Posts Tagged ‘wild speculation’

The importance of choosing the right model

December 13, 2009

I’ve had the idea for this post bouncing around in my head for several months now, but now that I don’t have classes to worry about I can finally get around to writing it. I want to talk about some “pathological examples” in computer science, in particular in complexity and computability theory.

I wanted to post a picture of William H. Mills with this post, in the spirit of Dick Lipton’s blog. Unfortunately I can’t find one! Mills was a student of Emil Artin in the ’40s; he finished his thesis in 1949 and promptly (as far as I can tell) disappeared until the ’70s, when we find some work by a William H. Mills on combinatorial designs. After this, there’s nothing of note until Dr. Mills passed away several years ago at the age of 85.

While his work (assuming it’s his) on combinatorial designs looks interesting, W.H. Mills’ small place in history is assured by a short and unassuming paper from 1947, published when he was still a student. In this paper he showed that there’s a constant, which he called A, such that [A^{3^n}] is prime for all integers n \geq 1. Nowadays it’s called Mills’ constant.

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Where do graphs live?

December 5, 2009

This post came out of some thoughts I posted (anonymously, but mostly because I didn’t feel like registering) over at nLab. I don’t think it’s a secret that I’m heavily interested in the relationships between category theory and combinatorics, and more generally the ways in which we can use “structured” algebraic objects and “continuous” topological objects to gain information about the unstructured discrete objects in combinatorics. That said, the folks over at the nLab work on some crazy abstract stuff, which seems about as far away as possible from the day-to-day realities of graph theory or set systems. And maybe it is — but I hope it’s not, and as far as I’m concerned, this is a windmill that deserves to be tilted at. (After all, it might be a giant.)

So as my jumping-off point, I’ll take my observation from last time that the relationship between graphs and digraphs is analogous to the one between groupoids and categories. I briefly mentioned something called a quiver, which can be thought of as any of the following:

  • Another name for a digraph, which categorical people use when they don’t want us combinatorialists stomping in and getting the floor all muddy;
  • A “free category,” i.e., one in which there are no nontrivial relations between composition of morphisms;
  • An algebraic object whose representations we want to consider; it’s worth thinking of this way mostly because of the “freeness,” although if you try to define it more formally you’ll probably end up with the previous definition;
  • What you get when you take (part of) a category and forget all the rules for how morphisms compose.

This last point is the most interesting one for our purposes, since it’s clearly an algebraic object but isn’t as restrictive as “free category,” and thus has a chance of capturing the unstructured behavior of the combinatorial zoo. But it’s tricky to turn this into a rigorous definition that actually includes everything we want to be a quiver… so we’ll just use “quiver” as a fancy name for “digraph.” However, there’s an important philosophical lesson to be learned from the final point, so I’ll set it off:

Philosophical lesson. The edges of a quiver shouldn’t carry any information except for the vertices they are incident to; more generally, paths in a quiver shouldn’t carry any information except for their sequence of vertices.

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In which I am late to the MaBloWriMo party

November 4, 2009

Hey, guys. Been a while, hasn’t it? Sorry for not blogging more; I blame having to “learn stuff” in “classes.” It’s very silly.

I didn’t find out about Charles Siegel’s MaBloWriMo until almost November 2 (and then there were more of the aforementioned “classes”), but it sounded cool, and gave me a reason to get back to blogging. So to make up for the late start, I’m going to try to write at least 30 total posts for the month of November, which should hopefully average in the range of 1000 words in length.

So, again because of school, I’m supposed to be spending November writing a final (expository) paper for introductory algebraic geometry. This means that I’ll probably be posting bits and pieces of drafts of said paper for MaBloWriMo, which means that there’ll be some unifying theme to some of the posts. The bad news is, I don’t actually know what it is yet, since I haven’t figured out my final paper topic, since I’ve been procrastinating horribly.

So instead I’ll talk about my usual craziness; over the next few days, I’ll ask the question: Is there a higher category theory for graphs? What is it?

So if you’re a combinatorialist, the easy way to think about categories (at least at first) is as special sorts of directed graphs with some extra structure (specifically, a way to glue together edges to make new edges.) To make this rigorous, we can think of every (small) category as having an “underlying digraph” where we just draw a directed edge from X to Y if there’s a morphism X \rightarrow Y.

It turns out that this construction’s functorial: it gives us a forgetful functor from the category of small categories to the category of digraphs. (A morphism between digraphs, by the way, is exactly what it “should” be; it’s a function between the vertex sets that respects the adjacency structure.) (more…)

A FOCS-related question

July 27, 2009

Actually, it’s not so much related to FOCS as to a specific paper in FOCS. I’m talking about the paper of Basu and Zell in which they formulate and prove an analogue over the real numbers of Toda’s theorem. I happen to know virtually nothing about real complexity theory, and even in standard complexity theory Toda’s theorem is pretty far from my usual interests of circuit complexity and derandomization, but their abstract nevertheless caught my attention for one single reason: the method of proof.

As far as I know, all of the proofs of (the classical) Toda’s theorem rely on heavy machinery from complexity and combinatorics, and tend to be quite involved. Basu and Zell’s paper also seems to be pretty long and technical, but their result is proved (and formulated) using tools from algebraic topology. For instance, the real analogue of #P is defined using the Poincare polynomial (the generating function of the Betti numbers, from a combinatorialist’s viewpoint). Basu and Zell are also kind enough to provide a definition of #P from a “geometric” viewpoint as well. My question is whether this topological or geometric intuition over GF_2 (or in classical complexity) can be extended to produce a new proof of Toda’s theorem.

By the way, if this post sounds like I don’t know what I’m talking about even more so than usual, that’s because I don’t know what I’m talking about even more so than usual. Apologies.

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