My last post was about Hee Oh‘s talk at CIRM from that conference I went to last month-it actually covered the first third or so of the first of four lectures she gave. Étienne Ghys gave seven short talks on his favorite groups, which was a huge blast, so I thought I’d try to share some highlights. This post is a surprisingly simple open problem in group theory, which talks about functions on a circle. A circle! Who would’ve thought we still don’t understand everything there is to know about circles?
You may remember that I once did a series of posts 1, 2, 4, on the homeomorphisms of the torus. You don’t need to read all the posts to get this post, I just wanted to point out that at one point I used the notation to indicate the homeomorphisms (continuous functions with continuous inverses) of the torus which are isotopic (wiggle-able) to the identity. In fact, is a group, very related to the group we’re discussing today.
Instead of homeomorphisms, we can also talk about diffeomorphisms: these are homeomorphisms which are differentiable, whose inverses are also differentiable. Rather than dive into a definition of differentiable here, I’m just going to give you an intuitive definition: differentiable functions are “smooth” instead of chunky.
Some functions are differentiable, and some aren’t (see illustration above). You can also take second and third and n-th derivatives, and we say functions are n-differentiable if it’s possible to take n derivatives. So in the above example, the red rectangle function is at least 1-differentiable (maybe 2 or 3 or more), but the blue function isn’t differentiable at all.
Notation time: we call a circle , the sphere in one dimension. So a hollow ball would be , and so on. In this post, we’ll be talking about twice-differentiable diffeomorphisms of the circle that preserve the orientation of the circle: so if a point x is clockwise from y is clockwise from z, then f(x), f(y), and f(z) are also in clockwise order. This group is written .
Great, now we know the group we’re talking about. Now let’s get into the nitty-gritty of the problem. First, a subgroup is a subset of a group which is itself a group. For instance, a subgroup of the integers, under the operation of addition, is the even integers, . This is because adding two even numbers gives you an even number (2+2=4). In contrast, the odd integers are not a subgroup of , since adding two odd numbers gives you an even number (3+5=8), which doesn’t lie in the set of odd integers.
Next, we need the concept of a normal subgroup. FYI, mathematicians really care about normal subgroups: they give us lots of insights about the structure of groups, and they help us cut up groups into smaller, more manageable chunks- lots of times we’ll prove things about normal subgroups in order to say something about the larger group. We start with a subgroup, call it N. Then N is normal if for every group elements x and y in N, is also in N. The means the inverse of x with respect to the group operation. So in the integers under addition, the inverse of 2 is -2, because 2 + (-2) =0. In the real numbers under multiplication, the inverse of 2 is , since .
In our example, the even integers is a normal subgroup of the integers (you can convince yourself of this). It’s pretty easy to find subgroups of most groups, but finding normal subgroups (which aren’t just the identity element or the whole group) can be a little harder. We say a group is simple if it has no non-trivial normal subgroups.
So here’s the open problem I promised at the beginning: Is simple?
And if you want, here’s another one: is simple? Those are homeomorphisms of the disk that fix the boundary circle and respect area.
I just find it crazy that we don’t understand everything there is to know about functions of a circle or of a disk! It’s amazing!
In terms of personal blog time, I did in fact bake last weekend, a lot (we were at a vacation rental house full of bakeware), but I didn’t take photos. Mostly I baked the cookies, plopped them down in front of our awake and lively friends, and went to bed every night- turns out I’m not great at adjusting to living at 9000 feet. Day one was those awesome salty shortbread cookies, day two were 3-ingredient peanut butter cookies with added peanuts and chocolate chips, and day three were double chocolate bacon cookies which I totally screwed up on but were still delicious. Ten people made it through a pound and a half of butter in three days, which is glorious.