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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A Letter From the Editors on the Executive Order on Immigration

In editing the AMS Graduate Student Blog, we hope to provide a platform for discussion about issues that affect and are important to the lives of graduate students. At times, this has meant publishing pieces that grapple with complicated and complex topics including sexism, diversity, and discrimination, both in academia and in our communities at large.

That said, until now we have never taken a specific political stance on an issue, nor called for our audience to do the same. However, in light of President Trump’s recent executive order placing immigration and travel restrictions on individuals from Iran, Iraq, Syria, Sudan, Yemen, Libya, and Somalia, we feel that there are certain issues that are too important not to take a vocal and principled stand against.

We, the editorial board of the AMS Graduate Student Blog, condemn—in the strongest possible terms—these actions by President Trump, and we ask that he repeal this executive order as soon as possible. Moreover, we implore our readers, our fellow graduate students, and the entire mathematical community to stand up with us and other protesters around the world in condemning the president’s actions and demanding change.

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