## Kissing numbers, current research in hyperbolic surfaces

30 Mar

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 and 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 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 path, and wiggle things around, you’ll see a four-holed sphere (two holes above” the curve, and two holes “below,” one of each inside a 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 and the 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.

## Fun-stuffed peppers

24 Mar

I don’t normally like making hors d’oeuvres, dumplings, or even cookies that require anything besides dropping, because I am lazy in this very specific respect.  But my advisor’s weekly “secret” seminar has treats (the speaker from the previous week brings treats for the following week), and I thought I’d try something crazy and wild.  So I bought a can of crescent roll dough (key to making the amazing easy cinnamon buns) and a bag of sweet mini peppers, and figured I’d figure out the rest the day before seminar.

I was surprised that I couldn’t find exactly what I wanted by googling, so here’s a Yen original recipe.  What actually happened was that I wanted to make pigs in a blanket, but then remembered we had a vegetarian in the seminar, so I bought the peppers instead.  These turned out AWESOME- the sweet peppers popped a bit, the paprika + pepper jack made it a little bit interesting, the cheesy filling was so creamy and offset perfectly the crispy buttery crescent roll dough.  I highly recommend.

I hope one day baby gets a tutor named Jack, so I can tell him to go pepper Jack with questions

There are two annoying things to do: one, seed the peppers.  I was lazy and just chopped off the tops to take out the seeds, but they’d look prettier if you left them on and cut a slit to pull out the seeds.  Two, fill the peppers.

Of course Jack won’t be useful for taking tests. Baby will just have to cr(e)am if he doesn’t study ahead of time

We’ll just want to fill his head up with knowledge

I used a regular knife to fill the peppers.  The filling was a block of cream cheese + a few slices of pepper jack cheese + paprika for color.  These would also be great with some sliced green onion, any other sliced cheese, and any other savory spice.  Using a mixer to whip the cream cheese was key for making the filling spreadable, and it also spread out the sliced cheese very well.

Third annoying thing that was less annoying because it meant we were close to done: wrapping the peppers in crescent roll dough.

But academics isn’t everything. I don’t want him to tear up if he messes up on a test. Just roll with the punches, baby!

Also I got to use our pizza cutter, which we never use!  I don’t know if we’ve ever used it on pizza.  Or made pizza.  Hmm I should make pizza!

I guess that’s what happens with kids: you just have to bundle them up and send them off into the world

These are both adorable and misshapen.  If you’re a fancy Pinterest-er who follows food blogs, I’m flattered you look at this and you should make these and make them more beautiful and tell me about it.  If you’re not a fancy Pinterest-er, you could make these and really impress people with them because they are so delicious and don’t need to be beautiful.  And they come out beautiful anyway!

Fun-stuffed peppers (Makes about 20 peppers, about 7 appetizer-sized servings.  The “about” is how many peppers come in your bag)

1 package of Pillsbury Crescent Roll Dough (or something similar)

1 bag of mini sweet peppers

4 oz cream cheese (half a block)

2 slices of cheese (I used pepper jack, but cheddar or gouda could also be good) [you could also grate some cheese]

1/2 tsp paprika (ideas for other seasonings: garlic salt, oregano, cilantro…)

Set the oven to 350.  Seed your mini peppers by slicing them the long way and scooping out the seeds w/knife, or just cutting off the tops (the seeds are connected to the tops).  You’ll still want them sliced the long way.

Use a mixer to beat the cream cheese w/cheese and paprika.  Use a knife to scoop a small amount of filling into each mini pepper.

Roll out the crescent roll dough.  Use a pizza cutter to cut each triangle into three slimmer triangles, or be lazy as I was above and just cut strips.

Wrap each pepper with a small triangle or strip of dough.  If there’s leftover dough, just press it onto a larger pepper.

Bake for 15-20 minutes, until dough is golden brown.

## LIME PIE IS SO GOOD

14 Mar

Pi day is almost over and I almost didn’t post!  Nothing exciting mathematically for this quick post; just a RIDICULOUSLY DELICIOUS AND EASY LIME PIE.  If you buy a graham cracker crust, it has THREE INGREDIENTS.  If you don’t, it has five!

Fun and cutely embarrassing fact: when I was in high school, my best friend and I would come up with codenames for our crushes- hers were a series of colors, and mine was a series of fruits.  Also, how we became best friends is I “stole” her “boyfriend” (we were 14), whom we later dubbed “orange” because it’s the only color which is also a fruit.  Anyways, I saved lime for a really serious one, because it was at the time my favorite flavor.  In fact I think “lime” never got assigned to anyone.  I wish I could go back to my teenage self and ask her if this guy I married is worthy of the “lime” title (hopefully she’d say yes!  I’d say I think he is, but it’s now unclear to me how good the title is.  In any case I like my husband a lot, but maybe not as much as I loved small green fruits when I was 15).

How pi-tiful would it be if I missed a pi(e) day post?

The not-secret ingredient behind lime pie is a can of sweetened condensed milk- this is also the not-secret ingredient behind Vietnamese iced coffee.  You too can make ca phe sua da or ca phe sua nong at home: pour some sweetened condensed milk in the bottom of a glass, add some espresso/very strong coffee, then add ice cubes or some hot water.  Stir.  I used to squeeze sweetened condensed milk over bread pudding as a sort of creme anglaise substitute.  Also, shaved ice + freshly cut chilled fruit + pour over some sweetened condensed milk = DELICIOUS.  And of course you need sweetened condensed milk for magic bars, which I will make and blog someday.  Just thinking about sweetened condensed milk is making me happy right now.

You may notice that this is not a key lime pie.  Key limes are small.  This pie would require juicing 20 or so key limes.  There are four limes in the picture above.  I am fairly lazy. You do the math.  (This is funny because there’s no math to be done here, besides maybe realizing the inequality 4<20.  However, in my meeting with my advisor a few weeks ago he told me that the key point that I had missed somewhere was the fact that 4/3 of three is less than five.  So there is math!  Or something something about me.)

This was my first graham cracker crust, and my friend actually made it (this may be the second time I’ve made a pie with someone.  First time was also in high school, with a friend who coincidentally shared my last name– we made a beautiful apple pie.  She showed me her secret of leaving the apple slices in a bowl of water to keep them from turning brown.)  We microwaved a stick of butter, crushed a bunch of graham crackers in a plastic bag, and mixed the two with a pinch of salt right in the pan.

I feel like graham crackers look much better than they taste- a fool’s gold of cookies. Just like pie, right? (according to this slate article: http://www.slate.com/articles/news_and_politics/assessment/2011/06/pie.html) [This is a pun on pyrite]

I didn’t realize graham cracker crusts were so easy!  To make it a bit prettier you can tamp it down with a measuring cup or anything flat.  Then bake it while you make the filling.

If you code in Pi-thon instead of Python, what comPiler should you use? (consultant on this pun: husband, programmer extraordinaire)

Step one of filling is zesting the lemons.  If you don’t have a microplane grater yet, I highly highly recommend buying one.  This is one of very few kitchen tools I’ve bought (rolling pin?  Why not a wine bottle? Any other gadget?  Why not a knife?).  Also, think about how long this would take if you used 20 key limes instead of four normal limes.

I wonder if anyone has analyzed the horoscope data of people who are really good at memorizing digits of pi. Are there more Pisces and ScorPios? Are Capricorns pretty good, but more likely to accidentally throw in some other digit?

Next, toss in some egg yolks.  You could of course use the whole eggs, but yolks make things more custardy/rich and they’re what I used.  Then I had egg whites for breakfast (you could also use the opportunity to make pavlova, the best thing I’ve ever made).  I used my stand mixer and mixed that up really well, til pretty light.  You could also just use a fork/whisk and a bowl for this recipe, but I do think that using a beater makes the texture super smooth.

I wonder if I can make it through this whole post only doing Pi jokes. No reason in pi-ticular. Just kidding, it’s Pi Day!

Now add in your can of sweetened condensed milk and beat that too.

Speaking of memorizing digits of pi, I’m horrible at it. I was pretty bad at P.E. as a kid, but I’m even worse at Pi E.

I guess I should add “memorizing digits of pi” to the big Pi-le of things I’m bad at (also, editing out weak puns is in this pi-le)

Ugh I’m using my husband’s stupid little computer and I accidentally published this post here.  Sorry!  Updating as fast as I can to finish it.

Finally, juice those four limes (or 20 key limes) and toss that in to the filling, and mix.  I have some asides about recipes here: generally, key lime pie recipes I found listed graham crackers and lime juice using cups as units of measure: so 1 1/2 c of crushed graham crackers and 2/3 c of lime juice.  The lime juice I understand, because limes come in different sizes/might be juicier or less juicy, and you want recipes to be repeatable.  But aren’t all graham crackers the same size?  Every time you crush X number of graham crackers, you should get 1.5 c of crumbs.  I’m just saying as an amateur graham cracker crust maker, I did not know how many graham crackers to use.

Pour your filling into your parbaked crust (it’s been in the oven for about ten minutes by the way, unless you’re really slow at juicing limes in which case maybe you took it out after ten minutes), and bake for another ten minutes until it looks not jiggly.

I’ve gotten myself into quite a mess. I should’ve brainstormed pi puns ahead of time, then selectively chosen from the list. Then I’d be out of this pickle by my pi-culling.

I want Weird Al to make a parody of that N Sync song, from the point of view of the White and Nerdy. (“Yo I know Pi to a thousand places”) It’d clearly be called “Pi Pi Pi”

We let this cool for a little bit, then stuck it in the freezer because we wanted to eat it.  You could also put freshly whipped cream on top if you wanted.  But it’s great on its own, and is SO EASY.  We literally went from “hey, do you want to make a pie?” to eating a pie in one hour, which includes the 15-20 minutes of freezer time.

If the singer of “Party in the USA” had been really into saying digits of the ratio between the circumference and diameter of a circle as a kid, would she be called Piley Cyrus? (instead of Miley, which is short of Smiley, rather than her given name. Thank you wikipedia)

SO GOOD.  SO EASY.  SO GLAD THIS POST IS DONE SO I DON’T HAVE TO KEEP COMING UP WITH PI PUNS.

Lime pie (adapted from the ever-amazing smitten kitchen)

1 sleeve of graham crackers

1 stick of butter (1/2 c)

Pinch salt

1 can of sweetened condensed milk (14 oz)

4 limes or 20 key limes

3 egg yolks (or eggs)

Crush the graham crackers (we stuck them in a plastic bag then pulverized with a bottle, but a rolling pin or something would work well too).  Melt the butter.  Mix the graham crackers and salt in a pie pan, then pour over the melted butter and mix.  Tamp down into a crust.  Bake at 350 while you make filling (set timer to ten minutes to be safe).

Beat the egg yolks with the zest of the limes for several (5 per SK) minutes.  Meanwhile, juice the limes.  Then add sweetened condensed milk and beat more (3 minutes per SK).  Stir in lime juice.  Pour into parbaked pie crust.  Bake for another ten minutes.

Let cool completely, then chill.  Then eat.

## Some thoughts on knots: current research

8 Mar

Over the weekend I went to the Third(!!!) annual Midwest Women in Mathematics Symposium (remember when I founded it?  Now it’s all fancy with funding and many attendees and event staff!)  As it turned out, not very much of the math in my parallel session was exactly up my alley, and also I was feeling lazy so I didn’t take many notes.  But here’s a small recap/introduction to knot theory from my memory.

Aside: I like using knot theory as an example when people ask me what math is for (this happened a lot as an undergraduate and less and less as the years go by).  I’m not even sure if this is true, but I tell people that mathematicians were studying knot theory for decades, and then biologists realized that they could use it to study how proteins fold and interact with other molecules.  APPLIED!  IN YOUR FACE, MATH DOUBTERS!  Unclear where I picked up this bit of folklore, but it’s my number one defense when people say that modern math research is useless.

So what is knot theory?  It’s certainly not not-theory, despite my claim as above that it can be applied.  Knot theory studies objects called knots.  A knot is some way that a circle is embedded in space- imagine taking a shoelace, knotting it up however you want to, and gluing the ends together.  (By space knot theorists mean $S^3$, but we can just think of it as $\mathbb{R}^3$, or the space we live in).  To talk about knots, knot theorists draw knots as diagrams using over and under crossings.  Two diagrams can represent the same knot, like in the picture below.

Even though they look different, these knots are the same.  They’re called the unknot.

If I didn’t mess up, the blue knot is the same as the orange knot- just follow the crossings and you’ll see that nothing is actually knotted; it’s just a pile of string lying on top of itself.  Below are some pictures of other knots.

It’s hard to tell if two diagrams represent the same knot.  Mathematicians can use a diagram to assign polynomials to a knot, and do it in such a way that if two diagrams represent the same knot, then they give the same polynomial.  Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial (which generalizes the previous two).  These still aren’t that great though, since two different knots can give the same polynomial (so while you can tell if your diagram ISN’T the unknot by seeing if the polynomial isn’t 1, you can’t tell if it IS the unknot if the polynomial gives you 1).

The first knot theory talk I saw connected knots to surfaces, so I was a fan.  It was given by Effie Kalfagianni, a professor at Michigan State.  One thing you can do with a knot is use it as the boundary of a surface.

From wikipedia: I was having a really hard time making my own pictures.

There are different ways to make a surface from a particular knot- draw a different diagram and you’ll get a different surface.  One thing you can study is the genus of a knot: this is defined as the minimum genus (# holes) of a surface bounded by that knot.  So for any diagram you draw, you can’t make a surface with a smaller number of holes.  The genus of the unknot is 0.  The genus of a knot using orientable surfaces is known, and there’s an efficient algorithm to find it.  BUT the problem is open for non-orientable surfaces (these are surfaces that don’t have two sides).

“Sometimes I feel like I can’t trust you… it’s like you’re two sided.”

So Kalfagianni’s research, joint with her student Christine Lee, puts a bound on the non-orientable genus of alternating knots, which are knots with diagrams that alternate between over and under crossings (alternating: the purple and red knots.  Not alternating: the unknot, either blue knot (there are two over crossings in a row)).  They use one of the factors in the Jones polynomial to do so.

So that was talk number one!  The second talk I saw was by Maggy Tomova, an assistant professor at University of Iowa.  I actually didn’t write any notes down for her talk, but I remember a cool concept from it.  A knot diagram is in bridge position if you can draw a line across the middle so that there are only local maxima above it and local minima below.

GET IT? It’s a visual pun!  The green is in bridge position.  The red is not.

One immediate note is that in general, bridge position is not unique: given a knot in bridge position, you might be able to find another diagram in bridge position that represents the same knot.  There are some properties that ensure that a bridge position is unique (this is a theorem that I don’t remember).  Tomova is working on some theorems that have to do with knots in bridge position, and I’m sorry that I can’t tell you more information.  She did her Ph.D. at UCSB though, with the same advisor as some delightful other people who are her co-authors on this project (the delightful only applies to the first link; I don’t actually know her other co-author but I really like Yoshi and the fact that he goes by Yoshi).  Also, one of her previous co-authors taught me abstract algebra when I was an undergraduate and he was a postdoc!  That link is to a piece he wrote on going to the “Dark Side,” a.k.a. leaving academia for Google.

So I am not a knot theorist, but there’s your post with thoughts on knots!

## What is geometric group theory?

2 Mar

I’m sitting in my bathrobe on the couch eating a bowl of chicken soup while husband watches baby, which is all to say that I apologize if this fever-tinged post makes less sense/is less factual than my usual math posts.

This post is the story of my fairly young field of math, informed by the folklore I’ve heard in the three years since I first heard the words “geometric group theory” and some wikipedia.

Pure math is, very very roughly, divided into three main areas: algebra, analysis, and topology.  There’s a whole bunch of other math that doesn’t fit in here (logic, set theory, category theory off the top of my head), but these are the three required core courses that every U.S. Ph.D. student studies in their first one or two years of grad school.  Geometric group theory lives between algebra and topology- “group theory” is the study of groups, which we’ve seen a few times before, and “geometric” means that we’ll be looking at shapes.  Geometric group theory (GGT for short) uses geometric/topological methods and ideas to come to conclusions about groups associated with shapes.

Fundamental group of this four holed surface is quadruples of the integers (the mouth is a hole but not the eyes)

There are a few main ways to associate groups to shapes: the first we learn is the fundamental group, which will get its own post sometime- this group records different loops on our topological shape.  There’s also homology groups, cohomology, mapping class group, higher homotopy groups, etc. etc.  These all record different info about the shape.  The fundamental group of a circle is the integers, of a torus is $\mathbb{Z}^2$, or pairs of integers, and of the n-holed torus is $\mathbb{Z}^n$, as in the picture above.

Speaking of segues, geometric group theory started as a way to answer some questions (as fields of math are wont to do).  In the 1910s, a mathematician named Max Dehn posed three questions about groups.  To understand them we’ll need to know about group presentations, which is just a standard (but not canonical) way to write groups.  So take your group, and look at the generators you have.  Label each generator by a letter in an alphabet- we have a good one, it starts with “a” and moves on to “b” but you could also do $a_1, a_2, a_3\ldots$ if you wanted.  Then write down all the true equations involving your generators.  This is best done with an example also I am done with my soup =(

Let’s take the group of pairs of integers, $\mathbb{Z}^2$.  We’ll use (0,1) and (1,0) as our generators, since any pair (x,y) can be written as x(1,0) + y(0,1).  Let’s label them by a=(1,0) and b=(0,1).  Then a true equation is a+b=b+a.  Since we’re in group-land, let’s skip the “+” sign and say addition is our group operation, and write ab=ba, or equivalently, $aba^{-1}b^{-1}=e$, where I used for the identity element (0,0).  Then our group presentation is $\langle a, b | aba^{-1}b^{-1} \rangle$.

Back to Dehn’s problems!

Word, dog.
But actually you pronounce Dehn like a great Dane. I don’t mean to say the mathematician was a dog, just that his name sounds like a dog. This was funnier before I started writing the caption.

One was the word problem: given a word in your alphabet, could you tell if it was the identity element?  It’s clear (0,1)+(1,0)-(0,1)-(1,0)=(0,0), but what about a word like $aba^{-1}b^2a^{-4}b^5$ in our group presentation?  Actually the question of whether a word is trivial (another way of saying equal to the identity) in our group presentation is pretty easy to answer: just count up the exponents of each letter.  If they sum to 0, then you’re trivial.  But in general this is hard.  Dehn’s other two problems were also hard.

Group theorists used combinatorial (roughly, counting) methods to try to answer Dehn’s problems, and wrote algorithms (Dehn did this, actually) to tackle them.  On the way they built up combinatorial group theory, and drew lots of pretty pictures of trees (graph theory) and planar diagrams (which is what I do a lot of).  According to wikipedia, in the 1980s GGT started appearing after Gromov wrote a thing on hyperbolic groups (hyperbolic post here).  That was around the time that people started realizing that you could generalize properties of groups: instead of saying oh, group A has properties 1,2,3, and so do groups B, C, D, etc., you could say all groups that are quasi-isometric to group A have properties 1, 2, 3.  Understanding groups up to quasi-isometry is one of the main goals of geometric group theory.  (quasi-isometry defined in this post).

And now geometric group theory is a thriving young field.  You can tell from this page that UIC is one of the big GGT departments in the world, and this page shows all the conferences going on about it.  In fact, I’d say all GGT mathematicians on the internet know Jon McCammond’s GGT website.

And that’s all I have to say about what geometric group theory is.  Back to very important work, napping.  Apologies again for lack of clarity, precision, sense-making…

## Ridiculously easy mini banana cinnamon buns

15 Feb

While I’ll always make those vegan (sometimes) pumpkin cinnamon rolls whenever I have three or four hours to wait, I now make these cinnamon buns every time I want cinnamon buns NOW (but really, within half an hour from start to finish, which is faster than going to the store to get them).  I’ve had this recipe for about three months and have made these at least six times- they were the first thing I baked after baby was born!  They were also the second thing I baked after baby was born.  And possibly the third.

No matter how many people you have eating these, they’ll be gone very quickly.  I think four is a great number, so everyone gets to have two and maybe wishes they had more.  But three is also wonderful, because eating three of these is wonderful.  Honestly I could eat all eight that come in a batch by myself.  What I’m saying is THESE ARE SO GOOD GO BUY SOME CRESCENT ROLL DOUGH AND MAKE THEM NOW.

They’re so good that I didn’t take a picture of all the ingredients beforehand.  If you bake at all you already have all of these minus the crescent roll dough.

I just did jury duty. Everyone had pat responses to everything

First put a pat of butter in a smallish casserole dish or pie pan, and stick it in the cold oven.  Turn the oven to 350.  I love when stuff goes in a cold oven, so you aren’t wasting gas by preheating.  Anyways.

Open your can of crescent roll dough (this was not intuitive to me and it took me awhile and I still think I do it wrong), and spread out the dough.  Pinch together the edges so you have one big rectangle of dough.

It wasn’t hard to get the witnesses to open up, you just had to press at the right moments

With the facts spread before us, we had to return a verdict

It’s OK if it’s not perfect; pinching just makes it easier to cut the buns.  Now stick another two tablespoons of butter in the microwave for 15 seconds (or if you’re one of those people who leave butter out, just take that softened butter.  I think I’ll start leaving butter out.  It’s so convenient!) and spread it on the dough.  Mash or slice a banana and spread it on top of the butter.  I’ve done it both ways and I think I prefer eating the mashed banana, but slicing is SO EASY.  Then mix some cinnamon into some sugar and sprinkle it over the whole thing.

We were following the letter of the law, ignoring ideas of sin. Amen! Church and state should be separate

Sometimes witnesses would sprinkle in random facts/observations, and we’d take that into account too

Now take the long edge and fold up an inch.  Roll up the whole thing this way (so you’re traveling a shorter distance and holding the longer side).

Taking a break from court- check out this natural log! It rules!

Cutting the roll is the hardest part of this.  I’ve used floss, as in the next picture, but a sharp slightly wet knife has worked the best.  No matter how I’ve done it, I smushed the rolls.  That’s OK.  Cut in half (the dough already does this for you), then cut each half in half, and each of those in half, so you end up with eight. (Yes, Yen, two cubed is eight.  Good job!)

Sometimes it was hard to cut between the lines and figure out what actually happened.

Now pull that casserole dish out of the oven- the butter should be nice and melted now.  Spread it around, and sprinkle with two tablespoons of brown sugar.

This was just a small civil case, not like Brown vs. Butter of Education. Oh it was Board of Education, huh. Mixed it up in mind for some reason.

Many things were mixed up in the case- who was where when, why and how what happened happened, sugar and butter (I brought cookies for day 3 of jury duty)

Now put the rolls, spiral side down, in the pan.  They’ll be a few inches apart, which is fine- they’ll spread into deliciousness.  This brown sugar-butter combination on the bottom makes a crust similar to pineapple upside down cake.  It’s awesome.

Bake for 20 minutes.

I love these plain, but you can also glaze them (powdered sugar + milk + vanilla) or frost them (cream cheese + powdered sugar + milk +vanilla).  I prefer the cream cheese and will include that below.

It was hard to look at the plaintiff when we returned the verdict in favor of the defendant- there was no way to sugar coat it.

Baby cinnamon rolls, adapted from kevin and amanda (theirs is also good but uses twice as much butter and sugar)

1 can Pillsbury crescent rolls

1/4 c butter (1/2 stick)

2 TB brown sugar

2 TB white sugar

1 tsp cinnamon

1 very ripe banana

(optional): 2 oz cream cheese, 1 c powdered sugar, 1 TB milk, 1/2 tsp vanilla

Cut two tablespoons of the butter into a pie pan or a small casserole dish.  Place into the oven and set oven to 350.

Spread out the crescent roll dough, pinching seams together.  Soften remaining two tablespoons of butter (microwave 15 seconds) and spread on dough.  Either mash banana and spread atop butter, or slice and lay in lines across the dough (like an American flag).  Mix white sugar and cinnamon, then sprinkle over the dough.  Roll up dough into a log.  Cut into eight slices.

Remove warm pan from oven.  Sprinkle with brown sugar.  Place slices spiral side down onto the butter-sugar.  Bake for 20 minutes, until golden brown.

(Optional): Soften the cream cheese (microwave for 15 seconds), then mash with a fork.  Add in powdered sugar, milk, and vanilla, and mash until you have a thick frosting.  Plop some onto each warm cinnamon roll.  Devour.

## Protagonists are male; I didn’t wear makeup as a kid

30 Jan

Apologies for a long delay in posting; we just came back from our meet-the-family/honeymoon vacation with baby. Here’s a quick post on neither baking nor math; both should return soon.

As a kid I played with my brothers all the time.  I distinctly remember playing with those little matchbox cars and having them talk like transformers to each other.  I always picked the pink car because she was the “girl” hot wheels.  This is ridiculous to me now, since we had 20 or so cars and they didn’t have faces or anything indicating their genders besides color.  And it’s not like 19 cars were blue and one was pink; they were all different colors, designs, etc.  But I was fixated on the pink one.  Looking back, did I just have a favorite car, or did I feel like the other cars weren’t for me?  If this post was just this anecdote, I’d say that I just had a favorite car which happened to be the pink one, and my brothers didn’t share the same obsession over any single car.  But.

We also played with legos.  We had two little hair clipons that you could put on your person to make them a girl, and we also had one head with lipstick and mascara.  I didn’t use that head because I didn’t (and still don’t) like makeup.  So every time I played, I’d put a hair on a person to make them female, which meant that our default Lego population were all male (comically statistically unlikely number of bald men in the Duong Legotown at any given time).  It also meant that girls couldn’t be firefighters, policemen, or pirates, since those all had separate hats and you could only wear one at a time.  This was zero percent a big deal to me as a kid, and is some percent a deal to me now.  I would be remiss not to link to Anita Sarkeesian’s video on this here.  And here’s part 2.

I was the youngest and did and liked everything my big brother did and liked, but I never got into Transformers, Teenage Mutant Ninja Turtles, or Gargoyles as much as he did.  I loved Sonic the Hedgehog and Tails (a two tailed fox sidekick, I think?), but then the cartoon came out and it turned out Tails was a boy too, and my interest waned.  We played Secret of Mana often on our Super Nintendo, which was a super fun multiplayer RPG, and my brother would be the Boy main character and I would play Girl or Sprite.  I also loved Super Mario Bros 2 more than the other installations of the series, because I could play Princess Toadstool (and she could fly which was badass!)

As a child, I wanted characters who reflected me, or who I could aspire to be, or who I could relate to.  I didn’t want to be the sidekick all the time, but I was, mostly because I was younger but partially because I was a girl, and by default girls are sidekicks or trophies and boys are heroes.  Girls are heroes in girl-oriented products/games, but protagonists are male in general audience products/games.

I wondered whether this last statement is true, so I looked at my three month old baby’s books.  Turns out animal heroes are also by default male.  For instance, the Very Hungry Caterpillar is a he.  So is the Moose and the host in If You Give A Moose a Muffin, Spot the dog, Max in both “Max Explores Chicago” and “Where the Wild Things Are”, Duck in Dooby Dooby Moo (and the two other books in the series), the baby in I Love You Through and Through, Bear in Bear in Underwear, the dinosaur in Thesaurus Rex, and of course Bruce the bear bully in Big Bad Bruce.  I’ll note that the premise of the Dooby Dooby Moo series is that Farmer Brown (male) has a bunch of cows (female) who type, but the main actor of the series is Duck, who is a he.  I counted 12 of Ian’s books with male protagonists.

Girls are heroes in girl-oriented products/games, but protagonists are male in general audience products/games.

How about both?  Where is Baby’s Belly Button, the Tickle Book, Cloudy with a Chance of Meatballs, Introductory Calculus for Infants, and Head to Toe all include both male and female leads characters.  The collection of Dr. Seuss Books, Count to Sleep Chicago, Hippos Go Berserk, Noneuclidean Geometry for Babies, and Possum Come a Knockin’ all don’t have male or female leads.  So that’s nine in the both or neither category.

What about female?  Nope.  None of his books have a female main character.

To be fair, a friend did gift us The Munschworks Grand Treasury of stories, which includes multiple female protagonists.  But that’s for when Ian is a bit older (also I haven’t read any of them yet).

My husband pointed out that this is what happens when we don’t have a gender neutral pronoun in English, implying that “he” is the default pronoun.  But that’s exactly my point.  “He” is the default.  So my son will get to be the hero, and use all the matchbox cars, and be Optimus Prime or Bumblebee or Rafael or Leonardo or Sonic or Tails or Ash Ketchum or a lego pirate or firefighter or policeman.  And if I ever have a daughter, I’ll have to figure out what to tell her so that she can be all these things too, and doesn’t feel like she can only be the pink car or the lego figures with hair or Princess Toadstool/Peach.  Or buy her a whole new set of toys catered just for girls, because boys and girls are apparently so fundamentally different that this face

somehow only reflects half the population, at least when I was a little kid trying to play.