFTs can be studied with homotopy theory and they are related to, well to the category of co-boardisms in D plus one dimensions and they're also if you understand them they are related to basically classifying space of diffeomorphism groups of M and they are related to manifold bundles. So if you wanna classify bundles over the manifold where the fiber is some given manifold or diffeomorphic to a given manifold this is like maps to bediff and somehow you can understand them in terms of, well they're related to this category of co-boardisms and basically all of these ideas will be the focus of lecture series number three by Soren Galatius. Okay, so this will be the more algebraic topologic lecture series of the bunch. Okay, so as I said, the concept of topological quantum field theory came from physics and in fact, yes many of the interesting examples that you'll encounter come from physics so let me explain in what ways. So basically if you want more examples more interesting examples, they come from quantum field theories and they come in two different ways. So there are two kinds of TQFTs. First of all we have TQFTs of Schwartz type. Okay, basically which are based on path integrals and the typical example of this, well basically what this says is that at least for closed manifolds in the dimension d plus one the numbers that you're supposed to get are given by an integral over an infinite dimensional space of paths, the kind of thing that you study in quantum field theory. So the typical example is Schoen-Simons theory. This is a two plus one dimensional TQFT and let me not tell you what it does to two manifolds but let me tell you what it does to a three manifold that's closed. It associates some number, well then I guess I called it X. So Z of X is the integral over the space of all connections in some given bundle over X. Let's say maybe the trivial SU2 bundle of E to the two pi I k Schoen-Simons, the Schoen-Simons function over the connection and then D of A. So this is a, well this is an infinite dimensional space and this integral doesn't make too much mathematical sense. I mean, does work trying to make sense of it but roughly that's what it is from physics so this is something written down by Whitton but then mathematicians manage to give a definition of this invariant kind of figuring out what properties they should have. They gave a definition in terms of surgery descriptions of three manifolds and this basically gave what's called the Whitton Reshetikin to arrive in variance. To arrive in variance of the three manifold. So these are usually called WRT. They are the beginning of what's called quantum topology and yes, and I think Pavel Putrov will talk about them in lecture five. Yes, which starts this week as well. Let me just say one important thing. You can also do the, you can add what's called Wilson loops. So you can add in here the trace of the holonomy of A around some loop gamma. This is called inserting a Wilson loop and well let's say some, some not. And this way you can put knots in the picture and you can insert more and you can get a link. So basically you also have WRT for links in three manifolds and in particular for links in R3 what you get is exactly the Jones polynomial. Well if the bundle is the SU2 bundle or other words are the SLN polynomials that again Jake Rasmussen talked about. So these are some famous invariants of knots which were discussed in lecture number two. So yes, so one way to think of them is coming from this path integral. You're doing okay, I have halfway through, great. Okay, so these are what's called TQFTs of Schwartz types. So then the other way that TQFTs come from physics are TQFTs of Witten type and this happens in the presence of supersymmetry. So basically some supersymmetric quantum field theories. Not all of them, you need some property. You can do something that's called topological twist and get some TQFT. Yeah, so I will give, this is what I talk about in the rest of the lecture. I'm not gonna explain much about physics but what I wanna do is, well just say what some buzzwords mean like supersymmetry and things like that and then I will give some examples, the ones that are of interest to mathematicians. So yes, okay, so let me mention some features of this kind of theories. First of all there, so they come from TQFTs so there's some quantum system involving some H bar and then when you set H bar equals zero you get the classical limit. So that's classical field theory which is usually much easier. It's in terms of differential equations. Now the topological twist, the TQFT, doesn't actually involve H bar. So you can actually read it kind of from the classical limit and everything is gonna determine. So the TQFTs of Witton type, yeah so the TQFTs are kind of determined typically by solutions to some partial differential equations and the typical examples I'm gonna come back to this in a moment are Cyborg-Witton or Yang-Mills equations. Yes, let me say okay what is supersymmetry? Okay, usually abbreviated as Susie. Well in physics you have two types of fields. You have bosons and fermions. Okay, so the typical boson is the photon and the typical fermions are the electron or the proton or the neutron. Okay, but mathematically, well okay, let me just say it like this. So they're all fields, they're sections of some bundle. This have integer spin so you should think of them or for example vector fields, just elements of Tx, just sections of Tx or scalar fields, that's another example. And these ones have half integer spin. So the typical examples are spinners and I think you've seen spinners in Haydish lectures. Right, I mean there's many theories of this type. The point is fields come into two out of these two categories and some theories involve both fields and then there is a symmetry between the fields and that's called supersymmetry. So basically when you have a theory of this type that exchanges the two, that's called the supersymmetry and I guess let me also mention the notion of supercharge. This is a generator of the supersymmetry group. Okay, so you have a supersymmetry group and yes, it typically is determined by a finite number of superchargers of the succinct transformations. Okay, another word you will hear in physics talks. I'm sorry for physicists in the audience. I mean, I know this is kind of very low level for you but yes, I think mathematicians, as a mathematician I had, it took me a while to figure out what these things mean. Okay, so there's a notion of BPS state, which you will encounter over and over again. So this stands for Bogomolny, Prasad and Somerfa. And what it is, it's a state in this kind of theory, supersymmetric theories, it's a state, I mean, yes, so a quantum physical theory is usually in terms of states that the theory can be and there are operators acting on them, so the superchargers are one of them. So these are fields that are annihilated by the supercharge. But okay, for mathematicians this might not mean much. The way to think about them is that they are given by solutions to a first order partial differential equations and they minimize some energy. And actually, all right, so let me maybe give an example. So there is Yang-Mills theory, which I'll come back to this, where do you have a connection, well, it's in terms of connections and the Yang-Mills equation says that D star of the curvature of the connection is zero. But in mathematics, people study something more specific, which is the ASD equation, anti-self dual. This is a particular case of solutions and that's just that star FA equals minus FA. So.