Categorification of Fourier Theory

so uh what I'd like to do is talk about first about the idea of categoric Asian so let me borrow a term from my colleague Dan freed and talk about category number as a loose measure of the amount of abstraction that is involved in a particular mathematical idea mathematical theorem mathematical construction category number zero refers to the most concrete type of mathematics so the mathematical objects that you would be interesting at this level are things like numbers and the kinds of theorems that I would call category number zero theorems are theorems that are statements about numbers like a typical theorem would say that two numbers are equal to one another so saying that that is category number zero is not to say that it can't be a very interesting or deep statement many of the most interesting statements in mathematics are theorems of the form two numbers are equal to each other it's just saying that this is a statement which has a very concrete meaning let me distinguish that from category number one where the objects of interest are not concrete things like numbers but there's slightly more abstract things like mathematical structures so there's all types of structures that we study in mathematics sets what you might think of as having no structure at all things like groups vector spaces etc and the kind of statements that you're interested in at this level are slightly different a typical theorem would not be that two numbers are equal to each other but you might want to say that two structures are the same and here you would say not that they are equal but that they're isomorphic and the way that you give proofs of theorems at this level have a little bit of a different character if you want to prove that two mathematical structures are the same it's not a yes/no question it's typically something that you prove by providing a construction a construction of an isomorphism between two groups or two vector spaces or something like that so I want to talk about going one step higher category number two the objects of interest are not individual mathematical structures but categories meaning things classes of mathematical structures and the typical theorem that you would consider at this level is a theorem of the form two categories are equivalent so I want to emphasize that if you're not familiar with the idea of a category well you might still be familiar with mathematics that happens at this level so for example when we teach our students linear algebra we typically first teach them linear algebra it's a subject which is about matrices which are these you know grids of numbers we teach them how to multiply matrices and do all sorts of calculations with them and then later we teach them a more abstract point of view that linear algebra is about vector spaces and linear transformations and things like that and hopefully we teach them that actually these two languages are equivalent there's just two ways of describing the same underlying structure so that's an example of an equivalence of categories and so typical theorems at this level are theorems that tell you that there are two different ways of looking at a same mathematical structure or there's two different mathematical structures that somehow are essentially the same so I'm going to talk in this lecture about the idea of category fication which is a term for taking some statement that you're interested in which happens at one of these levels and then studying a related statement that happens at a higher level so rather than give you some simple examples to show you what I mean I think I just want to take this entire lecture and devote it to one extended example and the example that I want to start with is the Fourier transform so let me remind you what the Fourier transform is if you start with a function f from the real numbers to the complex numbers then you can associate to it its Fourier transform which is given by the integral of f of X e to the 2 pi X Y DX so the Fourier transform is a construction it takes one function of a single real variable and produces another function of a single real variable and this is a construction which I would say happens at category level zero it's about very concrete objects a function is not just a number but it's it's a collection of numbers it's some fairy sort of concrete mathematical object and this is a concrete construction which takes one such object to another one now one of the things reasons that the Fourier transform is useful is that under appropriate hypotheses F and F hat are equivalent pieces of data you can translate questions about ask two questions about f hat and often the translation procedure makes those questions much easier to answer so let me describe a sort of proto theorem about the Fourier transform is that Fourier transform gives you a bijection functions from R to the complex numbers two functions from R to the complex numbers so I'm calling this a proto theorem because as it stands well if you interpret this literally it's false it's a proto three room because it's a sort of loose description it's an informal statement of a class of theorems that you might want to prove so there are various theorems that instantiate this proto pyramid so for example an actual theorem along these lines would be well you attach some precise meeting to what you mean by function what kind of functions do you want to consider well one of the nicest statements is to consider square integrable functions where integral functions on are isomorphic to square integral functions on R via the construction that takes F to it's Fourier transform so this is now a statement that I would say is happening at category level one it's a statement that two structures are isomorphic to each other in this case it's the vector space of square integrable functions on R is isomorphic to itself via a particular construction so I don't want this talk to be about the details of instantiating the proto theorems so if you like I'm about to switch to talking about the discrete Fourier transform and for finite groups and then it's then it's literally true but this is I'm just going to talk at the level of protocol so this Fourier transform is some construction it takes concrete things like functions produces other concrete things like functions and a category level one a statement about it is that it's giving you an equivalence between those two types of data and I think it is worth pointing out that well if you want to describe the construction you don't have to worry too much about exactly what you mean by function I you know function you can take a sort of eighteenth-century point of view a function is something I know it when I see it here's a formula for it but if you want to prove an example of a theorem of this type then you have to say exactly what do you mean by a function or maybe it doesn't mean function at all maybe it doesn't quite mean function in the usual sense to make this statement about l2 spaces but this is something which is typical I think of what happens when you move from one level to the next level it forces you to think through questions like that okay so this Fourier transform is sort of a story which is happening between these levels zero and one and I would like to talk about category fiying it that is finding analogous phenomena which happen at higher levels so let me start by mentioning one of the properties that the Fourier transform has which makes it so useful so let's suppose that you have a function from the real numbers to the complex numbers then you can always and you have some real number T then you can always take your function f and translate it by T so if T is a real number let's define f sub T of X to be f of X plus T now this is another function from the real numbers to the complex numbers and you can contemplate its Fourier transform and well you can just read off directly from this formula how that's going to be related to the Fourier transform of the original function f this is the Fourier transform of F times something times some simple function that depends on Y so this is in some sense the defining property that the Fourier transform has and the one of the things that makes it so useful is that it converts the action of translation on functions from R to the complex numbers to the action of multiplication by some functions that we understand very well or another way of saying that is if you think of functions from the real numbers to the complex numbers let me now specifically say I'm thinking of l2 functions this is a representation of the group of real numbers this group acts on l2 functions by translations via this procedure here and the trend that group of translations is an abelian group and therefore you would expect that you ought to be able to in some sense simultaneously diagonalize all of these linear transformations and this is in some sense what this Fourier transform does for you it convert it gives you some sort of something you can think of roughly as a basis for l2 functions on R in which this translation action is just diagonalized for you so this is something that you could contemplate doing for any group if you have any group G you can talk about functions on G so well you could talk and then you can study the action of G on that space of functions by translations and you might try to simultaneously diagonalize all of those translation operations so let me be a little more specific specialized to the case where this works particularly nicely so let's assume that G is compact and abelian so some examples would be G a finite group or G being the circle per pound so what those are particularly nice assumptions because well if you have a compact the compactness of G guarantees you that any representation of G can be broken up as a direct sum of irreducible representations and the commutativity of G guarantees you that all of the irreducible representations are one-dimensional so any representation of G can be broken up into one-dimensional bits and in particular you can study what that does for the to the space of l2 functions on G itself so let me introduce a notation so G check is going to mean the dual group of G so this is the collection of characters of G so an element of G check is a function from G to the unit complex numbers a function lambda which converts the multiplication in your group G to multiplication of complex numbers so why is that an interesting thing from this point of view well if you have any function yeah from G to the complex numbers suppose that it if you think of it as sitting inside this vector space of all functions on G if it were a simultaneous eigen function for all of these translation operators then it would have to be a scalar multiple of one of these characters lambda in other words if you knew if for all elements g and g you knew that there was a formula which said f of g h is f of h times some scalar that depends on g then well what does this tell you it tells you first of all that F of G is lambda of G up to a scalar times f of 0 so it's a scalar multiple of whatever this function lambda is and secondly as long as F is not zero that'll tell you that this lambda has to be one of these characters so the Fourier transform or Fourier series in a more abstract form or a more general form for a general group G says something like that the space of functions on G breaks up as a direct sum of one dimensional representations which are indexed by this set of characters in other word in other words every function on G can in the suitable sense be written as some linear combination of these characters and writing the function as this linear combination that's expanding a function in its Fourier series or at least that's what this means when the group G is s1 which is a perfectly respectable case to think about for everything that I'm going to say from now on so but this is something you can contemplate not just for the vector space of l2 functions on G you can contemplate this for any representation of G so suppose that G acts on a vector space V well what I said earlier tells you these assumptions on G allow you to break V up into a direct sum of one dimensional representations each of which is characterized is can just be described by a character so that decomposition is not quite canonical for example if your group G was trivial it's like choosing a basis for V as a vector space but what is canonical is what's called the isotopic decomposition so in the situation where you have your group G acting on a vector space V well for each character lambda you can look at what I'll denote by V lambda which is the set of the collection of those vectors such that for all G in your group G on V is just a scalar multiple of V where the scalar is given by lambda so this is a subspace of the vector space V that you started with and a sort of proto theorem well is that V can be recovered as the sum of these subspaces V lambda or let me state this more precisely as a or sorry less precisely as is sort of proto theorem that you can contemplate in this context which is that the construction that takes a representation of G and assigns to it these subspaces gives you an equivalence between representations of G and families of vector spaces parametrized by gene duel so it I want to think of this proto theorem as an analogue of the previous proto theorem so the previous part of theorem was about functions on G it was telling you any function on G can be well I stated it earlier for R instead of a general group G but it was telling you that any function could be written in some sense as a linear combination of characters so this is a statement not about functions on G but about representations of G and it's saying that any representation can be written as some sort of combination of representations that we very understand very well which come from the characters of G so I would like to think of this as analogous to the previous theorem but it's not just analogous to the previous theorem it's also closely related to the previous theorem because if you take this proto theorem and you wonder what happens when I apply this procedure to my favorite representation of G like the representation of G on this space of functions well what pops out is the theory of for the previous construction of the Fourier transform or a Fourier series Fourier series are what you get when you apply the ISO tippet decomposition to the space of l2 functions on G so this proto theorem I would describe as a category fication of the previous one this is a statement at a higher categorical level the construction is one step more abstract than the previous construction we're not functions to functions now we're taking mathematical structures on one side we have representations of G vector spaces with some additional data and on the other side we have families of vector spaces indexed by the dual group again essentially vector spaces with some additional structure so that's a slightly more abstract setting and the theorem the proto theorem is also at a more abstract level it's not a theorem that two vector spaces are isomorphic it's now a theorem that two categories are equivalent so I would like in the rest of the talk to do this one more time I would like to describe for you a theorem which is a category of this prototype so in order to do that I want to look at this right-hand side and think of it as analogous to something that we saw before so I want to think of these objects families of vector space is parametrized by G check as a kind of as a analogous to functions on the group G dual so I want to think of these as functions from G dual well not complex valued functions on G dual we're not assigning numbers to every element of this group instead we're assigning vector spaces to every element of this group functions from G dual into vector spaces so let's say we had such a thing it's just a collection of vector spaces indexed by G dual and now just like with functions there's an action of G dual buy translations so if we had some other element mu in G dual then we can take one of these families of vector spaces and just translate it by mu so let me define what let me define Mew of this should be another object of the same type it's another collection of vector spaces indexed by the dual group but now I'm just going to take each of these characters and multiply it by me first so this is just precisely analogous to earlier how we thought of in the group G acting on complex valued functions by translations so this construction meaning the construction that I just described gives you an action of G dual on well let's say the kind of mathematical objects that we're thinking about here families of vector spaces parameterize by G dual so I want to think this is saying that this these things here form a representation of G dual but no it's a representation in a different sense G dual is not acting on a vector space instead it's acting on a category it's acting on a class of mathematical structures whose individual members are like vector spaces or they're like vector spaces with some additional bells and whistles attached so this is something that you can contemplate more generally just like earlier we started by thinking about the first representation that I mentioned in this talk was the regular representation any group G acts on the collection of complex valued functions on G well this is kind of like the regular representation of G dual but it's not a complex valued function that we're thinking about but a vector space valued function that we're thinking about so this is like the regular representation in well in this new context but now instead of thinking about the regular representation let's think about an arbitrary representation so more generally one can consider representations of this group G dual on any category C so if you don't know what a category is what I'm talking about roughly is a category is something which is a collection of mathematical structures and saying that we have a representation of G check on this category C means that we have a rule which given any structure of the type that we're contemplating any object of the category C we can translate it by any element of G check to obtain a new structure of the same type so I've now erased my formula for how you do this with vector spaces which are parametrized by G but that's a typical example and maybe one of the simplest so I want to think of representations of G CH dual on categories as analogous to representations of G on vector spaces it's it's a category fication of that idea so if you have a group G which is acting on a vector space there's all sorts of things that you can do to it so one thing that you can do is look at the space of invariant vectors so V superscript G is defined to be the collection of vectors in your vector space V such that G V is equal to V for every element of your group G and this is a subspace of the vector space V it's the largest subspace where on which the group G is acting trivially it's the subspace of invariant vectors so this is one of the first things that you might ask if someone gives you a representation of G you might ask alright what are the invariant vectors are there anything so there's an analogous construction that you can do when you have a representation of a group on a category so in this case you can consider in variant objects see so by analogy here I'll denote the invariant objects by writing a superscript with the where that's the group that's acting and what are these well these should be objects of your category on which which are fixed by the action of this group so you want each of these objects when you act by an element of your group to be carried back to itself but now we're talking about categories that's no longer a condition it's a structure that you should contemplate so objects of Si together with an isomorphism lambda C with C for all lambda in your group so this isomorphism let me denote it by alpha lambda and these ice morphisms should themselves satisfy a condition sorry should be compatible with multiplication so this is a definition let me not worry too much about spelling it out but I want to emphasize something that's different between this construction here and the construction below that it's based on so if you have a group acting on a vector space and you look at the invariant vectors you make something simpler than your original representation this V superscript G it's a subspace of V and it's something that you think of as being smaller and easier to understand in this case that doesn't necessarily happen so let me give you an example let's suppose that you take C to be the category of vector spaces and I want to let this group act on it and I just want to take this to be the trivial action meaning if you have some element lambda the way it acts on a vector space V is just by taking it back to the same vector space beat without changing anything so then what do you get when you apply this construction well now you want to be talking about objects of your category in this case that means a vector spaces together with an isomorphism of lambda of your vector space with your vector space so what we are considering here our vector space is V together with plus a isomorphisms alpha lambda of V well here I said with lambda of V or lambda of C but I'm considering the trivial action where lambda doesn't do anything so these are isomorphisms of V with itself satisfying the condition that they the formation of alpha is compatible with products so what is this well the data of a vector space V together with those isomorphisms that's just what we mean when we touch that's a representation of G check so that the category of vector space is that's something that we think of is very simple something that we understand very well and when we apply this construction of taking invariance to it we get something which possibly we don't understand very well we get the category of representations of gee check which we think of as more complicated than the category of vector spaces so here we thought well the cat the invariant vector sit inside of V in this case well there's a there's a map from representations of G to vector spaces where you just take the underlying vector space and forget the action of G but that's no longer something that you think of as behaving like an inclusion it's something that in some sense forgets everything that's interesting if I give you a representation of G you're probably not so interested in what its underlying vector space looks like after all vector spaces the only interesting question to ask about them is what dimension they have what you're interested in is not in the vector space V but in what these isomorphisms are that's where the real meat of the representation lies habit r of that is that if you have a representation you can often form you can often find many representations that have the same underlying vector space if you have data like this you can often change you can often find a new representation that has the underlying vector same underlying vector space V but with different isomorphisms alpha lambda so let me return for a moment to the general case where we're talking about a category C which has an action of the group G check and let's suppose then I give you an invariant for this action so what does that mean well remember that means that we have an object of the category C and we have isomorphisms between it and all of its translates so then you can try to form let's try to modify this invariant object without changing the underlying object of C only changing these isomorphisms you can try to form a new object let's call it C Prime inside the invariance where the under I'm going to take the same underlying object of C but instead of using the isomorphisms alpha lambda I'm going to replace alpha lambda by alpha lambda times some scalar and that scalar well I don't have to use the same scalar for all lambda maybe that scalar will depend on lambda so question when does this make sense in other words when does this construction just multiplying all these ice and Wharf isms by a scaler give you a new element in this category of fix points well there's a condition that these alphas have to satisfy which I wrote down over here and that condition places a constraint on how these scalars should behave the answer is this makes sense whenever C lambda lambda prime is C lambda times C lambda Prime so for example if C lambda was given by evaluating lambda on G for some element of your original root G so what have I just described well just describe for you a construction which does the following it takes as input categories see acted on my G dual so this is like representations of G dual but again not on vector space is now on categories and it produces via the construction of taking c2 the invariant part produces for you a new category but that new category has some additional structure given any object you can rescale these isomorphisms to produce another one what you have one such construction for every element of your original group G this produces categories acted on by the group G and now let me state for you a sort of proto theorem about these this construction so this construction let's say it just say is an equivalence construction is invertible it's essentially its own inverse you get a category acted on by G if you do the same thing but taking G invariance you can produce a category acted on by G dual and to the sense it's essentially the category that you started with once again this is a proto theorem and in order to turn it into an actual theorem I would have to be very specific about what kind of groups I'm thinking about what kind of categories I'm thinking about and so forth but instead of doing that I have three minutes left according to my clock I just want to emphasize the analogy that I sketched for you I just want to bring out the sense in which the three constructions that I describe for you each one at a higher level of abstraction than the previous one are in some sense the same or they're all avatars of the same idea so let me remind you the first thing I talked about was the Fourier transform or Fourier series which took functions on G and identified them with functions on G check for functions from G to the complex numbers and related them to functions from G check to the complex numbers and the second thing I talked about was the isotopic decomposition sorry my chalk is and here you were talking about representations of G on vector spaces and on the other side you had functions from G check in to vector spaces this third most abstract thing that I described well we were now still thinking about representations of G but now on categories and I'm saying that those are the same thing as representations of G check on categories so first I want to bring out an analogy between passage from this line to this line and passage from this line to this line so on the right here you're thinking about functions on G check on top you're thinking about complex valued functions here you're thinking about vector space valued functions on the other side the replacement that matches with that is replacing thinking about functions with thinking about representations now passage to this line is based on the same procedure you replace thinking about functions from G check to vector spaces to actions of G check on categories replacing functions by representations and here well in some sense you don't change anything you're you were talking about representations before you're still talking about representations so there's a strong analogy between first these three statements and also how you pass from each one to the next and once again these are not just related by analogy they're related in a more precise way is that you get each of the stories above by taking the story below and applying it in the simplest example if you want the theory of Fourier series well you apply the I so typically composition to the representation given by the regular representation of G the representation on l2 of G if you want the theory of the I so typically composition you apply this to the regular representation of G on categories which matches on the other side with the representation of G check on acting Tribune the category of vector spaces all right I out of time so I will stop there so one way to so for every element of G you associate equivalence of the category with itself for every pair of elements you give an isomorphism of the equivalence associated to G times H with the composition of the equivalence is associated to G and H individually and forever and there's a diagram that needs to commute involving three tuples he lit some on the basement the first and third lines look more similar than one because the change happens with well here let me make this these other lines look more analogous look let me say it in a different way so the ISO typic decomposition describes a relationship functions from VG into the vector spaces two functions from G check in to vector spaces and down here we're relating functions from DG two categories two functions from PG check two categories so no dmg check have a map seem to see star you know by definition and eg cross gchat maps into the category of vector spaces as a sort of D looping there's a given a an element of G check there's a corresponding one dimensional representation of G and easy what's that they recover Oh recall what BG is so BG means eg is a space satisfying this was an answer for mathematicians not for its the BG of topology yeah yeah so maybe in this context it would be easiest to think of it it's a I mean it as a category it's the category with one object and the autumn or isn't room for that object is G okay so a functor from that BG into the category of vector spaces is a replication

Channel: Harvard Math

Views: 56,194

Rating: 4.914989 out of 5

Keywords: Fourier Theory, Category theory, Categorification

Id: w3f8KEcv4RE

Length: 47min 17sec (2837 seconds)

Published: Tue Apr 28 2015

Reddit Comments

[u/kfgauss]

He's really a first-rate speaker. I'd bet it's a colloquium talk, which is a hard thing to do well, and this is a great one.

Edit: having finished watching, maybe not even a colloquium - I'm impressed he did such a good job talking about categorification without going beyond advanced undergrad/intro grad school material.