• ## An excluded subobject theorem for nondistributive lattices

I gave a talk yesterday at the PMC’s SASMS. I typeset some notes, and I thought I would share them because I drew pictures. This typeset version has more details and less visual intuition than what I presented on the blackboard, which is why the whole thing fit into a half hour. Find a link to the notes and some flavour text under the cut. ...

• ## sl(2) for Combinatorialists

There is a long and terrific story to tell about Lie theory, and I wish I could do it justice, but there’s far too much to say in a single post. What I have today is merely one application of one Lie algebraic idea, which ends up being a useful theoretical and practical tool in enumerative combinatorics. ...

• ## Graph Homomorphisms and Cores

Today I’d like to talk about an open problem I’ve been interested in for the past couple of years. It’s a very hard problem, in that there are easier special cases that are famously unapproachable, but it makes for some rather pretty algebra, living at the intersection of graph theory and category theory. If you ask me, it’s too good to be false. ...

• ## A Few Translation Exercises

One of my favourite books is Gödel, Escher, Bach, by Douglas Hofstadter. If you’re patient enough to read it all, I highly recommend it: it’s a great book about mathematics and cognitive science. ¶ One of the main goals of the book is to motivate, and sketch a proof of, Gödel’s first incompleteness theorem. At one point, he provides some exercises in transcribing number theoretical statements in a specific implementation of Peano arithmetic he calls TNT. ...

• ## A Diophantine Contest Problem

This is a shorter post, about a silly little problem I came up with a few months ago. It’s not a very intelligent problem, in that it doesn’t really serve any mathematical purpose; nor is it altogether tough; but I think it has a certain aesthetic pleasantness to it. ...

• ## The call/cc Yin-Yang Puzzle

The call/cc yin-yang puzzle is an ancient piece of Scheme code, which was written—or more accurately discovered—by David Madore in the year 1999 upon his invention of the esoteric programming language Unlambda. It is a rite of passage for aspiring Schemers to grok these five lines, if they claim true mastery over the power of the continuation. ...

• ## The Groups of a Field

The following post is a digested version of a question I asked on math.SE a few months ago. ¶ To every field $F = (F, {+}, {*})$, we can associate two natural groups. These are the additive group $F^+ = (F, {+})$ and the multiplicative group of units $F^\times = (F \smallsetminus \{0\}, {*})$. A fun question to ask, especially of someone just getting started on basic group and ring theory, is whether or not these two groups are ever isomorphic for any field. ¶ If you haven’t seen this question before, feel free to try it yourself! ...

• ## Prime Filters in Distributive Lattices II

Recall from PFDL I, I introduced distributive lattices and filters, and we proved the easy direction of a characterization of Boolean algebras. Today I’ll detail a proof of the tougher and far more obscure converse—it involves some sneaky technology from formal logic. ¶ Theorem 1 states that, in a Boolean algebra, every (nonempty) prime filter is an ultrafilter. Its converse is as follows: ...

• ## Prime Filters in Distributive Lattices I

I’d like to talk about some results pertaining to distributive lattices. In particular, there’s this one interesting theorem about Boolean algebras I’ve been thinking about lately. One direction is reasonably famous, pretty useful and not very hard to prove, so I’ll cover that. But what I really wanna talk about is the converse direction, which is a result that almost nobody I know has ever heard of, and is impossible to find anything about on the internet. ...

• ## Why is a group?

Often when people talk about groups, they say something like: groups are objects that encode the notion of symmetry. After working a bit with groups and group actions, it’s easy to convince yourself this is the case, but this sort of a posteriori explanation might seem a little circular—at least, it does to me. ...

• ## Review of Integrals

Mathematics has a lot of integrals in it. ¶ Some of them are kinda hard to compute, like $\int (\sin x + \cos x) e^x\,dx$, where you have to use integration by parts and rearrange or something. Some of them are really hard to compute, like $\int \sqrt{\tan x} \,dx$, where you have to perform many unintuitive substitutions in sequence to obtain an unenlightening and kind of disgusting answer. ...