Hi everyone. Here's a quick announcement: the Applied Category Theory 2019 school is now accepting applications! As you may know, I participated in ACT2018, had a great time, and later wrote a mini-book based on it. This year, it's happening again with new math and new people! As before, it consists of a five-month long, online school that culminates in a week long conference (July 15-19) and a week long research workshop (July 22-26, described below). Last year we met at the Lorentz Center in the Netherlands; this year it'll be at Oxford.

Daniel Cicala and Jules Hedges are organizing the ACT2019 school, and they've spelled out all the details in the official announcement, which I've copied-and-pasted it below. Read on for more! And please feel free to spread the word. Do it quickly, though. The deadline is soon!

APPLICATION DEADLINE: JANUARY 30, 2019

Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Mathematicians do this all the time:

• When you have two integers, you can find their greatest common divisor or least common multiple.
• When you have some sets, you can form their Cartesian product or their union.
• When you have two groups, you can construct their direct sum or their free product.
• When you have a topological space, you can look for a subspace or a quotient space.
• When you have some vector spaces, you can ask for their direct sum or their intersection.
• The list goes on!

Today, I'd like to focus on a particular way to build a new vector space from old vector spaces: the tensor product. This construction often come across as scary and mysterious, but I hope to shine a little light and dispel a little fear. In particular, we won't talk about axioms, universal properties, or commuting diagrams. Instead, we'll take an elementary, concrete look:

Given two vectors $\mathbf{v}$ and $\mathbf{w}$, we can build a new vector, called the tensor product $\mathbf{v}\otimes \mathbf{w}$. But what is that vector, really? Likewise, given two vector spaces $V$ and $W$, we can build a new vector space, also called their tensor product $V\otimes W$. But what is that vector space, really?

Once upon a time, while in college, I decided to take my first intro-to-proofs class. I was so excited. "This is it!" I thought, "now I get to learn how to think like a mathematician."

You see, for the longest time, my mathematical upbringing was very... not mathematical. As a student in high school and well into college, I was very good at being a robot. Memorize this formula? No problem. Plug in these numbers? You got it. Think critically and deeply about the ideas being conveyed by the mathematics? Nope.

It wasn't because I didn't want to think deeply. I just wasn't aware there was anything to think about. I thought math was the art of symbol-manipulation and speedy arithmetic computations. I'm not good at either of those things, and I never understood why people did them anyway. But I was excellent at following directions. So when teachers would say "Do this computation," I would do it, and I would do it well. I just didn't know what I was doing.

By the time I signed up for that intro-to-proofs class, though, I was fully aware of the robot-symptoms and their harmful side effects. By then, I knew that math was not just fancy hieroglyphics and that even people who aren't super-computers can still be mathematicians because—would you believe it?—"mathematician" is not synonymous with "human calculator." There are even—get this—ideas in mathematics, which is something I could relate to. ("I know how to have ideas," I surmised one day, "so maybe I can do math, too!")

One of my instructors in college was instrumental in helping to rid me of robot-syndrome. One day he told me, "To fully understand a piece of mathematics, you have to grapple with it. You have to work hard to fully understand every aspect of it." Then he pulled out his cell phone, started rotating it about, and said, "It's like this phone. If you want to understand everything about it, you have to analyze it from all angles. You have to know where each button is, where each ridge is, where each port is. You have to open it up and see how it the circuitry works. You have to study it—really study it—to develop a deep understanding."

"And that" he went on to say, "is what studying math is like."

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Have you heard the buzz? Applied category theory is gaining ground! But, you ask, what is applied category theory? Upon first seeing those words, I suspect many folks might think either one of two thoughts:

1. Applied category theory? Isn't that an oxymoron?
2. Applied category theory? What's the hoopla? Hasn't category theory always been applied?
   

For those thinking thought #1, I'd like to convince you the answer is No way! It's true that category theory sometimes goes by the name of general abstract nonsense, which might incline you to think that category theory is too pie-in-the-sky to have any impact on the "real world." My hope is to convince you that that's far from the truth.

For those thinking thought #2, yes, it's true that ideas and results from category theory have found applications in computer science and quantum physics (not to mention pure mathematics itself), but these are not the only applications to which the word applied in applied category theory is being applied.

So what is applied category theory?

To help answer this question, I've written a little booklet—a collection of expository notes inspired by the 2018 Applied Category Theory workshop that took place in the Netherlands earlier this year. That booklet is now available on the arXiv, and you can access it by clicking the title page below!

As I mention in the booklet's introduction, the goal is to give a taste of applied category theory from a graduate student's perspective. In doing so, I've shared two themes and two constructions that appeared frequently during the workshop.

In this week's episode of PBS Infinite Series, I shared the following puzzle:

Consider a square in the xy-plane, and let A (an "assassin") and T (a "target") be two arbitrary-but-fixed points within the square. Suppose that the square behaves like a billiard table, so that any ray (a.k.a "shot") from the assassin will bounce off the sides of the square, with the angle of incidence equaling the angle of reflection. Puzzle: Is it possible to block any possible shot from A to T by placing a finite number of points in the square?

As I mention in the video, this is one of a number of billiard problems that folks studying dynamical systems have asked. It was mentioned in a 2014 lecture given by Maryam Mirzakhani, which you can watch in the video on the left! Maryam describes the puzzle but leaves the audience to think about the solution. And in that audience was category theorist Emily Riehl, who did indeed think about a solution on and off for a few weeks. (You can read Emily's reflection on her experience in this AMS tribute to Maryam.)

Fast forward to last fall, when I ran into Emily at a Women in Topology workshop at MSRI. During a bus ride up the hill to the MSRI campus, Emily suggested that the puzzle might be a good fit for Infinite Series. I couldn't agree more! The solution she shared with me is delightfully simple for what seems to be an incredibly complicated puzzle. It's the solution she believes Maryam intended the audience to find, and it's what I want to share with you today! In the video above, I set up some of the mathematics needed to work through the solution, so if you haven't watched it already, now would be a good time. In any case, I'll assume you're comfortable with the notions of flat tori, a tiling of the xy-plane with flat tori, and a lattice in the xy-plane.

Alright, let's get to it!

My sincerest thanks go to Emily Riehl for today's episode and blog post!