30 Jul

This talk happened in March and I still remember it (and I was super sleep deprived at the time too).  Immediately after the talk, another grad student and I were chatting in the hallway and marveling at how good it was.  He said something like “I feel like a better person for having gone to that talk.”

A few days later, I ran into the speaker and told her that I had loved her talk, and she said “I’m super unintimidating so feel free to email me or ask me if you have any questions.”

During the talk, at one point she said (again, up to sleep-deprived memory coarseness)

“It’s more important that you learn something than that I get through my talk.  There’s no point in rushing through the material if you don’t take something away from this.”

All of these quotes are to say that this was probably the best talk I’ve seen (and I’ve seen lots of talks).  Particularly because of that last quote above.  Speaker put audience before ego, and that is a rare and beautiful thing (the other contender for best talk I’ve ever seen was by someone who recently won a big award for giving incredible talks).

The good news is that this is something anyone can do – mathematics at this level is a matter of practice and good habits, and not “talent” or “genius”.

OK, done fangirling!  On to the math!

We’ll be talking about a property of groups, so brush up from a previous blog post or wikipedia.  First we need to define a  (total) ordering on a group: a binary relation ≤ that satisfies three properties (which you’d expect them to satisfy):

1. Transitivity: if a≤b and b≤c, then a≤c
2. Totality: for any a, b in the group, a≤b and/or b≤a
3. Antisymmetry: if a≤b and b≤a, then a=b.

A few examples and nonexamples:

• The usual ≤ (less than or equal to) on the real numbers is an ordering.  For the rest of this post, I’ll freely switch between using ≤ to denote being in the group, and being in the real numbers (it should be clear when we’re talking about real numbers).
• Comparing heights of people is not an ordering: it’s not antisymmetric (see picture)
• Ordering words in the dictionary is an ordering: if you’re both before/at the same place and after/at the same place as me, then we must be the same word.
• Consider the group $\mathbb{Z}_2\times\mathbb{Z}_2$, which you can think of as a collection of ordered pairs $\latex \{ (0,0), (0,1), (1,0), (1,1)\}$.  If we define an ordering by (x,y)≤(a,b) if x+a≤y+b, then we’d break antisymmetry.  If we defined it by (x,y)≤(a,b) if x<a and y≤b, then we’d break totality (couldn’t compare (0,1) and (1,0)).

Top: reals are good to go. Middle: just because we’re the same height doesn’t mean we’re the same person! Bottom: (0,1) and (1,0) don’t know what to do

• Can you come up with a relation that breaks transitivity but follows totality and antisymmetry?

Notational bit: we say that a<b if a≤b and a does not equal b.

Now we say a group is left orderable if it has a total order which is invariant under left multiplication: this means that a<b implies ga<gb for every g in the group.

Let’s go back to the reals.  If you use multiplication (like 3*2=6) as the group operation, then the usual ordering is not a left(-invariant) order: 2<3, but if you multiply both sides by -2 you get -4<-6, which isn’t true.  However, if you use addition (like 3+2=5) as the group operation, then you see that the reals are left orderable: 2<3 implies 2+x<3+x for every real x.  This is a good example of the fact that a group is a set and a binary operation.  It doesn’t make sense to say the real numbers are left orderable; you need to include what the group operation is.

Here’s an interesting example of a left orderable group: the group of (orientation-preserving) homeomorphisms on the real numbers. (Orientation preserving means that a<b means that f(a)<f(b), all in the reals sense).  If you don’t feel like clicking the link to prev. post, just think of functions.  To prove that the group is left orderable, we just have to come up with a left-invariant order.  Suppose you have two homeomorphisms g and defined on the reals.  If g(0)<f(0), then say g<f.  If f(0)<g(0), then say f<g.  If g(0)=f(0), then don’t define your order yet.  If g(1)<f(1), then say g<f.  And so on, using 2, 3, 4…  Looks like a good left order, right?  WRONG!

Pink and blue agree on all the integer points, but not in between

If g and f agree on all the integers, they could still be different functions.  So we haven’t defined our order.  We need a better left order.  What can we do?  I know, let’s use a fact!

FACT: the rationals (numbers that can be written as fractions) are countable and dense (roughly, wherever you look in the reals, you’re either looking at a rational or there are a bunch in your peripheral vision).

So now we do the same thing, but using the rationals.  Enumerate them (remember, they’re countable) so use $q_1,q_2,q_3\ldots$ in place of 1,2,3… above.  It’s another fact that if g and f agree on all rationals, then they’re equal to each other.  Let’s make sure we have an ordering:

1. Transitivity: If f≤g and g≤h, then that means there’s some numbers (call them 2 and 3) so that f(2)<g(2) and g(3)<h(3).  But since we had to go to 3 to compare g and h, that means g(2)=h(2).  So f(2)<h(2), so f≤h.
2. Totality: if I have two different homeomorphisms, then there has to be a rational somewhere where they don’t agree, by the second fact.
3. Antisymmetry: We sidestepped this by defining < instead of ≤.  But it works.

Here’s a “classical” THEOREM: If G is a countable group, then it is left orderable if and only if it has an injective homomorphism to $\text{Homeo}_+(\mathbb{R})$.

Remember, injective means that each output matches to exactly one input.  Since we showed that there’s a left order on the group of orientation preserving homeomorphisms on the reals, we’ve already proven one direction: if you have an injection, then take your order on G from the order of the homeomorphisms that you inject onto.  So if h is your injection and g, k are your group elements, say that g<k if h(g)<h(k) in $\text{Homeo}_+(\mathbb{R})$.

One thing Mann does in her paper is come up with an example of an uncountable group that doesn’t do what the theorem says (she also does other stuff).  Pretty cool, huh?  Remarkably, the paper seems pretty self-contained.  If you can read this blog, you could probably do good work getting through that paper (with lots of time), which is more than I can say for most papers (which require lots of background knowledge).

That brings me to the “also”: I’ve been quite tickled to be asked about applying to grad school/what grad school entails a handful of times, some of those times by people who found me via this blog.  So please email me if you’re interested in whatever I have to say on the subject!  I’ve applied to grad school twice and have friends in many different departments and areas.  I hear I can be helpful.

27 Jul

That’s really all I have to say about this.  It’s incredible.  So well written.  It’s the best piece I’ve read that explains math and doing math to non-mathers.  We talk a lot about analogies (caves or dark rooms with light switches or knives etc.) but this one just goes straight to it (the Devil’s game).  I loved this quote and also it made me tear up a bit as I was sitting in my windowless office with the door closed taking a break from a problem:

As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening.

Unfortunately, in the print edition it’s not very well formatted (too much wall of text, which the NYT Magazine has been doing lately).  But the online version looks great.

Anyway, go read it.  And if you want to know more about his childhood and see the way Australians write the word “pediatric,” read this one too.  The NYT one is a better piece of writing, but The Age one covers different ground.

Regular (very long) math post coming up on Thursday!  Left orderable groups!

Easy ga kho gung (Vietnamese braised chicken with ginger) [based on my mom’s recipe]

23 Jul

Slow-cooked chicken with ginger, garlic, and onions.  I love this recipe.  It’s one of those classic home-cooking recipes that you aren’t likely to see at a restaurant, but every family makes it.  In fact, when I was in Vietnam several years ago I made some friends and visited their village for one night.  They showed me how to grow rice in their paddy, and we walked around the village, and practiced driving a motor scooter while trying to avoid the water buffalo that hung out on the roads.  At night their mom made us a big feast for dinner, consisting of rice from said paddy, rau muong xao toi, and this braised chicken (freshly killed from their neighbor).  Very traditional, very delicious.

I also ask my mom for it every time I make it, so I thought I’d blog it so I could stop bugging her.  I’ll tell you how I do it and also make notes for where my mom takes more time and makes it more delicious than I do.  Also, it’s made with items that you probably have in your pantry (we buy garlic from Costco and always have onions and ginger and frozen chicken parts).

This is a RAW file. I’m just kidding it’s a jpeg.

If you don’t have fish sauce in your pantry and you’re interested in making Vietnamese food ever, then you should buy a bottle.  If you aren’t, then I’m not sure why you’re reading this post.  I like using coconut water/juice (I always keep it around because it’s all I drink when I’m sick), but water or chicken broth work great too.

My mom always soaks chicken in salted water for half an hour before cooking it, “to get rid of the smell.”  Brining does keep the chicken super moist, but I’m always too lazy to do it.  It’s good if you feel like it though!

Also, traditionally the chicken parts are chopped up into bite-size pieces for this dish.  Part of that is frugality and part of it is flavor- more surface area to soak in more of the sauce.  Plus it’s fun to bust out your cleaver!  I generally make the pieces baby-fist sized (so three or four bites) because I am lazy.  You could also not chop them.

I guess they had to give that suburban sitcom star a nickname instead of just calling him by his last name. Then it’d sound like a serial killer sitcom instead of a family one: Leave it to Cleaver!

Next, chop up some garlic, onion, and ginger.  A few thoughts on this: for our wedding someone gave us a mortar and pestle, and it is AWESOME for garlic.  I don’t even peel or smash the cloves, I just throw them in and smash them a couple times.  The paper falls off and you can pick it out.  This isn’t great if you care about uniform sizes, but if you want a ton of garlic quickly smashed into smallish pieces, this is definitely the way to go.

If the actress from Young Frankenstein comes up to you and wants to fight, try to walk away. You’ll get Teri Garr-licked in no time.

I am a total sucker for those stupid “17 life hacks that will change the way you sit on a couch!!!” articles.  I’ve seen “one weird trick” a few times for peeling ginger: use a spoon.  Unfortunately, this one actually works!  Especially if you have a fairly smooth/not-too-knobby piece of ginger.  Just push the spoon tip in at one end of peel, eating side facing the ginger, and pull down while pushing into the ginger.  I can’t believe this worked and now I’ll go nail polish my keys so I don’t mix them up and save my bread bag close-things to label cords.

I actually have naked ginger in my house a lot (my baby is a redhead!)

You’ll want diced onion, smashed pieces of garlic, and matchsticks of ginger.  Throw that in with your chicken (if you brined it, toss the brine), along with sugar and salt, and let it marinate for at least 15 minutes.

Just like in the human world, in horse races there are far more males than females. I’ve been to exactly one horse race in my life, and it was almost all stallions, but there was one lane with a female. There was a mare in eight.  [I just told my husband this caption, and his response: “our kid is going to love you.”  Not even a chuckle from him!]

My mom first browns the chicken in a little bit of oil, then adds in the garlic, ginger, onion.  I actually marinated it in the pot, and just put the pot on the stove and turned it on.  Like I said, super easy.  Put it on high, add some fish sauce for flavor, and then your liquid (I used coconut water).  Bring it to a boil, then turn it down and simmer for as long as you have.  The longer you simmer, the richer the flavor.  The chicken will be cooked after about 15 minutes so if you’re in a rush just eat it then.

How cute would it be if we called every adult animal like we call chickens? Kittenens? Puppyens? PUPPY YENS?!?!

A few minutes before you want to finish it, I like to add some cornstarch to thicken it.  To avoid lumps, put the cornstarch in a small bowl/ramekin and spoon some of the hot liquid into it, then whisk that til smooth.  Add the mix to your pot, and stir.  Then bring it back to a boil.

‘don, sob’, some’, shiratak': these are all rame’kin.

Serve this with rice and plain boiled vegetables to soak up as much of the sauce as you can.

Easy ga kho gung:

Bone-in chicken pieces (I like thighs, but drumsticks or a whole chicken are also great) [Enough for the number of people you are serving]

1/2 head of garlic (just a lot of garlic.  Like 7 cloves at least)

1 thumb-sized piece of garlic per 4 servings

1/2-1 onion

2 TB sugar

2 TB fish sauce (nuoc mam; we always get Three Crabs brand)

salt, pepper

2 C water or chicken broth or coconut juice

2 TB corn starch

Chop chicken into pieces.  If desired, brine in salt water for half an hour.

Dice onion, smash garlic, and peel and matchstick ginger.  Add to chicken (drain brine, if using) with sugar, and salt and pepper to taste (about 1 TB of salt should be fine, you can always add more fish sauce later).  Stir, and marinate for at least 15 minutes and up to overnight.

If desired, heat 2 TB of oil over medium-high, and brown chicken pieces, 5 minutes.  Then add marinade, and proceed.

If you didn’t brown, just cook the whole thing over medium high.  Add fish sauce and liquid of choice, bring to a boil, stirring a few times (at least two or three times).  Lower heat and simmer at least 15 minutes, or for an hour.

Five minutes before you want to eat, place corn starch into a small bowl.  Spoon in some of the hot liquid, and whisk until smooth.  Add corn starch mix to pot, and incorporate and bring back to a boil.  Boil for one minute while stirring, then turn off stove.

Serve with rice and boiled vegetables.

Efficient geodesics in the curve complex

15 Jul

I have a not-secret love affair with blogging the curve complex: I (intro), II (dead ends), III (connected).  I’m surprised I didn’t blog the surprising and cute and wonderful proof that the curve complex is hyperbolic, which came out two years ago.  Maybe I’ll do that next math post (but I have a large backlog of math I want to blog).  Anyways, I was idly scrolling through arXiv (where mathematicians put their papers before they’re published) and saw a new paper by the two who did the dead ends paper, plus a new co-author.  So I thought I’d tell you about it!

If you don’t remember or know what the curve complex is, you’d better check out that blog post I (intro) above (it is also here in case you didn’t want to reread the last paragraph).  Remember that we look at curves (loops) up to homotopy, or wriggling.  In this post we’ll also talk about arcs, which have two different endpoints (so they’re lines instead of loops), still defined up to homotopy.

The main thing we’ll be looking at in this post are geodesics, which are the shortest path between two points in a space.  There might be more than one geodesic between two spaces, like in the taxicab metric.  In fact, in the curve complex there are infinitely many geodesics between any two points.

It’s easy to get metrics messed up, but the taxicab metric is pretty straightforward- there are lots of geodesics between the red star and the starting point.  I guess if you’re an alien crossed with a UFO crossed with a taxi then maybe the metric is difficult (butI totally nailed portraits of UFO-taxi-aliens)

Infinity is sort of a lot, so we’ll be considering specific types of geodesics instead.  First we need a little bit more vocabulary.  Let’s say I give you an arc and a simple (doesn’t self intersect) closed curve (loop) in a surface, and you wriggle them around up to homotopy.  If you give me a drawing of the two of them, I’ll tell you that they’re in minimal position if the drawings you give me intersect the least number of times of all such drawings.

All three toruses have the same red and green homotopy classes of curves, but only the top right is in minimal position – you can homotope the red curve in the other two pictures to decrease the number of times red and green intersect.  I just couldn’t make a picture w/out a cute blushing square.

If you have three curves a, b, c all in minimal position with each other, then a reference arc for a,b,c is an arc which is in minimal position with b, and whose interior is disjoint from both and c.

Green is a reference arc for red, orange, yellow: its interior doesn’t hit red or yellow, and it intersects orange once.  Notice that it starts and ends in different points, unlike the loops.  (This picture is on a torus) [Also red and yellow aren’t actually in minimal position; why not?]

Now if you give me a series of curves on a surface, I can hop over to the curve complex of that surface and see that series as a path.  If the path $v_0,v_1,\ldots,v_n$ is geodesic, then we say it is initially efficient if any choice of reference arc for $v_0,v_1,v_n$ intersects $v_1$ at most n-1 times.

The geodesic $v_0,v_1,\ldots,v_n$ is an efficient geodesic if all of these geodesics are initially efficient: $(v_0,\ldots, v_n), (v_1,\ldots,v_n),\ldots,(v_{n-3},\ldots,v_n)$.  In this paper, Birman, Margalit, and Menasco prove that efficient geodesics always exist if $v_0,v_n$ have distance at least three.

Note that there are a bunch of choices for reference arcs, even in the picture above, and at first glance that “bunch” looks like “infinitely many,” which sort of puts us back where we started (infinity is a lot).  Turns out that there’s only finitely many reference arcs we have to consider as long as $d(v_0,v_n)\geq 3$.  Remember, if you’ve got two curves that are distance three from each other, they have to fill the surface: that means if you cut along both of them, you’ll end up with a big pile of topological disks.  In this case, they take this pile and make them actual polygons with straight sides labeled by the cut curves.  A bit more topology shows that you only end up with finitely many reference arcs that matter (essentially, there’s only finitely many interesting polygons, and then there are only so many ways to draw lines across a polygon).

So the main theorem of the paper is that efficient geodesics exist.  The reason why we’d care about them is the second part of the theorem: that there are at most $n^{6g-6}$ many curves that can appear as the first vertex in such a geodesic, which means that there are finitely many efficient geodesics between any two vertices where they exist.

I DID NOT MAKE THIS PICTURE IT IS FROM BIRMAN, MARGALIT, MENASCO. But look at how cool it is!!!

Look at this picture!  The red curve and blue curve are both vertices in the curve complex, and they have distance 4 in the curve complex, and here they are on a surface!  So pretty!

If you feel like wikipedia-ing, check out one of the authors on this paper.  Birman got her Ph.D. when she was 41 and is still active today (she’s 88 and a badass and I want to be as cool as she is when I grow up).

Mmmm rummmm cake

9 Jul

When I started this blog I couldn’t imagine the directions my life would take.  Two and a half years ago, I did a post on a semi-homemade rum cake and titled it “Shame on me” for being semi-homemade instead of from scratch.  Rereading it, I feel like that’s the cake I should’ve made the other night.  I was feeling bad because I’d found a fundamental problem in my research (still unresolved…) after wasting away my morning and before picking up the baby late from daycare because I’d forgotten an umbrella.  So after putting the baby down and eating dinner, I decided to make a rum cake.  During this time I was chatting with my husband and offhandedly asked him the last time he was proud of me.  He said “I’m proud of your right now, you’re making that cake.  You’re so capable.”

Hell yes I am capable!  My research is stagnant; last time I timed myself it took me 13.5 minutes to run a mile; I keep not getting the oil changed on my car.  BUT I’m trying research, I am active, and my baby is still alive.  AND I made a rum cake.

Some days, you need pep talks from those who love you.  Some days, you need to bake a cake.

Semi-homemade doesn’t hurt the eggsecution of this dish

I didn’t have a Bundt pan (we don’t know what happened to that old one but rum cake is the only thing I make in a Bundt pan so it seems silly to buy one), so I used a 13 x 9 pyrex instead.  Generously grease and flour it, then sprinkle with nuts (I used peanuts do not do that.  Use pecans or walnuts).

Peanuts, equality, and granite- doesn’t quite have the same ring to it as liberté, egalité, et fraternité. I guess France can keep its flag-they’d be nuts to adopt this one, even though it rocks.

Then you make your doctored cake by adding pudding mix, an extra egg, and rum. Yum.  This makes the cake a bit lighter and bouncier while somehow super moist (that’s the pudding mix).

When you whisk upon a star, you’ll probably burn up no matter who you are. Because stars are really hot and why would you be whisking anyway anything you wanted to bake would already be burned to a crisp/nothingness

I thought this label was hilarious on the box:

IF I PUT THINGS IN ALL CAPS YOU WILL READ THEM. (google translate:) Si pongo COSAS EN MAYÚSCULAS USTED puedan leerlas.

So then I took this picture:

Breaking all the rules! That’s me!  Or illiterate! O analfebetos!

Anyways, while that’s baking you can doodle around the house for half an hour, then make the glaze- it’s just butter + sugar, boil it, then add rum.

I ran out of white sugar so I substituted in brown for the remainder. Insert appropriate racial joke here? It’s hard because multiple ethnic groups claim “brown” which doesn’t make sense because isn’t everyone some hue of brown? I’m staring at my white husband’s skin right now to try to figure out the color. Pinky-peach but not like a sunset. He just held up a piece of paper but white people aren’t all albino. I don’t know. I’m not good with colors. That’s why I subbed in the sugar.

After the butter and sugar have boiled, turn off the heat.  Make sure you use a deep sided pot, not a pan, because when you add the rum things get exciting and it fizzes up.

This isn’t a zero-rum game

After the rum

After the cake comes out of the oven, poke it all over with toothpicks or skewers, as much as you can until you get bored (so maybe 40 times?).  Then drizzle half the glaze all over it.  Wait a bit, invert the cake onto a plate or baking pan, and poke again + drizzle again.  It’s great warm, but it’s awesome about 12 hours later, when the rum has soaked in a lot and everything is moist and rummy.

Buttery rum cake, adapted from allrecipes

cake part:

1 c chopped nuts (I prefer pecans)

1 box yellow cake mix

1 box vanilla pudding mix

4 eggs

1/2 c each water, vegetable oil, rum

glaze part:

1/2 c each butter, rum, white sugar, brown sugar

1/4 c water

cake part: grease and flour a pan (Bundt if you have it, 13×9 if you don’t, two 8″ or 9″ rounds if you don’t have that either, if you don’t have those I’m not sure why you’re looking at a cake recipe)  Sprinkle with nuts (toast if you so desire)

Whisk together cake mix and pudding, then add remaining cake ingredients and mix.  Pour into pan, bake at 325 for one hour.

glaze part: fifteen minutes before cake is done, melt butter in a pot with sugars and water.  Bring to a boil, constantly stirring.  Take off the heat and stir in rum BE CAREFUL IT BUBBLES UP.

When cake is done, poke holes all over it.  Dribble half the glaze over it.  Carefully loosen sides of the cake from pan, then cover with a serving plate and invert the cake.  Poke holes all over the nutted top half, and dribble remaining glaze all over it.

Not a sociologist or ethnographer, but I am a curious person (about gender and race)

2 Jul

Inspiration for this post: this tweet.

So I’ve written before about being a woman in math, and this will not be my last post on the subject either.  First, some background.  One really, really awesome thing about my field (geometric group theory) is its webpage.  Some time ago, a great professor at UCSB made this website which includes a list of all active geometric group theorists in the world (self-reported), a list of all departments in the world with said people, lists of publishers and interesting links/software, and most importantly for me, a list of all conferences in the area.

Long aside: said professor once gave me some great advice which I have since forgotten/warped in my memory to mean: do what you want to do.  This is probably not what he said, but he did use this amazing website as an example: at the time, people said that making the site was a waste of his time, and now its a treasured resource for researchers around the world.  Everyone in GGT knows this site (because they or their advisor is on it!)  So that’s part of the reason I have this blog, and started that women in math conference- it’s maybe a “waste” of my time, but it’s something I want to do and now people are starting to know me for it.  At both the Cornell and the MSRI programs I went to these past two months, a graduate student has come up to me and told me she reads my blog, so yay!  I love you, readers!  Also, side note in this aside: the video lectures from the summer graduate school in geometric group theory are already posted (in the schedule part of this link), so if you like videos and GGT I’d recommend them.  Lots of first and second year graduate students in the audience, so they’re relatively approachable.

Back to topic: I went through the list of conferences that had occurred so far this year and “ran some numbers,” by which I mean I divided.  I did this because I noticed that at the past few conferences I’ve attended, there seem to be disproportionately many female speakers (in a good way).  For instance, at this summer school I counted 12/60 female students (though later someone said there are 14 of us so don’t rely on my counting) and 1/4 female speakers.  But the numbers at that level are so low that the data is essentially meaningless: 25% vs. 20% isn’t that meaningful when the other choices are 0, 50, 75, or 100% female speakers.  But if you collect enough data, it probably becomes meaningful.  See my table below.*

If I were a sociologist or ethnographer, I would do this for all the conferences and interview a random sample of attendees and organizers in order to come to some data-backed conclusions about the phenomena here.  I’m not, so I’ll just make some guesses.  It looks like American conferences artificially inject more gender diversity into their invited speakers lists, while foreign ones don’t (YGGT in Spa a notable exception).  I’d also guess that conferences that target graduate students have more women speakers than conferences that don’t.

Three things that support my “artificial diversity” theory: to attend an MSRI summer school, graduate students are nominated by their schools.  Schools can nominate two students, and a third if she is a woman or an underrepresented minority.  The NSF, which is a huge source of funding for American conferences, is really into “Broadening Participation”, which means including participants who are women, African-American, Native American, Hispanic, or disabled.  And, as seen in table above, the percentage of female domestic speakers is twice that of foreign speakers.

I think this is great!  It’s much easier to do something if you see someone who looks like you/has gone through similar struggles doing so.

A response to myself from a few years ago, when I felt feelings about the burden of representing all women at a table full of men: I felt bad recently for wanting to ask a Hispanic female graduate student what she thought about increasing numbers of Hispanic women in math, because I thought I was placing this exact burden on her.  I was expecting her to speak for all Hispanic women.  But another graduate student solved this conundrum for me- her experience is invaluable in trying to understand the plight of her demographic, but we shouldn’t be too hasty to generalize from it.  And more importantly, someone needs to ask these questions.  My discomfort is relatively stupid and small compared to the issue at hand- we should try to solve these problems together and respectfully, but there’s bound to be missteps along the way, and that’s OK.

I don’t have solutions, and I’ve barely stated the problem or why we should care about it, but at least I’m trying to ask questions.

Universal acylindrical actions

25 Jun

I’m at a fantastic summer graduate school at MSRI (the Mathematical Sciences Research Institute, a.k.a. “math heaven”) right now and re-met a friend I’d seen at a few earlier conferences.  I saw that she’d posted a preprint up on arXiv recently, so I thought I’d try to blog about it!

Remember that a group is a collection of elements paired with some kind of operation between them (the integers with addition, rational numbers with multiplication, symmetries of a square with composition).  For that operation, you put in two group elements and get another group element out.  You can imagine different functions with different inputs and outputs.  Like you might have a function where you put in Yen and late night, and it outputs pumpkin.  Or you could put one group element in, and a location, and get a different location [like if you put in the group element -2 to the location (3,3), maybe you get (1,1)].  More precisely, a group action on a space is a homomorphism* which takes in a group element and a point in the space and outputs a (possibly different) point on that space.  For instance, you can give an action of the integers on the circle by saying that rotates the circle by $n/2\pi$.

Each integer rotates the circle by pi/2 times the integer. Looks like circle is getting a little sick of the action…

In the picture above, if you input the integer 2 and the original purple dot, you get the new location of the dot (180 degrees from its original location, aka pi away).  If you say the original purple dot is location and the new location is y, the notation is that 2.x=y.  A homomorphism is a function that respects this: f(xy)=f(x)f(y).

We say a space is hyperbolic if it locally “looks like” hyperbolic space (there’s a particularly nice function between it and hyperbolic space).  The title of Carolyn’s paper is “Not all acylindrically hyperbolic groups have universal acylindrical actions,” so we need to learn what “acylindrical” means (look, we’ve already learned what groups and actions are, and we know the words “not”,”all”,and “have”!  We’re doing great!)

Here’s the precise definition, and then I’ll break it down:

An action of a group on a hyperbolic space is called acylindrical if, for any $\epsilon >0,$ there exist numbers M,N>0 such that for every pair of points x,y with distance d(x,y)>M, the number of group elements that translate both x,y by less than epsilon is bounded by N: $|\{g: d(x,g.x)\leq \epsilon, d(y,g.y)\leq \epsilon\}| \leq N$.

Here’s the non math-y intuition for this: if you have a pool noodle and you spin one end around, the other one generally will fly away from where it used to be.

Here’s the math-y intuition for this: choose two points that are M-far apart.  Make a little $\epsilon$-circle around each, then connect the two with a cylinder.  The condition says that only a few group elements preserve the cylinder (that means that when acts on all the points in the cylinder, it maps them back into other points in the cylinder).  So if you have a bunch (perhaps infinitely many) elements that preserve one circle, most of them send the other circle/rest of the cylinder away.

A group is called acylindrically hyperbolic if you can find a hyperbolic space on which the group acts acylindrically.  In practice, such groups actually act on a whole bunch of different spaces acylindrically.

Now suppose that you’ve got an element in G and you want to see how that particular element acts.  We say is loxodromic if you can find a space and a point in it so that the map $\mathbb{Z}\to X$ that sends an integer to the orbit of the point $n\mapsto g^n.s$ is a quasi-isometry– roughly, if you draw all the points that gets mapped to if you apply over and over again, you get something that looks like a line.

The older tree is the same as the younger tree up to scaling (multiplication) and adding some constants (the leaves). This is an example of a quasi-isomeTREE.  [Also pretend both trees go on forever.]

Just for fun here’s a picture of something that’s not a quasi-isometry:

The ribbon on the right goes on forever in both directions, so it’s not quasi-isometric to the tree

You might’ve noticed above that we say an element is loxodromic if we can find space on which it acts in this particular way.  But we also said that a group can act on several different spaces.  So even if an element acts loxodromically on one space, that doesn’t necessarily mean it acts loxodromically on another space (even if the group acts on that other space).  We actually call an element generalized loxodromic if there exists some space on which it acts loxodromically.  Then if you can find an action so that all generalized loxodromic actions are, in fact, loxodromic, you’ve found a universal acylindrical action.  So this paper gives an example of an acylindrically hyperbolic group that doesn’t have such an action.

Blog notes: For the summer I’m going to blog every Thursday (day was chosen arbitrarily).  Also, I went back and tagged all the gluten-free recipes as gluten-free.  And you should know that whenever I mention a person in this blog by name or link to them, that means that I admire them/am inspired by them.