Category is…A Tour of Math3ma’s Blog

I am a huge fan of Tai-Danae Bradley’s blog Math3ma. Why? In her blog, Tai-Danae explains concepts related to Category Theory and many other fields of math with beautiful illustrations in an accessible way. Math3ma was launched in 2015 when Tai-Danae was in her second semester of graduate school and has been going strong ever since. Tai-Danae, was also a co-host in “PBS Infinite Series”, has written a collection of introductory, expository notes on applied category theory, “What is Applied Category Theory?”, and was featured in the “Mathematically Gifted & Black” website. A while back on Twitter, I read a post asking for advice on how to start a blog as a graduate student and was inspired to revisit Math3ma and chat with Tai-Danae about her blog. In this tour, I hope to give you a glimpse of the blog’s style, content, and insights from Tai-Danae herself!

As she describes in her post “What is Category Theory Anyway?”, Category Theory provides a bridge between different areas of mathematics and its objects (see Figure 1). Informally, “each has objects in it (set theory has sets, group theory has groups, topology has topological spaces,…) that can relate to each other (sets relate via functions, groups relate via homomorphisms, topological spaces relate via continuous functions,…) in sensible ways (composition and associativity).”

Figure 1: Example of categories showcased in Math3ma’s “What is Category Theory Anyway?”

Similarly, Tai-Danae’s blog and its great content is a bridge between many wonderful areas of math and it’s readers.

VRQ: Can you tell our readers a bit about yourself and your blog?

Tai-Danae Bradley: “I’m a 6th—and final! —year PhD candidate inmathematics at the CUNY Graduate Center. My research interests lie in the (non-empty) intersection of quantum physics, category theory, and machine intelligence. I also have deep admiration for ideas within algebra, topology, and homotopy theory. Some quick facts about my blog: Math3ma is pronounced “mathema” as in mathematics. Mathema is a Greek word that means “a lesson.” The domain mathema.com was owned by an Italian tech company. I didn’t want to give up the name, though. That’s why the “e” became a “3.” The blog’s official logo is an “M” surrounded by little electrons. I double-majored in math and physics as an undergrad, so the logo is a nod to my interest in both subjects.”

Figure 2: Math3ma’s Logo.

VRQ: What is the inspiration behind your blog?

Tai-Danae Bradley: “Math3ma began as a tool to help me adapt to graduate school. I knew that the transition from an undergraduate program to a PhD program would be a challenging one for me. The blog’s About page opens with this idea: The learning process often involved writing. That is, I learn best by putting my thoughts on paper. That’s because the process of writing helps me to slow down and think carefully about the mathematics. This is when the ideas have a chance to slowly simmer and marinate to eventually become “aha moments,” which are hard (impossible?) for me to get otherwise, without putting in this concentrated time and effort. I developed writing as a study skill pretty early on—in middle school, I think—so it naturally carried over into high school and college and later to graduate school. In particular, after my first semester I’d already amassed lots of mini expositions I’d written for myself. So I had all these little essays laying around and eventually decided to share them online in case other students could find them helpful, too. “Math3ma was originally created as a tool to help me transition from undergraduate to graduate level mathematics. Quite often, I’d find that the ideas of math are hidden behind a dense fog of formalities and technical jargon. Much of my transition process was (and still is!) learning how to fight through this fog in order to clearly see the ideas, concepts, and notions which lie beneath.”

Figure 3:  From “Learning How to Learn Math”, Tai-Danae shares: “In the third line, I’ve written, “…ask yourself…what is it that they all have in common?” I suspect this may be the origin of my affinity for category theory, a unifying language in mathematics.”

VRQ: Has blogging changed how you view math? If so, how?

Tai-Danae Bradley: “Blogging hasn’t changed how I view math, but my view of math affects how I blog. This goes back to the idea mentioned earlier: “Quite often, I’d find that the ideas of math are hidden behind a dense fog of formalities and technical jargon.” The articles on Math3ma are my personal attempts at fighting through the fog. I believe that much of advanced mathematics can be made accessible to a wide range of people, even though it may feel inaccessible because of the sophisticated language. So I spend a lot of time distilling ideas for myself. As I mentioned above, this takes a lot of work that often involves putting my thoughts on paper. Occasionally, I’ll polish up some of these writings for a blog post. Blogging, then, ultimately began as a tool to help me gain access to the simple ideas that lie beyond the misty fog. This is one of the main reasons I wanted to go to graduate school—to learn how to see mathematics clearly. Math3ma is a public record of this personal endeavor.”

VRQ: Any advice for other mathematicians interested in creating their own blog?

Tai-Danae Bradley: “The blog’s most frequently asked question is how I make the illustrations. It’s super easy! I draw them by hand with pen and paper, snap a photo with my phone, Airdrop to my computer, and edit (add color, etc.) with Photoshop. In the early posts, I used a pen tablet but that didn’t last long. I like the personal touch of a real pen. So, I’m not sure if this counts as advice, but if you want illustrations then old-fashioned pen and paper is an affordable place to start! Another thought I often hear is, “I really want to start a math blog, but I just don’t have time to post regularly.” I don’t have advice on time management, per se, but I will share something that I’ve personally found helpful: don’t worry about frequency! Or rather, I don’t worry about frequency. I blog only when I have time, and I think this has worked out well so far. My blog articles are really written for me, so there is no internal (or external) pressure to publish on a certain schedule. If you enjoy mathematics, and you enjoy sharing it, then I think others will be just as delighted to share in your joy, whether it’s once a week, once a month, or once a year.”

Some of her favorite blog posts include:

1.”A First Look at Quantum Probability (Part 1 & Part 2)”, where she shares some simple mathematics that’s a combination of linear algebra and elementary probability theory (see Figure 4).

Figure 4: Comparison of classical and quantum probability from “A First Look at Quantum Probability”.

2.  “The Yoneda Perspective/Embedding/Lemma“ (a three-part series), which is one of her favorite theorems,  and says, informally, that a mathematical object (a set, a group, a vector space,…) is completely determined by the set of all mappings into or out of that object (see Figure 5).

Figure 5. Illustration of the relationships showcased in “The Yoneda Perspective/Embedding/Lemma”. 

3. “The Tensor Product Demystified”, where she illustrates through examples and pictures, how you can make new vectors from old using the direct sum and the tensor product (see Figure 6)!

Figure 6: Illustration from Math3ma’s “The Tensor Product Demystified”.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)