 So recording, got it. So today I'm going to be giving a broad overview about stellarators. I'll start from the very basics about how we can get good confinement with 3D magnetic fields. And I'll talk a little bit about the history and the current status of where our stellarator research is today. So let's see. Here's an overview. So as I said, I'll talk about how we get good magnetic confinement with these 3D fields through stellarators, what ingredients we need to get good confinement, and then how do we put those together through numerical optimization. And then at the end, I'll talk about the context of some current research in stellarators. So we're broadly interested in turtle confinement for nuclear fusion applications. We want to confine a hot plasma for sufficiently long periods of time to get net energy gained through nuclear fusion. But there are other applications that could be considered. For example, there are astrophysical plasmas that are composed of non-neutral plasmas, such as electron-positron plasmas. And these could also be studied with magnetic confinement. Magnetic confinement has also been used to produce hot laboratory plasmas used for basic plasmas studies. And the basic ingredients you need for turtle confinement is to, first of all, get field lines which lie on continuously nested turtle surfaces. Because the guiding center motion largely follows the magnetic field lines on average, if the guiding centers stay close to magnetic surfaces, then they will stay confined. So the most naive way we could think of doing this is taking something like a cylinder and then wrapping it into a torus. So we get no escape of field lines from this turtle region. And so we get something that looks like a hybrid between an onion and a donut. So we get these continuously nested magnetic surfaces. The field lines lie on these surfaces. And the guiding centers largely follow the field lines on average. And we call the short way around the torus, the poital direction, and the long way around the torus, the toroidal direction. So as you may have heard, just getting field lines that lie on magnetic surfaces is not quite enough. And that has to do with the fact that we have magnetic drifts off of field lines. So the lowest order guiding centers follow the field lines on these magnetic surfaces, but then to the next order, they start to drift away. And one of the drifts that we see in magnetic confinement devices is the grad B drift. So if you have a torus, then the field strength is naturally larger on the inboard side than on the outboard side. And that gives rise to a drift that is largely vertical. So if we have a field line or a guiding center that's lying on the surface, the guiding center is going to feel a net drift in the vertical direction. But we can try to cancel this out by getting the guiding center to go from the outside of the torus to the inside of the torus. So if I project this motion onto a plane, it's going to look something like this. So by getting the guiding center to move around the torus, we can get a net cancellation of this drift on average, and that will give us good confinement. So the way that we quantify this is through a number called the rotational transform that tells us the number of colloidal turns of a given field line per toroidal turn of a given field line. And the typical width of the guiding center orbit in comparison to the magnetic surface goes inversely with iota. So we naturally want larger values of iota because this gives us better confinement. We stay more closely to magnetic surfaces. So there are different ways that we can get this iota or rotational transform. And the way to look at this problem is to see the expression for the rotational transform when we're at the very core of the torus. So we're close to the magnetic axis, which is pictured here. So we have this formula, which is from theory by Mercier, and it tells us the ways that we can generate rotational transform if we're close to this magnetic axis. So the most obvious way to do this is by driving some current in the plasma, and this is the way that Togomex generate rotational transform. So if we have a current that lies on the magnetic axis, that gives us a component of the field that goes politely around just from the right hand rule. And so this gives us some rotational transform. The tricky piece about driving current in the plasma is that it's difficult to do in steady state. So you typically have to use some kind of transformer coil using induction. And so this can't be done in steady state. And also the current can give rise to instabilities, which can lead to sudden losses of confinement. And so this talk is going to instead be focused on the stallerator, which uses an alternative method to get rotational transform. So if we take a look at this region that's very close to the magnetic axis, then you can show that the shapes of the magnetic surfaces are actually ellipses. So like this ellipse that's pictured here. And the ellipse rotates as you go around the axis. And this is one way that we can generate rotational transform. So in this expression, delta prime, delta is the angle between the curvature vector and one of the semi minor axes of the ellipse. And so if delta prime is non-zero, this tells us that the ellipse is rotating with respect to the curvature vector. The second way that we can generate rotational transform is by having torsion of the magnetic axis. So torsion is a measure of the non-planarity of a curve. So if you have a tokamak, the magnetic axis lies on a plane. So that doesn't give you any additional rotational transform. But if you have a magnetic axis that wobbles in and out of a plane, this can give you rotational transform. And this was the mechanism that Lyman-Spitzer came up with when he invented the first stolarator. So Lyman-Spitzer was the founder of the Princeton Plasm Physics Laboratory where I am now working. And you can see here a diagram of the figure eight stolarator that he invented. It's fairly simple. It's produced by some planar circular coils that give you this non-planarity of the torus. So if we want to drive rotational transform without having current in the plasma, you can see that there's generally two ways to do this. And they both require breaking of the toroidal symmetry. So if you have a current in the plasma, you can keep your torus axis symmetric. And as soon as you want to try a configuration without current, then you start to break the toroidal symmetry of your device. And that's what we call a stolarator is a device that generates rotational transform without driving current in the plasma, but through 3D shaping instead. So one way to think about the rotational transform is it's analogous to a geometric phase that arises through the motion of the curve through parameter space. So in this animation, what I'm showing is an example magnetic axis. And the purple arrow is pointing in the direction of a nearby field line. So you can see as this arrow is rotating about the curve, it generally doesn't come back to the same position. So if you follow it around, you can see that it's coming back. In some places it rotates very quickly, but then when it gets back to the start, it's generally rotated by a finite amount with respect to the initial condition. And this is what gives rise to the rotational transform. So the stolarator concept was the basis for some early experiments at PPPL. So I discussed the figure eight experiment. This was the basis for the Princeton Model A device in the early 1950s. There were some later experiments at Princeton which tried to improve over some of the limitations of the figure eight by introducing magnetic shear. And that led to this Princeton Model C stolarator. This is a simple race track design where you have straight segments at the ends and then a helical segment at each of the endpoints of the torus. So the earliest stolarator experiments at Princeton exhibited some poor transport properties. So they found that there was some misalignments of the coils in the Princeton Model C stolarator that gave rise to bone-like diffusion. So they weren't necessarily declared an immediate success. And around this time, the Tokamak started to have some early success. So there were measurements from the Soviet, Soviets that they had been able to achieve high temperatures in their Tokamak experiment. And at this point, a lot of the energy and magnetic confinement community began to focus on the Tokamak instead of the stolarator because they had this early success. And at this point, the Princeton Model C stolarator was then turned into a Tokamak experiment. So since this time, so here's a quick timeline. So in the 50s and mid 60s, there were several stolarator experiments at Princeton, but then they were transitioned into largely Tokamak experiments at PPPL. And I think there's some resurgence in interest in the stolarator concept within the fields. And I'll talk about why that is toward the end of my talk and some new developments that we're seeing. Okay, so how do we actually get good confinement in the stolarator? We saw that some of the early stolarator experiments perhaps did not have great confinement. And we'll begin to probe how we can do better as we design a magnetic confinement configuration. So one of the basic ingredients we need in stolarator confinement is magnetic field integrability. So if we take a set of coils that I'm showing here, these are coils that produce the NCSX stolarator configuration. This was a design that was partially constructed at Princeton, but it was never completed. We can just look at a Poincaré section of the magnetic field lines. So here we follow the field lines many times around the device. And every time it intersects a plane at constant throttle angle, we place a mark. And this allows us to see the motion of the field lines. So we can see that we do have some nice magnetic surfaces, but we also have some magnetic islands and some stochasticity at the edge. And here's a large view of what some of these magnetic islands look like. So if we want to have good confinement, as I outlined in the introduction, we want to have a large volume of nested magnetic surfaces because this allows us to maintain a large pressure gradient in the plasma. If we have a large region of chaotic field lines, then the pressure tends to flatten, so it's not as good for confinement. And we call the presence of continuously nested magnetic surfaces integrability, and this is because of the connection with the Hamiltonian nature of field line flow. So it turns out the magnetic field can be written as a Hamiltonian system. So we can write the vector potential in this form. We have some function chi that will act as our Hamiltonian. And we're going to use a coordinate system where rho labels constant toroidal surfaces, but these are not necessarily magnetic surfaces. We're just considering this to be some background coordinate. So we set up some surfaces of constant rho, which are toroidal surfaces, and then we have two angles which parameterize the position on this surface. So then if we write down the equation for the motion of a magnetic field line, where we're parameterizing the trajectory by phi, then we can see that the trajectories are analogous to Hamilton's equations, where chi is acting as our Hamiltonian, theta is our coordinate, rho is like a momentum, and the toroidal angle phi is taking the place of time. So this tells us that if we have a autonomous Hamiltonian, so chi is independent of the toroidal angle phi, then this tells us that we're going to have integrability of our system. So it's a one and a half degree freedom Hamiltonian. If it's autonomous, we get energy conservation, and that energy conservation corresponds to having magnetic surfaces of constant chi. So chi in that case is a surface label. So in the case of axi-symmetry, where all physical quantities are independent of phi, this tells us that we automatically have magnetic surfaces. If we are in a three-dimensional system where the physical quantities can depend on phi, then integrability is no longer guaranteed. And this is the reason that we have to start to worry about the presence of chaotic field lines and magnetic islands. So if we start with a Hamiltonian that is integrable, this first piece in green, and add a non-integrable perturbation that is scaled up by epsilon, you can see that we can start to develop a magnetic island at this 1 over 2 rational surface, and the island grows as we scale up this parameter epsilon. So the phase purchase looks very similar to that of a pendulum because of the Hamiltonian nature of the field line flow. We start to add in multiple perturbations, then we get multiple islands that exist at these different rational surfaces, the 1 over 4, the 2 over 4, and the 3 over 4. And as the widths of the islands increase, they can begin to overlap, which gives rise to a volume of chaos within the magnetic surfaces. So as we get farther away from symmetry, this is something that we have to worry about in designing the magnetic fields to preserve the necessary surfaces that we desire. So there's another benefit of axi-symmetry that we lose as we go toward stellarators, and this comes from the single particle Lagrangian picture. So if we consider the motion of a charged particle in a general magnetic field, we can study it through this general Lagrangian, and if our Lagrangian is independent of the territorial angle phi, which comes through the axi-symmetry property, then this tells us that we get a corresponding conserved momentum, which is p phi. And you can show that if you have a sufficiently strong magnetic fields that p phi is approximately constant on magnetic surfaces. So this tells us that if we have a purely axi-symmetric magnetic field, then we automatically get confinement of our charged particles, and they're essentially stuck to magnetic surfaces. So this is one of the benefits of axi-symmetry, but as I'll discuss, there are ways that we can try to preserve the interability of charged particle motion, even when we go to three-dimensional systems. So in general, we have to be careful to optimize magnetic fields in order to confine three-dimensional particle orbits. So what these pictures show on the right are the orbits of guiding center motion in a tokamak if you add on some error fields. So you start to get some particles that can come trapped in the local minimum of the magnetic field strength. That's the trajectory in pink here. And you can also get charged particles that follow these banana-like orbits in purple, but the banana tips begin to wander because of the symmetry breaking. And there's much more complexity when we go to fully three-dimensional systems on the left. This is a, you can see some particle orbits on a stalerator. Again, you can get some local trapping features and you can get transitions between different types of orbits between the trapped and the passing. And so in general, this is something we like to avoid because then it can give rise to larger diffusion. And this is something that we can study through neoclassical theory. So the drift kinetic equation allows us to study the drift, the guiding center drift motion when we add in collisions. So this is an equation for the distribution function f. And in general, stalerators tend to have larger neoclassical transport than tokamaks. And this is especially true as we go toward lower collisionality. So what this figure shows is a typical calculation of the neoclassical diffusion coefficient for a tokamak in this dashed line and a stalerator. And you can see that there are different regimes where the diffusion scales differently with respect to the normalized collisionality. As we go to higher collisionality, the scaling for the stalerator and the tokamak are pretty similar to each other. But at lower collisionality, we get much more transport in this stalerator configuration. So we get this, the scaling with one over the collision frequency. And this is especially problematic if we want to operate at high temperatures and experiment because it tells us that as we increase the temperature, then the diffusion actually gets worse. So this one over new regime comes because we have particles that get trapped in local wells and they de-correlate through collisions. We can estimate the scaling of this diffusion coefficient by using a random walk like argument. So we can take the characteristic radial step size to be something like the radial drift over the collision frequency and the typical timescale to be one over the collision frequency. And this gives us a diffusion that scales like temperature to the seven halves. So this is definitely a regime that we want to avoid when we're designing a configuration. So there's lots of ingredients we would like to include if we want to design a stalerator. So we want it to be MHD stable. We don't want any large scale instabilities that will degrade the confinement. We want to think about the bootstrap current. This is an effect that also comes from neoclassical theory. So while stalerators are designed to have to not need current in order to get rotational transform, there is a small amount of self driven current that comes through pressure gradients. And this is called the bootstrap current. So some configurations try to minimize the self driven current in order to improve some of the stability properties. We also have to think about the confinement of the most energetic particles. So these are typically the ones that are hardest to confine because they have very low collisionality and they can also experience different resonant properties that we have to take into account. The equilibrium beta limit is something that is a little bit different in a stalerator than in a tokamak. So while in the tokamak, the beta liven is often driven by stability considerations. In a stalerator, it can also be driven by degradation of the magnetic surfaces. So as you increase pressure, you can get some losses of intergrability, which has to be accounted for. We have to think about what the diverter might look like in a potential configuration. And then one of the major challenges is how feasible it is to build the coils. So to get the rotational transform in a stalerator configuration, we have to get the twisting of the field lines by twisting the magnetic surfaces. And this generally requires some magnetic coils that are quite complex in shape. And it's non-trivial to actually construct them in practice. So we have to take into account these type of engineering considerations in our design as well. Okay, so these are the basic ingredients we would like. And so how do we actually put these together? So we want to get intergrability of the guiding center motion. And the surprising thing is that we can actually design our magnetic field to get intergrability, even if we're far away from axisymmetry. So the way to see this is that if we take our standard Lagrangian for charge particle motion and take the limit of strong magnetization, then we get this modified guiding center Lagrangian that only depends on the position on a magnetic surface through the magnetic field strength. So here I'm using a set of coordinates where psi labels magnetic surfaces, and then theta and phi are some set of coordinates on these magnetic surfaces. So if this tells us if we have a symmetry of the magnetic field strength in this particular coordinate system on surfaces, this again gives us a corresponding conserved momentum. So note that the symmetry of the field strength on the surface does not mean that the magnetic surfaces themselves have to be symmetric. It's a separate property. So again, we can show that this conservation of momentum tells us that the charge particles stay confined to magnetic surfaces on average. So there's two terms in this P phi expression. The first is relatively small if we're strongly magnetized. And so we again get confinement to magnetic surfaces analogous to the axi-symmetry case. And we call this property quasi-symmetry. So there are several different varieties of quasi-symmetry. One is quasi-polatal symmetry. So here we have contours of the field strength in this coordinate system that wrap around the torus poloidally. So this this figure that I'm showing is the QPS experiment that was developed at Oak Ridge. So you can see that the shape of the torus is nowhere close to being symmetric, but we still get near symmetry of the magnetic field strength in this coordinate system. And the coils can be quite complex in order to confine this magnetic field. We can also get quasi-helical symmetry. So these are contours of the field strength that wrap around in a helical direction. And the black lines here indicate the approximate symmetry direction. And this is a visualization of the helically symmetric experiment at the University of Wisconsin. So this is currently the only operating experiment that is testing the quasi-symmetry idea. We can also get quasi-axi-symmetry. So this is more analogous to axi-symmetry as in a tokamak. So we get the field strength contours that wrap around the long way around the torus, toroidally. But again, you can see that the shape of the surface is very far from being symmetric. So we know we want to get this quasi-symmetry property, but actually finding magnetic fields that have this property is not in trivial. So to find these magnetic fields, we have to use optimization tools. So typically this takes place in two steps. So in the first step, we try to optimize an MHD equilibrium to get confinement properties that we would desire. So we specify the shape of the boundary of our plasma, like this one that's pictured in yellow. And given the shape of that boundary, the magnetic field everywhere inside is completely determined. So we can use the shape of the boundary of the plasma as our degree of freedom in the optimization problem. So we optimize in this space to try to find magnetic fields that have good confinement properties. So once we find this boundary of the plasma that we would like to build, we can then go and find the coils that support this equilibrium. So this is again an inverse problem. It's an optimization problem. So we want to find the coils that give us b normal on the surface, which is close to zero. So we want the magnetic field to be approximately tangent to the surface when it's provided by these coils. And in the second step, we also have to take into account how complex these coils are, how how buildable they are in practice. So this two-stage optimization approach has been tested experimentally and verified. So I mentioned the helically symmetric experiment to HSX at Wisconsin. Here's another picture of the device. And they've tested that with the quasi-symmetric configuration, you can actually improve the neoclassical confinement. So they have a mirror configuration where they add in some additional coils, which degrade the quasi-symmetry. And here you can compare the mirror configuration to the quasi-helical configuration. And you can see that the thermal diffusivity is actually reduced in the experiment with the optimized configuration. They can also compare with the theoretical calculations of the neoclassical transport. These are the lower lines here. And you can see that in the experiment, there's generally enhanced transport in comparison to theory. And this has to do with microturbulance and turbulent transport that gives rise to additional diffusivity in practice. But nonetheless, we're able to confirm that using this two-stage optimization approach, you can find equilibria with this quasi-symmetry property. So one of the largest advances in the Stellarator is the design and construction of the Wendelstein 7x experiment. This is an experiment in Greifswald, Germany. And it was designed not with quasi-symmetry, but with a related optimization approach called Omnigenity. And so this device just came online in 2015. It was designed for several criteria. So they wanted to have stable equilibria up to high beta. They wanted to have improved fast particle confinement in reducing the plasma bootstrap current. So I'm just going to play a short time-lapse video so that you can see this impressive feat of engineering in the design of W7x. Let's see if this will go through. So here you can see, this is the vacuum vessel for the main chamber. You can see some of the modular coils they're putting in place now. So this is just one half period of the device. You can see the various ports that they used to have experimental access. So on W7x, now that they've turned the machine on, they've been also able to demonstrate that the optimization actually worked. And so what they did is they measured in the experiment the heat flux. And they normalized the heat flux that's computed theoretically from neoclassical theory to the total heat flux that's actually in the experiment. And you can compare the actual W7x experiment, which is neoclassically optimized to configurations that aren't neoclassically optimized like LHD. So in this picture, you can see that this normalized heat flux is actually less than one. And again, this has to do with the large amount of turbulent transport in the experiment. But what you can see is that if you try to support the same temperature and density gradients in an unoptimized configuration, there'd be no way that these gradients could be supported in the experiment because the neoclassical heat flux largely is much larger than what they actually measure in the experiment. So this tells us that it's actually, it paid off in the end, all the work that went into designing this equilibrium and engineering it. They're able to now achieve record-breaking discharges due to this neoclassical optimization. So now I'll place accelerators in context a bit. So there are two leading approaches to magnetic confinement, the Tokamak and the Stelerator. So while the Tokamak enjoys automatic particle confinement because of its Tertile Symmetry, you get automatic guiding center integrability. You also get automatic integrability of the magnetic field because of the Tertile Symmetry. The coils are relatively simple in a Tokamak. You can see they're planar. That doesn't mean they're necessarily extremely simple to build, but they're at least simpler than something like W7X. The problem with Tokamaks is that they require a very large plasma current, and this can give rise to instabilities, sudden losses of confinement. And writing this current in steady state is challenging from a technical point of view, but it's also inefficient because that means in a reactor that you have to take some of the energy from your fusion reactions and put that back into driving a current. And so the Stelerator tries to address some of these problems by avoiding the current. So the advantage is that you get low recirculating power in a reactor. You also don't have to worry about plasma terminating disruptions due to current driven instabilities. Stelerators also have the advantage that they have a softer pressure or beta limit. So it's often due to degradation of the equilibrium rather than a large scale instability. But the challenges that we require careful optimization of the confinement and the shaping is produced by rather complicated magnets. So on this plot, I'm comparing some large Stelerators and Tokamaks with respect to the triple product. So the triple product tells us that if we have this critical value, then we can get ignition. So the losses from thermal conduction balance the energy that's produced through fusion. And so the vertical axis in this plot is the triple product and the horizontal axis is the duration of the pulse. So really we would like to be in the top right corner of this plot. So what you can see is that for the Stelerator discharges LHD and W7X, we can more easily operate toward the right because it's easier to go in steady state. And we'd also like to push to to hire triple product in the Stelerator. So currently the Tokamak has the the record for the triple product in a turtle confinement device. But the Stelerator is not too far behind. So these yellow dots show some of the recent discharges from the early campaigns in W7X. In an upcoming campaign, OP2, they're helping to push up a little bit further and get closer to the envelope of what the Tokamak has been able to achieve. So what are some open questions and new frontiers in the Stelerator community? So as I've emphasized, we're now able to reduce the new classical transport of the collisional losses in the Stelerator. And there's still a lot of losses that occur due to micro turbulence. And since we have a lot of degrees of freedom in designing a Stelerator, we would like to use those degrees of freedom to reduce the turbulence. We'd also like to understand better the properties of Stelerators as they go to high pressure or high beta. And then we'd also like to take advantage of some innovations and optimization to improve confinement. So how can we design a Stelerator with reduced turbulence? So one approach to this is to reduce this trapped electron mode that MJ discussed in the previous talk. And so one way to do this is by by manipulating the sign of the derivative of J parallel, which essentially tells us the direction of the drift motion, such that the trapped electrons cannot resonate. And it turns out that W7X was able to be constructed with this so-called maximum J principle. It wasn't actually part of the design, but they realized it after the equilibrium had been finalized that W7X has this property. And so in the experiment, they can show that W7X can operate in a so-called stability valley. So there's a valley here where you can see that the growth rate is minimized. So as you start to increase the temperature gradient, the ITG drive gets stronger. And as you start to increase the density gradient, the TEM gets stronger. But in the middle, you can operate in a stability value. And favorable conditions can be achieved for confinement with this maximum J configuration by using pellet injection. So as you increase the density gradient, this can stabilize the ITG. And you can see that the largest triple product shots that have been achieved so far have been using the pellet injection technique. So this plot is showing the triple product as a function of the density in some W7X shots. And this green dash line is the record that was achieved in LHD. So at the top, these are the shots that have been achieved with pellets and the stars. And you can see that they're pushing up to very large triple products. So this is a potential scheme to reduce one of the modes that gives rise to some transport and magnetic confinement. But of course, there are other modes that we have to think about when we're designing a configuration. And so there's a lot of work in the community on developing reduced models that can be used for designing a configuration that can operate at reduced turbulent transport. So one of the surprising features of a stalerator is that when you cross a predicted pressure gradient boundary that's predicted from MHD theory, that we don't see a sudden loss of confinement. And we don't see a large scale instability in the plasma. And this has been seen in the large helical device in Japan. So what this diagram shows, the pressure gradient and the pressure on the horizontal scale and this black dash line tells us the predicted mercier instability thresholds. And you can see that there's a bunch of discharges that we're able to operate above this threshold. However, when the growth rate gets too large, then you can see that there's not as many discharges that are able to operate. So it appears that the stability threshold is more like a soft boundary. There may be some increases in transport, but it doesn't give you a sudden loss of confinement. And this is quite different from what you would see in a tokamak. And one of the reasons for that is as you go to increasing beta, then you get losses of integrability, the pressure can flatten, and this reduces some of the drive for the MHD modes. But we still need to do some more theory to understand this basic feature of accelerators and understand how we can design a accelerator to be able to self heal at high beta. So that means as we increase beta, the currents that come from the plasma would actually heal the islands rather than degrading the integrability of the fields. And this would give us a configuration that could potentially operate at higher beta. So a lot of improvement has been made in the past years by taking advantage of innovations in optimization. And one of these is in finding magnetic fields that can get much closer to this quasi symmetry property. So in the bottom right, I'm showing some figures of the field strength contours in a bunch of devices that were designed to be quasi symmetric. So if we had perfect quasi symmetry, all these contours would be exactly straight in the plane. And you can see that they're approximately straight, but they're not exactly precise. And here's just another way of visualizing this data. So on the y axis, this is a measure of the quasi symmetry as a function of the radius. And you can see it's on the order of 10 to throughout the device. Just within the past year, we found some new configurations that get much closer to quasi symmetry. So if we look at the field strength contours, they by look exactly straight. And when you actually measure the quasi symmetry, you can see that it's improved by orders of magnitude over the previous configurations. And this property is actually really important when we want to confine the most energetic fusion boron alpha particles. So with these new configurations, we can show that the confinement of the guiding center losses is analogous to or is comparable to what you would get in a tokamak. And so this has been able to demonstrate that accelerators really do have the confinement that's needed for a future reactor. And one of the ways that we've been able to improve in the optimization of stellarators is by taking advantage of tools that have been developed in related areas of physics and engineering. And so as I've highlighted, the stellarator optimization problem is really a problem of optimizing a shape. So we want to optimize the shape of the boundary of the plasma or the shape of coils. And there are very related areas that arise in, for example, aerodynamic engineering, you want to optimize the shape of a car or an airplane to reduce the drag. So in these fields, there have been tools developed to more accurately and efficiently evaluate these kind of problems. So we've been able to take advantage of concepts, which are called the shape gradient and the adjoint method, which allows us to very easily get sensitivity information without directly preserving the shape, but by taking advantage of some underlying self-adjointness property of the equations. So what I'm picturing in this figure is the shape gradient for the drag. So this is a measure of the local sensitivity to perturbation. So it tells us if we were to push the car inward in the red region and outward in the blue region, we could reduce the drag. And we can get an analogous figure for the stellarator. This tells us how we would perturb the boundary of the stellarator in order to push the rotational transform towards some target function. So by taking advantage of these adjoint methods, we've been able to more efficiently push configurations toward quasi-symmetry. So there's a graduate student at Princeton who was working on this project with me. So for obtaining these precise QH and QA configurations, I showed a few slides ago. This can take on the order of 10 to 4 CPU hours, but by taking advantage of advances in numerical methods, we can push this down to something like 10 CPU hours. And so we can more easily scan the space of quasi-symmetric configurations to find some interesting designs. And I think there's a lot of excitement in the field of stellarators now. So there's a design that's currently under construction, which will be likely the first quasi-symmetric design. This is the CFQS device. So it's in collaboration with NIFS. And so the physics design is complete. And I think they've started breaking ground and they're putting the device together, and hopefully we'll see some of the results coming out in the next few years. There's also an influx of private money into the fusion community. And this has had an impact on the stellarator as well. So there is a new stellarator company coming out of Princeton, which is taking advantage of some innovations in magnet design. So their idea is to not have complex modular coils, but to have some small dipole-like coils that give you the confinement. So you can have very simple toretal field coils that are easy to construct and large sets of small dipole fields that give you the confinement and are not too challenging and can also give you some experimental flexibility. There's also a private company based out of Wisconsin that is also taking advantage of some advances in magnet technology. So they're planning to use high temperature superconductors to construct their coils. And they may produce a quasi-helical or quasi-isodynamic device in the next few years. And so I think this is one of the areas where there's a lot of area for development is seeing how we can use improvements in magnets to improve confinement in the stellarator. So I'm going to be moving to Columbia in the new year and I'm also going to be developing my own little experiment to be a future testbed of optimization. And the goal of this experiment will also be to demonstrate some new magnet technology through HTS tape and demonstrate that we can use simple coils to produce a confined equilibrium and maintain some simplicity in previous designs that were at Columbia. So if you're interested in learning more about the background of stellarators and a little bit of the physics, I wanted to point you to a text that I've been working on with some of my collaborators. And so the goal of this document is to really start from first principles and to build up to the concepts that you need to understand the confinement in the stellarator and how they're designed. So we're working on publishing this as a book, but for now it's available on the archive. Anyone is welcome to check it out and please give us feedback or questions if there's anything that comes up. So I think I'm coming to the end of my talk. I want to make sure that I have plenty of time for questions. But I think as a community, we're really looking toward pushing the stellarator toward a fusion power plant. And so the community recently put out a strategic plan for the US, which is pushing toward a national program that is leading to the design of a compact power plant. So there have been previous attempts at designing stellarators that can be pushed toward a fusion power plant. This is the schematic of the area CS design. And there have been many innovations since this time that could really push the stellarator further. So I'm excited to see what progress will be made in this area in the coming years. So thank you for your attention. I'm happy to take questions at this point. Thank you very much, Elizabeth, for an absolutely excellent talk. And I'm sure that the strain and interesting world of stellarators is introduced to the participants, maybe for many of them for the first time. And I'm sure that there was a very high transformation and translation of information from your talk to the people. So I encourage anyone of you to ask a question. I mean, I'm sure that many of the things you have heard for the first time, so it's not very easy to ask question, but if there is anything you can ask. So Elizabeth, I'm going to ask you in the afternoon, we normally get together and have an informal session where the participants can ask questions. Today, would you be able to join us? What time is this? We are meeting at two o'clock? You are two thirty. Somewhere around between two and two thirty. Yeah, two o'clock. Around two thirty. Okay, I will put that on my calendar. I think I should be able to join, yeah. Okay, so then I mean, I think that I will try to give them a joint perspective of various things just in more or less in the way that you started, but maybe a little bit more simply with a little bit more of algebra and then we can all try to kind of make some kind of an impact on this entirely new audience to this field. But I mean, they are already, to some degree, introduced to tokamak confinement, but not really, not in any fundamental sense. So we'll go through the whole notion of confinement in a in a somewhat general manner. Okay. And I'm sure Professor Paul will greatly contribute to that. And as she said, she's moving on to Columbia to build an experiment. Congratulations for that. And we'd be looking forward to the results that you produce. Thank you. Yeah. So I will be sure to join in the afternoon and also please feel free to send me an email if you have any questions that come up later on. Okay. Wonderful. Thank you. Okay. Bye.