 of the morning, Aisha Koreshi from Istanbul in Turkey, and she'll talk about algebraic and homologic for the first time, if I start again. So I'm happy to introduce the last speaker of the morning, Aisha Koreshi from Istanbul, Turkey, and the title is up there, please. Okay, thank you very much. So first I want to thank the organizers for organizing this wonderful school and workshop and for the invitation. I will talk about polyomina ideas today. So the idea is, so basically the motivation is starts from the ideas generated by T-minors of a matrix, of a genetic matrix. So in other words, the letter, sorry, the determinant ideas and then their quotient rings which are known as the determinant rings, more generally, letter determinant ideas and the letter determinant rings. So as we already saw in the morning in Alessandro's talk, a little bit of reference of ideals of T-minors of a certain matrix. So I'm sure that most of you are already familiar with that. There are some nice references here. So what we do is that, okay, we take a matrix X where the entries are, we can just simply call them variables, they are indeterminates. So let's call it X and then for any fixed T, for any admissible T, we are taking, we are considering all the T-close-tree sub-metrices looking at their determinants, in other words, T-minors. Okay, and then we are generating ideals with these T-minors and these ideals are called determinant ideals and then the quotient ring. So this is denoted by ITX. When we take the quotient here, this is the determinant ring, usually denoted by RTX. More generally, now we also know the definition. I mean, you might have seen it before, one-sided letter. So what is the idea? Now we fix a shape inside this matrix, looks like a letter, so it's called one-sided letter and now we are just restricting ourselves to those T-minors or in other words, those T-close-tree sub-metrices that lie within this shape. And then we generate the ideal, ideal of T-minors called ladder-determinant ideal. We can make it a little bit more general, we can make this shape look like this. So now we have the staircase on both sides, known as two-sided letter and so if I call it Y, so this is the ladder-determinant ring. Okay, so I will put it here. As a generalization, so this one-sided letter and two-sided letter, okay, they were introduced by Abayankar, later it was shown that, okay, these ladder-determinant rings, they are domain by Nara Simhan, Kohan Makore property by Herzog and Chiung, Konka then proved the normality for these shapes. Now the idea is that, okay, we do, we generalize it a little bit, these shapes. And let's say now I am talking about this shape inside the matrix, yeah, and now I want to study the ideals of T-minors, such that the corresponding matrices, they are contained inside this shape. Okay, so just going from here to here, considering a little bit more arbitrary shapes inside the matrix. So here now the words poliomino come from because just to identify these shapes, calling them ladder here and calling them poliomino here. And in the beginning what we are doing, we are restricting ourselves to just T equals to two. So we will be talking about ideas generated by only two minors, so only two cross two sub-metrics because that's the easiest case to deal with, but of course one can define it for any admissible T, you know, as long as there's a T cross T matrix contained inside. So in other words, we are talking about quadratic binomial ideals in the polynomial ring whose variables are coming from the entries of the matrix. Okay, so now talking about the binomial ideas, quadratic binomial ideas, we have seen some other classes also. So the ideas generated by, for example, adjacent two minors, motivated by the applications in algebraic statistics, some of the references I've shown here. And then ideas generated by arbitrary set of two minors in a two cross N matrix. These are very well known as binomial ideas. We saw them multiple times last week in several talks in the school. So for binomial ideal, yeah, we are restricting ourselves to only two rows, two cross N matrix, and we are taking any collection of two minors here. We define all these poliomino ideas. So poliominos are plane figures obtained by joining unit squares that we call cell H to edge. In a cell, they look like this. So what we have here, for example, in the first picture, we can see that we have unit squares. They are joined together with each other. So this unit square, we are calling them cells. And we have five cells. Actually, in all of these poliominos, we have five cells. So they appear basically in recreational mathematics, mostly in combinatorics, talking about their enumeration problem, tiling, also their relation with diapers and most converts appearing in the study of these algebraic languages. But we did here, basically, we just used these shapes of poliominos to put a name here. So the idea is just to study the ideal of two minors and now just to identify these shapes since they look like this. So okay, we just put the name here, use this word here poliominos and call these ideas that later we will generate here as poliominos ideals. So but this is poliominos themselves, they actually appear in mostly in combinatorics. And the most well-known problem about here is motivated by some application in physics, cell growth problem. Given a fixed number of cells, let's say we want to construct all possible poliominos of cells, let's say up to 10. So how many distinct poliominos one can write up to translation, rotation, and reflection, and so on. So this is one of the major problem in interior poliominos studied in combinatorics. And now we will define the poliomino ideas. Okay. So as I said, so what we are going to do, we are going to look at these shapes and we will associate some quadratic binomial ideals, which are coming from the ideals of two minors of a matrix. So first we put some terminology and some notation. We are taking a partial order in N2. When we talk about N2, we will just restrict ourselves to the positive integer points only. Because what is the idea? Why we are talking about N2? Because we can identify the entries of this matrix as points in N2 with the positive indices. Okay. Then we have the definition of an interval in N2. So we take two points and then we write an interval just like as all the integer points within these two fixed points. So for A to B, we are taking everything that is in between them. So our interval will look like this. If I have this point and this point, A and B. So I take all the integer points and it will look like a rectangle in general. Okay. So now if I have an interval of size one, it's a square. So this is what we are calling a cell. For example, this is a cell from ij to i plus one, j plus one. These pink points are vertices of this cell, the corner points. From here we can identify the diagonal and the anti-diagonal corner points. And then the edges are the boundaries of this cell. Okay. So what is a polynomial? The formal definition is we are taking bunch of cells in N2, this unit squares. And if we can reach from one cell to the other cell via connected path, then we say that, okay, it's a poliomino. In other words, we take bunch of these cells. If the shape looks like connected, it's a poliomino. So we want to be able to reach from one cell to the other cell. So for example here, a path from C1 to C8 is shown. So we are moving edge by edge from one cell to the other. So this is a poliomino. Okay. We fix a poliomino, a bunch of cells in N2 that can be also viewed as an increase inside the matrix. And then we write the vertex set of P as a union of all the vertices of all the cells that are contained in our poliomino. And then we assign a polynomial ring to it. So in this polynomial ring, the indices of the variables are coming from the vertices of poliomino. Okay, so just like when we are writing this ring Xij, the variables that are coming from the entries of the matrix. Inner interval. What is an inner interval of a poliomino? When all the cells of that interval belong to our collection, belong to our poliomino, we call this an inner interval. So for example here, this red, the interval shown with the mark with the red lines, this is our inner interval because all the six cells of this interval is contained in our collection. So, I know, what did I do? In this poliomino, for example, this is the cell that I'm not putting in my collection. Okay, here I have an example of an interval which is not an inner interval because this cell in this interval is not contained in the collection. Okay, so what we do now, we take our poliomino, we fix a shape and then we, for every interval, so we have now this diagonal and anti-diagonal entries, so basically we have an hour two cross two sub-metrics, we take is determinant and then this is what we are calling here inner to minor, pressing some wrong button again and again. So, given this interval, we take the determinant of the corresponding matrix and we are calling it an inner minor of the poliomino P. Okay, and now we take all the inner minors of P and we generate an ideal, it's a quadratic binomial ideal, we are calling it a poliomino ideal and the quotient ring denoted by Kp, coordinate ring of the associated poliomino. Okay, so this is the structure that we are going to talk about. So again, the idea is just to study the ideal of two minors of more arbitrary shape inside the matrix, but later on along the study it became, I mean, we noticed that combinatorics of poliomino, apparently it made it very easy to study most of the algebraic and homological properties of this quotient ring. So one can actually describe many invariants in just, in terms of the combinatorics of the associated poliomino. So we will see it in a moment. So I mean, I'm sure that okay, the definition because of letter determinants and ideals and rings, it might be clear, but just to be more precise, I put another example here. So for example, we are taking this red entries and we only want to have all the two minors that we can create from these red entries. So given that one, I am assigning this poliomino with these four cells. So these red marks, these red entries can be identified as the vertices of this poliomino. And then we are writing all the two minors that we can construct here. So this is the poliomino ideal associated to this poliomino mark in the red entries. Okay, so this is the ideal that we are talking about. Okay, good. So of course it generalizes the class of one sided and two sided letter. And also, if you are familiar with hebe rings that are toric rings associated with the distributive lattices. So if we have a planet distributive lattice, we take the hebe ring that can also be identified as a coordinate ring of a certain poliomino of certain shape. Okay, so when we have an ideal generated by an arbitrary set of two minors, of course, one first one would like to know that the quotient ring defined by them if it is a domain, if the ideal is prime, if not, if it is radical or not, or what are the primary components? So this, we just start from the very basic question. We just want to know, okay, when these ideals, they define a domain. In case of two cross N matrix, the case of so-called binomial ideal, we know that they are always radical. You just pick any bunch of two minors, you generate an ideal is always radical. We know the primary decomposition, we know many nice properties. And this is not true as long as we just increase the rows to three. If you take just randomly some two minors, generate an ideal, it needs not to be radical. One can write many easy examples. So we start with this question that, okay, when these ideals of these inner two minors, poliomino ideals, when they are prime, when they are radical, and so on. So I will just now talk about some nice shapes of different poliominos and then what sort of properties that we know so far. So here we have what we call a row convex poliomino. So it means that if you have two cells in a row, you have the whole row inside your poliomino. So this is not column convex, for example, because we have these two in one column, but the middle one is missing. So this is row convex. This one is column convex. If a poliomino is both row and column convex, we just simply call it convex poliomino. So for example, here, this kind of a shape inside the matrix, it can be identified with a convex poliomino. So for these shapes, so this was one of the first results, that these ideas, they define domain and in fact, Kohan-Makohli normal domain. We know the dimension in the terms of the vertices and the number of cells within the poliomino. In fact, they are causal also. More interesting shapes that we can create inside these matrices. So simple poliomino, rather than giving the more combinatorial definition, I've just put some pictures where one can actually understand maybe better. So what do we mean by simple poliomino? So simple poliomino here just means that there is no any hole embedded inside the poliomino. So for example, here, this cell is not inside our collection. So there is like a hole topologically embedded inside the poliomino. So this one is not simple. However, this is simple. There is no hole embedded inside. Okay, so these are kind of like the, because once we start having holes, things become much more complicated. For simple poliominos, it remained the open question for a while, but then Herzog and Madani, and then separately with my collaborators that okay, we were able to show that, in fact, in this case also, we again get a causal Kohan Makole domain of dimension described in the terms of vertices and the cells of the poliomino. The idea is actually that once one can, first we show that this poliomino ring here, this is isomorphic to a semi-group ring with quadratic generators, where these ij's are coming from the vertices of poliomino. So once we identify it in this form, this is much simpler than talking about this more complicated structure. After this identification, so it becomes a little bit easier to talk about it because here we just have quadratic binomials in our toric ring here. Okay, now some other known classes, list of prime poliominos where we know our quotient ring is integral domain. So okay, convex and simple poliominos we know. Then now we allow holes. So if the hole is itself a convex poliomino, here one can again argue with some localization techniques that we have again. Our coordinate ring of poliomino ideal is again an integral domain. Then grid poliominos and thin poliominos, I haven't put the definition, but the names are exactly what one would imagine, grid poliomino, something that looked like grid, and then there are a bunch of holes. Thin poliomino, it really looks thin. So it means that we don't have four cells embedded inside. So we have parts of sort of like this form. So just cells joined together with each other and we do not have sort of like a tetromino for cells in this form inside. So with this thin structure again, these are known to be prime. And poliominos, if they admit a quadratic grammar basis, by using that or identifying them with the suitable lethices that okay, they are again, they define a prime ideal close path poliominos with certain zigzag walk combinatorially that one can define inside poliominos. They're also known to be prime. And then Maskia, Rinaldo and Romeo recently they conjectured that okay, if poliomino does not have the so-called zigzag walk, then the poliomino ideal doesn't matter how many holes are embedded inside, but the poliomino ideal will be prime. But this is a very recent work. And zigzag walk is again something defined completely combinatorially in terms of the language of poliominos. Okay, and there is no any known example of poliomino ideal that is not radical, which is kind of nice because when we are taking this arbitrary minors, even in a three crocent matrix, we know that they are not radical. But as long as we fix this poliomino, we fix a shape on them, apparently, we expect them to be always radical. We do not have any known examples so far. Very recently, Sisto, Navarra and Vir, okay, this is the first paper where the radical property of the poliomino ideals have been discussed. They, in the case of this closed path in poliomino, when they are not prime, they show that it is radical and they also gave the primary decomposition of poliomino ideal in this case. And this is, I think, the only paper where the radical property is discussed. Okay, some other properties that we know for these rings. Characterization of convex poliomino when they have linear resolution and when the first CZG module in the resolution is generated in a, is linearly generated, so it's linearly related poliominos. And then some bounds for the regularity and multiplicity and characterization of convex poliomino that satisfy green laser field condition, NP properties. And then, Czerny-Davies conjecture for simple thin poliominos and they are golden signs, so this conjecture is discussed. So, mostly for convex and simple poliominos and very particular classes of these poliominos. We know some nice, some discussion of some nice homological and algebraic invariance and some other properties, but most of the things, because they are relatively new class of ideals, so most of the things are still unknown. Okay, so I have time, right? Maybe 10 minutes or so. So, I will just say now a little bit about that, how these combinatorics of poliomino happen to be very useful in the study of these ideals of two minor. So, one can actually really use it to describe some, for example, regularity and Hilbert series and other nice properties of these ideals. Okay, convex poliomino. In theory of poliominos, the convexity of a poliomino is measured by the terms of the parts that you define in them. So, in other words, if you can reach from one cell to the other cell inside the poliomino with that most K term, so every time you change the direction, it means that you are taking a turn. So, if you can reach from one cell to the other cell in at most one turn, this is called one convex. If you can reach from one cell to the other cell in at most two turns, okay, it is two convex and so on. So, there is a notion of K convex poliomino. So, this one is the one convex poliomino because of one turn in the shape of L. So, in combinatorics, they are calling it L convex poliomino, which is studied extensively and many nice properties of these poliominos are known, which actually helped to study the algebraic properties, basically, ideals that are generated by miners of these shapes. So, when we have a L convex poliomino by using most of the combinatorial results that are known, apparently one can reconstruct them in the shape of a one-sided leather. So, one can deform them by using these results, identify them as a one-sided leather, show that they define the same algebra, and then we can use what we know from the one-sided leather, translate these properties back to L convex poliomino, and then we have some very nice consequences. So, for example, non-attacking rooks. So, this is a poliomino. By saying non-attacking rooks, just like in the chessboard, we are choosing at most one cell from a column in a row. So, in chessboard of size eight cross eight, we can put maximum eight non-attacking rooks. So, for example, here, if I put these two rooks in this pruned chessboard, I cannot put a third rook, because if I will try to put another rook, it will be either in the same column as these two or in the same row. So, basically, I'm choosing cells in distinct rows and distinct columns. Okay. Now, if I choose a different configuration, I can actually put four rooks in this pruned chessboard. Okay. Given a pruned chessboard, there are very few classes of chessboards that one knows what is the maximum number of rooks that you can put. Okay. For eight cross eight, we know we can put eight maximum, but when you cut it, you create something new. How many rooks can you put? It's a very nice recreational problem with some nice application, but mostly studied in combinatorics. So, let Rp be the number of maximal rooks that we can place on our poliomino P. And then, apparently, for L-convex, regularity is exactly equals to the rook number of the poliomino. And this results then later motivated. That's okay. So, there is this nice interpretation. Can we use more combinatorics of poliominos to describe some other properties of these poliomino ideas and that happened to be also true. So, most of the properties, okay, in case of L-convex poliominos are known, but for example, if you go to two convex, that has not been studied yet, for example. Okay. So, I think I have five more minutes, right? Yeah, okay. I'll try to be quick. So, we take a poliomino P and we denote Rk to be the number of ways of arranging k non-attacking rooks on cells of P. Okay, in how many ways I can put two rooks, in how many ways I can put three rooks, for example, and so on. So, this is the oldest reference that I could find for these rook polinomios that are also related to matching polinomios. I mean, okay, they have been studied, I think, for centuries, these kinds of problems, but the formal definition with association with the matching polinomial up to my knowledge, this was the earliest reference that I could find, but yeah, but this is my own literature service, I'm not so sure. Okay, so, the rook polinomial is defined in combinatorics as following. So, it's a generating function. So, for T to the power k, our coefficient is Rk, which is the number of phase of arranging k non-attacking rooks, okay. And the degree of this is, of course, is then the rook number, the maximum rooks that you can place on your poliomino. So, for example, here, if I take this poliomino in how many ways I can put no rooks, it's like empty, doing nothing, only one way to do it, how many ways you can put one rook, it is exactly equals to the number of cells, which are five here, in how many ways we can arrange two rooks, they are given here, all the possible ways, six of them. In how many ways we can put three rooks here? So, this is the only unique way to do it. Then, okay, so here, the coefficient of T cube is one. So, this is the rook polinomial. So, as I mentioned before, a poliomino is said to be thin if there is no any sub poliomino of this shape. So, there is no square tetromino embedded inside it. So, we have only thin strip of cells just going along. So, for this shape of poliominos, so this is a simple thin poliomino, for example, there are no holes embedded, and there is no any square tetromino embedded inside. So, if we consider this kind of shapes in our matrix, we apparently have the following result. So, we look at the reduced Hilbert series, we look at the H polinomial, the coefficient of the, so actually the H polinomial coincides with the rook polinomial of P. So, this is something which seems a little bit strange, that why? But yeah, apparently, because when we are taking, when we are putting the rooks, we are choosing the distinct cells in the rows and column. And we know, I mean, I haven't mentioned much about Grobner bases, but we know actually quite enough about the Grobner bases of these shapes. In particular, in most of these cases, the set of generator gives you the Grobner bases with some nice monomial orders, where the initial terms can be recognized as either coming from the diagonal terms of a cell or the anti-diagonal terms of a cell. So, for example, if I have something like this shape, a very simple one, this poliomino with these four cells, I take all the minors, so let's say my diagonal terms are coming from this, from the diagonal, sorry for the initial terms are coming from the diagonal corners. And if I choose, for example, this, I put a rook here and rook here. In other words, this diagonal has nothing to do with this diagonal. So, in the initial ideal, we have certain regular sequences embedded inside, which can be identified as an edge ideal also. So, basically, I mean, the proofs, the ideas are coming from this theory. So, it is not so actually unnatural that these kind of results are true, that rook polynomial actually really plays an important role in the properties, algebraic and homological properties of these poliomino ideas. Okay. So, yeah, here the Hilbert series, as I said before, it coincides with the rook polynomial here. Does the similar result hold for any simple poliomino? So, let me actually just start with the definition, sorry, with an example. So, this is a simple poliomino. Now, we have this shape, this square tetromino inside. Right? And so, in how many ways I can put zero rooks? It's an empty set. In how many ways I can put one rook? I can put it on the cell A or B or C and so on. In how many ways I can put two rooks? So, either I put in AD or I can choose B and C and so on. So, this is all the possible arrangements and this R3 is all the possible arrangements to put three rooks. Okay. Once we have these kind of shapes, we have to fix something about the rook arrangement and that we do in the following way. So, this is the rook polynomial here, huh? So, what we do, A and D, they are diagonal corners of an interval such that the anti-diagonal cells here also, they are also inside the poliomino. So, we identify this AD with BC. So, now whenever we have a rectangular interval, so if you are putting rooks in the diagonal cells, we can always put them in the anti-diagonal cells also and obtain another rook configuration. And so, we identify them. Okay, this switching operation, sort of a switching operation. And so, we say that, okay, one configuration can be obtained from the other one. If we can perform some switches and then starting from one, we obtain the other one. This is our equivalence relation. So, we mod out all such configurations and then we define this new polynomial that we are denoting by r tilde. So, up to this equivalence, we are counting all the distinct ways of putting rooks with a certain number of rooks on our poliomino. So, after this, a little bit modification, we gave the conjecture that, okay, for any simple poliomino, when you look at the reduced Hilbert series, the H vector will correspond to this equivalent through configurations. The proof is done for the so-called parallelogram poliominos, which are, in fact, just two-sided letters. L convex poliominos, which is, again, a very wide class, but again, the question remains still open that once we are outside L, we talk about K convexity, so if K is bigger than one, we don't know, and all the poliominos with cells up to, I mean, that have 40, up to 14 cells. And this is, that was then by using Macaulay, and yeah, so this is because of the computational evidence. Okay, I will finish here. Thank you very much. Okay. Are there any short questions? Thank you. Not in all cases. Only for simple poliominos, when there are no holes. And then, so once we allow hole, so there are some classes where we know, but in general, no. Yeah, so for simple poliominos, then some other particular classes, but not always. So this, in case of simple poliominos, to write this conjecture, we are identifying them as non-attacking. Yeah, but in combinatorial references, what I've seen is that depending on the author, sometimes they consider it as attacking, sometimes not. But for simple poliominos, if they are not in this, I mean, there is a gap between, so okay, they are non-attacking. Thank you very much again.