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PROFESSOR: Any questions about where we left off-- up to where we left off? OK, what I'll do then is give you a few more examples of the combinations in 22 to show which ones we have to retain as frameworks for crystallographic point groups and which ones exist as groups but which involve rotational symmetries that are not permitted to a lattice. So we've seen a combination of three orthogonal twofold axes and then projection that would look like this. And the international symbol for that point group is just a running list of the different axes that are present, 222. The next group in the sequence would be 3 2 2, where we took a 120 degree rotation. We combine that with a twofold axis perpendicular to it and the new twofold axis comes out and reminds you again of things that are quite clear but which are easy to forget-- that this angle here is 1/2 of 2 pi over 3. Don't forget that 1/2. So the neighboring twofold axis is 60 degrees away and then if we allow t...

GERALD SCHNEIDER: This is where we were at the end of the last lecture. I think I said a little bit about how the cerebellum can fit into this kind of scheme and the corpus striatum. Now, concerning the corpus striatum, in this book of readings, there are some very interesting chapters here at the end about pathologies of corpus striatum. It starts out with the chapter called "Still Smiling." It's in the book Newton's Madness, Further Tales of Clinical Neurology by Harold Klawans, a neurologist. They're very interesting stories. It gives you a real feel. It's very easy reading. I'd like you to read those chapters. And there will be some things about these dyskinesias, abnormal movements, and not just Huntington's chorea and Parkinson's disease, but some other pathologies of corpus striatum, also. I want you to remember that the neocortex in mammals-- it's the defining structure of mammals in the brain. We have a neocortex, even if...

So, let's start with where we were. We were talking about exponential growth in populations. And, we said we could describe this as one over the dN/dt equals some growth rate, r. And, in this case, we're talking about, let me ask that is a question. As a model for population growth, what's wrong with this? What does this project? This is N. This is time. There's no stopping it. I mean, we'd be knee deep in everything if populations grew according to this model, OK, because it just goes off into infinity in terms of density. So, we know that this is inadequate. In fact, some people describe the entire field of population ecology as a field that tries to determine why real populations can't grow according to this model. In other words, the whole field is trying to understand what the mechanisms are in populations that limit their growth. So, they don't grow exponentially forever. So, in this case, this is really a maximum growth rate. We can call...

PROFESSOR: Here is where the power of this comes when you decide that you're going to invent all possible Hamiltonians at this moment. You've reduced the infinite dimensional space of functions in the line to two points, so you have a two-dimensional vector space, dramatic reduction. So here we decide, OK, here is the Hamiltonian. And it's going to be a two-by-two matrix, and it better be Hermitian. So what options do I have? Well, Hermitian means transpose complex conjugated gives you back the same matrix. So let's try to parametrize such a matrix. I could put a0, a real quantity here, and another real quantity in the bottom size. And if they are real, the transpose complex conjugate will remain the same. That's OK. So I could put a0 and a1 here. I'll do it in a little different way. I'll put a0 plus a3, and a0 minus a3 here. Now, the thing is that a0 and a3 have to be real. So I'll use a0, a1, a2, and a3. And they all should be real. So h...

NANCY KANWISHER: So seeing where animals are going, so you can avoid them if they're coming after you or so you can catch them if you're going after them, right? One of the arguably uniquely human abilities is precision throwing, right? No other animal can do that. That's a very human thing. Although, visual motion is shared with lots of ability to see motion is shared with lots of animals. What else did you notice? What else seemed funny or harder to discern with stop motion? Yeah? AUDIENCE: We care about small details like [INAUDIBLE] to understand what the person is seeing. NANCY KANWISHER: Yeah. Yeah, so I was making notes to self. I haven't done that demo before. But in future, it would be really good to have the audio quality terrible. Because if the audio quality is terrible, you would lean more on lip reading. And we might have noticed more. But it's really hard to do that probably even at relatively fast flicker rates because that motion infor...