Quivering with Anticipation

A lecture on some advanced math, half way digested.

So, I went to a lecture at UC-Riverside entitled 'Generalized Cluster Categories. What the hell does that mean, you ask? Me too. But its good to try new things, even if they are a tad over your head.

I've learned this much, cluster algebra's and later cluster categories were invented in the last 10 years and are intimately related to something known as a quiver representation. A quiver is basically a graph with a bunch of directed arrows between each of the points (note the picture). Quiver representations are often used to describe and prove things about Lie Groups. Lie Groups are themselves a slightly less obscure branch of mathematics that are used by physicists to help explain the relationships (and symmetries) between the fundamental particles of the universe.

All in all, the relationship between Cluster Algebras, Quiver Representations, Lie Groups and Fundamental Particles is pretty damn far away from anything you would ever need know in order to windsurf, buy groceries or finish your taxes, but you might find these mathematical systems useful if you were trying to unify physics. Actually, its reported that quiver groups are quite useful in a broad range of mathematical applications ranging from linear algebra to finite dimensional algebras. If you're not quivering with anticipation, yourself, by now, you must have a heart of stone.

But I digress, when a quiver group is 'mutated', there are certain rules that are followed. Basically, one chooses a point on the graph, reverse the direction of all the arrows that touch it, then add arrows to any sub-quiver group (confused... me too) and remove any instances where multiple arrows have emerged that go in the opposite direction between the same two points. Because opposing arrows are clearly against the rules.

From these obscure but simple foundations, the Quiver Lecture I attended spun out into some sort of awful deluge of nomenclature and references purportedly attempting to categorize the ways that quiver groups can mutate, but that left me clamoring for some sort of reasoned ground to stand on.

Which brings me around to the moral of this lecture. When attending advanced mathematical talks without sufficient background, be sure to stay awake long enough to pick up the simple stuff. You never know when you'll find yourself at some applied mathematics gala and need some party favors to throw around.

If you all want to learn a little more about quiver groups or just play with them in child-like naivety, like me, there is a really cool website here where you can make your own quivers, mutate them and have all sorts of fun.