 Thank you very much. It's a great pleasure to be here at the ICTP where for me over the years much mathematics has happened and Jason yesterday gave an absolutely splendid lecture provoked by a question I asked him in 2009 when we were here in this very room. But anyway, enough reminiscing. Let me see if I can actually make this move. So I want to just share with you a couple of small ideas and observations that we've developed or come up with over the last few years. This is, if this worked, yes, joint work with Eleni Panayotu, who unfortunately can't be with us. She's busy teaching hordes of students mathematics in Santa Barbara. We became, we started working on sort of nodding and linking and entanglement in polymer gels some years ago and in one of those and in the principal model or for simulation, of course, one uses periodic boundary conditions. And so I'm just going to focus on one-dimensional, let me, one degree of periodicity. So in fact in those settings I think of them as being sort of a long tube. Here I've drawn it where the cell is actually like a cube but in other situations I like to think of it as like a round tube with a disc cross section. All right, the mathematics is roughly the same, the simulation is similar, but and this example or this setting I think is maybe a little bit nicer, a little more natural. So when I think of a tube though, the tube could be something with a square cross section or could have a disc cross section. Olympic systems are those that consist of ring polymers inside or rings. Yunnan, is Yunnan here someplace? Yeah, wasn't it you and Javier Arsoaga had some papers on like Olympic systems, mitochondria? Yes, okay, good. The memory is still working. Okay, so I was reading one of these papers and a student came into my office, a very bright student and needed a project. And so I was looking at your paper and I said, oh okay, let's look at systems of this character employing the methods that Eleni and I had developed. And so I'm going to tell you about the work with Spencer Igram who's now starting a doctoral program at the University of Indiana. So okay, so the context is these tubes and they're going to contain either, let's see here I'm still struggling, filaments that is open chains or closed circular chains, okay. And we're, they're going to be periodic, okay. And we're going to vary the cross sectional scale. So here in these cases here I've got sort of rectangular square scale. So we're going to make, we have a persistence length, the length of the edges in our polygonal chains. And we're going to vary the geometry of the tube cross section in terms of that length scale, okay. So we'll start from very skinny ones to ones that are basically infinite. We're going to vary the cell length. So that has to do with sort of the density. How much stuff we pack into a single cell that we then translate to create our periodic system. So that's a question of varying density. And then alignment control. So we could have no alignment, and that would be a totally random system, let's say, or we could imagine that there are some natural phenomena occurring that tends to make the direction, these are somewhat oriented, if you wish, moving in a way parallel to your axis. So we're going to look at entanglement or linking. In particular, we measure our entanglement in terms of linking with these kinds of things varying. All right. So let's see, did I advance that cell or not? I can't tell. Try it this way. Yes. Okay. So the objective is to understand in the degree of entanglement of these filaments, either rings or arcs, filaments closed or open in terms of the cross-sectional scale, the density, and this alignment control. And I forget what I'm supposed to stop, but I'm going to talk very slowly so that I use up all the available time. Because I only have two or three observations to share with you. I like teaching a calculus class. If you tell them more than two or three things, it's hopeless. So I like to use that motto. Let's see here. I don't know where to point this. Oh, there we go. Okay. Let's go back. Right. So linking. So I like to use the Gauss linking definition for linking numbers. And that's because of an observation that's occurred to many over many years. Namely, it doesn't depend. All you need to have are two oriented filaments in space. They don't need to be closed rings. You need to learn to be happy with real numbers as being a linking number. And this has the wonderful property that they depend, oops, shoot, hit the wrong button, continuously on the data. Namely, the filaments. You move the filaments a little, the linking numbers move a little and so on. So that's a very useful formula. In fact, there is a, for polygonal filaments, there's a closed form that's very rapidly calculated. The periodic linking number is then to say take one of those filaments and then calculate how it links with all the other, with the translates of all the others. Well, the point is our filaments are always compact. And since they're compact, then the other filament eventually is going to translate so far into the distance that its contribution should be irrelevant. And Eleni in her doctoral dissertation proved that that's exactly right, whether you're in one, two or three dimensions, that this periodic linking number is actually well defined. It converges and satisfies this, it's symmetric and so on. Turns out to be a really, really challenging problem. And that was really the, one of the, that's the major accomplishment, I think, of one of that part of her doctoral dissertation is the fact that these things that seem obvious in fact actually are well defined. So we're going to use a periodic linking number of the linking between two, two chains. But we also want the self-linking number. And the self-linking number is similarly defined in terms of a Gaussian writhe term and a torsion term. And also those are well defined in a periodic system. If you take a chain, let's call it filament L, you take the self-linking number of one of its component constituent pieces and add the linking number of itself with all translates of it other than itself. Okay? And so then you have a periodic self-linking number and it satisfies all those wonderful properties. Okay. So we have self-linking numbers and we have linking numbers. Now what? We've got a gel. We've got a system with a periodic system in which we've got some finite number of filaments generating the entire system. So we want to, we want to quantify the entanglement of the system or if you wish the linking present in the system. And so what we do is we create then the sort of linking matrix. And so our linking matrix is just the matrix of those terms. Okay? So that's a matrix that encapsulates then all of the entanglement present in the system. And so it's the derived quantities of this linking matrix that we're going to focus on eventually. Well, not exactly, but that's going to, this plays a principal component in this and I'll explain how. Right. What is next? Hmm. I don't know. There it is. Ah, okay. So just a quick reminder about, a quick reminder. Maybe you weren't aware of this. So some simple observations. That is, if you start with a system where you have basically a cell with a, a single chain in it, then you start, so you can get the sort of linking matrix. If you glue M of these cells together, as we did here with two, see here, if I look at the periodic system, I'd say, well, it's going to be generated by a single chain because this piece gets matched up with that piece and I've got one ring. Here, I've got two rings. I've got this one that's intact. If I imagine now this whole thing here is going to be my basic cell, I've got two, I've got this one, then this one and this one and so on. So if I put together M cells into a single cell, gluing these M copies together, then the linking matrix that I get from such a system is a symmetric, centrosymmetric matrix. Now, I imagine, oh, there is chalk here, okay. That you all know what a symmetric matrix is, but you may not recall what a centrosymmetric matrix is. So a centrosymmetric matrix basically has a center. So it has an entry in the center, which means it has odd dimension and it's symmetric around the center. So this is the same as that. This is the same as that and so on. Okay. So it's symmetric around the center. I'm sure you can't read that. It's too small, but such matrices have absolutely have some very curious properties that one can exploit and have consequences for things like eigenvalues and eigenvectors. So I'm not going to go into that. Just point that particular surprising feature out to you. There we go. So in fact now what we do is we often we will take larger and larger blocks and this is in the hope of understanding what happens asymptotically to the entanglement as we increase the scale of our of our cell size. And basically what we see here is encapsulated in the fact that the linking matrix for the first part is a separate block in there which tells me then that the eigenvalues or the linking that's entangled that's encapsulated just in that first cell persists to infinity. So it's a stable part of that and the rest becomes rather more complicated and is hidden away down in this elf. But the single cell linking that you get persists as you sort of increase the scale of your system. So that's a curious feature that you might be able to see here later on. Now okay so Olympic systems. So one there's a sort of simple point here that I wanted to make. Namely with Olympic systems when things are very well not very dense there's no linking. Okay and then as you make them more dense at a certain point suddenly everything is linked together. This point is about if I remember right 0.08. So right in here is where you go from no linking to total linking and then the sort of the sort of amount or the calculation of the amount of linking present and it's quite stable thereafter. Where this phenomena is important and here's another way of expressing that is in the area of percolation and I can't recall if you guys use that term or not but that's a sort of like a universal kind of phenomena that arises in the study of percolation or of such systems. Okay so that and what I'm measuring these so since it's one-dimensional my matrix let me see here. Linking of two chains right so I just have a two by two sort of system here that we're measuring the linking between that's present in the system. So this is sort of observed point in the one-dimensional system then is sort of like zero and then you get you can't see it but this is the first place where you get a non-zero value and then as you look across the entire population of such systems you see the probability of linking keeps increasing as the function of the density. Now not all that interesting or surprising and if this were the whole story with you I wouldn't really bother sharing this with you but in fact with Spencer Ingram and Eleni we went to look at two-dimensional systems so now rather than one p periodic boundary condition now we have a laminar kind of system which is much like which is really like what Arsuaga and Diao and others have been looking at related to the biological system in the mitochondria it's a two-dimensional system and so we did our analysis and our linking matrix in the two-dimensional system and we discovered as we look at the the eigen values of our matrix that in fact you can detect before you get a two-dimensional saturation you get a one-dimensional saturation that is to say if you look at the matrix and you look at its eigen values and you get a first non-zero eigen value so the eigen vector then gives you a direction in which the the rings in that these are all unnotted and small unnotted rings they're very much like like circles that are bent and twisted a little bit so they were not doing random stuff so much but they in some dimension they start hooking up and so there will be a one-dimensional linked infinitely linked system parallel another one another one and another one so you see a one-dimensional kind of filamental linking system that occurs and dominates up until another point when a second eigen value comes into play and then creates a two-dimensional saturated system okay so you can see an evolution in the underlying structure of the saturate of the entangled system in one case it starts looking just like parallel lines in some direction parallel strands and things like and then it becomes a big like a like a blanket okay well of course having seen that in two dimensions we ask okay what happens in three and by golly the same sort of phenomena happens in three let me go back I can so the at 0.08 is when you first get the when you get the first infinitely entangled subsystem and so in some vocabulary that's percolation that's the point of where this universal percolation occurs but what we're saying is that it's a little one can look at this and see a more subtle picture if you wish but where there's a first sort of infinite system and where is one-dimensional and then later you start getting this two-dimensional system kicking in and then similarly in three dimensions so there is I think a very powerful utilization of the periodic linking matrix analysis and the associated eigenvalues to try to characterize capture the character the capture the character of the nature of the entanglement present in the system in a way that's maybe not accessible from other perspectives at least to my knowledge so that's one idea so I have another 20 minutes or something for a second idea and so here's just another graph of again valence right what do I mean by valence it's the number if you take one of these rings it's a number of other rings that are linked with it so valence zero means no linking at all and here is this this universal point that I mentioned 0.08 density where you start where you get the linking of two rings and that persists for a while and then you start getting higher and higher density by the way these numbers of course are not you know the variation goes from 0.08 to to one terms of the density but you see this linking goes up you know quite how to say rate not regular but bounded fashion okay controlled again by the the character of the the size of these rings okay so second theme so this was stimulated then by papers of I'm reading this work of Renzo on sort of fluid flow and entanglement and fluid flow and felicity in the relationship that Eleni and I have been interested in between helicity and linking in entanglement is measured by this sort of periodic linking and so I'm going to focus our attention on filaments single change or filaments of length 25 okay so these are open they're going to occur in a tube of not unit radius in these pictures it's units unit radius but these are really round tubes that just haven't drawn in the circles of some radius and here the radius is one and then we have this alignment constraint when I when I have zero means no constraint at all so it's totally random so this is like a random walk unit step size consigned to a tube of radius one okay and then we take one of those well you see multiple colors there I yeah they're all the same so this is multiple colors represent different translations and then if I invoke and this alignment constraint which scales from zero to being totally random to one being a straight rod so total constraint okay if my constraint is point five then this is what you will see and so you'll see different slopes and sometimes you'll see it can come it can go back a little bit and so on so we're going to look at again an analysis of the entanglement that we see under the variation of both the tube size and the alignment constraint okay so just to give some sort of geometric sense of this one of the things we should look at is a radius of gyration and the other is the diameter and I should say what I mean by diameter if you have a chain by diameter I mean the largest distance between two vertices in the chain okay so it's like a graph diameter of the graph okay so let's look at this when the alignment is like close to one then it's pretty close to straight and so the diameter is basically 25 okay so it's sort of basically stretched out when the alignment is zero then the diameter is maybe a less than maybe about eight or so and as I increase the radius of my tube so this is when the tube is really really tight so this is probably like the diameter that you would expect to get if you were doing a random walk on a line of step 25 as the radius gets larger so this is now 5 so the radius is 5 so that sort of diameter is like 10 then you're sort of approaching a random walk in three space and so you'd expect to see a diameter that's as close to the expected number there okay and then as you vary the alignment of course you should see differing diameters one feature of this as you'll see this it's quite stable the dependence on the radius of the tube is really quite sensitive here notice that the radius of gyration of the random one is the smallest and of course the radius of a gyration of the aligned one is the largest because it's stretched out as far as it possibly can be so the order sort of changes okay well does this make sense and so here is are some pictures then of length 25 chains with alignment constrained at 0.25 but where I vary the radius from a quarter up to one so here the pictures aren't really to scale otherwise I'd never get them on a screen but you can see that as the the radial confinement in how to say it de diminishes I wish if the radius gets larger and larger then the character of these filaments evolves sort of closer and closer to the sort of free space kind of behavior similarly here I vary the so the radius is a quarter in the alignment scale is from random to about 0.75 ran again the scale there the scale is not the same because we know here this is going to be to be relatively confined right because it's random and as I increase the the alignment then they stretch so here this tube is probably at length about 25 and so to get it on the same scale that's what we had to do so this is there's a little back and forth motion in there but it's mostly linear okay so all of this seems quite banal actually no surprises so why would I bother telling you about this so there's the pictures that we had and our question is okay what this what is the consequence of this for the entanglement of a system then can subject to these constraints so okay absolute self-linking so remember we just caught we describe the periodic self-linking we take the absolute value of the self-linking because we're going to do an average and if we do the average just because of the fact that we could do mirror symmetries the average would always be 0 so we take the absolute value of the self-linking and see how that depends on the these parameters so once again I see not a huge amount of variation as I change the radius of my tube which we find somewhat curious also somewhat curious is that when I look at random configurations I get the smallest self-linking whereas the largest self-linking comes with the largest alignment force so these things are getting stretched out and yet and and have and have the largest self-linking okay granted we're going from 1.1 to 1.6 so it's not like an earth shattering amount of change but it's really quite persistent evolution from a random self-linking which is maybe a baseline but as it stretches out the self-linking increases so this I would propose to you it's a little curious at least it wasn't what I had expected and so cause me to well actually what it caused me to think is we screwed up somehow that this couldn't possibly be right and so we spent some time checking things and then trying to figure out why and so what I'd like to share with you this morning is well why is this the case ooh okay it's the torsion that dominates I think here is the torsion for the the torsion for the random one and here's the torsion for the alignment and the aligned one has more torsion to it now recall somebody a few days ago defined how to think of the torsion you take three consecutive edges the first two determine a plane and then the last two determine a plane and it's the angle between those planes well is measured by the angle between their normals but anyway it's the angle between those two planes so it tells you that as you're sort of going you imagine your chain sort of going your filament going in this direction it's sort of going like this right it's sort of doing and we keep track of the sign by the way it measures the amount of twisting present in the chain the other contributing factor right to the self-linking is the Gaussian right coming from the curvature oops wrong thing there we go and notice that as one expects that term is very very tiny for things that are aligned and isn't all that large for things that are random so in fact what's happening is that the portion is what's dominating the self-linking that's so the torsion's dominating the self-linking and so why is that and the our answer is well it's due to the fact that when so this is going to happen with the linking to so it's going to be the same answer that the stretched out ones link with or if you wish entangle with longer segments of the other filaments when you're compact and you're sort of random you're only you're only entangled with those that are right in your immediate neighborhood okay the ones that you have contact with but as you stretch out you have contact with more and so on the net then you get the entanglement increases with alignment and persists with alignment because you are in fact interacting with more filaments and so it would seem then that a more aligned system would be more powerfully entangled than a random system for that reason okay let's see what happens and here it is then it's exactly what happens with the eigenvalues right you'll see that the smallest eigenvalue comes from the random system and the largest is from the more aligned system and then the variation between and it varies you know not a great spectrum here but a significant spectrum so this is the largest eigenvalue in we're looking at a three-chain system three independent filaments of length 25 well somehow okay I wonder do I have another option oh I did second largest right so the second largest really drops down quite a bit and but a similar relationship the third one even smaller but still an important three eigenvalues and they're really quite bunch together so if we look at just a sort of you know the random ones we can see that the same sort of separation between the smallest and the and the and the largest eigenvalue gives you a sort of spectrum of the quality of the entanglement or the linking that's present in the present in the system here's what happens that was a random case why right so this is the random case right alignment is at 0 0 what happens when I stretch it out everything shifts up a bit and even more when I get really closer right so so contrary at least to my intuition I'm expecting largest entanglement if you wish for aligned systems sort of fibrous systems okay so now someplace here just another slide that shows you this kind of how how they relate when I change the tube radius the stability there and so basically what I wanted to show you share with you is how Eleni Panajotu's periodic linking matrix can be used to study both Olympic systems and tease out from that the evolution of the sort of nature of if you wish entanglement in one two and three-dimensional systems and also how this bears it gives one an ability to study the sort of entanglement in a one-dimensional periodic boundary system of filaments so those are the folks I start we started working of course some time ago with Doris the Adoro and Christos Zerminakis Spencer is now off to graduate school and then of course Laney is back in Santa Barbara teaching I'm having trouble with this so anyway thank you for your attention