 Vincent from Purdue University, we speak on one of the fantastic joint contributions of the orchestra and very clinically, tight closure. Thank you all. So I will talk about the work of Hoxster and Hewnecke mostly and how tight closure came about. And this tight closure is really the inspiration for this conference. And I thank Jugal Irma, just like Claudia, thank him. Jugal was really the main force behind this conference and more than year-long events in honor of Mel Hoxster and Craig Hewnecke. So thank you, Jugal. And we're getting more emails from him for further events and happenings. Yeah, so Craig is probably online and we really appreciate his presence, but Mel has not been able to log in. Instead, he is sending us this message and I will read it out loud. I would like to thank the organizers and the participants for listening. I count myself lucky to have had so many wonderful collaborators, colleagues, students, and mathematical friends. I wish everyone much joy in their pursuit of mathematics. This is a very Mel statement, and he has been a great inspiration to most of us here mathematically and also personally. And this is just another one of those evidences. So there is so much to tight closure. It's impossible to cover it in 60 minutes. So I will make some assumptions. I will not define very much. And most of the things that I'll talk about will be in characteristic P. Just it will make things simpler. So if I write some statement and I just say R is a ring, it's probably a ring of characteristic P, maybe I won't say it. That's standard notation there in tight closure. You've seen it many times this week, last week, and in your life. So here's the definition of tight closure. I won't read it out loud. But why would somebody, so, algebraists, we solve equations. That's what we do. But why would somebody make you solve infinitely many equations simultaneously? That's really hard. Well, it turns out that this gives a really useful notion. This notion didn't come out of nowhere, as has been pointed out many times, the work of Pesquina's Peron homological conjections in the 1960s, the proof of Hoxton and Roberts on rings of invariance of linearly reductive groups acting on polynomial rings, being co-McAuley. That also used similar methods. Heunica's proof of the, Heunica eto theorem and power, integral closures of powers of ideals also used some methods. And what I learned on Monday from Craig Heunica's talk, there was also a preprint of, never published preprint by Mel Hoxter of deep local rings that also use tight closure type proofs. So this definition, we have to solve infinitely many equations at the same time, has unified some of those old proofs. But it does actually a lot more. So that's the whole power that it was, didn't just write one proof for all those previous theorems, but it also gave really quick, elegant proofs to some of the new theorems. And then there has been lots of activity forever after. And I will touch on some of those things in the next hour. There were several announcements and published statements before this first paper in the journal of the American Mathematical Society. This one was published in 1990. I studied from the preprint. It was a thick preprint, not teched. So when this came out, I didn't really have to use it, but that was really a masterpiece. And it still is a masterpiece. I keep using this paper for reference for all sorts of things. But what was so beautiful about this paper, amazing about this paper, it's not only it gave these basic definitions and it gave basic properties of how tight-closure behaves. If I is containing J, then the tight-closure of I is containing the tight-closure of J and all that. Not only it gave all those basic definitions that whenever you define something, you have to prove the basic things. And not only it proved, gave really quick proofs of those old results, but it proved so much more. And while it was proving so much more, it developed not just things in tight-closure proper, but there were things that never saw tight-closure in the statement or don't possibly don't need it in the proof. So I learned a lot of mathematics, not just about tight-closure from this preprint, it was very rich. And furthermore, when it came, at least when it came in this published version, they already had seven thick papers lined up ahead of time. I brought one of those thick papers, 80-some pages in the Phantom Homology that appeared several years later. So they already had a map, they already knew what was coming up. Anyway, I had a front seat, I was a student of Craig Hewnekes, and later I was a postdoc with Mel Hoxter, and some of the lessons that I learned from the two of them, and even from just their papers, not necessarily from their saying. So understand your methods and literature deeply. And it's worth taking the time to understand the methods really well. It will pay off in the long run, you'll be able to use them somewhere. And then once you understand the method, just play with it for a while. If you don't play with it, well, then it's just that stale thing somewhere in the back of your brain, but play with it, try to see where else it applies, and then apply it somewhere else when you can. And then again, the power of this is that it applies, these methods apply, not just in tight closure proper. And such theory doesn't happen all that often. Okay, so the first result is that if R is a regular ring, then for all ideals, I, the tight closure of I is the same as I. So for this, what you need to know is that the Frobenius functor is faithfully flat on regular rings of characteristic P, of course. Everything is characteristic P. So in particular, that element C is in the Frobenius power of I colon, the corresponding power of X. So that's in I colon X, Frobenius power of that I colon X. And if you take higher and higher Frobenius powers of I colon X, well, there aren't very many C's like that in the regulating. The only possible C is zero, but C is not supposed to be in any minimal prime. So then I colon X, if it's, cannot be a proper ideal. So I colon X, so whatever is in the tight closure of I has to be in the ideal. All right. So next, so this is already a beautiful result and maybe I should say, oh, I even say, rings in which all ideals are tightly closed as everybody in this room knows are called F regular, or sorry, weekly F regular. The next result, this one was not in the mountain of theorems that the tight closure simplified. So this was a new result that was getting proved with tight closure. Sorry with an L generated ideal or maybe it doesn't have to be L generated but it must have an L generated reduction which we hope L is strictly smaller than the number of generators of I. Then for all non-negative integers N, the inter-closure of the N plus L power of I is contained in the tight closure of N plus first power of I. So if we start with a regular ring then you can forget that tight closure there and so that is generalizes the Brians' code of fear from the rings of convergent power sears that was done earlier. Here's a Hoxter's question and this is one of the beauties that interplay between elementary, basic algebra and tight closure. So here is a fact. You start with a polynomial ring in two variables. So that's a regular ring and start with three elements, arbitrary elements, F, G and H and let I be the ideal generated by F cubed, G cubed, Z cubed. That's a three-generated ideal but in a polynomial ring, this ideal may not have a two-generated reduction but at least locally it will have a two-generated reduction. So if we can prove that inclusion that F, G, H raised to the second power is in this ideal locally then we'll be done. So it suffices to prove that inclusion locally and locally L is equal to two and then if we take N is equal to zero and then we have the, oh and we also need to know that F times G times H is integral over I so the square of FGH is in the integral closure of I squared. So this is an element, yes. Oh, oh, sorry, excuse me, Z is supposed to be H. I'm sorry. I saw, I read this many times. Okay. Yeah, Z equals H. So, so Hoxter's question is can you find an elementary proof of this fact? So if for any FGH, surely you can do that but can you give an elementary fact without using tight closure methods of this inclusion for an arbitrary FGH in the ring. Okay, so I won't teach you anything new today so but maybe by the end of the hour you can have a proof of this. Okay, another big thing that they proved with tight closure is colon capturing. So not every system of parameters is a regular sequence but if you are in a pretty good ring so module finite over a regular domain and if you start with elements that actually are in the base ring but in the bigger ring they form a system of parameters or part of the system of parameters then the colon of n minus one of them with the nth one well it may not be contained in the original ideal of n minus one elements but it will be in the tight closure. So that's, so at least if you know that you're in a weak left regular rings then you can forget the tight closure but in generally can't. And then here's another version of colon capture. Again you have a base ring A that's regular and R is module finite over it but in addition also torsion free and if you start with ideals in the ring they don't have to have anything to do with parameter ideals but if you do any kind of colonning or intersecting with extended ideals in R well it suffices just do your colons or intersections down in the base ring and then extend so that makes it quite a bit simpler. These are not the two most general statements that we can make about colon capturing they're much more general statements but suffice it to say over here. And this has some strong consequences so monomial conjecture is the easiest one that so if we start with a system of parameters of R then we want to show that the teeth power of a product of all those system of parameters is not in the ideal generated by T plus first powers of the parameters themselves. So you might as well go to completion to prove that and you might as well go mod a minimal prime so then we are in the theory in local ring that's complete and domain and if you need or maybe we should have passed on infinite residue field at some point as well or but then you just take a regular sub ring that's generated over the base field or sorry over the coefficient field by X1 through XD and then there that colon ideal so there the inclusion definitely doesn't happen and therefore it cannot happen by the colon capturing in the bigger ring. So monomial conjecture is very easy to prove once you have colon capturing and it's well known that the direct summon conjecture is equivalent to the monomial conjecture so we won't prove that. So there are these easy proofs with tight closure. I repeated the monomial conjecture and direct summon conjecture and then one of the consequences here so this is the Hoxter-Roberts proof that the rings of invariance of linearly reductive groups acting on polynomial rings like Colin McCauley they that proof and also that's the direct summon that proof use these basic tight closure methods and so what you can prove very easily with tight closure that direct summons of regular F regular or weak left regular rings have the same property and then all such rings are Colin McCauley so that gives the Hoxter-Roberts rings of invariance are Colin McCauley. Then I probably shouldn't read all these there are lots of vanishing theorems this vanishing theorem that I wrote here and I will not read is already appeared in that amazing paper but just sometimes you can guarantee ahead of time that certain tour maps are zero the modules themselves need not be zero but the maps are zero and this vanishing of tour this theorem in particular can also implies the direct summon conjecture there are much more general vanishing theorems proved with tight closure and subsequently and then there's the phantom intersection theorem that is also touched on here so the original theorem by Paul Roberts that was proved is if you start with with a complex of free a finitely generated free modules and you know that all the homologies have finite length and the zero of homology is not zero then the Paul Roberts proved that the mention of the ring is less than equal to the length of this complex and that can also be proved with tight closure and we'll see later with tight closure you can even do more of these intersection and generalize new and improved vanishing intersection theorem. All right, more that they did in that first paper if proven of the scissor theorem of Evans and Griffiths so let R be in a theorem a local ring of characteristic P let M be a finitely generated case scissor theorem of finite projected dimension and let X be a minimal generator of M then the depth of the order ideal of this element X so that's the images of in R of all the of X under all the possible homomorphisms from M into R so the depth of that order ideal is at least K the original theorem of Evans and Griffith had proved that the height of that order ideal is at least K but with tight closure that they proved more that's even the Kull McCauley the depth is at least K and one of the consequences of this is that any sub module of this M that's generated by up to K minimal generator so M has some minimal generators take a set of minimal generators and if you choose a subset of up to K of them that degenerates that is a free sub module and in this not quite this form in some of this that form what I just said was already proved by Auslander and Bridger and by Brunson Fetter but not in the greater generality that Hoxton Hewnecker did actually this scissor theorem of Evans and Griffiths was proved by Hoxton Hewnecker not by tight closure but something they called Kull McCauley tight closure so in their papers they introduced more than just tight closure proper but there are various other notions and I will not give their definitions so Kull McCauley tight closure is one in particular that was used for the proof of the scissor theorem and then there's also tight closure with respect to family ideals the original family is just the set of all principal ideals generated by elements not in any minimal prime but you can take more general families then one thing I did not define and I'm not planning to defining is tight closure of modules in characteristic P or otherwise and for modules you don't necessarily need to assume that modules are finitely generated so in that case you might need some kind of approximated approximation so finitistic you can do the regular tight closure definition but you may want to bring it down to something that uses finitely generated modules so that was finitistic tight closure or also absolute tight closure and then another thing that I will not talk about is tight closure in characteristic zero we need some there's a lot of machinery that goes into this and it's the one other thing is maybe you don't necessarily care about tight closure in characteristic P or characteristic zero but the machinery that brings tight closure questions from characteristic zero to characteristic P is worthwhile learning and there's a lot of really amazing tricks that help you understand what is really going on and reduce the questions into the bare bones what is going on just like what Craig Hewnecker was saying in his talk on Monday this is typical Mel but also typical Craig reduce your question is what is the true question what is the gist of it and so what does it mean to be in the tight closure collect the data translate it somewhere see how you can play with it there okay and and then the other thing that comes up in the the other thing I want to say about tight closure in characteristic zero there are two possible definitions of reducing from characteristic zero to characteristic P one is where C's just appear when you are in various mod P's and one is where you're also collecting information about C those two definitions of tight closure turn out to be the same but that brings into the uniform notion so one thing that tight closure really forced us to force us or enabled us to do is to think about uniform results can use one test one C in the definition for more than one ideal for infinitely many ideals and I'll talk more about that in a little bit but there are a few more other closure operations that were happening at the same time Hoxter was developing the theory of solid closure in any characteristic that was a it was a hope that solid closure would be equal to tight closure and then you could prove things in equal characteristic rings but it pretty soon it came up that no they're not the same and then Hoxter with his student Juan Villes was doing diamond closure and Adele of Ratcho another student of Hoxter was doing special tight closure and there were other closure operations and okay and then Neil Epstein developed many closures in very fun abstract way all right oh there's also inter closure regular closure plus closure yeah they're all interrelated that's right regular closure also appears in Hoxter and Hewneck as even I think even in this maybe in this one of these two the white one all right thank you plus closure Karen Smith did a lot of work on plus closure that's another beautiful theory that came out of tight closure with test elements all right so in the original definition of elements of tight closure of an ideal for every element X that's in the tight closure you need to come up with that test element C that makes C times X to the Q be in the appropriate for binias power of I but can we find C that works for every ideal and for every X in the tight closure well that sounds really nice right there is no such thing there's no such notion of test elements for integral closure so for integral closure it is also true that an element X is in the integral closure of I if and only there exists C away from minimal primes such that C times X to the N is in the nth power of the ideal and that's very easy to prove with evaluation theory for example but there's no one C that would work for all ideals I and all X so I will say more about that maybe it doesn't work for ideals but then they're asymptotic if you are willing to change powers that's a different but it turns out that these test elements for tight closure do exist under some fairly mild assumptions on the ring and let's see the, oh before I get there so in the proofs of existence of test elements that's another rich part of mathematics that Hoxton Hewnecke wrote up so beautifully again even if you don't care about tight closure the existence of test elements well maybe they're technical but there's so much to learn there there's you'll have to learn about excellent rings and Lipman Satay theorem and generic smoothness and powers of discrete valuations amazing things that they do what trace can do there are all these things that come into play and there are many different proofs of this test elements because there are different assumptions and then different existence theorems for these test elements but one of the consequences of test elements is the persistence of tight closure under ring homomorphisms so already early in the theory of tight closure you can prove that tight closure respects inclusions of ideals that tight closure of tight closure is the same as tight closure it has all those properties that you expect you expect of a closure operation that are really easy to prove but it is not easy to prove that if X is in the tight closure of an ideal I in the ring R and you pass to an algebra that X is still in the tight closure of the extended ideal so what you do need is something like a test element and some other clever tricks that again are really good ring theory whether you want to do into tight closure or not and so when you do those tricks with you get that, oh yeah elements of the tight closure in the ring stay in the tight closure of an extended ring all right, now phantom homology and phantom acyclicity criterion so I assume here that most of you or maybe all of you have seen the Buxom-Eisenbad criterion for exactness so this is, I will comment on that suppose that R is a homomorphic image of a Koma-Caulley ring and is locally equidimensional and then we start with a G of finite free complex so finite complex of finitely generated free modules and this complex satisfies the standard rank and height conditions after tensoring with R mod the nil radical so standard rank and height conditions so rank of a matrix of a map is what is the large, at least if we're in the domain simplistic one, rank of a matrix is the largest integer such that the largest integer t such that t by t minors of that matrix don't vanish so that's over domain so and then you want the rank of a module to be equal to the rank of nullity of the next one minus the rank of the previous map so suppose you have, well we have these standard rank and height conditions in Buxom-Eisenbad you instead assume standard rank and depth conditions which is stronger than height conditions and then the conclusion here with the weaker hypothesis is that all higher cycles of the Frobenius power of this complex are in the tight closure of the boundaries in this cycle so in other words, that's the terminology homologies of all Frobenius powers of this complex are phantom Buxom-Eisenbad concludes that the homologies are all zero but here with tight closure it's well it's zero up to tight closure so this is a pretty powerful result and this first result is though for each one of the elements in each of the cycles you wanna prove that it's in the tight closure of the boundaries so you need the C for each one of those elements but with a few more assumptions on the ring you can even get this phantom acyclicity with the denominator as they call where you find one C that annihilates all of these homologies you do need some conditions but again it's beautiful commutative algebra developing those and so Ian Albertbach developed the phantom homology and modules of finite phantom projected dimension to make it into a theory that kind of resembles the theory of projective dimensions modules of finite projective dimension that turned to be so then there are more notions related to tight closure and so Coal McHallifier so this is like in this phantom homology things are homology is zero up to tight closure but maybe these Coal McHallifiers make them zero and if you multiply by sufficiently high power or maybe there's a uniform power or that makes the the converts the height condition into a depth condition so it converts the height into depth and then it end up with exact complexes so that's a powerful notion to two of Coal McHallifier's then why is that on the same line? I guess it was used there too R plus so R plus so we started with the domain and in the theory in a local domain characteristic P and we take the integral closure of R in the algebraic closure of its field of fractions so that is called R plus it's a hugely non-netherian ring and I see, yeah, I need to start with an excellent or maybe a complete local domain so R plus is a big Coal McHalli algebra so lots of good things are happening and I already mentioned there before if you if you start with an arbitrary ideal I in your ring extend it to R plus and then contract that's called the plus closure of the ideal and it's always true that the plus closure is contained in the tight closure but and then there was Karen Smith did a lot more with plus closure and proved equality in many other connection many other connections between plus closure and tight closure then these test elements in Coal McHalli Coal McHalli fires played also a large role in the uniform properties so Craig Kuhnike wrote the paper very influential paper Uniform Art and Ries Theorem in which he developed a huge amount of theories so in a pretty good, not such under mild assumptions character C0 character Cp if you take there exists a K such that for all ideals I and all powers N the inter closure right to the N is contained in the N minus kth power of the ideal I so that was a huge thing earlier I said that there is no test element for integral closure but here what we have is all right so it's not that C or one multiplies the integral closure of I to the N into I to the N it multiplies it into C being one multiplies it into the lower power of the ideal that was pretty amazing what else in the same paper there were other results and I guess I'll and uniform art in these that's the whole title of the paper uniform art in these results so if you take two finitely generated modules M and N then for all ideals I there exists a K such that for all ideals I I to the N times M intersect with N is in I to the N minus K times the the module N so that was also an extremely powerful result and there are many people who worked on then art and wreaths alike or uniform type properties and Hewnecker did a lot of that Ian Aberbach I don't know why I didn't write and I'm a little blanking out right now what else is who else was proved and I forgot to say who else worked on Brians and Skoda theorem because there are now uniform versions of the Brians and Skoda theorem as well there are many people who worked on that and I don't want to forget anybody why didn't I say that I didn't I did it I'm blanking out well with there will be a test at the end of this lecture and then what's the next thing on my list so huge amount of singularity theory also arose out of tight closure there have been several talks earlier today and previous days so singularity theory we're talking about F rational F regular F pure F injective lots of these notions and some of these notions correspond to the course geometric notions and characteristic zero lots of work here was Smith Smith Harrah Karen Smith Harrah Nobo Harrah Mehta Srinivas so rational log terminal singularities and there are these analogies going back and forth Smith and Harrah did a lot of work also in geometric interpretation of test ideals and then another huge part of that got boosted by tight closure and Hilbert Kuhn's functions they started before tight closure but I think they got second wind or more with after tight closure we already saw some talks same for Hilbert Kuhn's multiplicity so and we already heard about it today Monsky did a lot of work and then Hoger Brenner did a lot more work and then also the I should mention at this point Hoger Brenner and Paul Monsky they prove that tight closure does not commute with localization so this is one of the basic properties that you would hope that tight closure does commute with localization but no it doesn't but fortunately it doesn't destroy theory the theory is still extremely powerful even if that basic property doesn't hold and then F signature and F threshold we already heard more about that earlier today so I'm finishing a little early I don't think I have any more no I'm finishing a little early but here's a test quiz I couldn't possibly do cover all the things that are in tight closure that tight closure influenced and this was definitely influenced by my preferences and what I know but the quiz is please chime in what should I mention or what would you mention if you were giving this talk Craig probably has a lot of things but I want to hear from this audience first I mean for me for me personally this notion of solid closure was the true starting point so I so the the many P to the power E that was a bit too much for me and for me solid closure seems more conceptually and for me that was the starting point which so that is a great paper of Hoxter I think even if it doesn't work in equal characteristic exactly I mean in positive characteristic it gives really a nice interpret and surprising interpretation of tight closure by local comology like local comology behaves when you add an element to the ideal in the sense of forcing algebra so for me that was a very important point so I in fact so I could go around some some of the really tight closure stuff I see good thank you somebody else Ian you were a front seat driver too so I'm putting you in the spot well yeah I was definitely near the beginning of this I got to pick low hanging fruit it was nice I've been trying to think of something else because what I'm going to say is I feel like I'm just tooting my own horn I I think that the for instance the Brienne since go to theater theorem which look like in some sense it was mostly about regular rings and Craig and I were able to prove it for for a rational rings and generality I felt very good about that and there had been results about rational singularities but they mostly applied to ideals generated by parameters I guess I'd add also something that Craig really originated was this idea of the Branson's go to theorems with coefficients that led to for instance led to early proofs that over regular rings the great associated graded being cone Macaulay and gave it gave the re-sring being cone Macaulay I think so those reduction basically that the reduction numbers the local reduction numbers had to be small I think in the regular case that Libman eventually solved without tight closure but I think that he only solved it because Craig open the door yes so this type of Branson's go to here with coefficients this again has to do with interaclosures of powers being contained in tight closures of ordinary tight closure powers or ordinary powers time some ideal coefficient and so this anytime I saw the theory of evolution of Eisenbaden-Mazer I thought of this Branson's go to theorem with coefficients but in a different way I would say that tight closure theory always sort of carried a lot of dramatic tension and excitement so like there was always that there are all these questions of like okay so you've got tight closure can you can can you characterize it in some other way and so like for instance with the you know Karen Smith's amazing theorem that it's well for parameter ideals it's plus closure well is it plus closure in general dot dot dot and then you know many years later of course you know Holger and you know and Paul Monsky showed that it couldn't be because that you know because that it doesn't commit with localization in general and plus closure does and but then there were all these other operations and I you mentioned some of them and but the thing is that a lot of them kind of are tight closure but they're just different at least in you know over complete local domains like the you know the diamond closure if you do it in characteristic P and the or are really close to tight closure so it's sort of created this this way of thinking about about the framework of closure operations as as kind of a thing in itself and then people were keep kept hoping to do this in mixed characteristic and I guess maybe people kind of are by now because these yeah perfectoid techniques but you know I mean I think so you were saying that like okay you learn tight closure theory it's gonna help you have a lot of techniques to do other things and that's true but I think it also you know always provided a lot of inspiration to try things which is you know also really valuable thank you so so one of the really nice theorems which you have not listed but it was mentioned in the week from by somebody is the relation of the symbolic powers and the ordinary powers I think this is really a very astonishing theorem and it's I mean there are proofs in other settings by I in Smith and Lazarus felt but so the much more general statement by by Unik and Hochstor about the uniform bound that is also one of my favorites I have to say so that should definitely be on the list okay and Alexander had a question so what I want to use this as also way to say thanks to the organizers for both the some the school from last year and also this what I'm about to say I didn't know until I had the lectures from Ian last year when we were talking about this uniform parents on school at some point you said that there was applications to the chemical property of these algebras so in 1996 there was this two results one from Johnson and Ulrich and one from Goto Nakamura and Ishida that were finding sufficient conditions for the resal to be going Macaulay and the one from Johnson Ulrich I'm very familiar with the uses reciprocal intersections they use different techniques instead based on this local reduction number that come ultimately from the study of uniform parents on school and rely on on this type closure theory and I didn't know about it until Ian talked about it last year and so the best outcome for me personally was going back to that paper and reading and understanding the last piece of my thesis that I was missing so thank you oh the man so one more I mean we still have we still have time so I mean the regarding so it's more maybe more philosophical remark I mean regarding the uniformity so my experience in my life is rather that there are also many cases where you do not have uniformity and so tight closure itself they are really infinitely many equations and you cannot really reduce it to finitely many equations so that's in that case you do not really have uniformity so that I think that was the hope somehow that you would have here a more uniform behavior but I think the generic case as soon as the ring and the ideal gets more is more on a difficult side of the spectrum of the rings we we have then you will not have that uniform behavior so I think in general you will really need a good reason for uniformity so don't expect too much uniformity in your life yeah so I didn't realize I would finish early I saw skipping a few things one of the things I was going to ask is so in light of the Ananyan and Hoxster's proof of the stillman or count example to the no sorry proof of the stillman conjecture is there some bound so if you're in a polynomial ring mod an ideal and you want to understand if x is in the title of java and maybe you even know your test element c it works for all of them so you know c so all you're looking for is c times x to the q contained in iq so you there are infinitely many equations but maybe you can solve this equation for the first 17 q's but then is there some pattern is there some uniformity or is there some bound on the degrees of the coefficients that you can take in front of your generators they're given x and the generators of the ideal can you bound in some way in the sense of greater herman or abraham sideberg the degrees of the coefficients so i that's wide open as far as i know any other what else would you have done craig probably has some craig are you there i'm here yeah i can say all things with um oh i don't think uh for this uh very nice talk and i wanted to thank the organizers one more time they put in a tremendous amount of work jugal in particular to organize this and i really appreciate it i'm sorry i'm not there i've been enjoying all the talks and i i'm glad they're on video because i'm able to see the ones i miss because it's in the middle of the night for me for a lot of the morning talks um the one well let me say a couple things when mel and i you know first started tight closure that was certainly one of the most exciting times in my own life and it's it's very good to be lucky in math i think it helps a lot but it also uh to me there's so much other work that uh tight closure connected to the f-split school which of course shrinivas is here knows very well the multiplier ideals and algebraic geometry and in a sense you're lucky that it connected but the lesson for me is if you follow problems which are interesting and develop things which will help you solve them it almost surely in mathematics will connect to other things and the timing just happened to be right for tight closure but as much as i appreciate the retrospective math is all about moving forward and i'm very gratified to see all the beautiful work being done on many different things that the speakers have done in this in the conference and in many cases tight closure has has disappeared from what's going on and is maybe only a faint noise in the background but that's the way it should be that's that's what you want is as math moves forward one problem that i would always wanted to solve using tight closure is already known it's a beautiful theorem of dale kukowski's about characterizing rational singularities by the square of of integrally closed ideals being integrally closed and i was always sure there should be a tight closure proof of this i've never been able to find it but i always felt if you could find it you would learn something new and that's something i've always personally wanted to do and of course the many problems still remaining about localization at one element or weak implies um F regularity or strong F regularity is very much an area of a lot of interesting problems i think which come up and i think ian's speaking about that tomorrow partly so thank you again it's uh it's been great so i can take any questions if there are any for me personally but thank you craig and said dale good to see you craig i can't literally see oh there you are what would you have said about tight closure um you know somehow i've i've missed out on this except for a just one vague comment that it looks a lot like almost mathematics that's i mean this whole thing about the test elements and then going to the plus closure and all of this i mean that it it's i remember the last msri that paul roberts was really excited about this uh this paper by um fault faultings where he had started this and and i'm i have no idea how he saw that this was the thing which would lead into the proof of the direct summit conjecture but somehow somehow he did he knew that was that was what was necessary that's right well height and of course had just done his right three dimensions yeah which is fantastic of course yeah and which is also very much actually it's one of the great regrets of my life in a sense of that a time constraint i was asked to uh sort of go over the manuscript of of gabber and maras and almost mathematics and i would have gotten in on the ground floor so to speak but i was so busy i turned it down that was real mistake on my part actually you probably probably was you probably wouldn't have had time to do anything else if you undertook that yeah that that was the problem we're going to enter into that you have to commit a huge amount of time it's always a question in math you know for everybody in the audience you know this very well that when you see a problem that you're interested in there's always a question of whether it's worth the time commitment to do it and you have to make choices and judgments about what the most valuable thing is and what you can expect to get out of it so i know that's something every mathematician has to do in general for um mathematicians and artists also especially for more senior ones there's always this trade-off is whether you should try to get into the latest latest hot thing that's happening or continue to develop what you've been doing and it's it's a hard trade-off and it's very hard i think i always i i always tried to do a lot of different things but it and then i couldn't go into as much depth sometimes as as i should have but it's a trade-off as you say i saw the comments that i found that conversation between dill and crick very interesting but i want to emphasize maybe um the point that that there's many ways to have fun in math and and and have a decent career in math and you can try to go deep in one topic or you can go broad in different things and i think it's it's a big 10 so you can choose what that is suitable for your personalities i think that's and the community of algebra is actually one of the um a good area for that because people are very friendly and and very tolerant of different styles i think that's i think that's very true and you have to understand who you are as you say that's two things that are comfortable for you and you have yeah fun doing that's right you have to do it the way you you want to do it yeah i agree totally yeah craig i've really enjoyed all your comments that you've been making this meeting and i see that you're becoming a very sage like uh like a person nobody ever called me sage before so i'll put that on a little plaque i think thank you getting old has some advantages i guess not many but i am really gratified to see all the young people in our field and they're doing so many wonderful things that it's just a joy for the people in the audience of which there are many who are sort of my age if you think back of what community of algebra was like 50 years ago i mean it's just had tremendous explosion of ideas and beautiful theorems and i expect the next 50 years to bring even more things like that so unfortunately i won't be alive to see them all but i can i get this almost tingly sense when i see things sometimes about oh there's a lot more there to do and it's a good feeling thank you very much craig and thank you