## 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.

## Soda-unders? Nope, popovers!

17 Jun

I wonder if the author of The Circle is really into baking. He must get that a lot.

When baby was four months old I was really itching to start baking again, but I needed things that required very little time/effort and preferably had lots of reward.  Turning to my trusty Moosewood Cookbook (affiliate link), I paged through until I saw a super easy recipe that included the word “or” in the ingredient list.  I don’t think I’ve said enough how much I love this cookbook- the recipes are so good and so forgiving and I actually follow them.  I love that this recipe says “2 or 3 or 4 eggs.”  And despite my love of excess amounts of butter (see super easy french toast souffle), Moosewood in general uses little butter to great effect.

So these are popovers, which I hadn’t had before but I guess are a thing.  A DELICIOUS thing!  First, butter some muffin tins- I have these little ceramic ramekins and I microwaved two tablespoons for thirty seconds and stirred it up.  Also, we have pastry brushes now (thanks, Crate and Barrel wedding gift cards!)

I haven’t greased pans a lot in this way- guess I have to BRUSH up on my skills

Butter, flour, eggs, milk, salt.  That’s it for these crisp-on-the-outside, custardy-on-the-inside soft rolls.  And it’s just one bowl.

I feel like I’m running out of puns… guess I should milk my head for what I can

After whisking up the eggs, add the flour and salt and beat that.  Then pour into your buttered muffin tins.

My lovely ladle lumps (I realize that is a whisk but I am tired let’s just pretend it’s a ladle and that ladle sounds like lady)

I used to always get confused when people said “in” after a “y” sound because I thought it sounded like my name (Minnesota accent?). I wonder if that famous blind musician got confused a lot when people baked muffins and said oh look, an ar-RAY of muffins

That’s it!  Half an hour from start to finish, even faster than lime pie!  These are so good fresh and hot with nothing, or with a bit of jam or butter on them.  After they come out, poke the top sides with a fork so the steam can escape.

Moosewood Popovers

2 TB melted butter

4 eggs (moosewood says 2 or 3 but I love eggs)

1 1/4 c milk

1 1/4 c flour

1/2 tsp salt

Preheat the oven to 375, and grease a muffin tin with the melted butter.  Whisk the eggs with the milk until mixed, then add the flour and salt and whisk together.  Pour into the muffin tins and bake for 35 minutes until the top looks dry.  Prick with a fork and eat!

## I’m sexist (and “so is everyone” isn’t an excuse)

10 Jun

Over the weekend we hung out for a few hours with some of my husband’s coworkers and their kids.  One wife is a very pregnant stay at home mom of two toddlers, and one husband is a stay at home dad of a toddling soon-to-be older brother.  I’ve hung out with the female coworker and her husband and child more, and their child is closer in age to our baby.  I am very impressed with him for staying at home with the kid.

When I first went back to work I had baby in day care three days a week and watched him for two, hoping that’d ease my transition back.  But those two days were SO HARD- it’s constant, mundane, brain-draining, frustrating physical work that’s incredibly, ridiculously rewarding.  (See photo)

And being alone with a little one all day with no adult interaction is rough- it was hard on me and it was hard on our marriage (which is awesome I highly recommend marriage by the way).  By the afternoons I was itching to work, but when I was at the office I was aching to be with my sweet little baby.

Aside: I have no thoughts on “having it all” except that the phrase doesn’t make any sense to me.  My hormones and heart want to be with my baby ALL THE TIME, and my mind and exercising body do not, and unfortunately all these things go together so it is impossible to have all my desires met.  Probably my wants will evolve as my child ages, and as I get more children, and I get older and my career moves, but right now it is impossible for me to have all the things I want.

So, my sexism.  When I’m with the SAHM, I take it for granted that she stays at home, watching two handfuls and running the household (she does EVERYTHING in that home) and even gardening and raising chickens.  We chatted about cooking and pregnancy and adapting to our new bodies and making friends.  Whenever I talk to the SAHD, I feel in awe that he stays at home, and we talk about the frustrations of hanging out with a little one all day and strategies to not go crazy.  Thus there are two sides to my sexism:

1. I do not feel in awe that she stays at home.  I assume that she does not go crazy or feel frustrated or feel any sort of internal struggle with all the things I said above about having little ones.  This is totally unfair to her and speaks deeply of my cultural assumptions (women can stay at home and don’t feel all the things that I feel and also I am a woman so this really doesn’t make sense).  Also, she’s got two kids so she has it way harder than him.
2. I do feel in awe that he stays at home.  This is unfair to him- it implies that I think staying at home is so hard on him, and further implies that maybe I think men can’t handle it. Also, it’s the only thing we talk about vs. a wide variety of things I can talk to her about.  I’ve now put him in a box with one interesting thing to discuss (his dadhood) vs. being a full human being with other thoughts.

I realized this on our way home from their place.  What do I do to fix this?  The clear answer is that I need to treat every person as an individual of individual circumstances, and treat each person with respect.  But while that abstraction is well and good, I need more concrete action items to get better.  When I talk to SAHDs, I won’t say things like “I’m so impressed that you stay at home” and instead I’ll talk to them like human beings.  When I talk to SAHMs, I’ll try to invite commiseration on how difficult raising kids is (you have to tiptoe here depending on how close you are with a person, b/c I’m not a SAHM but I could be if I chose to so anything coming out of my mouth could be seen as judgmental).  Hopefully these actions and saying these words will eventually change my internal attitudes too.

This all reminds me of a great essay I read two years ago, which I highly recommend.  Also, if you change one letter in the dude’s wife’s name, you get my name:

“Meanwhile, Jen is always wrong. At home with the kids, she’s an anachronistic housewife; at work, she’s ditching her kids to nurture selfish professional ambitions. Somewhere, lurking at the root of this all, is the tenacious idea that men should have a career, whereas women must choose between a career and being at home.”

Other thoughts that should make their own blog post but aren’t because the next posts will be on baking (hopefully) or math (that’s ok too): I was very eager to read the NYT op-ed titled “What Makes a Woman?” but it was a bit more defensive and less full of brainstorming of collaborative solutions between cis and trans women than I was hoping.  On a similar note, my friend recently posted about her friend’s blog about being a trans*dude , which I’ve started reading and I agree with what she said “I’ve learned so much reading it!”  I still have a lot to learn (like I don’t know why that asterisk is there in trans* but I’ll find out).

## Cool earrings and the Pythagorean Theorem

4 Jun

I’m so bad at suspense/surprises.  I wanted to write this post and say LOOK AT THE COOL THING AT THE END SURPRISE but instead I put the cool thing in the title.  In any case, last weekend I went to my college reunion and saw a dear friend who is doing an incredibly cool summer project about teaching and very thoughtfully teaches math to high school students during the school year.  She was wearing an amazing pair of earrings, which are the topic of this post.

A photo posted by Yen Duong (@hipsteryet) on

Let’s look at them separately and then together.  First, the top, with the square embedded in the larger square.  We’re going to use some variables (which we talked about in this old post).  Notice that the triangles on the four sides are all identical right triangles.  Let’s label the sides of them: a for the short side, for the longer leg, and for the hypotenuse.

I LOVE visual proofs.  Personally I find them much more convincing than lists of equations with no pictures.  Spoiler alert that’s what we’re doing right now (visual proof, not a list of equations)!

Since the inner gray square is a square, and all of its sides are labeled by c, the area of the inner gray square must be $c^2$.

Once, during a calculus test, a student asked me for the formula for the area of a triangle.  I got mildly upset and said to think about it for a few minutes/come back to the question.  Eventually he got it, but I think it’s because another TA told him the formula.  Anyways, the area of the blue triangle is half of the base times the height, so it’s $\frac{1}{2}ab$.

With four blue triangles, we have a total of 2ab area from the triangles.  So the total area of the larger square is $c^2+2ab$.  But I can also look at the outside square.  Its sides are made up of one short blue leg and one long blue leg, so the large square side length is a+b, which means the outside square area is $(a+b)^2$.  Then the first earring gives me the equation $c^2+2ab=(a+b)^2$.

Now let’s look at the second earring.  I’m going to use the same variables since they’re the same triangles.

Here, all four pink triangles are identical.  The orange square has for all of its sides, and the green has sides, so we know their areas.  We also still know the area of the pink triangles.

If, like that calculus student, you happened to forget the formula for the area of a triangle, you can see it here: two triangles together form a rectangle with base and height b, so the area of the rectangle is ab.  As the rectangle was made up of two identical triangles, you can see the area of each individual triangle is 1/2ab.

Again, the length of the side of the larger square is a+b, so the larger square’s area is $(a+b)^2$.

Summing up the orange square, green square, and four pink triangles gives another expression for the area of the larger square.

So the earrings are both pretty cool separately- we got to prove that binomial expansion works in the second earring, and we got to play around with this technique with the first earring.  What if we put them together?  Both of the earrings have the same large area, which we decided was $(a+b)^2$.  So we can set the other sides of the equations equal to each other, and…

Party time!

WHAT IT’S THE PYTHAGOREAN THEOREM WHERE DID THAT COME FROM THIS IS SO COOL!  Just look at how happy the squares are.  That’s how happy I felt when I saw Shira’s earrings, and also how happy I hope you feel after seeing them too.

Side note, I need to bake something.  Unfortunately those lime pies from a few months ago were so delicious that every time I feel like baking, I make those pies.  Another dear friend who does not do math asked me to try making pretzel salad, so that’s on the agenda, but my in-laws just visited and left a box of graham crackers.  Yes, we could feed them to the baby, but they’re just sugar and honey (which you shouldn’t feed to babies), so I’ll clearly make a graham cracker crust and might accidentally make another lime pie unless inspiration strikes.  We’ll see!

## What is the fundamental group?

29 May

I’m at Yale for my fifth reunion, and it’s my birthday!  Happy birthday to me!  I’m a little overwhelmed by seeing all the old friends/catching up/showing off pictures of the baby so I’m hiding in my suite and pumping milk and writing a quick post about the fundamental group.

We’ve talked before about what a group is– a set of elements with some operation that takes two elements to another one (like addition with the group of integers takes 5+3 to 8 or multiplication takes 1*9 to 9) which satisfies some group axioms.  Given a geometric or topological object, we can associate a group with it my defining these elements and an operation, and making sure that they satisfy the axioms.

First we fix a basepoint of our space, which means that you pick a point and say that’s the one, that’s the special one I want.  Then our group elements with be isotopy classes of loops (this means you can wiggle loops to be the same, as in the red ones below) that go through the basepoint.

Red curves are homotopic to each other; blue curve is not

The group operation is concatenation– first you do one loop starting and ending at the basepoint, then you do the next loop starting and ending at the basepoint.  You can homotope away the middle connection to more clearly see the resulting loop.

Here’s the example:

Red and green make… more red. I didn’t want to make a brown curve

I don’t want to FOMO my reunion (I already ran away to take a nap) so we’ll make this super fast and just look at the fundamental group of the circle and of the torus.

How can I make a loop around a circle?  Well, there’s one obvious way- make one full circle and end up where you starting.  You can homotope to something that backtracks for a bit and then comes forward again, and you could go around two different directions (counterclockwise vs. clockwise).  So let’s call these +1 and -1.

Red goes around once counterclockwise, even though it backtracks a bit, and orange goes around once clockwise.  Imagine the colors down on the black circle.

If you put the red and orange curves together, concatenating like we did above, you’d fully backtrack over yourself, which means you could homotope to just a point.  Let’s call that 0 (can you guess where we’re going here?)

This blue curve goes around the circle three times.

You can go around any integer number of times, but no fractions because you won’t end up back at the basepoint.  So this is a rough schematic of why the fundamental group of the circle is the integers.

I want to go back to my reunion, so I’ll just tell you that the fundamental group of the torus is $\mathbb{Z}^2$, a.k.a. ordered pairs of integers, as I hinted in a previous post using the following picture.

Left: follow the numbers to see the knot. Right: look at the bottom-most green line.

Sorry for the short and delayed post.  It’s my birthday, YOLO.  (I’m not that sorry about this post being delayed but I am sorry if it is unclear/too short please comment/let me know if you need more explanation).

## Basic, soft, chewy chocolate chip cookies

17 May

I haven’t been feeling very inspired baking-wise lately, so I was happy that a friend brought over some frozen chocolate chip cookie dough balls before we left Chicago.  They were so tasty that she emailed me the recipe.  She then emailed me a few more times about said cookies vs. another recipe, and sent me the delightful line: “And now I’ve sent you too many emails but cookies: serious business”

What if the general of the Confederate States of America actually transmogrified and moved to Scotland? So everyone’s been saying the Loch Ness monster, or Nessy, but really she’s a he and is a Ness Lee monster?

I baked a half batch of these a few weeks ago and dutifully melted the butter, which really does turn the cookies from a normal soft cookie to a delightful chewy cookie.  But I was lazy this time and wanted to finish off my Earth Balance (from when I had a potentially vegan person over and made some awesome vegan mac and cheese) so that’s what I did, and ended up with a basic, solid chocolate chip cookie.  This blog is usually pretty adventurous, but everyone needs a basic chocolate chip cookie recipe!

First, cream the butter with the sugar.  I love how much brown sugar is in this recipe.

I’ve worked out a code to transmit messages via rolls and dairy products. But you can trust me with your secrets; I won’t butter a word.

Make no mixtake- I’ll make puns no matter who reads this

Finally you add the dry ingredients.  Since I use salted butter I rarely add salt to cookies, but I am incorrect and you should add salt.  It really brings out flavor!  I’m just lazy (also note that I didn’t use one egg + one egg yolk, because why would I separate out one egg?)

Mix that up, and stir in the chocolate chips.

I also bought parchment paper the other day because I am not good with silpats despite my professed love for them.  I made cookies on them about a year ago that tasted like dishwashing liquid because I was putting the silpats in the dishwasher.  Parchment paper is so easy! (But so wasteful.  But so easy!  Dang it I feel bad enough for switching from cloth to disposable diapers when we moved.  Now I feel guilty about parchment paper.)

To freeze cookies, just scoop them out into balls on a baking sheet and shove the whole thing in the freezer til they’re hard.  Then dole them out into freezer bags, and label them.  Bake for about 2 minutes longer than normal.

If Tom Petty was a ball of chocolate chip cookie dough: “Cause I’m freeeeeeeze, freeze fallin”

Puns were worse than usual this time around.  But the cookies were still good!

2 c flour

1/2 tsp baking soda

1/2 c + 2 TB butter (1 stick + 2 TB)

1 c brown sugar

1/2 c white sugar

1 TB vanilla

2 eggs

2 c chocolate chips

Cream together the butter and sugars.  Preheat oven to 325.  Prepare a baking sheet (parchment paper or silpat or nothing).

Add in the vanilla and eggs and cream.  Add the flour, and sprinkle with baking soda (this is because you are lazy like me and didn’t sift them together earlier).  Mix.  Add the chocolate chips and stir.

Drop by tablespoons onto your cookie sheet (I fit a dozen per sheet).  Bake for ten minutes, or until the edges just start to look toasty.