[MUSIC] All right, you've presented your problem. It's very exciting, the audience is on the edge of their seats, they want to know, how did you solve this problem? That's what we're going to talk about next, how do you present your technical solution? By the end of this video, you'll get started on articulating your technical solution in a way that's clear and compelling, so the solution is the fun part. That's what you did. That's your chance to say, hey, look at this cool solution I came up with for that problem I told you about. Isn't it exciting? So you want to convey that to your audience. And you want them to see how awesome and elegant and creative your solution really is. So let's take a look at some of our examples of how we talk about our solutions and then we'll ask you some questions about our presentations. >> We looked at a particular measure of complexity called Scott rank after Dana Scott, who's a computer scientist. And this measure of Scott rank tells us how complicated the objects that are being described are. And we know a lot about the Scott rank of computable structures and computable objects. And we know that the Scott rank can get pretty high up there. And so we were thinking, well, maybe the Scott rank of the automatic world is going to be bounded quite a bit lower than the computable world because they're so much smaller and weaker and these tiny little things. But the amazing thing was that we proved that you can find structures that are arbitrarily complicated in terms of Scott rank, that they have really, really, really high Scott rank but that are still automatic. And so even though these machines are way smaller and simpler and stupider than the full-blown computers, they can still code up objects that have arbitrarily high Scott rank, as high as any computable structure. >> That's really neat. >> It's so cool. And the key technical insight that we had to have was that we had to use really clever codings and representations of these really complex problems. So we had the really complex objects that had the high Scott rank, and we had to code them up in clever ways so that the automata could still somehow represent them. >> So the way we did that was we had to define what is exactly an intersection. How would we identify an intersection, given this graph structure. And after a lot of thought, we went back and forth, how does an intersection look, roads coming in, roads going out, what we realized was that it was easier to identify what an intersection isn't. So, in other words, these points in here, you can see those aren't intersections because there's just a single road that passes right through them. Now to make this slightly more complicated, we have both one-way streets and two-way streets, so there are a couple different scenarios where we look at a point in the graph and determine that it's not an intersection. And here's what these look like. So on the one hand, we have this situation, where we have a point, this could be pink or yellow. And a road coming in and a road going out. And this road has the same name. Say, this is Main Street. So this represents a one-way street going through this point. In this situation, this node right here, that's not an intersection. So we can rule that out. >> So it would be yellow. >> Exactly, that's a yellow point. We can ignore it. The other situation where we don't have an intersection is just the two-way street version of this. So we have a road passing through in this direction, but it's a two-way street. We have edges in both directions, all of these road segments have the same name. This is also not an intersection. >> Right. >> So you saw that even though we're talking about different problems, we have different solutions, and our talks are tailored to different audiences. It's still relatively easy to pick out what our solutions were, and how to relate the solutions to the problems that we were trying to solve. Now, again, hopefully we made it look easy, but it's not always this easy. In fact, every single one of us rehearsed that a lot. The first few times we tried to do this, it was not clear at all. We went down wrong paths. We fell into many of the common pitfalls that people run into when they're trying to present their solutions. So we'll take a look at those pitfalls next, how to identify them, and then how to avoid them.