What is a Manifold? (6/6)

In posts 1-3 we were able to reduce all of the geometry of a curve in 3-space to an interval [a,b]\subset \mathbb{R} along with two or three real-valued functions. We also discussed when two sets of such data give equivalent (overlapping) curves. This enabled us to patch together a collection of such sets of data into one unified spatial curve.

We then studied the specific example of re-defining the metric on the plane so that its geometry is precisely that of a 2-sphere. We saw that for measurements of angles, lengths, and areas, all we need is a dot-product on vectors. Given an open domain in the plane, once we have a dot-product, we will be able to make such measurements. Our goal in this post is to make the following definition of a manifold more tangible.

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Ordered Fields and When You Can’t Order Them

The real numbers \mathbb{R} have an ordering on them–given two numbers x and y, we can tell whether x = y, x > y, or y<x. So as math people, we like to generalize this to other sets–when can we say that a general set is ordered? In this post, we’re going to explain the explicit of idea of what it means for a field to be ordered, and then show that the complex numbers \mathbb{C} can’t be ordered–no matter what ordering you put on it. (If you don’t know what a field is, just think of the real numbers \mathbb{R} or the complex numbers \mathbb{C}.)

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On “Imposter Syndrome”

Here’s how it happens: You’re in graduate school and were one of the best people in your major from your school. Honestly, that’s how you got into graduate school in the first place. You go in the first few weeks, you meet your new peers, and you engage in mathematical discussion. It’s really fun, being with people who are just as excited about math as you are. But then, a horrible thing happens. Someone, in conversation, mentions something you don’t know. And not only that, but the way they talk about it suggests that anyone who knows anything about anything knows what they’re talking about. Or maybe, in an even worse turn of events, this person is a professor. What are you going to do?

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The Science of Moving Dots

A guest post by Allison Kotleba:

When most people think of basketball, they picture the tall players, the fast-paced plays, and the seemingly impossible shooting skills. However, spatiotemporal pattern recognition does not Allison Kotleba_925519_assignsubmission_file_Kotleba Blog Post Ted Talks Picturecome to most people’s minds when discussing the game. In his Ted Talk titled The Math Behind Basketball’s Wildest Moves, Rajiv Maheswaran discusses the use of spatiotemporal pattern recognition in analyzing the players’ movements and using this analysis to help coaches and players create effective game strategies. This up-and-coming science aims to understand and to find patterns, meaning, and insight in all of the movement in our world today.

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What is a Manifold? (5/6)

In our last post, we invented a new geometry by re-scaling the inner product of the usual Euclidean plane. This modification did not change any of the angles in our geometry, in the sense that if two curves intersected in a particular Euclidean angle, then in our new geometry they still intersected in the same angle. However, distances and areas had shrunk and had done so significantly at points away from the origin. For instance, we found that the total area of the plane under our new metric was \pi – a finite value.

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