 That's the next session. And so I am very, very glad to introduce Krem Huneke. And before his talk, I'd like to say something of gratitude to him and also Merhokster. So I needless to say, so they are great works, very so fundamental and so it's anyway a great thing for commutative algebra. And so especially for this conference, it is that notion of ingenious notion of tight closure is wonderful and so I or we are very grateful to him. So OK, please. Thank you. Thank you very much, Kichi, for that very nice introduction. And we owe a real debt of gratitude to Merhokster for what he's done for the field, as you'll see in my talk. Let me ask, first of all, can everybody hear me all right? Is it OK? I can't really see reactions, but OK, OK. It's OK? OK. And could I ask someone I can't? I love to answer questions or have comments or corrections or other comments people want to make. So please interrupt me, but you'll have to do it through Kichi, I think, but please do. I plan for that in the talk. So there should be time built in for it. I know there's lots of Mel's students and postdocs and co-authors in the audience. So there's probably lots of room for corrections as I go. So I'm talking on 50 years of mathematics, Mel's work. It's already a misnomer. The title a little bit, as you'll see in a second. But let me start. It's not letting me move. I tested it before, and it made me move the screen. Let's see. There we go. So first, let me thank the organizers, to Chung and Merlina and Jugal for all the very hard work they did to organize this workshop. And also, I want to thank the local organizer, Lothar. And I also finally want to thank all the other speakers and the participants whose presence is crucial to the success of the workshop. So let me begin. Here's an outline in my talk. I'll have a short introduction to Mel. I'm going to talk about a lot of his papers in what I call the early years, from 1969 to 1985. Then I'm going to skip ahead to after 2008, for reasons you'll see. And finally, if I have time, I'm going to go into more depth about Stillman's question. I wanted to not just mention papers, but actually talk a little mathematics. And that part of the talk, if I run out of time, I can shorten. I can't really, well, I could lengthen it, too, for that matter, but probably not with slides. So here's the main character, Mel. I don't know if you can see my pointer, but Mel is second from the right on the first row. This is the math team at Stuyvesant High School in New York City. Stuyvesant High School is a very famous school in the United States. It's got graduates who are four or five Nobel Prize winners, Wolf Prize, Opel Prize, Fields Medalists, several Academy Awards. So it's really a distinguished group. Among those, Mel was one of the best in his years. So he got, after attending Stuyvesant, he got a Westinghouse scholarship to go to Harvard. And he got his BA degree there in 1964. He then went to Princeton. His official advisor was Shamora, although I don't really know how much contact they actually had. And he got his PhD in 1967. So you'll notice that's only three years. And his thesis is really incredible. So that was already a sign of things to come. He's had 48 PhD students. The first one at the University of Minnesota in 1972. And he had three finished in 2021. I didn't see the list of people coming to this conference, but I know for sure several of his students are here, of his many, many students. He has over 100 papers. I didn't count exactly how many. Because of that, of course, I can only cover a very small fraction of his work. So what I chose were my own favorites, together with ones I thought were very important for the field, a little combination of both. He had 25 co-authors. Also, I think many of them are here. And he supervised many postdocs. I don't know how many, but certainly well over 20 postdocs over the course of his career. And this shows the misnomer of some kind of reverse pigeonhole principle. He couldn't have just done 50 years of mathematics if he's published in seven different decades. But that's really amazing. I think, however, there are at least maybe two other people present, either by Zoom or in the audience, who can say the same thing. And maybe more, I don't know about. But I find that incredible. Won't happen to me, I know that. So someone who has awards, he won the American Mass Society Coal Prize in algebra in 1980. This is the highest prize given in algebra by the American Mass Society. He was a Guggenheim Fellow shortly after that, which he took at MIT, 82. In 1992, he was elected to both the American Academy of Arts and Sciences and also the National Academy of Sciences. I wanted to list just a few talks of his that I thought were really important. So the first one is one I unfortunately missed. I was a first year graduate student in 1974. And I really wish I could have gone to this conference. It was his first CBMS conference. He gave 10 talks about homological conjectures and a wide variety of topics. And these notes of this conference were for me like a Bible for quite a few years. He was an invited speaker at the International Congress of Mathematics in Helsinki in 1978. And then the last one I wanted to mention was he gave another CBMS conference. These are conferences where the main speaker gives 10 lectures to a day for five days with supporting talks as well. And this was entitled Analytic Methods in Community of Algebra at George Mason University. It was one of my first main conferences. I met many people there for the first time. And I think this conference is widely viewed as one of the most important conferences of that decade, well, of the ensuing decade to come. It's sort of odd because he was showing, a lot of it was his interest in the Branson-Skoda theorem. And the proof by analytic methods. So the actual talks in some sense have never been used a lot by Community of Algebra. But it's set an agenda which had enormous consequences. So let me start talking about some of his papers. His thesis had two different topics in it. And the one that he's most famous for, he published later, he published several papers, but this is maybe the main one, called Primaideal Structure and Community of Rings. And I'm just going to quote a passage from the paper because he said it better than I can. So bear with me as I just read this. We call a space spectral if it's T0 and quasi-compact. The quasi-compact open sets are closed under finite intersection and form an open basis. And every non-empty irreducible closed set has a generic point. Spec A, where he's a ring as a community ring, is well known to be spectral. So it has all these properties. And in fact, these properties do characterize the spaces in the image of spec. So he viewed spec as a functor from rings to topological spaces. And so what he proved in this paper was that these properties that are in blue actually completely characterize topological spaces, which are the spectrum of a ring. And then he goes on to say, as a byproduct of this study, remarkable facts about the structure of the prime ideals in a community ring come to light. For example, given any ring A, there is a ring whose prime ideals have precisely the reverse order of the prime ideals of A in the lattice of prime ideals. So this paper of his, he once told me he thought it was the hardest thing he ever did. I don't know if he still feels that way, but it's actually his second most cited paper. So it played a big role in those years, especially at the school that Roger Wegan studied it very deeply. But he had another part in his thesis about symbolic powers. And this influenced me a great deal. My favorite one of a series of two or three that he published in those years is called Symbolic Powers in the Ethereum Domains. It's in the Illinois Journal in 1971. He defines a strong control for an Ethereum domain to be a function from the non-zero elements of R to the natural numbers such that for any two non-zero elements A and B, the product is not in the symbolic VA plus VB plus 1th power, symbolic power of every prime ideal of R. So it gives a uniform bound on how deeply products can vanish. And that's actually called a control, but he shows that there are two more properties that the value of a product is at most the sum of the values. And A itself is never in the VA plus 1st symbolic power of any prime. So what he shows in this paper, and it's in usual Hoxter fashion, he just breaks it down and he keeps breaking it down till it becomes almost trivial. Every local domain has a strong control, as well as every ring finally generated over a field. And this result is, to me, clearly strongly related to Chevrolet's theorem on symbolic powers and the maximal ideal topology. It's related to the Zarisky-Nagata theorem, which he actually gives another proof of in his thesis. And then to later work on uniform arteries. Actually, about every month or so, I believe that this should prove a uniform property for symbolic powers, but I can never make it work. So he began his academic career after his thesis at the University of Minnesota, where he was there from 67 to 73. And Jack Egan was there. And if you remember Egan, Egan had just fairly recently done the Egan-Northcott complex and proved like the maximal minors of a generic matrix form a prime ideal. And I think Egan gets all the credit for sparking Hoxter's interest in Cohen-McAuley rings. And this resulted in a really remarkable paper there. Cohen-McAuley rings invariant theory in the generic perfection of determininal loci, again in 1971. So in this paper, they prove that the ideal generated by the k-sized minors of a generic n-by-m matrix are a prime ideal defining a Cohen-McAuley ring. Let me say a few things about this. Of course, nowadays, this is well known. There are much better proofs available at this point. In this paper, they also talk about so-called generic perfection, so they actually prove a lot more. They prove that the minors of any matrix in a suitable regular ring, which have the biggest height they can, the generic height, do give you a Cohen-McAuley ring. And then it's easy to determine whether it's prime or not as well. This is really easy now from things like the books from Eisenberg criteria. Resolutions, but at that time, all of this was really completely new, as far as I know. Only a few special cases like the Egan-Northkite work had preceded it. The method they use is what they call principal radical systems, which is just sort of like a brute force method where they kill one variable at a time, starting at the top row of the matrix and create this whole class. It starts creating rank conditions on sub-matrices, and they build up this entire class and prove they're all self-radical. And once you know it's self-radical, it's easy to show that the miners are irreducible. So that gives the primeness, and the Cohen-McAuley follows as well from this principal radical system. So it's a real tour de force. Please interrupt me if anything. So after that, he started seeing Cohen-McAuley rings everywhere. He studied them for a wide variety of rings. He proved that the normal monomial rings, the invariance of torrid groups, are Cohen-McAuley. That was in the Annals in 1972. He proved Krasmanian's or Cohen-McAuley in 1973. And through his student Reisner, Reisner gave in his thesis a topological characterization for when Stanley Reisner rings or Cohen-McAuley one or another result. That's why their name, that's why Reisner appears in the name of Stanley Reisner rings. And during that time, he had a conjecture that all these Cohen-McAuley results were really a special case of saying that invariance of reductive groups always give Cohen-McAuley rings. And this led to his work with Joel Roberts. Joel Roberts was also at the University of Minnesota, not to be confused with Paul Roberts. And in their great paper, rings of invariance of reductive groups acting on regular rings are Cohen-McAuley. This is together with Joel Roberts. They proved the following main theorem. Led G be a linearly reductive, affine, linear algebraic group over a field K of arbitrary characteristic acting K rationally on a regular netherian K algebra. Then the ring of invariance is Cohen-McAuley. So all of these previous results are in some sense special cases of this. The determinational one is a little different in the sense you have to prove that the invariance or that the determinational ideals are defining on the nose a ring of invariance. That is, they give you all the equations. So there's more to prove. So I wanna quote from this paper. Again, the method is just this tour de force. And I'm gonna give two quotes from the paper to just give you a sense of it. When G is linearly reductive, the ring of invariance, R, S upper G is a direct sum of S as R modules. The Reynolds operator gives an R module retraction from S to R. And this fact plays a crucial role in the proof. Briefly, there is a reduction to the case of a graded minimal counter example. In quotes. And then a contradiction is obtained by utilizing two separate reductions to characteristic P positive while preserving finitely many consequences of the fact that the ring of invariance is a direct sum. And that finitely many consequences is related to the fact that they couldn't preserve the fact it was actually a direct sum. So they go on to say, they introduce, they study purity. The study of pure subrings is forced upon us because in the reduction to the graded case, we lose the direct sum and property, but retain purity. In characteristic P, the situation is simpler in many respects, and we obtain the following short results. So this theorem appears in their paper. If R, if sorry, if S is a regular the theory and ring of positive characteristic and R is a pure subring. For example, if R is a direct sum, then R is in fact Cohen McCollum. So this was this tremendous paper which led to many, many other things. Let me stop here in case there are any questions or comments that I haven't noticed. Anybody? Or additions to what I've said? Nothing? Kitchie? Are you keeping track or is Jugal? I can't hear anybody. So I don't know. I'll continue, I guess. So, are there questions? Okay. Wait, okay. Thank you. Oh, sorry, there is a question or there is no question? No, no questions. Okay, thank you. That's too bad. So he continued his work with Jol Roberts in the important paper which focused on the purity of Frobenius, which was, of course, a big theme for what's coming later. So in this paper, the purity of Frobenius and local cohomology, they proved many results about purity of Frobenius. And I wanna say that Richard Federer proved his famous criterion for F purity a few years later, but Federer was a student of Hoxton. It was part of his thesis. It was while I was there as a postdoc at Michigan. So all this, in 78, he wrote an article in the bulletin and he has this quote, life is really worth living in an Ethereum ring and all local rings have the property that every system of parameters is in our sequence. So he fell in love with Colin Macaulay rings and as you'll see, he did even more in a second. So this, however, was BTC. So he did other work at the same time. Another significant result of Hoxton's, which was based on the thesis of his student, is in the paper, Colin Macaulay rings, combinatorics and simplicial complexes. It's the proceedings of Oklahoma Ring Theory Conference in 1976. This is where I actually first met Hoxton. In fact, maybe I should tell a story. I think I'm allowed to tell some stories along the way, although I probably shouldn't. I was a grad student, but my hometown was in Oklahoma and I went back for this conference and the organizers asked me to pick people up at the airport because I could use my father's car and I knew the area. And he asked me, is there anyone you wanna pick up? And I said, yes, I'd love to pick up Mel Hoxton. Well, Mel did, sir. And okay, so he sent me to the airport but of course I picked up other people too. So I was waiting and I had a group of people around me and we were waiting and waiting and waiting. And finally somebody said, well, why are we waiting here? We're all here. And I said, well, I'm waiting for Hoxton. And of course, Hoxter was in that group. And it just wasn't anything like I thought he would look like. In those days, there was no internet. I didn't have any picture of him. And so I had built up another picture in my mind. That was embarrassing. So let me tell you about his work on simplicial complexes. So if Delta's an abstract simplicial complex with finitely many vertices and K as a field, then the Stanley-Ryster ring, K Delta's the ring obtained by the quotient of the polynomial ring where the vertices are variables. On the vertices modulo, the ideal generated by all square free monomials whose support is not in the solution compounds. So this is the definition of the Stanley-Ryster ring. And what Mel did in this paper is gave a formula which is used a great deal for the bedding numbers. So they're topological invariance. It's not so important to read the formula other to realize that it links the topology to the actual bedding numbers. So I may not even bother to read that. You can read it yourself. The terms are defined below the, there. I'll leave it up there just for a second. But again, some sense, I mean, of course the details are important, but it's the idea that was the most important. Well, in his beautiful paper, contracted ideals from integral extensions of regular rings in Nagoya. Oxford considered the question of direct summons. I mean, I think this was motivated by his work with Roberts and proves that in equal characteristic, a regular local ring splits off from module finite extensions. And he gave a criterion for this, which he'll recognize as the monomial conjecture. So I want to read it. Let R be be a regular local ring and let X1 through Xn be a regular system of parameters per R, so they generate the maximum ideal. Suppose R is inside S, where S is a module finite R algebra. Then R is a direct summand of S, if and only if, for every positive integer, the kth power of the product of the kth power of the Xs is not in the ideal generated by the k plus first bound. So he does this, if you've never read this paper for the students who are here, you really should. It's a very Hoxter-like paper. It just passes to the injective hall and builds up splittings in the artinian case and then lifts them back. So this led, we're sort of going back in time, to a very famous preprint, which has never been published. He did it in Argus, where he was doing things called Deep Local Rings. I hope you can read the abstract, because I want to read that also. So in this paper, it proves the intersection conjecture of Pesky and Sparrow, Alzheimer's zero divisor conjecture, and Bass's conjecture, all in the equicarteristic case. Of course, all these we now know are true in mixed characteristics because of all the wonderful work being done, which was partly spoken about in the last week. So a key point is that if you have a system of parameters for such an R, then there is a module, M, R module, not necessarily a finite type. In fact, definitely not, pretty much, such that the X's don't generate the whole module and such that they are a regular sequence in that module. So the colon ideal behaves as a regular sequence. So this is his construction of Big Cohen College modules. And the way he did it, again, it's just such a simple idea. And he had the ability to carry it out. So you start with an Ethereum local ring. You have a system of parameters. You have a module and an element, A, M, M. So suppose the X's are not a regular sequence on M. Well, then there'll be some sort of relation, X1, M1, plus dot, dot, dot, plus X, K plus 1, M, K plus 1, equals 0. And the M sub K plus 1 won't be in the previous X's, X1 through XK, times the module. That would break the regular sequence. So what he does, he just forces it in. He gets a new module. He just takes the module and adds a direct sum of K copies of R, and he goes mod an element. And that element, when you go mod it, forces M sub K plus 1 to be in the X1 through XK times the new module. So simplest thing possible. And then he lets A1 be the image of A in this ring. You'll see where this A1 and A are coming from in a second. And he calls that a first modification of type K. And then he iterates it to just talk about a modification. So you can iterate this process. And what he proves is that there is a module, M, such that the X's form a regular sequence on it. And you have this non-degeneracy condition that M is not equal to I times M, or I is the ideal generated by all the X's. If and only if. This is if and only if. It's not just if. For every iterated modification, M comma A of R comma 1. So that's the starting point. You start with R and 1, which is certainly not in the ideal of X itself. Such that you can prove that this A is not in the extension of I to M. So this gives a criterion for when there is a big corn macaulet. And then he proves that this holds by reduction to characteristic P. And here's one of the first really tight closure arguments. The way he proves that this A is not in any of these is a characteristic P argument. Well, that wasn't all he was doing. He certainly was focused on Cohen-McAulay links. There's no doubt about that in modules. But he did other things along the way. The Zariski-Lippmann conjecture states that if you have a field of characteristics 0 and R is a finitely generated reduced K algebra, and P is a prime of R, then if the derivations are RP free after you localize, then it has to be rigged. This is like saying the dual of the module of differentials is freed and the ring is rigged. Of course, if the module of differentials is freed, that's it was known. And what Mel did was he proved this in the case R is quasi-homogeneous. That is, it's homogeneous with respect to some weighting of the variance. So that was in 1977. Just sort of along the way. Oh, by the way, I can prove this too. And before leaving the 1970s, I want to mention one other work, which played a very large role in my own interest. I'm glad David is listening, because he can correct anything I say. So Nausstellensatz with Niel Plotin sends Zariski's main lemma on holomorphic functions, states the following, you have a finitely generated ring over field and I in ideal, then I is actually the intersection of I plus certain powers of E is fixed here. And you run it over the intersection over all maximal ideals containing I. And E depends on the Niel Plotin C degree of I. So the Nausstellensatz is the case where I is prime or self-radical, basically. And there you just take the intersection over all the maximal ideals containing I. So E is 1. So this is a generalization of it to allow Niel Plotin. So they do it for modules. They do other things. But they also bring up uniformity questions in this paper. And I want to say this. I believe this work started at Nagata's CBMS conference. I was a student and I was invited to lunch with both of them one day. And they were walking and started talking about this. And I just could barely keep up with the ideas going back and forth. It was just like bing, bing, bing, bing, bing, bing, bing. And suddenly they had the result. I was very amazed at the time. So here's Mel at a Taniguchi meeting in Japan that's Matsumura Banquet seated right next to me. Now in the 1980s, I want to mention one other work of his. Many of the homological questions are still open today, in fact. They often concern how many of the good properties of modules over regular local rings can we expect to preserve if one or both of the modules have finite projected dimension? So it's a pretty obvious question to ask. And one of these questions is concerned intersection multiplicities. So let S be an Ethereum local ring. Two M and N, two finitely generated modules, such that the length of the tensor product is finite. So then, Sair defined the intersection multiplicity, chi of N to be the alternating sum of the tors. So all the tors have finite lengths, sorry, the lengths of the tors. So this makes sense. And it corrects what the sort of obvious thing would be, which would just be the length of the tensor product, which doesn't work well. And in the eco-characteristic regular case, I think even for the un-ramified eco-characteristic case, Sair proved that this length condition on the tensor product forces that the sum of the dimensions be at most the dimension of R, which is what you would think if you think about it geometrically. And moreover, if they meet properly, so the sum of the dimensions is the dimension of R, then this weather characteristic is awesome. And the question was at that time, well, what if you just assume M or N has finite projected dimension, or both? Shouldn't this still be true? And in a really amazing work with his student Dutta and his colleague McLaughlin at Michigan, what they did, they constructed a module over the hypersurface, first one you write down, basically, you want cross P1, with finite projected dimension and finite length. So it has finite length, so all the tensor products also have finite length, such that the Euler characteristic is actually negative. So what it showed was it's not enough to just assume one of the modules has finite projected dimension. Whether it's true, if you assume both, as far as I know, is still an open question. But maybe I'm not up to date on that, and somebody can correct me if that's wrong. They do this by brute force, by the way. They do it by thinking of the module as a vector space. I think M squared or M cubed denylas, I can't remember. I think it has length 24, maybe. And they just write down what you need. It's pretty amazing, and prove it exists. So the reason I'm going to skip ahead is from 1986 to 2007. That's what I think of as the tight closure years, when both of us were heavily involved in thinking about tight closure. This is a picture in my driveway at Purdue, my home at Purdue. And certainly, Mel did other work during those years, but I think it's fair to say that this really was most of what he concentrated on during those years through his own work and through his students' work as well. So this is a good place to stop again for just a second. I have about, I think I have time to do what I wanted. Any questions or comments from people that I've mentioned? I want to apologize, by the way, for people whose work I've not necessarily mentioned. Mel did so much that I'm picking and choosing. Questions? Sorry, no questions. No questions. This is very disappointing. You all know all this too well. So I do want to give his quote later. I told you about his quote BTC, but his quote ATC is life is worth living, which I always liked. So 2,000 feet beyond, I'm just going to pick a couple papers, actually, because I want to take the rest of the time talking a little bit about Stillman's question. So in the paper he wrote with Florian Inescu, also a student of Mel's, called the Frobenia structure of local cohomology. They prove a rather amazing fact about families of submodules of an arbitrary module. There's a whole build up to this with other people's work, but I'm just going to quote this one. So their result states, let M be a module over an excellent local ring and consider a family of submodules, which are closed under sum. This is the infinite family. Blows under sum, intersection, and primary decomposition. If the set of annihilators only consists of radical ideals, then it has to be finite. So it's a very strong, finite result. And they apply this to local cohomology modules. And there's a whole story there that I'm not going to tell it today. But I've always liked this result a lot. And while I'm on local cohomology, I want to mention another one of paper I like a lot. It's still an open question whether the support of the local cohomology module is always closed. So this is the local cohomology of an Ethereum ring with a support in some ideal. And it's not known whether or not the support is closed. I'm not even sure what's right. I sort of believe it. I certainly tried to prove it. But it could be there's something very strange happening. But in his paper with Louise Nunez-Battoncourt, called the Support of Local Cohomology over Hyperservices and Rings with FFRT in 2017, they proved the theorem that says that basically in characteristic P for hypersurfaces, the support is always closed. So this is a very nice result. It was also proved, more or less at the same time, by Modi Katzman and Wen Liangzhong. Proofs are not the same at all. So I know myself and other people tried to generalize this, even to complete intersections. And so far, we fail. But I expect this to be solved one way or the other. So I'm going to end. I think I have only taken about 45 minutes so far, so that's good. I asked for possibly more time because I wasn't sure how fast this would go. And I was hoping there would be some questions I could talk about. But maybe you'll save some for the end or comments. So let me go to Stillman's question. So I'll start with the question. So you fix positive integers, d1 through dn. And you consider all ideals in an arbitrary polynomial ring in capital N variables, which you think of as varying, which are generated by N homogeneous polynomials of degrees d1 through dn. Then Stillman asks, is there a bound to the projected dimension of such ideals, which only depends on their number and the degrees? So let me define b of d1 through dn to be the supremum of the projected dimensions of all ideals, as described in the question. Then Stillman's question is just asking if that supremum is finite. If these can always go up to infinity or somehow there's only finite data there that bounds. I always like this question a lot. So let me remove all suspense. I don't think there probably is any, but there might be somebody in the audience who doesn't know the answer. But in 2020, Ananian and Hoxter prove that Stillman's conjecture is true. So what I want to do is talk about the conjecture a little and the meaning of it, and talk then a little bit about their proof. And that'll be the rest of my block, basically, however long that takes. So their proof actually is just, I've said the word amazing a lot with mouse working. I mean, that's just the way it is. It's an amazing multi-tiered induction, and it proves a lot of other things along the way, which I'm not going to talk about. But I do want to mention that in recent years, the proof has definitely been made conceptually simpler and generalized in recent years by the work of Daniel Ehrman, Stephen Sam, Andrews, Snow, and there are some other proofs now, too, as well, I should mention. So let me just talk about the actual question for a while. So you might, when you start considering this question, you might, you want to try examples out right away to see if it has a change. So the simplest or most special in a way you could possibly imagine would be when they're generated by monomials. But in her Chicago thesis under Kaplan-ski, Diana Taylor proved that the projected dimension of n monomials of arbitrary degrees can be at most n. So in this case, the degrees play no role at all, and nor the number of variables. And you get that the projected dimension is bounded by the number of genera here's the fact, what I call little n. Big n is the number of variables. Little n is the number of genera. So once you do this, you might say, well, what about the other extreme? What if I sort of consider random polynomials? Because random should be really bad, somehow. But no, that's not true. If you fix the degrees and choose random elements, homogeneous elements of those degrees, then the projected dimension is, again, exactly n. I mean, not even less than or equal to n. It's exactly n. And it's just because they have to form a regular sequence. So randomness was not the right idea. Basically, if you just choose random polynomials and you have enough variables, they're always going to be a regular sequence. Prime avoidance assures that. So in both those cases, it's just sort of the most special and random, whatever that means. The projected dimension is always bounded by the number of them. And that's still true in low dimension. That's the next thing you might try. You might say, well, what about low dimension? Well, if n is 1, of course, oh, I'm sorry. I meant if capital n is 1. That shouldn't be little n here. These should be capital n. If the number of variables is 1, then, of course, the rings of PID, every element is generated by a regular sequence, and the projected dimension of the quotient is at most 1. If you go to two variables, then the fact that polynomial rings are unique factorization domains shows that you can factor out the GCD, and then the projection dimension is at most 2, which is sharp again. Well, the first case not covered by these things is when you have three polynomials and they're all quadrics. So the d's are 2. And the ideal should have co-dimension 2, because you don't want to be able to factor out a common divisor. And if it's co-dimension 3, they would form a regular sequence, so the causal complex would be the resolution. So here's a specific ideal that has a projected dimension 4, in fact. And I give it the resolution. I didn't put the twist in. So that shows that this supremum of the projected dimension of three quadrics has to be at least 4. But in fact, David and I, over lunch one day, long lunch, proved that b222 is 4. And this was long before Stillman did ask this question. In fact, I visited Mike in about 2000, and we talked about three cubics at that conversation. More generally, there is a better theorem now for n quadrics, which is one of the few sharp results. If you have n quadrics generating an ideal of Still co-dimension 2, however, the projected dimension is at most 2n minus 2. So in the example given, it's 2 times 3 minus 2 is 4. And this is sharp. This was work of Montero, Jason McCall, Alessandro, Siciliano, and myself, which we did at the last big MSLW. But of course, as most people in the audience, if not everybody knows, pre-generated case is already very general. Bruins prove that essentially every possible free resolution is the resolution of a free-generated ideal. And you can make it homogeneous if you want. I mean, if the resolution you started with is homogeneous. Up to shifting degrees. So when you do this process, the degrees of the generators have to increase a huge amount as you increase the length of the resolution. So it tells us that despite all the examples so far, that the degrees of the generators have to play a role. And in fact, amazingly, as far as I know, there is a maximal because of the Inanian-Hochster theorem. The maximal projective dimension of three quartics is not known. I don't even know if there's a good guess. Maybe David or somebody knows what the guess is. Three cubics was done eventually. Montero and McCall approved that the value is 5 can be 5, but no big. So that's sort of the background to Stillman's question, and sort of what was known some since. No, it wasn't what was known, because there's one more case, which is very important geometrically. And that's when the polynomials actually define a smooth projective variety. So I suppose I have in homogeneous equations. Well, they're not equations, polynomials. Giving a homogeneous ideal and capital in variables. But I assume they form a smooth projective variety, or saturation as a projective variety is smooth. So it means it has an isolated singularity at the origin. What about this case? This would be a very natural case to consider. And it turns out the answer was already given in a 1981 paper of Faltes, a few years before Stillman asked the question. And what he proved, I love this paper. Again, it's another paper you should read if you haven't. He proved that either such an eye is generated, so eye is to find a smooth projective variety. Either it's generated by a regular sequence, in which case of course the projective dimension of the quotient is just in, because the causal complex is resolving it. Or the number of variables actually has to be bounded by at most three n. And then of course the projective dimension is at most three n by Hilbert Sissich there. So this is another case weirdly enough where the degrees don't play any role at all, just the number. But it also points out, a really important point is that, well, let me read, I guess I should follow my notes here. So as I just said, there's a bound depending only on the number. And here's what I wanted to say. So something much stronger is actually proven. If eye is not generated by a regular sequence, then the number of variables, which is because of the isolated singularity, that's the height of the ideal defining singularity. The Jacobian matrix gives, I mean, minors, appropriate size minors give that locus, is bounded absolutely by a function of the number of the f sub i. And it's this dichotomy of either knowing an ideal is generated by a regular sequence or bounding the height of its singular locus, I think is a really crucial insight that an onion and hawkster had. They didn't exactly say it this way always, but if you look at their proof, this is to me a very crucial piece of it. So I wanna do one more case where we can prove something. So suppose you have a regular sequence, the f's. So of course then the projected dimension of the ideal quotient of them is just in. But we can consider the subring they generate inside the polynomial ring. And the containment, because they're a regular sequence, the containment of a into s makes s into a flat a module. So for the students, of course, you can see that because this subring, a, is actually a polynomial ring itself, isomorphic to. And the f's generate its homogeneous maximal ideal, or it's a, well, they are graded, but maybe I should just say that maximal ideal. And so the resolution of the residue field over a is given by the causal complex. And then when you shift because they're a regular sequence in s, the causal complex is still the resolution. So the torres vanish. So s is a flat a module. I've always liked this quote of David Mumford. The concept of flatness is a riddle that comes out of algebra, but which technically is the answer to many prayers. So of course, I think I'd better not say anything more about that quote. This observation has an important consequence for the solution of Stillman's conjecture. Not Mumford's quote, but the observation. Namely, suppose you have an ideal in this subring. So by the Hilbert-Syzygy theorem, the projected dimension of a mod j is at most n. So there's an a resolution of a mod j if length at most n, the number of the f's. When one base changes by tensoring, the free resolution becomes an s free resolution from the flatness. So the conclusion is, if we're given polynomials in s, they don't have to form a regular sequence, but if they're polynomials in a regular sequence, much smaller cardinality, we can solve Stillman's conjecture. And that observation is really the basis of their group. So here's their strategy. Given homogeneous polynomials, f1 through fn of degrees d1 through dn, they find auxiliary polynomials, g1 through gp, with p bounded by a function depending only on the degrees, and well, the number n and the degrees, but n is sort of tacitly in there. Such that the original ones you started with are actually in this subring, generated by the g's, and they're a regular sequence. Then, by what I just discussed, that forces the projected dimension to be at most p, this number of g's. So that's what I just pointed out was an easy case. And what they had, there's an insight to say, maybe this is always true. So here's what they discovered. This is a theorem. You fix integers d1 through dn, then there's just an integer m, such that if you start with homogeneous, if you're given homogeneous polynomials of those degrees d1 through dn, then one of two things has to happen. Either they're a regular sequence, which case, of course, you're done, or some non-trivial homogeneous C linear combination of the FIs can be written as a sum of homogeneous polynomials of strictly smaller degree. And that m only depends on the degrees. So let me just let you soak that in before I give the slide. You can think of this as, how can you think about this? It's reducing the degree, basically. It's giving a construction to reduce the degree. And this work is based on a new notion, which they call strength. It's pretty technical. I think I'm gonna skip this slide because it doesn't really add much unless you've really digested the... But once you have this theorem, there's sort of a reasonably straightforward induction. You can replace some linear combination of the Fs by homogeneous polynomials, which have smaller degrees. And even though the number of them has increased, the degree has decreased, and continuing this process eventually puts the Fs in a subring where they are a complete intersection. And S is bounded by some enormously huge function of D1 through Dn. Oh, I meant to say something. Let me back up. That's why I was pausing. I meant, I was pausing for myself. I meant to say something. You know, they're homogeneous, so you always have Euler's relation. You can always write up to multiplying by the degrees. You can always write the Fs as a sum of the variables times the partial derivatives. And those are homogeneous of smaller degree, of course. Or you could just say, oh, well, they're in the ideal gender by the Xs and just write it summing. You don't have to use Euler's relation here. But of course, the point is, the number there is the number of varities. That we don't ever want to use. So what they essentially do in this is they sort of look at the Jacobian and use Euler's formula and the polynomials there, and they more or less show that either they're a complete intersection or a generalized row of the Jacobian has to have a very small line. They can't be too non-singular. It's this faulting dichotomy a little bit coming. So a one-sentence summary of the answer to Stillman's question, based on their work, is that polynomials of fixed degrees and number, no matter how many variables always seem to act in a very strong sense as if they are polynomials in a relatively small number of variables. More precisely, the polynomials of fixed degree and number, no matter the number of variables, can always be written as polynomials in a bounded number of auxiliary polynomials, which themselves form a regular sequence. So that's what they proved. It's a wonderful theorem. It has been worked over a lot and improved at this point, but they had this fundamental insight which allowed them to solve it. So finally, that's the end. I wanna thank Mel personally and for I think the whole field or wonderful inspiration or friendship, guidance and all the mathematics he's done. Our field would be very, very different. Thank you, that's it. Thank you very much for a very impressive and interesting talk. I have some questions. I would like to make two comments if I may. Can you hear, can other people hear me? I can hear you fine. Can the audience hear me? That's harder to tell. Yeah. Keiji, is this audible? I guess not. Oh well. You can tell me and I can... Yeah, yeah. We can hear you. I think that's talking right now if you can't hear. Okay. We can hear you. So I wanted just to mention that Mel was also extremely generous to young people in the field. He did have a tendency to think that what he was working on was the most important thing in the moment, but nevertheless, if somebody else did something that was interesting or worthwhile, he was really very generous to the person and really gave his ideas, I think, to his students very freely and very generously and that influenced our field a great deal. I think fields aren't changed by a few senior people in them and the tone of the field in commutative algebra has always been particularly friendly. One other remark about this principle that Craig mentioned that of polynomial ring is flat over the ring generated by a regular sequence. I've been working with Jeremy Gray on history of mathematics a little bit lately and I've only very recently learned that this was proven by Ostrowski who's famous for extension of valuations. In 1922, he was interested in when free resolution, which was a big new idea at the time over a subring could be extend, become the same free resolution over a bigger ring and he proved exactly in the case of polynomial rings over rings that it had to be a regular sequence that generated the subring. So I'm not a sport runner of all this work. So Kitchie, did you all hear what David said? Yes. Yes. Okay, so I don't have to try to repeat any of them. I had no idea about Ostrowski, that's very interesting. So I have a question. Hi Craig, it's Neil. Hi, Neil, hi. Hi. So I suspect what your answer will be here but I wonder about the periodization of 1985 to 2007. What marks 1985 and what marks 2007 in your opinion? Well, 1986. Well, I was just trying to leave out tight closure and we really began work on tight closure in 1986 and I think our last paper together actually appeared in 2009 but our last paper on symbolic stuff connected with tight closure appeared in 2007. So that seemed like a good date to cut that off. I don't want to wrong mail. It certainly did lots of other work during that time but I think if you look at his papers then most of them in those years were about. I looked at all of his top-sided papers by the way and I think I mentioned every one of his top-sided papers that doesn't have to do with tight closure. His second most cited paper is from his thesis on characterization of the apology. Spectrum. Does that answer your question and is it what you thought? Okay. Thank you. Okay, okay. Yeah, thank you very much. Okay, okay, so okay. So I think every audience is very useful and so every audience has lots of, yeah, program to consider and may I ask one question? So what is your first impression to Mel? Say that again. What is your first impression of Mel? Oh, of Mel? When? You mean when I met him at the airport? Yeah. Well, I don't know what I had pictured in my head but it wasn't what he looked like. But you know, at that conference, I mean everybody was sort of, everybody in my own sphere was buzzing a lot of the work he had done. And the work of Booksbomb and Eisenberg was coming out at the same time. It was very exciting time for me. Okay. Thank you very much. Mel has a way of looking at you sometimes and you sort of feel like he's trying to figure out a way to say something nice after you just said something really stupid. With me, it wasn't always so easy. Okay, thank you very much. You're welcome. Another questions? Okay. One more question. Hello, Professor Muniki, this is Arindam. Oh, hi. Hi. I just have one question about Professor Hochster's thesis, the characterization of topological spaces which are speck. Is there any progress on that? Like characterization of topological space which are speck of, say, Gwen Macaulay ring or any progress in that direction? I'm not an expert. So certainly he did other things along that line and other people have also. That's a very good question. I think in his, in the Nebraska notes in 73 that I mentioned, he mentioned some open problems in those notes. I just don't remember the answer. I think Colin Macaulay has not been. There is, I think that's unsold. That's a great question. Thank you. But I could be wrong. I'm not an expert on it. That's one of the few papers of Mel I've never gotten through. Thank you, Greg. This is Irina. Thank you for a very nice talk. This is more of a public service announcement. Mel always says, a reasoner, not Reisner. And he insisted on that, that he would know. So yeah, Stanley Reisner rings. Yes, okay. Thank you. I knew I'd make a mistake on that. I never, it looks like Reisner, and I always say a reasoner. Thank you. That is an open service. Okay, thank you very much. So we'd like to continue this forever, but we have to finish. And okay, thank you again. So.