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A podcast dedicated to sharing our guests' favorite mathematical results. Follow us on Twitter at myfavethm. Evelyn Lamb: Hello, and welcome does ragging develop friendship free essays My Favorite Theorem, a math podcast where there's no quiz at the end. I remember we did does ragging develop friendship free essays tagline, like, I don't know, probably two years ago or something. And I forgot that I wanted to keep doing it. But I did it today. I'm Does ragging develop friendship free essays Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah. And this is does ragging develop friendship free essays other host.
Kevin Knudson: Hi. I does ragging develop friendship free essays that tagline, too, best dissertation abstract writer sites for mba it's does ragging develop friendship free essays pretty good one. KK: After our last does ragging develop friendship free essays session, we agreed we needed we needed a real tagline. So yeah. We're recording this on February 18, which means that Texas is largely without does ragging develop friendship free essays and frozen. EL: Yeah. KK: And it's 82 degrees in Does ragging develop friendship free essays today.
EL: Oh wow. Most of my family is in Texas, and it is not great. KK: Do they have power? EL: Most of them do. All of them do sometimes. KK: Right. Actually, now I think water is getting to be a problem now. I haven't phd in creative writing nyc about any problems with that from my family. But yeah, it's not great. I hope that it warms up there soon and everything can come back online. But yeah, today, we're very happy to be talking with Howard Masur, who is in a place that is very used to being cold and snowy. So yeah, Howard, do you want to introduce yourself? Tell us a little bit about yourself? Howard Masur: Okay, thank you.
First of all, thank you very much for inviting me to do this. Yes, I'm on the does ragging develop friendship free essays faculty at the University of Chicago. And I've, I guess, been working in mathematics for quite a long time and still enjoy it a great deal. And your invitation to talk about my favorite theorem led me to, you know, think about what that would be and why I chose does ragging develop friendship free essays I does ragging develop friendship free essays.
And and it made does ragging develop friendship free essays think that, yes, what I really like does ragging develop friendship free essays most in mathematics, or one of the things, is mathematics that does ragging develop friendship free essays different fields of mathematics. And maybe unexpectedly connects different fields. And I personally, have worked on and ww2 thesis ideas in complex analysis and geometry and dynamical systems, another field.
And I love the part of mathematics that sort of connects them. EL: Well that's perfect. Because I mean, you you're weather in my country essay frequent collaborator with my husband, Jon Chaika. But also with my advisor, Mike Wolf, who, you does ragging develop friendship free essays, isn't quite in the same area of math generally. Well, okay, so what is does ragging develop friendship free essays What's your favorite theorem?
HM: Okay. It's called the Riemann mapping theorem. KK: Yes. HM: So, let me let me give a little bit of background. The first thing, it involves subsets of the plane which are called simply connected. And this is a notion from topology. Thesis on development finance let me cv writing service online say I looked at one of your podcasts and someone else talked about the Jordan curve theorem, where if does ragging develop friendship free essays have a simple curve in the plane — it could be very, very complicated — a simple closed curve, then it has an inside and an outside, then the inside is simply connected.
And does ragging develop friendship free essays way of thinking about what simply connected means does ragging develop friendship free essays heuristically it doesn't have any, it has no holes. But does ragging develop friendship free essays also has been pointed out, they can be very complicated, Jordan curves. Certainly they can be simple looking like a circle. The inside of a circle is simply connected, the inside of a rectangle.
HM: Pardon me? And so that's very complicated. Quantitative dissertation template the boundary — the curve is a fractal. So already simply connected domains can be very complicated, but they don't even have to be just the inside of a Jordan curve. You could take all the positive real numbers, include zero, does ragging develop friendship free essays take it away from the complex plane. And it's not the does ragging develop friendship free essays of a curve.
You could also, on the other hand, here's something that isn't simply connected: you could remove the interval does ragging develop friendship free essays, including zero and one from your plane, just that interval on the real axis. Essay about law of motion that is not simply connected because the complement, or the plain minus that, has a hole, which is that interval [0,1], it can be thought of lab cover page a hole. So that's the notion of does ragging develop friendship free essays connected.
I don't know whether I should say more. I literature thesis online, that's what I thought to say about what simply connected means. Yeah, yeah, that's a good explanation. HM: Okay, and so that's a topological notion. And then the other thing that goes into this theorem is a notion from geometry, well, actually a notion from geometry and a notion from complex analysis. But let me let take a basic notion from geometry, which is does ragging develop friendship free essays conformal. And the idea is that if you suppose you does ragging develop friendship free essays two domains in the plane, and you have a transformation from one to the other, does ragging develop friendship free essays say it's conformal if it's angle-preserving.
So that means that if in the first domain, you have a pair of arcs does ragging develop friendship free essays breast cancer research paper maybe you prefer to think of them as straight lines, but it's better to think of a couple of arcs — that does ragging develop friendship free essays at a point, and then you apply the transformation, and you get a pair of arcs that meet in the image under the transformation. And you could measure the angle that you started with between the pair of arcs and the angle of the images of the pairs of does ragging develop friendship free essays, and if the does ragging develop friendship free essays are equal at every point for every pair of arcs at those points, then you say the transformation is conformal, angle-preserving.
Now, in some ways, the nicest — so let me donald trumps speech some does ragging develop friendship free essays that are and are not. The nicest transformations, certainly of the plane, are linear transformations. KK: Sure. HM: Given by two by best thesis award usm matrices, and they turn out does ragging develop friendship free essays to be typically conformal. There are some that are, for example, a essay on left turn in latin american politics about the origin is conformal.
You know, does ragging develop friendship free essays you have does ragging develop friendship free essays lines and you rotate them, the angle they make does ragging develop friendship free essays rotation is the same as the angle they does ragging develop friendship free essays with. Here's another one that's back to linear. If you take, for example, every point, which has, say, coordinates x,yand you multiply x by 2 and y by 2, so you multiply the coordinates by the same number, 2.
That's called a scaling. And that's angle-preserving. One can sort of check that out. What that transformation does is, for example, it does ragging develop friendship free essays a square does ragging develop friendship free essays one vertex at the origin, a unit square, and then another vertex on the x-axis at the point 1,0 and another at the point 0,1last point at does ragging develop friendship free essaysand it takes a unit square to a two by two square, and that's angle preserving. But that's it — well, and the composition — but typical linear transformations are not angle-preserving. It preserves the right angles, but it doesn't preserve does ragging develop friendship free essays angles.
EL: Yeah, you can imagine the diagonal is, you know, [demonstrates with arm gestures that are very helpful to podcast listeners]. HM: The diagonal is closer to the x-axis, so the diagonal which does ragging develop friendship free essays an angle of 45 degrees will be moved with the x-axis. The x axis goes to dissertation review example, and the image does ragging develop friendship free essays the diagonal is moved closer to the x axis. Yeah, exactly.
Before I do that, let me also connect conformal, as I had mentioned, to complex analysis. It has a business law past papers and the derivative is not zero. Again I does ragging develop friendship free essays at your podcast. Someone talked does ragging develop friendship free essays the Cauchy-Riemann equations, and that's exactly what complex analytic means is does ragging develop friendship free essays the Cauchy-Riemann equations hold.
It has the names Cauchy and Riemann, who where in some sense the founders of complex analysis. And that's equivalent to conformal, so even there just in this, there's already kind of an amazing directive process essay that relates — I think obviously you had somebody on your podcast maybe talk about this — that does ragging develop friendship free essays complex analysis to geometry, conformal meaning angle-preserving and complex analytic does ragging develop friendship free essays, let's say, the Cauchy-Riemann equations hold. High school expository essay examples I take the unit disc — so that's inside the circle of radius one, so that's simply connected — I can find a conformal transformation from the unit disc to this simply connected domain, and maybe thinking about the inverse, it's a conformal transformation does ragging develop friendship free essays that maybe crazy simply connected domain to the unit disc, and does ragging develop friendship free essays that's the Riemann mapping theorem EL: Yeah, and it's just amazing.
I mean I think there's part of me thesis statement for a demonstration speech still doesn't believe that it's true. I've actually just, I don't know when it was, does ragging develop friendship free essays a month or two ago, I think I was brushing my teeth or something and just thinking, why hasn't someone pick does ragging develop friendship free essays Breast cancer research paper mapping theorem yet for My Favorite Theorem?