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.

fourholed

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.

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

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

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

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

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

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!

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.

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.

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)

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.

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

Screen Shot 2015-01-30 at 10.51.54 AM

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

The EGGNOGstravaganza part II: bacon stuffed eggnog french toast

9 Jan

This is a ridiculously decadent and delicious breakfast.  Crispy salty bacon, sweet maple filling, and rich eggnog.  You can make this without the leftover eggnog (use eggs+milk+sugar instead), or without the bacon.  It’s so good.

Pros to this dish: all delicious ingredients, all delicious.  There's no way this is BAd, no CONs at all, even if you forget the bacon

Pros to this dish: all delicious ingredients, all delicious. There’s no way this is BAd, no CONs at all, even if you forget the bacon.

I did not know about the best way to cook bacon until I made this dish!  There’s even a website devoted to it.  Just put your bacon into a baking pan and put into the cold oven.  Set to 400, and set the timer for 20 minutes.  That’s it!  Magical crispy bacon every time.

Bakin bacon

Bake in a pyrex

Don't bake on the stove!

Don’t bake on the stove!

And that way you can prepare the rest of the dish while the bacon is cooking.  Make the filling: mix cream cheese with brown sugar and a dash of maple syrup.  I used half a block of cream cheese, and a little less than that amount of brown sugar.  You can’t really have too much of this filling.  It’s SO GOOD.

If you needed to translate the first person direct object pronoun into the most widely spoken aboriginal language in Canada, you could say to a speaker, "Cree me"

If you needed to translate the first person direct object pronoun into the most widely spoken aboriginal language in Canada, you could say to a speaker, “Cree me.”

And if the largest group of First Nations people in Canada made their own concoction a la Mississippi pie, they could call it Cree Mud

And if the largest group of First Nations people in Canada made their own concoction a la Mississippi pie, they could call it Cree Mud

The big plus of using eggnog for french toast is that you don’t have to measure anything, really.  Just whisk (number of people) eggs in with some eggnog (1/2 c per person or so).  A pie tin is the best vessel for this, so you can dip your bread.

Then spread your bread with the deliciousness, and put a piece of bacon in every other piece of bread.

This bread is like the CBS daytime lineup, ready for Phil-ing (though CBS uses the doctor and we'll use bacon)

This bread is like the CBS daytime lineup, ready for Phil-ing (though CBS uses the doctor and we’ll use bacon)

Make little sandwiches of wonderful, and quickly dip them in the eggnog mix.  Fry for about 3 minutes on each side, or until they’re nice and brown and yummy looking.

Filled!

Filled!

If this bread was wearing underwear would it be a diphthong?  Unfortunately I'm pretty bad at telling what a diphthong is (Vietnamese is full of them)

If this bread was wearing underwear would it be a diphthong? Unfortunately I’m pretty bad at telling what a diphthong is (Vietnamese is full of them)

Everyday is FryDay!

Everyday is FryDay!

No need for makeup here, we certainly don't need any brownzer

No need for makeup here, we certainly don’t need any brownzer

Usually people serve french toast with syrup but this basically has syrup inside it.  It’s very good with a side of bacon.

IMG_20141221_112446087 IMG_20141221_112658982

Bacon stuffed eggnog french toast.  This is ingredients per person.

Two pieces of good bacon-one for stuffing and one for eating.  If you’re using the thin little slices double this.

Two pieces of white bread (white soaks up the egg better)

One egg

1/2 cup eggnog

1 oz cream cheese (so I made four servings and used half a package)

~ 1 TB brown sugar

~1 tsp maple syrup

A little bit of butter (for frying)

If you’re literally making this for one person, don’t dirty another pan and just cook your bacon in the pan you’re planning on using.  If more, though, use the bacon method- line a pan with bacon and put into a cold oven.  Turn to 400 degrees, and set a timer for 20 minutes.

Beat the egg and eggnog together in a pie pan.

Mix the cream cheese, brown sugar, and maple syrup together.  Spread on the bread slices.

Break a piece of bacon in half and put into a sandwich.  Dip both sides of the sandwich into the eggnog mixture, and fry in a skillet over medium heat until nice and brown on both sides.  Serve with the other piece of bacon.

Back to basics: algebra and variables (nytimes and sony)

3 Jan

I’ve had many students get tripped up over the concept of variables, which is the key idea in algebra.  Unfortunately these were calculus students, so we had to try to catch up on algebra while learning calculus (also needed to review trigonometry and geometry.  No joke, once a student asked me how to find the area of a rectangle while I was proctoring a calculus exam.  I was speechless.  I checked back later and student had figured it out.)  But it’s not just calculus students who have trouble with algebra.  I ran into this tweet last week:

Snark aside, I want to solve the problem in this article.  First, let’s go way back to basics and talk about variables.  It’s absolutely a leap to go from the concrete world of numbers to something more representational.  Instead of jumping straight into abstraction, let’s use analogies!

First, note that variable implies something that varies.  While we used to use just numbers and equations like 8+2=10, now we have expressions like 8x+2 or x*x.  The value of this expression varies depending on what value we assign to x.  This is the tricky part, because we’re used to things that stand by themselves and don’t vary depending on some other context.  OR DO WE?

Let’s take this expression:

I had a yen for cookies.

By itself, the sentence is ambiguous.  Are you saying you are buying extremely cheap cookies in Japan or Hawaii?  Or do you feel like eating some scrumptious small baked goods?  Or am I hanging out in your kitchen helping you bake?  In this expression, “yen” could mean three different things.  And depending on what value you put in for “yen” you’ll have completely different meanings.

whichone

How can we resolve the ambiguity?  Well, we could add some context.  For instance:

  1. He offered me cake, but I had a yen for cookies.
  2. I couldn’t afford a real meal, but I had a yen for cookies.
  3. I have lots of helpful culinary friends named Yen.  I had a yen to make bread; I had a yen roasting meat; and I had a yen for cookies.

By adding context we narrowed down the sentence meaning to be unambiguous.  This is like adding an = sign to an expression.  So with 8x+2, the value of the expression could’ve been 10 (if x=1), or 0 (if x=-4), or 3000 (if x=374.75), or anything really.  But if we add an = sign and have 8x+2=18, then we’ve declared that 8x+2 is unambiguously 18.  So is unambiguously \frac{18-2}{8} = 2.

Sometimes we still might have multiple solutions to an equation.  My sentence could be “I was in Japan, and I had a yen for cookies.”  In fact, sentences like that basically make up all the captions in this blog.

So I could have an equation like x*x=1, and we’d have two solutions: x=1 works, and so does x=-1.

That’s that for the concept!  Let’s apply some algebra to that tweet.

We have a total of $15,000,000 brought in.  Some of this is from $6 digital rentals, and some is from $15 sales.  And we have 2000000 transactions in total.  So let’s figure out how many transactions came from rentals, and how many from sales.  Then we can figure out how much money sales brought in, and how much money rentals brought in.

That was a heavy paragraph, so let’s break it down.  What do we want in the end?  How much money came from rentals, and how much came from sales.  How can we calculate how much money came from rentals?  Well, we can figure out how many rentals there were, and then multiply by 6, since each rental is $6.  Similarly, we can figure out the number of sales, and multiply by 15.  So let’s figure out the number of rentals and sales, using algebra.

First, we name our variables.  Let’s use for the number of rentals occurred, and use for the number of sales.  We know that R + S = 2,000,000 because there were two million transactions.  Also, we know that 6R + 15S= 15,000,000 because that’s how much money was brought in.  To make this a little easier to read, let’s cut out the 000,000 and add it in at the end.  Our system of equations is now:

  • R+S=2
  • 6R+15S=15

Now let’s solve.  If we solve the first equation for R, we’ll have R=2-S.  Notice that we have a variable in our solution for R, since the value of depends on the value of S.  But we can resolve the ambiguity by plugging this equation into our second equation.  Then we have

6(2-S)+15S=15

So, distributing, we have

12-6S+15S = 15

and grouping like terms gives

9S=15-12=3

Which means that S=3/9=1/3.  Going back to that solved equation earlier, this means that R=2-S=2-1/3=5/3.

So there were 1/3 million sales and 5/3 million rentals.  Going back to our “what are we looking for” paragraph, this means that $10,000,000 came from rentals, and $5,000,000 came from sales.

Of course, you could’ve just put this into wolfram instead of breaking it down into all the pieces like we did.  This post was mostly aimed at my old math 34a students at UCSB.  Likely the people who usually read the math posts on this blog won’t be interested in this post, but maybe you can pass it on to someone else!

an EGGNOGstravaganza! Part I: eggnog

22 Dec

I remember the first time I had eggnog- it was at my childhood best friend’s house, back when we lived in Minnesota.  I must’ve been ten or so, and I immediately loved this ridiculously rich, creamy, mildly spiced drink.  I loved it so much that I’ve only had it five times or so since.  Including a disastrous attempt to make homemade eggnog in high school, back before we knew how to use allrecipes and followed an old Joy of Cooking cookbook I had lying around.  That tasted like raw eggs sitting in milk, which is what it was.  That’s also what my first attempt at eggnog this year ended up as, which explains the coming posts of eggnog bread pudding, eggnog loaf, and bacon-stuffed eggnog french toast (no need wasting perfectly good eggs and cream!)  But my second attempt was AMAZING.  I know I raved about how great ambrosia is during my last post, and that I’ve talked about how pavlova is the best thing I’ve ever made, but damn.  This eggnog.  Definitely the best drink I’ve ever made.

So let’s start with the eggnog, shall we?  I did a comically awful job taking photos of the eggnog because it was SO GOOD that I totally forgot to take a picture of the finished product.  I had to go to the store to buy cream, but other than that I had all the ingredients on hand: butter, milk, nutmeg, cinnamon, vanilla, sugar, and (optionally) rum.  Actually I didn’t have cloves so I used star anise instead and I HIGHLY RECOMMEND USING STAR ANISE.  It’s so sweet and so spicy and just a great, underused ingredient.

“Illicium verum 2006-10-17″. Licensed under CC BY-SA 3.0 via Wikimedia Commons -

Also note that this is an overnight recipe.  It’s really really good as soon as you make it but it’s even better the next day.

First, steep your milk with the spices.  If you had a cinnamon stick lying around now would be a good time to take it out.  I just put in ground cinnamon and a few star anise, then heated the milk on low while I separated the eggs.

This would make an at first delightful smelling STEEPlechase, but then the milk would go sour and everything would be bad

This would make an at first delightful smelling STEEPlechase, but then the milk would go sour and everything would be bad

The recipe I used called for a dozen eggs, but I got tired of separating eggs so I used seven.  It still turned out great!  I think six at minimum.  Honestly a dozen eggs might be too rich, though not if you add rum (I made some non-alcoholic by omitting rum).  Anyways, for a note on how to separate eggs see this old post.  Separate your eggs.

Very clearly separate and not equal

Very clearly separate and not equal

Once everything is separated, turn the milk heat up to high.  Cover the whites and put them in the fridge or freezer.  I put them in the freezer and it was a pain in the butt to thaw them the next day so I’m not sure why I did that.  You will use the whites the next day though so hang on to them!

Whisk the yolks with the sugar until nice and fluffy.  I was thinking about using the stand mixer for this, but the hand whisk worked just as well.

This is a labor oF LUFF. Alternate caption: Instead of naming planes after numbers like F-10s, maybe they should include the airline name too.  So you could have a F-LUFFthansa

This is a labor oF LUFF.
Alternate caption: Instead of naming planes after numbers like F-10s, maybe they should include the airline name too. So you could have a F-LUFFthansa

Now pour some of the hot milk (which is now hot because remember you turned the heat up?) into the egg bowl and whisk some more.  You’re tempering the eggs so they don’t turn into scrambled eggs sitting in milk (oh high school).  This is a mildly technical task but you know me- I don’t care too much about details (it still turns out awesome), so don’t worry.

I want some of those tamper evident stickers but change them to say temper evident.  Then I could put them on custards and on my baby

I want some of those tamper evident stickers but change them to say temper evident. Then I could put them on custards and on my baby

Just keep adding a little hot milk and whisking until the mixture looks pretty smooth.  Dump it back into the pot with the rest of the milk and stir constantly over medium heat until it gets thick, or until you get tired of mixing (which is what I did).  It’s about three minutes either way.  Let it cool for an hour, then add the cream, vanilla, nutmeg, and rum if you’re using it.  Also take out the star anise.  Throw it in the fridge overnight.

We want a pitcher, not a belly itcher!  Well, we have one.  Actually, if you think about how my stomach felt during pregnancy, we also have the latter

We want a pitcher, not a belly itcher! Well, we have one. Actually, if you think about how my stomach felt during pregnancy, we also have the latter

The next day, take out your egg whites and let them come to room temperature.  Throw them in a stand mixer or use an electric one and mix them until they’re frothy and stiff, like in the next photo.

If I get a dog, maybe I'll get a couple of small greyhounds- I always seem to need two whippet(s) This is  both an old photo and old caption

If I get a dog, maybe I’ll get a couple of small greyhounds- I always seem to need two whippet(s)
This is both an old photo and old caption

Use a rubber spatula to gently fold the egg whites into the chilled eggnog, and serve.  SO GOOD SO FLUFFY SO TASTY.  Sorry for lack of photo.

Amazing eggnog, adapted from allrecipes

6-12 eggs, to your taste (12 makes it super rich, 6 is light and delightful.  I went with 7)

1 qt of milk (2 pts, 4 cups)

1- 1 1/2 c sugar, also to your taste (1 1/2 is way sweet, 1 is light.  I did a bit over 1)

2 c rum, also to taste

3 c cream (I used a cup of cream the previous day so I had this much.  You could also use 2 or 4)

1 TB vanilla

1 cinnamon stick and/or 1 tsp ground cinnamon

3 star anise pods

1 tsp nutmeg

Over low heat, combine the milk, spices, and 1 tsp of the vanilla.  Let it steep for five minutes, or however long it takes you until we mention this again.

Separate the eggs, putting the whites into a container for the fridge and the yolks into a medium-large bowl.  Refrigerate the whites.  Turn up the heat to high on the milk.  Whisk together the egg yolks and sugar until light and fluffy.  Add in a little hot milk (I did about 1/2 c at a time) and whisk until smooth.  Do this about three times until you have a very smooth and warm egg-milk mixture.  Pour the bowl of egg milk into the pot of milk, and cook over medium heat, stirring constantly, for three minutes.  Let cool for an hour.

Take out the star anise pods.  Add the cream and remaining vanilla.  If you’re using rum, add it.  Refrigerate overnight.

About half an hour before you want to serve the nog, take out the egg whites.  About ten minutes before serving, beat the whites on high until foamy and stiff.  Then use a rubber spatula to gently fold the whites into the nog.  Best served in a punch bowl, as a pitcher will leave the foamy stuff out.  Garnish with some more ground nutmeg, fresh if you can get it.

I am [not a] bro (sia)

5 Dec

The first time I had ambrosia was in my college’s dining hall salad bar, senior year.  I saw the label “ambrosia” and noticed mandarin oranges, so I grabbed a scoop.  FOOD OF THE GODS.  Who knew mini marshmallows, mandarin oranges, pineapple, and coconut went so well together with whipped cream?

Since I’m sometimes interested in health (or since I always dislike the idea of cool whip and sometimes dislike the idea of washing my mixing bowl and beater), I used full-fat yogurt + honey + vanilla instead of whipped cream.  Honey makes everything better!  The tang of yogurt + honey really complements the pineapple and sweetness of everything else.  I also added some green grapes and toasted pecans for color and texture.

I had to make an extra broceries trip since I don't usually have mini marshmallows in my house

I had to make an extra broceries trip since I don’t usually have mini marshmallows in my house

This is one of those cut up everything, mix, and let sit sort of “salads”.  More like a dessert which happens to live in the salad bar.

The bodega downstairs is called Go Grocer, but only men seem to work there so my friend calls it Bro Grocer

The bodega downstairs is called Go Grocer, but only men seem to work there so my friend calls it Bro Grocer

I wish I could tell you how to cut pineapple.  If I wasn’t operating on such sleep debt I would have looked up several links/researched, then taken step by step photos of the process.  But instead I just hacked at the pineapple and tried to cut out the eyes using a paring knife, which didn’t work great.  The point is, you need your pineapple in bite size chunks.  Canned pineapple would also work, but theoretically fresh will be better (because it’s not soaked in syrup).

I wonder if the star of The Pianist was in a frat.  They'd call him Adrian BROdy.  Oh they already call him that.

I wonder if the star of The Pianist was in a frat. They’d call him Adrian BROdy. Oh they already call him that.

My husband prefers ambrosia without pecans- I’ve made it three times since having the baby, with/without grapes and with/without pecans.  I like the nuttiness that helps weigh down the salad, but that also keeps it from tasting like angels are breathing into your mouth, which is maybe the point of ambrosia.  If you opt for pecans, chop em up and throw them into a frying pan on medium heat for a few minutes.  I left them for as long as it took me to cut the pineapple and grapes.  You can tell when they’re done because they’ll start smelling toasty and nutty and yummy.

According to urban dictionary, I am not a bronut- I am not your friend willing to share my donut.  Nor am I a crazy friend.

According to urban dictionary, I am not a bronut- I am not your friend willing to share my donut. Nor am I a crazy friend.

So you’ve got half a pineapple, a big handful of grapes, a handful of pecans, about a cup of mini marshmallows, a can of mandarin oranges, and a cup of shredded sweetened coconut tossed together in a bowl.  Now stir up your yogurt + honey + vanilla, and toss the whole thing with the dressing.  Let sit for at least an hour for the flavors to meld, and enjoy.  I think it’s better if you let it sit overnight so the marshmallows and coconut melt a little bit due to the acidity of the pineapple.

IMG_20141112_162530443_HDR

If these captions aren’t enough I can just tell you straight up I know zero about bro culture.

IMG_20141112_162945823_HDR

Health-conscious frat guys love their brogurt!

This dish just bros me away

This dish just bros me away

Ambrosia- cobbled together from the internet

1/2 fresh pineapple or a can of pineapple

1 c mini marshmallows

1 c green grapes, halved

1 c shredded sweetened coconut

1 c pecans

1 can mandarin oranges

3/4 c plain yogurt

1 TB honey

1 tsp vanilla

Toast pecans in a frying pan over medium heat until fragrant, maybe five minutes while you cut up the pineapple and grapes.  Mix yogurt, honey, and vanilla in a small bowl.  Add all ingredients to a large bowl and toss.  Leave in fridge for at least an hour, preferably 6-8 or more.

 

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