 Every day life, things are always flowing around from pouring a cup of coffee, you know, magnetic materials to large fluid-like substances like astrophysical plasma, whatever that is. So to describe the time evolution of such objects, we'll call fields. One tries to write down how strong and in which way they move everywhere in the region of interest. This is an example of that. So if you consider placing an imaginary ball as small as you like inside the field such that over time the amount of stuff in is equal to the amount of stuff out, it's called the no flux condition, then this starts to explain what zero divergence means. In what follows, we're going to focus on fields that are defined on some compact and simply connected regions, whatever that means, which satisfy the preceding conditions everywhere and we'll call them incompressible. So that's a soccer ball that I lost at one point from Florida. It's gone. It's not swirling around, so it's not rotating. We're going to come back to that. Okay, so from here I'll keep illustrating the important ideas allowed and then additional details may be found on the bottom of each slide. You have a six second intermission. Okay, so going back to the field of interest, one often wants to assume conservation of the relevant charge, so in this case for fluids it's mass. The expression called the equation of continuity up there is a way to express that we're not creating or destroying any extra of it. So if the field on a simply connected domain, which we'll get to, then there's another one in that same region which we can write the original one in terms of and we have this condition on the bottom, which we're going to get back to. So when you fix a particular potential, they're often interesting consequences. So for our 2D field of interest, one may fix A that I'm calling it purely in the Z direction and then its axis of rotation is going to only be in the perpendicular xy plane. Then if we also assume the original field is irrotational and this is the 2D domain simply connected then we can rewrite the field in terms of the smooth function phi while maintaining the irrotational function. So let's talk more about which domains work for this construction. So if you take a loop in 2D, you can squish it down to a point on the left then it works, if not it doesn't work. That's about how much we're going to go into that. Then in three dimensions now, if you go up to 3D you consider loops in there. Let's look at a sphere, the inside of a sphere and you take out a cylinder. Then the construction working or not is going to depend on of course how big the cylinder is. So if it goes through the whole space it doesn't work and the construction doesn't work. So let's recall the important parts of what we've done so far. We want incompressible fluid 2D and simply connected. Now we have somewhat of a feeling for what that means and then by doing all these things we get interesting stuff out of it because you're even in equations. Where did that come from? So maybe we can look at this in a different light. Maybe we could look at analytic functions, maybe we should be looking at forms or geometry instead and with that if we have all these things then we have these musical isomorphisms which are one of the things that first interested me in geometry which take you between different languages of expressing these things. So, so far we've been over some nice results but there are quite a lot of assumptions too. And while assuming cows are spherical and fly over the moon is a good idea, maybe, is anything so... So it turns out the answer is yes. And a couple applications include looking at 2D fluids for the clouds of Jupiter and then the things on the right are vortices and turbulence for atmospheric physics. And then also additionally my advisor and I are trying to use such a connection between gauge theory and fluid mechanics if there is one to give an interpretation of 2D turbulence and critical phenomena like this picture which was experimentally verified. So in closing I think it's always nice to see that amid lots of abstraction and notation math appears everywhere in our daily lives from in the sink to pouring a cup of tea. And thank you very much.