 The morning session is Professor Heilong Dao from Kansas University of Kansas and the title is Categorical Approaches to Singularities. So please. I just want to make sure you can hear, everyone can hear me loud enough. Well, so first of all, I somehow it doesn't seem to, first of all, I would thank the organizers of the conference for this wonderful opportunity. I mean, it was very inspiring to hear about the past, the work, the transformational work by Mellon Craig, but also to observe the present and the future. It's great to see many wonderful young researchers here from all over the world. So I think I would start with some maybe non-mathematical comments about not just Mellon Craig and Mellon Craig for example. I will explain it, but oh, okay, okay. So what is CIE? Community Antebra was GG, yeah, great, golden generation. So Community Antebra has been blessed as we all know with, so Community Antebra as we all know and appreciate have been blessed with a wonderful pioneers class of people who are great at everything, who are a fantastic researcher, teacher, but also very good at community building. And I think it's one big part of our success nowadays. Unfortunately many of them are going into retirement already, retired, and we even lost a couple of people recently, like Professor Goto. So I want to somehow this thing just either way to do it, oh, okay, I can do it manually. So I want you to take some minutes to talk about maybe a little bit about the future of the field, how to best learn from what they did and preserve the legacy that they have built. And here I'm just not talking about Mellon Craig, but of course David, Lucho, Keiji, and many other people, maybe a slightly younger who are present in this audience. So what can we learn from them to highlight a few things? And many of these things I think are very happy to report that people are already the new generation already doing, but I do want to mention them so that we can all learn from them together. So the first thing I think we can learn from the people is try to be brave. It's not just about trying to prove some big conjecture or some super hard problem. Everyone can be a little brave and try to go just a little out of comfort zones. In fact, the project that I'm going to describe to you next will be something like that, so we can all, it doesn't have to be something big, but we can all try to push about real a little bit of what we know, learn something new. And being brave or so I think, very beneficial to the field to make conjecture. I used to joke that you should make conjecture when you're young because you have no reputation to lose. But of course it can help the field to make conjectures, so don't be shy to advertise what you think is interesting. The next lesson I think we can all learn from these people, this generation is to be brought and I think that's very important for the future of the field as well. So one thing is quite interesting when you look at the history, you don't know what will become very popular. And so if you look at Stanley Risner's theory, which Mel didn't even write out, he didn't even publish in it, but it becomes now such an industry, it spans thousands of papers and hundreds of wonderful careers. Matrix factorization, I think when David wrote it, I don't think he has application in mathematical physics in mind, but it turned out to be the most cited number of his papers come from mass physics. Symbolic powers against something that I think Craig worked on very early in 1970, have seen such an explosion of activities. So you never know what will become popular, so be broad, try to broaden your horizon. You will never know what may become great 50 years from now. Yeah, and it's always good to communicate with the wider community, not just in our field, but in terms of how interesting and friendly we are. Whenever I travel, I've been told by people from outside that we actually have a reputation for being one of the very nice community, I think that's a great advantage and we should try to keep that and not, again, not shy in advertising that. And be able to make new connections to other areas of mathematics, I guess I already touched on that, but I think community of algebra and maybe many young people didn't realize it, but we actually have a great advantage because it's easily connected to many other areas of mathematics, whether it's algebraic geometry or combinatorics or algebraic topology, you've seen some of that in the talks. Okay, and the last, well, the third thing I think David mentioned yesterday is one lucky thing we had from the top people in the field that they are very nice and so we should keep doing that, be nice to, especially to young people, I guess you should be kind to old people too, but they, you know, in some sense they can take care of themselves better. You know, mathematics is comparative, it's hard and I think young people need a lot of support. And then, yeah, don't, if you're a senior, generally chair, good idea, question, career advice. Actually career advice are very good for young people. I think a lot of people don't get enough career advice from there, from there, sometime from the advisors. Be social, you know, try to talk to people, yeah, you know, be nice to people even though they may not understand what you say or even if they forget to cite your papers or thereby they hate you, you know, still be nice to them. In the end, the future of the field is bigger than anyone, any person of feeling. And if you're young, don't be shy about approaching senior people, go talk to them. Many senior people may seem, you may think that they are very busy and don't have time for you, but I think most of them are lonely and can't wait to hear about your new theorems. Just don't overdo it. All right, so, and then, I guess, work hard, yes, this is a, as I talk, a photo, and this is actually the only photo of both Craig and Mel and Ray Hyman. I was a fourth in this. And I've been feeling very lucky to be able to develop my career in contact with this gentleman. So anyway, let me switch now to, I'm not sure if I need to turn this off or something. So let me switch now to the math, and today I want to talk to you about how that algorithm approach to singularity, and we're trying to make this, perhaps this is somewhat a new topic to quite a few people in this audience. So I'll go slowly, so the first part is on the background, and the second part will be on some new work on sub-function of X, and what we call a split ring. And this join with the new result, and join with Mona Lisa Dutta, Suvik is going for a postdoc in Prague, both a fantastic student that did most of the work in here, but I want to talk about some background first, singularities. I'm using too much of course. So the talk is about categorical approaches to singularity. So first let me remind you of what singularities are, and of course, this is something that the commutative algebra is very familiar. So we've learned since graduate school, this hierarchy of a singularity. So we have hyper-regular ring, and then hyper-surface, which are regular ring mode R1 non-zero divisor, and then you have complete in-section, Gorenstein, Cohen McCauley, right? And these are things that we are all very familiar with. Perhaps slightly less familiar, but you've seen it in some of the talk. Are the singularities that arrive from the minimum model program, and that leave sort of related but quite different in nature. So we have things like rational, canonical, log version of this, and so on. As you saw in the talk of Mateo, and you've seen, I guess, log-terminal singularity in the talk by Porto Sato. So we also have this characteristic key singularity, which is very much the Hoxler-Hunicke style of mathematics that we've seen a lot. So this singularity defined characteristic key that use high closure technique. So we have something like f-rational, f-regular, and then let's go on and on. Still being added as we speak. And the amazing thing is that over the course of many years, the work of many fantastic people, we have this, we now establish fairly tight connection between the these two classes of singularity. Of course, Miao and Craig play a part, but Karen Smith and the Japanese school of community of Anjabra, many other people. So this connection has been a huge part of community of Anjabra that related to Anjabraic geometry. By the way, please stop me anytime if you. So categories, and for this talk, categories are not very technical. We just, it's more like a philosophy, right? So in category theory, we look at a category which is usually consists of a set of objects. And then we have the set of morphism. So maps between objects, plus some axiom. And the focus really about trying to understand the map between objects. The object itself may not be that important, but the map between them are the really what make it interesting. And for example, you probably have seen this, right? So the definition of projective module is whenever you have an M and you have a subjection from N to M, then you can always leave it to a map from P to N so that, so this is, you can define it for module, but the point is that once you have this definition, you don't need to talk about module, you just need to talk about letter and arrows, right? And that, the, I think, the power of category theory. Okay, so the goal of this part of my research is to, and of course, other people, but is to establish connection between singularities, so our nice singularity, perhaps this appear on the left hand side to, to nice property of related categories. And, you know, this, in this talk, I will mostly focus on subcategory of mod R. Of course, you can play this game with a lot more fancy and technical subjections as that. You saw it in Tony's talk. The derived category can happen a lot this nowadays. And perhaps after this talk, you can go back and watch Tony's talk and get more appreciation for what he's doing. This is a lot more technical, but for subcategory of mod R, I love it because this retains sort of the sort of in notion and naive feature, right? Because very concrete, you just talk about module. And I want to convince you that it's still interesting enough and perhaps that's a lot of work to do. So I will start with a very simple warm up, warm up question, or warm up perhaps a resolve, which is the simplest case possible. So R, M, K is a local ring. And I'm going to assume that the dimension of R is zero. So it's a Athenian case. And so we're going to try to start with this familiar. We're going to try to take this familiar hierarchy of singularity and try to put some categorical spins on it, on them, right? So how do you describe regular ring? You know, regular local ring of dimension zero, right? Of course that just means that R is a field, right? And studying module category of R is just studying linear algebra, which is actually quite a big subject. But the key thing is that, of course, all finally generated, well, all modules are, all finally generated modules are free. So they are direct sum of K. Of course, this K is also equal to projective. Sometimes I prefer to use this because it's also a non-local setting and projective is more easily described categorically. But I make no difference, of course. It's good to keep that in mind. So hyper-service, anyone know? Guarantine, so okay, so regular is obvious. The rest of the list is not so obvious. And of course there are no unique answer either. Maybe the easiest one is going to call it, right? So of course everything is going to call in dimension zero. How about Guarantine? The one way to describe Guarantine ring is to require that all modules are reflexive. And so reflexive modules are subcategories of mod R. And if they coincide, in this case, then it's precisely that I, Guarantine. How about hyper-service? Can you distinguish Guarantine ring from this guy? So of course there are no unique way to describe it. But you can describe abstract hyper-surface dimension zero as ring over any module is a self-dual, right? Or equivalently every reflexive module is self-dual. I guess this is, I should say. So Guarantine everything is reflexive. Hyper-surface equivalent to anything in mod R, not just reflexive but self-dual. Of course if it's self-dual then it's reflexive, right? Because if M is isomorphic to M-dual then it's, I think it's a cute characterization of hyper-surface that maybe you have not seen before. It's easy to do. I don't know a good characterization in this case, completing section. Of course, like David mentioned yesterday, you can say that the cutting number of K has polynomial growth. And that's a perfectly fine and interesting characterization. But it doesn't have the same feel as it's more homological. So there's no, of course, strict definition of what it means by category. But I hope by this example you've got a feel for it. Any question at this point? Craig was begging for a question and he never got any. Okay, so now let's look at dimension in any dimension. And so there's really, we have some amazing triumph over the year. Of course the first theorem is that if R is regular then it's equivalent to the partial dimension of anything is less than or equal to b. And so of course you can write in a way that will become more relevant later that the category of DCG of mod R is free or projective. This is a classical theorem by Oslo and the books by Amser. And then we have the results that say that by books by and crow and other people. I think it's built over the work of many years. But so this ad singularity, so this is a hypersurface of ad type equations. Of course are equivalent to, and here you will need some assumptions. So this result sometime need extra assumption. So if for the sake of time, I will sometime forget them. Please accept my apology. Even only if it is a guaranteeing no more plus the category of CMR has finite type, okay? Finity here just means that there are only finally many in decomposable objects, all right? So this is a very classical, very classical. Now let me move on. So maybe this is something that you may not have seen so much. So if r has two dimension rational singularity. So the definition of rational singularity is something like this. If you have y over x is a resolution of singularity. So you want to, so x in x has rational. I just run it here to sort of give you some reference. This is not going to be if the, right? So that higher direct image function to vanish at the point x or higher, all right? So again all of this singularity here when you of the minimum model program you have to use resolution of singularity, define them. Now, but now I want to give you a very communicative algebra is very actually simple characterization of rational singularity. And that is the following. So I guess here I should say as normal. And you have to assume the first classic zero, I think, but anyway. So one way to categorically define rational singularity is to say that the category of third CSG of any module is a finite time. This is a work, this is again a result of work by many people. So, Artyn, Vedya, Wundram, Enor, Vivek, and Yama and Wims. Perhaps the last people who put a definite result in on paper. But it's built on work by many people. Perhaps even start with Liebman where he proved that the class group is finite. It's a characterization of rational singularity. And in fact, you can for dimension, at least bigger than three, you can show in a joy in work with Takahashi and Vya that if n mod r is a finite type for any n bigger than equal to d, then r has rational singularity. I need, sorry. Okay, so I hope you will get a sense that these things are quite interesting. And, yeah, the other one is just no, yeah. So, singularity in high dimension are much more complicated. There's no such elegant, well, okay. Maybe if you're willing to look at certain special subcaterials, there actually might be. But it's get very complicated in dimension three already. I think you can write down some sort of category that characterized dimension three singularity. It will look a little technical and ad hoc. Yeah, so I hope I convinced you that this is something worth looking at, right? So the advantage of this approach is that is usually the result look very elegant and I mean short, right? Also surprising and it doesn't, at least on the face of it, it doesn't refer to characteristic, right? So if you want, you're willing to use this to define singularity, you can easily use the definition. As more functorial, obviously with the module category, you can more easily do things like complete localized, cutting down hyperplane and it has a lot of connection with other area. So that's some of the advantage. The disadvantage is that sometimes it's not easy to prove such statements. And of course, as Mel himself has pointed out, not to me, but I like the quote that he wrote sometime that, in the end, community enterprise really about understanding equations. So in the end, if you want to study deeply, you need to somehow understand equation. And this may look a little far from that. And I think, of course, studying equation has a lot of advantages, much more concrete and much more, you can do a lot more powerful. Nevertheless, I think because of the reason that I wrote down over there, I think it's worth pursuing, at least until we run out of things to do. And there are a lot of things to do. So let me describe you some result, some recent result. Actually the one recent surprising result is on the topic of AppRing, which is a very classical object study by, well, App and Zarecki Liebman, many others. So here, App is a local coin Macaulay of dimension one. And I would assume the ring is infinite to make it simple. And so the definition of AppRing is that due to Liebman, I guess I need analytically unremifined to be safe here. Probably you don't need, but. So if Y over Z and X over Z, sorry. Y over X and Z over X belong to the integral closure, then Y Z over X is in, equivalently, so this one is concrete, but maybe a little strange. Equivalently, you can say it more every integral close, close and primary ideal is stable. Namely, you have reduction number one, I C for I squared for minimal reduction. So the theorem that I have recently is that in dimension one, AppRing's precisely the analog of the Goranstein property, sorry, the hyper surface property. Well, not precise because not everything is not, well. OK, we will see. So the following equivalent, I is half. Any reflexive M is self-dual. So one part of the, so it looks very similar to the hyper surface case there. Of course, one class of AppRing's a hyper surface of multiplicity two, but we have a very pleasant characterization. And in fact, for AppRing, so in the complete case, when R is complete, furthermore, if R is complete, I can describe all reflexive module in decomposable reflexive module, just one to one with infinitely what the near points of R. So what does this, so this is a local ring obtained from repeatedly blow up the Jaco-Sundradical, blow ups of Jaco-Sundradical. Every time you blow up the Jaco-Sundradical, you get a semi local ring. But in the complete case, it decompose as a product of local ring. Each of them will correspond to an decomposable reflexive module. So it ultimately related to resolution of singularity. Because in dimension one, if you just repeatedly blow up the Jaco-Sundradical, you get to the integral closure. So these are all sort of intermediate singularity between R and the resolution. And it leads to, so this leads to some question about the connection between R singularity and rational singularity. And Janoschkola told me that actually, they are true. But if you look at the rational singularity, oh, right. Sorry, I should say one more thing that also another corollary of this is that the category of reflexive module, which is also, in this case, COG of CMR as finite type. So our ring may be considered as an analog of rational singularity in dimension one. So I have about 10 minutes left. I will try to describe my work with most recent work with Morileza and Solvik. So part two is sub-function of ax and auric split rings. So of course, let me remind you of what auric modules mean. For this, I assume that the field is infinite. The maximum quantum corollary module over R is auric. The number of generator of M is equal to the multiplicity with respect to the maximum ideal. Of course, equivalently, this has a very concrete description, namely, therefore, some system of parameters is just isomorphic to a bunch of direct sum of residue field. That's the only place I need infinite field. So this has been a tremendous subject of study. Of course, Bernd started it. And he called it maximally generated maximum quantum corollary module, maximally generated maximum quantum corollary module, which proves this lemma that if you want something named after you, you give it a terrible first name. And maybe I should call this MGM CM split rings. But so there has been tremendous body of work on auric modules and ships and vector bundles. And it has many, many wonderful applications. Unfortunately, I don't have time to even mention any of it. But you can easily delete it from many sources. So people have studied. So for example, so let U of R, the auric module. And people have studied things like finite representation type of subcategory, both on algebraic and geometric setting. However, I want to, the motivation for a project come from the following. So when is every exact sequence in the category of auric module split? OK. That's our starting motivation. And well, again, if you come from representation theory, many of this idea from this area come from representation theory, because there's also people who don't have commutivity. So they focus a lot on categories. Because of lack of commutivity, you usually study ring by study module or functor over ring. So Auslander is a very strong presence in that. But yeah, so a category such as everything's an abelian category where every sequence split is called semi-simple in that sort of the motivation here. Now, this is not an abelian category. So technical problem, but also things that are actually interesting, that forces to do a bit more. When you look at CMR or the subcortical auric, these are not abelian. Of course, they sit inside a billion category of mod R. But if you take a map between two, maximal kremacoly modules, a kernel or a cocoon or may not be maximal kremacoly. So that's a serious problem. So you can't really use a theory of homological algebra for a billion category. That's one of the issue. Maybe let me just move here. So let me just give you the main result. And then I will say a few words about what go into the proof. But let me just show you the main result. So just k local kremacoly dimension equal to d. Oh, we say that, OK. We say that R is US, if it's auric speed, if every is exact sequence in all R bits. So if d equal to 0, any R is US. Because the auric module are just vector space. And so every sequence of vector space is split. So things start to get interesting in dimension higher. In dimension one, if in all, if the blow-up, the maximum ideal is regular, it's not very surprising for people who know. Because a set of auric modules are just, so you still have to prove something. But it's not some compatibility. But the set of auric modules are basically auric, sorry, just maximum kremacoly module over the blow-up, but the maximum ideal. Something that Ben pointed out a long time ago. And d equal to 2. And now things really, again, as things go. So here I assume that R is complete. I have to start somewhere normal. And then, and also, if the field is corrected with 0, actually, I just do C. And so R is US, auric speed, if and only if R is a cyclic quotient singularity. And the number of indie-composable objects in the auric module is at most 2. So example of this, for example, is a suffering of a ring of two-dimension, and many more, actually. But it's hard to give a complete description. I'm one minute over time, but I didn't have time to get to this part. So in order to even talk about this sensibly, we need to develop some sort of homological algebra over non-Abelian category, as you can see here. When we prove that, so one of the technical points is that I want to study a sequence of auric module. But if the two ends are auric, so for quantum collie module, if the two ends of a sequence are maximum quantum collie, then automatically you get the third one is quantum collie. So just lifting this from the Abelian category is fine. But for auric module, you have to assume that the third one is auric as well. And so you prove that the collection of such a sequence forms some sub-functor of the X, the usual X between, and that's one of the points in the proof. And that takes one more paper to develop it. And there we can recover some of the results by Tony and Jeanette. I think we also have some in the thesis. And we can give criterion for when a collection of exact sequence give you a sub-functor of X and use a sub-functor to detect regularity and all that. I don't have time for more, so. Thank you. So any questions, Jones? I didn't have any question as such, but I just wanted to thank you for the first 10 minutes of your talk. As a new graduate student, I found it very motivating. And I think I can speak for many of my colleagues as well. So thank you. You are the future, so I hope to see many of you, many more years to come around the world. Any questions or comments? Ma, you seem to. I think this is a very nice question. But can't you turn, for instance, I mean, Eisenberg's vectorization, or his term with Piva about structure of a complete intersection into a categorical characterization? Sorry, Eisenberg and Piva. Yeah, for instance. Oh, you mean like the higher matrix factorization. Yes, but first of all, there's a uniform bow. So one of the things that I draw a lot, that you see a lot, is that somehow there's this number. So when you take higher CG enough, then the category become nice. And for rational singularity, if you take third CG, anything, it will be a direct sum of, finally, many incomposable, that you know how to describe. Now, the category of matrix factorization, for a complete section, yes, you take any high CG, you get this special matrix factorization that David and Elena described. Unfortunately, there's no uniform bow for when you need how many stuff you need. And in fact, you can prove that there's no. So you cannot write like CG 1000 is equivalent to, yeah, so that's a subtle. So for the hypersurface case, that's a uniform bow because of matrix factorization, D plus one is a bow, right? But no such theorem exists in complete section. Any other questions? Yes. I think in general, if we don't have, I mean, if the category is not abelian, how we talk about exactness, I mean, ingenious. That's a beautiful question, and that's exactly so. Yes, so the key word is exact category. And it has a glorious story starting from, actually, you know, Booksbomb, I don't know the name here somewhere. Many people have worked on it, and in fact, Quillen used exact category to, it's a fundamental tool to define this higher homotopy group that earned him the Fuse Medal. So yeah, so the point is that you don't have abelian category, so you describe your category with subjects, and then you describe all the exact sequence. So the data given to you is a bunch of object and a bunch of structure sequence. That satisfies certain axiom, right? So exact category I come with, basically, well, you don't even write a sequence, but they call it kernel, co-kernel pair, because technically, on its own, it may not have exact sequence, right? So to say exact sequence, you have to have kernel, co-kernel. But anyway, so you can write out the axioms. And it took a page, actually, to write that, but that turned out to work in many cases. So, yeah, CMR, Ulrich, R is exact category. It's not obvious, it's not obvious, you need to prove it. But once you have that exact category, then you can import some tools. So people have studied them and they have some tool, but to translate into an efficient version, you need to do some extra work as well. So people have very general theorem about when this becomes sub-functor, but it's hard to check. So we prove a couple of results that say, give very concrete numerical condition for when become sub-functors. Thank you. Any other questions? So, oh, okay. If this category is not abelian, is it possible that it's strangulated? I mean. Ah, that's another good question. No, so this is in between. If you look, for example, if I go in steam and you look at the stable category, then it's a triangulated category. And it's closely related, but not quite the same. But exact category is the right keyword here. So, if there is no question anymore, then let's thank the speaker again.