I just got back from the fantastic Graduate Student Topology & Geometry Conference, where I gave a talk and also brought my baby. I tried to google “bringing baby to academic conference” as I’ve seen one baby at a conference before (with his dad), and I knew this kid would be the only baby at ours. But it was cold enough/uncomfortable enough that I just had him stay in the hotel with my mom, and I ran back during breaks to feed him. Also, it was my first time being “heckled” by both of these two brother professors famous for “attacking” speakers- they happen to know just about everything and are also suckers for precision, which I am not (and should be). But I got a lot of good feedback on my talk, and I’m generally a very capable speaker (though I was not as prepared as I would’ve liked, thanks to somebody who likes to interrupt me every five minutes…) Anyways, this is not about me, this is about my friend who gave one of the best talks of the conference and more importantly, her research. This post is based on notes I took during her talk + skimming her paper (joint with her advisor) on which it is based.
Remember that we had our introduction to hyperbolic space. This research is focused on hyperbolic surfaces, which are shapes that locally look like hyperbolic space- this means that if you look at one point on the surface and just a little area around it, you think you’re in hyperbolic space. A good analogy is our world- we live on a sphere, but locally it looks like flat space. If you didn’t know better, you’d think the earth is flat, based on your local data. So how can we build a hyperbolic surface?
While hexagons in flat space always have angles that sum to 720 degrees, that’s not true in hyperbolic space. In fact, you can make right angled hexagons, which means that every single corner has 90 degrees. If you pick three lengths a,b,c>0 and assign these lengths to three sides of the hexagon like the picture, you’ll fully determine the hexagon- hyperbolic space is wacky!
Now glue two copies of a hexagon together along those matching a,b,c sides. You’ll have a funny shape with three holes in it, and those holes will have circumference 2a, 2b, 2c. This is called a pair of pants in topology.
You can glue together a bunch of pants to form a hyperbolic surface, by gluing them together along holes with the same length. Any hyperbolic surface, conversely, can be cut up into pairs of pants (this pants decomposition is not unique, as you can see below).
You could also set one of those lengths equal to 0, so you’d get a right angled pentagon as one of the hexagon’s sides would collapse. You can still do the pants thing here by gluing together copies of the pentagon, but instead of having a hole with circumference 2a like we had before, you’ll have a cusp that goes off to infinity- it’s like an infinite cone with finite volume.
Now we’ve built every hyperbolic surface (there are some more details, like how you glue together pants, but let’s just stick with this broad schematic for now). As long as the expression 2-2*(number of holes)-(number of cusps)<0, your surface is hyperbolic. So, for instance, a sphere isn’t hyperbolic, because it has no holes and no cusps, so you get 2 which is not smaller than 0. And a torus isn’t hyperbolic, because it only has one hole, so you get 2-1=1. But all the surfaces in the pictures in this post are hyperbolic- try the formula out yourself!
One thing you can ask about a hyperbolic surface is: how long is its shortest essential curve? By “essential,” we mean that it isn’t homotopic (this is a link to a previous post defining homotopy) to a cusp or a point. This shortest curve is called the systole of the surface. Systolic geometry is a whole area of study, as a side note. But we’re interested in the question: how many systoles can a surface have? This is called the kissing number of the surface.
A few notes: a “generic” surface has Kiss(S)=1, that is, there’s only one shortest curve if you happen to pick one “random” surface (scare quotes because no precise definitions). And it’s relatively “easy” to make a surface with Kiss(S)=3*(number of holes)-3+(number of cusps). Check for yourself that this number is exactly the number of curves in a pants decomposition of a surface. Using some hyperbolic geometry you can prove that there won’t be any shorter curves if you make all of the pants curves very “short.”
So what Fanoni and Parlier do in their paper is come up with an upper bound on the kissing number of surfaces with cusps. I won’t go into that, but I will try to explain part of a lemma they use on the way.
If your surface doesn’t have any cusps, then systoles can pairwise intersect at most once. But if you do have cusps, then Fanoni & Parlier prove that your systoles can intersect at most twice (and they build examples of surfaces with cusps that have systoles that pairwise intersect twice).
First they show that two systoles which intersect at least twice can only intersect in the way pictured to the left below, and not as in the right:
This picture from the Fanoni-Parlier paper I did not make this!
This matters because it implies that two systoles which intersect at least twice must intersect an even number of times. In particular, if two systoles intersect more than twice, then they intersect at least four times.
So assume for contradiction that two systoles a and b intersect more than twice. So they intersect at least four times. That means that there’s some intersection point somewhere such that the b-arcs coming out of it make up no more than half of the b systole length (see picture below)
If the green arc is more than half the length of the circle, then the blue one is less than half the length of the circle.
So if you look at these short b-arcs, plus the a path, and wiggle things around, you’ll see a four-holed sphere (two holes “above” the a curve, and two holes “below,” one of each inside a b arc).
Left: a schematic of how a and b intersect. Black dots represent holes or cusps.
Center: the short b arcs plus the full a path
Right: the short b arcs plus the a path, after moving four dots to be holes of a 4-holed sphere
This four-holed sphere has a curve on it, determined by part of a and the b arcs, which is shorter than the original systoles. This contradicts the definition of systole, so our premise must be wrong- two systoles can intersect at most twice.
This was proposition 3.2 in their paper- tomorrow I’m going to share propositions 3.1-3.3 with my advisor’s small seminar. Hopefully I don’t get heckled too badly this time!
OOPS I ALMOST FORGOT: life update. We bought a house and are moving to Austin, TX. I’m still planning on finishing my Ph.D., just virtually. [Up to a finite-index subgroup, obviously. Bad math joke]. I’ll probably be flying up to Chicago every so often to meet with my advisor/eventually defend my thesis. But yes, we’re driving in our minivan to Texas on Thursday. So… we’ll see when we get the internet set up in the new house. I’ll try not to make too long a break until my next post.