 Okay. Great. Good afternoon, everyone, and thanks to the organizer for giving me the chance to talk about my work. I work on high-density crowds. There are several reasons why high-density crowds are interesting. One is that you can spend your afternoon watching rock concerts on YouTube and while you're at work. And the other one is that at high-density, that's when you're most likely to see crowd disasters. And this video was taken during Love Parade Music Festival in 2010. In this area that you see in the video, and pretty much at the time the video was taken, 21 people died and more than 100 reported injuries in a collective phenomenon that's known as crowd turbulence. Crowd turbulence is something that we see when crowd density is really high, like above 6 or 7 people per square meter. And it's described as basically people losing the ability of controlling their movement. And density waves, like small fluctuations, have been amplified into density waves that move people away from their position, crashing them into barriers, or they make them fall. And very high pressures can be developed by this kind of phenomena. And sometimes in this occasion people die basically standing on their feet, crashed by the crowd itself. This video is studies from concert by Oasis in Manchester in 2005, 60,000 attendees. And you can see what I mean with people losing the ability of controlling their motion really. And now you're going to see what I mean with a density wave. No, this is a density wave. So it bounces at the stage and it bounces back, there is people falling here. Yeah, very pleasant. So at high density like that, I don't know if you've ever been in a concert like that, but it's very hard to even scratch your nose. So the good thing is that we can kind of discard psychology and physics of the contact between bodies is what really matters, is what dominates this situation. So what I do in my research is try to understand what is the connection between physical interactions and the way people are packed, and the way small fluctuations get amplified into these density waves and uncontrolled collective motion that you see in the video. And I try to find measures from physics again to try to understand this and to try to predict these events and ultimately to prevent them. The general framework is the equivalence between crowds and granular materials. This is used a lot, for example, for modeling crowds, like force models. But in particular, in the field of general granular media, there is a method that is called mode analysis. And it does exactly the job of linking the local structure of a granular packing with its response at large scale to perturbations, for example, like vibrations or shear compression. And the core of this technique is it's basically to take the trajectories of each grain, to compute the correlation matrix between the fluctuations of the particles around an equilibrated position, and then to extract the eigenvectors and the eigenvalues of this matrix. The eigenvectors are called eigenmodes, E of M, and they make a basis for the dynamics, so you can compose the dynamics on these modes. And the modes are basically displacement vector fields that describe motion for each particle. And each vector field corresponds to an excitation frequency at equilibrium. This is proportional to the eigenvalue. And if you believe me, the eigenvectors corresponding to the lowest excitation energy describe collective motions, describe the preferential directions of relaxation of the system, and the collective response to perturbations. So we took this methodology, and we tried to apply it to high density crowds. So a couple of years ago, we showed that for a simple model into dimensions like a force model, it was possible to extract these eigenmodes, and they were, they were, we were able to predict some collective motion occurring in crowds, and to see, to explain density waves, and to find unstable areas where people were most likely to fall. And this work was in collaboration with Jesse Silver, very good Harvard, and with my PhD advisor, David Sumter. However, well, people kept dying and getting crushed. So that was the starting point to look at real crowds. So this video is again from YouTube, is the same concert in Manchester. I don't know if you can see the little arrows, but we used PIV to, to track motion because it's almost impossible to track individual people. And so we have some frames where we see turbulence, and at some point we have the wave that goes from here to come now. Yes. So there is this wave that bounces to, to the stage, and then some part of the way is deviated down. So you know everything about a particle image velocity by now. And the thing is we divide basically our frame, our movie, into, into set of frame. One is just turbulence. And from that, we treat each point, each point of the grid of PIV as if it was a grain, basically. And we extract the modes, again, modes and again values. And the question is, are we able from this to predict the emergence of a wave in real crowds? So we, when we plot the modes, we see that actually mode number one describes a collective motion towards the stage and then along the stage. And if you remember, this would be the mode, like the collective response of the system that would appear at the lowest excitation energy. So it's also the most likely collective motion to occur. And you also know everything about correlation function, the velocity-velocity correlation function. This is mode number and this is distance. And we see that the first mode is quite long-distance correlated. A person is about four pixels and this arrives to 150 pixels. So it means I push someone here and someone dies over there and I don't even notice. Another thing we can look at is how to reconstruct the dynamics from the Egan modes. So the first thing we do is to project the dynamics on the set of Egan modes. We find the projection coefficients and then we are constructed dynamics with an increasing number of modes and we find that about 190 modes out of a thousand are almost satisfactory to reproduce what we see in the real dynamics. So as regards the spatial properties of our system, we can see that mode analysis gives us an explanation of density waves in real crowds and we can extract from turbulence. So from the motion of the system before the waves the fact that a wave is likely to appear and how it will be directed. And also we are able to reconstruct the dynamics both during turbulence and during wave propagation. However, if we want to prevent these kind of things from happening, we need to look at time properties, temporal properties. So for example one wants to know how mode number one gets excited and possibly when it's going to be excited can we predict really in time these things from happening. And so I started looking into time series and temporal signatures of density waves. There are two good candidates to look into temporal behavior. One are the projection coefficients and one other it's called modes explanatory power. They are basically projection coefficients but they are normalized by the size of the displacement by the dynamics. So they take values from zero to one and they're a bit of a nicer quantity to look at because basically A square it's telling us at each time step to what extent, what is the proportion of dynamics of motion that is explained by mode for example by each mode. So we plot A square this is mode number and this is time and we have some time frames for turbulence and some for the density waves. So during turbulence we see that this phase is characterized by a superposition of modes and low energy modes which describe collective motion are more active. They explain more dynamics than higher modes and what is really interesting is the behavior of C square of the square coefficients which are behaving linearly in the eigenvalue. And this is interesting because at the equilibrium this is what the behavior we would expect but there are no real reasons to believe that our crowd is an equilibrium system. So this quantity is basically telling us that we see some kind of signature of equilibrium during turbulence. When we look at the wave we see that mode number one is basically explaining most part of the dynamics and at the other modes are much less important and also we see that the wave it represents a deviation from this equilibrium like behavior that we see during turbulence. So then I looked into more detail in this C square distribution and I tried to fit a power law over time and this is the coefficient the exponent of this power law distribution and what we find is that there are actually two more occasions where there we see some deviation from this one over omega square behavior and we see that this corresponds to the stronger activation of one number one. So go back to the movie I don't know if you believe me but this is something that happens here there are two fluctuations at these times that are aligned with m equal to one and we could barely see by eye like we're looking at the movie we wouldn't notice that. So this is nice because mode analysis picks up this kind of fluctuations but now the question is what is the difference between small fluctuations along m equal to one and a wave so why some fluctuations they just disperse why the wave becomes so big and gets to the stage and moves a lot of people. So it looks like the lowest energy modes the first modes are the most important one so I looked into detail into a square alpha square and so we find the wave here and then we find also these two fluctuations which we can call in a fancy way like energy injection induced by a human activity so just people pushing and then what we see is that this energy injection seems to trigger some some kind of pattern into the lowest energy mode that look really like cascades and we see that the wave starts exactly at the half of the last cascade and so the question is are these cascades some kind of mechanism that allow the wave to propagate do they create some some positive interference some some constructive interference so I've been looking into all the possible time series here we see the peaks for mode number one but what is very interesting to look at is this back line up here which is the mode's total power so modes are basis for the dynamics so we would expect this quantity to be one all the time but what you see is that it actually drops three times mostly during this energy injection and during the wave so what this quantity is telling us is that we are seeing non-harmonic behavior that's because the agar modes are actually extracted in a harmonic approximation of the dynamics so the particles should not be moving too far away or in a weird way just as springs around an equilibrium position and so yeah so what we are seeing here is non-harmonic effect and this something like this has been observed also for granular media and this happens when granular media are perturbed with an amplitude which is larger than a certain critical amplitude and non-harmonic effects are actual power transmission between modes so what we are seeing the cascades are power transmission between modes excited by this energy injection so by people pushing so all together with mode analysis we are able to identify two phases for high density crowds one is turbulence as a superposition of modes and has some equilibrium like signature and then we can see that we found that the wave is basically the excitation of the first agon modes and looking at time series we can see some some temporal patterns that seems to come before the wave and to allow the wave to spread and these are those power cascades that we see these situations like it's characterized by two main features one is the fact that the first modes modes increase the activity before the wave propagates and this we see it from from the deviation from equilibrium like behavior and the second feature is non-harmonic behavior that we capture by a square so an actual question is here is this pattern that we see before the wave and those features some kind of conditions for a wave to propagate and if so we could actually use these measures for forecasting what would happen in a crowd at in real time so we would just need to film the crowd and do some analysis and and find out sometimes series that can give us early warning signals so what I would like to have on my screen if I would be one of those people monitoring crowd one is for sure a square the other measure I was thinking about is a measure that detects deviation from equilibrium like behavior so I would compare the distribution of a square from the average distribution during turbulence and then sigma the standard deviation up to mode M would just basically tell me about how heterogeneous is distributed without heterogeneously is the power distributed among modes so if gamma is high basically it means modes are exchanging power so if I try to plot this for our case I can do it for the first four modes and I see that this picks up basically power cascades and also I can compare mode number the first four modes with the first hundred and ninety modes and I see that there is kind of a transition when the wave propagates so of course this is just one movie we would need to look at many more crowds and compare them the problem is those events well it's also a good thing that those events are kind of rare but still a very big constraint on this research is the lack of high quality data and I guess this was everything I wanted to say thanks thank you very much I would like to ask