 is Manohar Kumar from IIT, Kharagpur, and he will talk on the regularity of symbolic and ordinary parts of weighted-oriented graphs. Thank you. First of all, I would like to thank the organizers for providing me this opportunity to present my work. And this is a joint work with my supervisor, Dr. Ramakrishna Nanduri. And the title of the talk is On Regulatory of Symbolic and Ordinary Powers of Weighted-Oriented Graphs. OK. So let me begin with some definitions and notations, which is required for the talk. And then I'll be talking about regularity comparison of symbolic and ordinary powers of weighted-oriented graphs. So here is the definition of symbolic power. So let R be an Eutherian ring. Then kth symbolic power of an ideal is nothing but take ordinary power and extend it to the localization Rp and contract it back to the R and take intersection over all the minimal primes of R mod i. Here one can take all associated primes also. But I'll be talking only about symbolic power in terms of minimal primes. And we have seen about the Castileo-Mamford regularity in previous talks. So here I'll be talking about regularity of symbolic powers and ordinary powers of some classes of non-square free monomial ideals via age ideals of weighted-oriented graphs. So let us see what is weighted-oriented graph. So take a simple graph and put orientation on its edges and weights on its vertices. You will get the weighted-oriented graph. And then we have an age ideal. So what we do, we identify all the vertices as the variables of a polynomial ring. Then we have a age ideal of a weighted-oriented graph which is nothing but ideal generated by monomials x i, x j reach to weight of x j corresponding to age x i to x j. So here we are denoting the age with the ordered pairs. It means that we have our age from x i to x j. Okay. And here we have one notation that B plus, B plus is collection of all the vertices which has weight greater than one. And here is one more important thing that if B plus are things. So I'm talking about those vertices. So like C, this is a graph. Suppose we have orientation like this and suppose we have one more orient age. If I say age is towards this, then I will not take weight greater than one. Weight means here I will not take any weight, like two or anything. So we have to take here one. But if I change the direction here and here, so all the age is going towards the vertices. So here we are allowed to take any weight. You can take even one also. Okay, so that's what. So these vertices are called sink, okay? And all the vertices which is going outside, we call it source, okay? Now, so here we have, if B plus are sinks, then ideal age ideal has only minimal associated primes. So now onwards, wherever I will say B plus are sinks, you are free to take symbolic power definition either as a minimal prime definition or associated prime definition, right? Okay, there will be some few notations. I'll be talking where it will be required. So, okay, we are ready to talk about regularity comparison of symbolic and ordinary powers of weighted oriented graphs, okay? So this is means conjecture. It says that for age ideal of any simple graph, regularity of kth symbolic power will be equal to regularity of kth ordinary power, okay? So the one advantage of this equality is as Professor Dipankar said that regularity of powers of homogeneous ideal age asymptotic linear function, but in case of regularity of kth symbolic power, it is not true. So it will be like, in case of monomial ideal, regularity of kth symbolic power will be bounded by linear function, but not linear. So if we know this regularity, at least for those class of ideals, we can say that regularity of kth symbolic power will be asymptotically linear function, okay? So in 2022, Min, Nam, Phong, Tui, and Wu proved this mean conjecture for k equal to two and three. So still we don't know, in general, for four, five, six, but for particular classes of graphs, many researchers have proved this mean conjectures. In case of simple graph, means there would be no any weights on the graph and the orientation. That will be only square free monomial ideals, which is quadratic. So if we ask same question, in case of weighted oriented graph that for any age ideal of a weighted oriented graph, regularity of kth symbolic power will be equal to regularity of kth ordinary power or not. So in this direction, in 2021, Mandala and Pradhan proved that if D is a weighted oriented odd cycle, such that V plus R sinks, then regularity of kth symbolic power will be less than equal to regularity of kth ordinary power. And even they have given one sufficient condition that if cardinality of this V plus is equal to one, then this equality will hold. So means we are saying that in ideal associated with such class of graphs, there will be only one monomials which has degree greater than equal to three, then the equality will hold. So similar results we have proved that regularity of kth symbolic power will be less than equal to regularity of kth ordinary power. For all k greater than equal to two, if a weighted oriented graph have at least one induced subgraph of this form and here I am saying weight of X is greater than one. So this vertex is neither a source nor a sink vertex. There is a one outward degree and one is inward degree. So such induced subgraph, we will have in a very animated only graph for which this inequality will hold and such a nice corollary we will get from this which is this inequality will hold in case of any weighted oriented bipartite graph. Now we will see by one example that whether sometimes the inequality will be strict or not. So here is one example. So I have taken this Cohen Macaulay weighted oriented tree. This was characterized by Jimney, Simi's, Villarreal and others in 2018. So ideal associated with this graph is unmixed ideal. And so here you can see for these type of weights. So if you say to write the ideals, okay. So the ideals associated to this graph will be Y1 X1 raised to weight six, Y2 raised to X2, raised to weight four, Y3 X3 raised to weight seven, X1, X2 raised to, okay, so we have a weight four and we have one more edge, X2, X3. So for which we have a weight seven. So this, for this ideal, you can see in the slide that equality of, particularity of symbolic and oriented power will not be equal. But if I change the weight of X3 as a three, then the equality of the relative symbolic power will be equal. So I have not changed the underlying graph and orientation, still we can see this. So totally depending on the weights. So still we are not able to figure out what kind of sufficient condition we have to put on the weights so that we can get the equality. But now we'll talk about those weighted oriented graphs which, I mean, which has all the B plus R sinks. Because in case of B plus R sinks symbolic power we have very nicely, I will be listening in the next talk by Dr. Mandal about that how nicely it will be have. So let us see first that what we have seen. So take any weighted oriented graph assuming B plus R sinks means those vertex which has only in degree will take the weight greater than one. For those vertex, I mean, for those graph we have this inequality of K symbolic power will be less than equal to equality of K thordinary power for K equal to two and three. As far as for higher power, four, four, five and six is concerned. So I have taken the complete description of a second and third symbolic power of age ideal of simple graph. So it is not known in case of simple graph even for higher powers. If we know those things also what I have used to prove these theorems, that degree complex of a monomial ideal which was used by Takayama to give combinatorial formula for dimension of local homology. So far that will not work if we know the description also. So now we'll talk about whether for K equal to two equality will hold or not. So here is one, like here we have a notations. So here F will be a family of, but this is of a triangle which will be in the underlying graph. And here N H is nothing but the closed neighborhood of a sub graph of underlying graph. Or even you can say weighted unit graph. And which is nothing but the union of all the closed neighborhood of a vertex of a sub graph. So here is one SAR proper bond for the regularity of small symbolic power which is K equal to two, which says for some class of graphs which says that take any weighted unit graph such that B plus R sinks and underlying graph G has no triangles, induced triangles. Or if triangles are there, then triangles are at most at a distance two from every vertex. So for which this inequality will hold. So this is nothing but regularity of idea square is less than equal to maximum of regularity of second symbolic power comma this term. So there will be so many terms like this because if we have triangles more than one then we have to see accordingly. So what is this term? So some of all the weights of the vertices which is belonging to this closed neighborhood of T which is nothing but the vertices of a triangle minus cardinality of a closed neighborhood of vertices of a triangle plus one. And again, we have to take some of weight of the vertices of a triangle. And that could be for other triangles also same. Okay, so let us see one example. Okay, so this is also a Cohen-McCole weighted unit graph which was characterized by Tyre, Lean, Mode, Reyes and Villariel. So ideal associated to this is also a unmixed ideal and here we can see these are the non-trivial weights and these are the sinks only. And these are like we can see, I have taken the one. You can once I have taken the one you can take any orientation here. It will not affect the ideals. So for this we have seen that the probability of ideal square is equal to 23 which is exactly equal to this sum. So we from here we have got that this is a sharp upper bound and using Macaulay 2 we have got this the probability of second symbolic power which is 22. So from here we have seen that in case of V plus R sink also equality will not hold. So we have given some condition on the weights for which it will be equal. So all the assumptions are same like previous theorem. There is one assumption I have put that whatever the triangles we will have their vertices of a closed neighborhood have only one vertex which has weight greater than one then this equality will hold. And for us such a class of graphs like G has no induced triangles or triangles are at a distance two. In this class of graphs this gap free graph will fall for which it will be also old. And gap free graph is nothing but take compliment of a graph which has no four induced cycle. Okay so I want to summarize the talk which is remaining part of the some classes of graphs which is remaining. Okay so here I have talked about V plus R not sink. You can see this is a greater than one and it is not sink vertex. So in this class one more class is remaining for which I have not able to show the inequality which is you can take X, Y, X, Z. Okay so okay this okay so this way. Here you can take any weight any direction I am taking random both you can choose. So just I am taking if you attach this one then for this class of graphs only remaining means there will be no class of graphs for which induced graph is like this. Then there will be only triangles which has way to greater than once vertices of the triangle. So this class of graphs it is remaining and if we talk about V plus R sink then we know inequality for K equal to two and three we don't know for and we have seen the equality of regularity of symbolic power will not be equal in general. Okay so these are the references. Before we break for lunch just a reminder for all those who gave the talk short talks you have to give a you have to prepare a zero poster. Okay so we'll be back at two o'clock for the afternoon session.