AMS Notices Spotlight June/July 2017

As a grad student, life can be hectic—with classes, homework, teaching responsibilities and more—so taking the time out of an already busy schedule to read the AMS Notices “cover to cover” is not always something that seems easily attainable. In fact, if you are like me, often when you get the email notification that the Notices is published and you quickly file it away into a folder to check out at a later date, but let’s be honest, you rarely enter that folder to peruse them after you click the button to file it away. Now as a mathematician, staying up to date is important and filing away the Notices for some future date may not be the best habit to have. That’s where this post and future posts come in handy. Every time that the Notices have a new edition available, we will be spotlighting an article that we think is of particular interest for us as graduate students. We encourage you to take a moment out of your busy schedule to read the article or, if you aren’t interested in the topic we are emphasizing, to peruse the other articles and find one that catches your eye. So without further ado, our first spotlighted article.

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Experience at an AMS Mathematics Research Community

The AMS Mathematics Research Communities (MRC) is an NSF-funded program to help graduate students and postdocs jumpstart their careers. Every summer the AMS runs three sessions, each on a specific subject. Budding researchers in that area work intensively with each other, alongside expert advisors, on open problems. Many of the 2016 participants already published their MRC results. The AMS offers some further funding for collaborations that began at an MRC and helps cover costs of travel to the annual Joint Mathematics Meetings (JMM), where the MRC topics have special sessions. Select senior participants become organizers for the JMM special session, providing a further organizational experience opportunity. During the program, mentors also offer workshops on professional development to assist attendees in navigating the space of job applications and grant proposals. Women and members of underrepresented minority groups are especially encouraged to apply.

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Matrix Multiplication, the human way!

Having to do copious calculations by hand when preparing for an exam, I came to realize that there was an alternative way of interpreting a matrix multiplication. This new insight would allow me to instantly guess the following product without ever doing any numerical multiplication:
\[\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & -8 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 0 & 0 \\
0& 1 & 0 \\
1& 0 & 0
\end{bmatrix}
=
\begin{bmatrix}
3 & 2 & 0 \\
6 & 5 & 0 \\
0 & -8 & 0
\end{bmatrix}\]

Was there a way to have known that the first column of the product would be the third column of the first matrix?

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Optimal Control Theory to Settle Reinhardt’s Conjecture

The 2010’s are a Golden Age for packing problems. In 2014, Hales announced the long-awaited completion of a high-profile machine proof project called FlySpeck, which verified his proof of Kepler’s conjecture. Johannes Kepler, in 1600, conjectured that the densest way to pack spheres is in cannonball stacks. This was Hilbert’s 18th problem. The final proof was quite involved and its correctness needed massive computer checking. For example, verifying the nonlinear inequalities took 5000 hours on the Microsoft Azure cloud. Other optimal packing problems have been solved recently too—with simpler methods. Cohn–Kumar–Miller–Radchenko–Viazovka [3] and Viazovka [8] found optimal sphere packings in dimensions 8 and 24. Viazovka was awarded a 2017 Clay Research Prize for this work.

 

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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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