Picture This!

Three cat pictures illustrate the difference between using tensor and matrix approximations to compress photographs. Image: Anna Seigal.

I doubt I’m the only person who sees the front cover of a math book or a conference poster and wants to know more about the picture. That’s why I was excited that when the Society for Industrial and Applied Mathematics came out with their new journal on applied algebra and geometry (SIAGA), Berkeley graduate student Anna Seigal published a series of posts illuminating the mathematics behind seven images on the SIAGA poster.

Seigal is half of Picture This Maths, a blog she and University of Aberdeen graduate student Rachael Boyd use to talk math with each other and, in the words of their “about” page, “shed some light on what doing a PhD in maths actually involves.” 

As the name implies, Picture This Maths tends to use pictures as a focal point of a post. Because I’m a good internet user, I’m especially partial to cat pictures, so Seigal’s recent post on using tensor approximation to compress images was welcome. She notes that tensor approximation is, as far as she knows, not currently used to store photographs, but it’s important to be prepared with this option in case the ongoing cat picture deluge requires us to get creative about how we store them.

Many of the posts at Picture This Maths look at what I would describe as applied abstract math—the math used is often more on the theoretical/”pure” side than what people typically think “applied math” is about. Seigal’s posts about the new applied algebra and geometry journal fall under this category, as do Boyd’s posts about persistent homology.

Homology is a tricky thing to explain, and I generally only think about it in an abstract, theoretical mathematical context. Homotopy is more intuitive for a lot of people, but especially in high dimensions, homology can be a more useful and more computable algebraic object to assign to a topological space. I like the way she explains why we use it. “So why do we do this? We might want to know something about a topological space, but maybe we can’t simply draw the space as it lives in a very high dimension. But the homology of a space is a sequence of groups which tells us about holes of all dimensions: and we know lots about groups!” Sometimes it seems like everything in math is figuring out how to ask a question we can answer. What is this manifold? I don’t know, but has this many holes in these dimensions.

The applied version of homology is persistent homology. I’ve encountered the idea before, but I never felt like I understood how it would be useful in practice. I still don’t think I could spot a good place to apply persistent homology in the wild, but Boyd has a nice post that describes how it can show up in viral gene transfer, Twitter connections, and non-transitive dice. If you’re a theoretical mathematician who can only handle low doses of applied mathematics, her post on Coxeter groups and the Davis complex (and its adorable diagrams) will make you feel at home again.

All in all, Picture This Maths reminds me of Math3ma, another favorite graduate student math blog. I think undergraduate math majors, math grad students, and other people who like looking under the hood in math will enjoy reading this blog.