The n-Category Café

Alex Hoffnung

Hi!

I have the unique position here of being the only graduate student host at the café (at least until June), so I guess I should present myself from that perspective.

I began studying mathematics relatively late in life, but nonetheless I am here now studying away and enjoying it very much.

In college, Noson Yanofsky was teaching me about quantum computing and suggested going to Riverside to study with John Baez for my graduate studies. At that point, my interests were in artificial intelligence, mathematics and mathematical physics, but I was certainly in need of some guidance about what I would pursue in graduate school.

Off to Riverside I went quasi-intent on studying quantum gravity. I showed up, started reading John’s lecture notes and eventually this blog. I told John I wanted to be his student. I think I pestered or guilted him into letting me sit in on his private meetings with his students. Months later, and after writing a short paper to pass the “I want to be your student test”, I was working with John. (In case you are wondering, I was the only student subjected to such a test!) But we didn’t work on physics for long — as John has enjoyed pointing out on occasion. Anyway, even though the blog seemed incredibly intimidating at first, it helped me get a grasp on some ideas that were interesting to me.

What have we been doing? Well much of it is recorded in some way or another here, on the nLab or on the arXiv. In brief, I have been studying higher category theory, topos theory, representation theory, and trying my hand at some geometry of sorts, all with an eye towards physics.

My thesis seems to be officially on a categorification of Hecke algebras. So there is a paper of this sort in the works along with some others on whatever seems most fun at the moment. In particular, I am investigating relationships between incidence geometries, buildings, Lie theory, higher categories, topological invariants and field theories of sorts.

So, I have a bunch of fun ideas and a nice bit of work to do over the next few years, which I will eventually begin discussing here. So I hope the next few years will be as exciting as the last few.

I should say something for all the other students, who, like me, are applying or will be applying for jobs soon. The advice that you should start early is good advice. While sometimes stressful, applying to jobs is fun. You get to sit down with an excuse not to be producing papers and just think and write about what interests you. So start early and spend lots of time doing this. Of course, take all my advice with a grain of salt — I haven’t even applied to any jobs yet! So I’d better get back to my applications.

Before I go, I will just give an idea of what I see as my role on this blog. As I said, in my first years of graduate school, this blog was a great way for me to see what was out there in terms of current research ideas and also who was out there. I have come to know a lot of great people through this blog (most notably the hosts and a few regulars) and have had the chance to meet and talk with many of them. Since talking math is up there with my favorite things to do, its great to have a lot of people to talk to. Even better is the chance for tons of collaboration. So as this thesis of mine gets closer to finished, I hope to start posting more, and I hope these posts can reach a wide audience. It would be especially nice to bring in some young graduate students and also some undergraduates to the conversations. This a place where people can come learn new things, meet new people and have fun. So a first goal would be to make it a little less intimidating to mathematicians who are just starting out and felt as I did.

See you soon!

Tom Leinster

For the first seven years of my research life I was obsessed with $n$-categories and other structures in higher-dimensional category theory. Then I got drawn into something different but still categorical, which is new and mysterious enough that I don’t even know what to call it. From one point of view it’s about self-similarity; from another, it’s about recursive definability; from another still, it’s a categorification of the theory of linear simultaneous equations.

Most recently this has led me into an investigation of the notion of size. That might sound like a hopelessly general thing to be investigating, but category theory makes it possible. So far the biggest advances have been in finding new and useful notions of the ‘sizes’ of categories themselves and of metric spaces.

The $n$-Category Café has played an important part in the story of the ‘size’ of a metric space — also called its ‘cardinality’ or ‘magnitude’. The first outing of the idea was in a post I wrote in February 2008. Then it turned out that there were close connections between the size of a metric space and two other concepts that were very new to me: entropy and biodiversity. I wrote a couple more $n$-Café posts about that (1, 2). Following that, Simon Willerton and I found ourselves having to learn some asymptotic analysis in order to understand how the size of metric spaces really behaves. So this stuff keeps taking me to different and unfamiliar places.

There are a couple of other guest posts I’ve done: Linear Algebra Done Right, and How I learned to love the nerve construction. I don’t have a plan for what I’ll post on in the future. But whatever it is, I’m sure I’ll learn a great deal: there are so many knowledgeable and talented people here, and the quality of interaction is so high, that I know it will be an education.

Mike Shulman

Hi everyone! I’ve been hanging out here off and on for some time, but I’m excited to now be one of the hosts. $n$-categories were one of my first loves, mathematically, ever since I first learned about them from Eugenia Cheng 7 years ago. Since then I’ve branched out into other aspects of category theory and higher category theory, especially as it relates to homotopy theory, algebraic topology, and (more recently) set theory and logic. One theme that seems to keeps popping up in my research is that $n$-categories are really just one corner of a largely-unexplored zoo of higher categorical structures. In particular, I’ve become a big proponent of double categories, equipments, and related structures. Here’s my web page.

I’m currently a postdoc at the University of Chicago, one of the few universities in the U.S. boasting a “research group” in category theory. Right now that group consists of Peter May, myself, and three of Peter’s graduate students. I won’t mention their names, but you may have seen them lurking around here before.

I also love explaining all sorts of cool math, and I’m also looking forward to doing some of that here at the Café. I’m really excited and honored to be a part of this virtual community that attracts so many smart people from all over the globe.

Simon Willerton

I am probably best known to category theory minded folk as ‘the other Catster’ — unlike Eugenia Cheng, however, I am not a pure category theorist and, if pushed on the matter, would probably describe myself as a quantum topologist. Looking on the arXiv, you will find seven consecutive papers from me with seven different classifications, though removing category theory and quantum algebra, you are left with various kinds of geometry and topology: that gives a fair idea of how I see myself.

As a graduate student I worked on connections between knot theory and rational homotopy theory. Since then I have wandered through topological quantum field theory looking at geometric interpretations of baby quantum groups via gerbes and thinking about links with algebraic geometry via derived categories.

I had the joy (not to mention the other emotions) of supervising Café regular and guitar man Bruce Bartlett on his thesis about categorical representation theory coming from topological quantum field theory.

Most recently I have been thinking about many examples of monoidal bicategories in different areas of maths such as representation theory, algebraic geometry, topology, logic and analysis and how they are connected via topological quantum field theory and possibly Khovanov homology of knots. I have also, as is well known around these parts, been thinking a lot about “applied category theory” in the form of the magnitude of metric spaces with a certain Dr Leinster.