Proof of Scott’s Criterion for separability (hard math) (UPDATED)

13 Sep

UPDATED: Thanks to my dear friend Teddy (who hasn’t updated his website and is at Cornell now, not UCSB), I’ve made the converse direction of the proof more correct.  There’s definitely still a glaring defect, but that’s entirely my fault.  

This is out of character for this blog- it’s not accessible for most people.  If you have taken a course in algebraic topology, you can read this post and I’ll explain everything.  Otherwise, I’m not offering enough background to understand it.  Sorry!  Blame pregnancy!

I’ve been spending the past few months slowly slogging through a big paper that my advisor recently cowrote, on an alternative proof of Wise’s Malnormal Special Quotient Theorem.  In the paper they spend a few paragraphs explaining Scott’s Criterion for separability, from Scott’s 1978 paper (need access for this).  I did not understand it when reading, but after meeting with my advisor and drawing some pictures it made a lot more sense!  So I’m going to draw some pictures for anyone trying to understand this- probably other graduate students.

Here’s the theorem as it appears in the MSQT paper.

Theorem (Scott, 1978) Suppose X is a connected complex and  H \leq \pi_1(X).  Then H is separable in \pi_1(X) if and only if for every finite subcomplex A \subset X^H, there exists an intermediate finite degree cover \hat{X} such that A embeds in \hat{X}.

Okay let’s unpack the theorem.  First we need to say what it means for a subgroup to be separableis separable in if, whenever you pick an element which is not in H, there exists some finite index subgroup of so that H is contained in and isn’t contained in K.  Intuitively, you can “separate” from via some finite index subgroup.  There are other equivalent definitions, but this is the one we’re going with.

Recall that a finite degree cover is a covering space where each point in has finitely many preimages.

Left side is an infinite cover, the real numbers covering the circle.  Middle is a happy finite cover, three circles triple covering the circle.  Right is a happy finite cover, boundary of the Mobius strip double covering the circle.

Left side is an infinite cover, the real numbers covering the circle. Middle is a happy finite cover, three circles triple covering the circle. Right is a happy finite cover, boundary of the Mobius strip double covering the circle.

Notation wise, X^H just means the cover of corresponding to H, so that \pi_1(X^H)=H.  Also, recall that an embedding is an injective homeomorphism onto the image of the map.  So, for instance, a circle definitely embeds into the middle cover above, but not into the infinite one.  You can map a circle injectively to a subset of the real line, say to [0,1), but it’s not a homeomorphism where the two ends meet.  And by connected complex let’s say CW-complex.

Okay it’s proof time!  For notation we’re going to let G = \pi_1(X).

Suppose that the condition is met, and we want to show that is separable.  Take an element not in H.  Since is the fundamental group of Xcorresponds to a (class of homotopy equivalent) loop(s) in X.  Since isn’t in H, it isn’t a loop in $latex X^H$- let’s say it’s a line segment.  By the condition, since this line segment is a compact subset of X^H, there exists some intermediate finite degree cover \hat{X} so that the line segment embeds into it.  This finite degree cover corresponds to a finite degree subgroup K.  Since \hat{X} is intermediate, is contained in K.  So is separable.  

Here’s a schematic:

I feel like this picture is self-explanatory (this is a joke)

I feel like this picture is self-explanatory (this is a joke)

Okay let’s do from the other side now.  Suppose is a separable subgroup of G.  Pick a finite subcomplex of (the actual criterion just says compact, but we’re sticking with finite).  Look at all the elements g_i of which have preimages in A- since is finite, we only have finitely many of these.  For any given $latex g_i$, since is separable we have a finite index subgroup K_i which doesn’t include g_i and which contains H.

I guess we still need to show embeds- do you believe me that it does?  I’m not sure I believe me.

Pick a compact subcomplex of H.  Since it’s compact, there are finitely many open sets that we need to consider, which cover A.  And since it’s a subcomplex of a CW-complex, this means we’re only looking at finitely many open cells in H.  These open cells project down to X, say in a set D.  Look at all the preimages of up in X^H- there’s infinitely many, since we assumed X^H is an infinite cover.  And is one of the preimages of by construction.  (Also let’s make D, small enough so we have a “stack of pancakes” instead of batter all over the place).  Again, schematic:

I know it looks like three, but there are actually Infinitely many preimages of the image of A

I know it looks like three, but there are actually Infinitely many preimages of the image of A

Now if we want an intermediate cover into which embeds, it can’t include any elements of that send to itself- that is, if g.D\cap D \neq \emptysetwe don’t want g in \pi_1(\hat{X}).  Because then wouldn’t embed.  How many bad are there?  Well, since deck transformations act properly discontinuously, for any point in X^H there’s an open neighborhood that never gets sent to itself (besides when is the identity, of course).  And we’re in CW-complexes, so we mean an open cell.  Look at the other cells of this particular component of D.  Again by proper discontinuity, there’s only finitely many g that’ll send this to some other copy of D.

Since is separable, for any one of these we have a finite index subgroup that doesn’t include and which does contain H.  Take the intersection of all these subgroupssince they all contain H, this intersection (call it K) does too.  And K doesn’t include any of the bad g.  Back in topology-land, K corresponds to a finite degree cover of X, since the intersection of finitely many finite-index subgroups has finite index.  And this cover is intermediate by construction, and embeds in it since there aren’t any bad g.

And that’s a proof of Scott’s criterion!  My next blog post will either be baking/cooking or a reasonable math post.

My mom’s rau muong xao toi (Vietnamese-style morning glory with garlic)

2 Sep

Since my mom’s thit kho recipe is one of my most popular posts, I thought I’d share another traditional Vietnamese recipe.  While I was in Boston, I got to visit the new small Korean grocery store, H Mart, in Cambridge.  So I looked up if they existed in Chicago, and lo and behold there’s one in the suburbs!  We took a trip out there last weekend- I love this grocery store!  There’s a food court in it with delicious Korean food!  And they sell marinated meat, and lots of other goodies that are hard to find elsewhere (the best instant Ramen, enoki mushrooms, Korean melons, lychees, kimchi… I’m just listing stuff we bought.)  In particular, they sell a vegetable under the name ong choy, which is also known as water spinach, morning glory, and in Vietnamese, rau muống.  If you ever go to Vietnam, Rau muống xào tỏi is pretty much the cooked vegetable side dish you’ll get.  Maybe some veggies in a soup, but overall there it’s a lot of fresh veggies with whatever you’re eating, or this garlicky tasty side dish.

I'm not really spoon-feeding you this recipe (muống means "spoon" in Vietnamese)

I’m not really spoon-feeding you this recipe (muống means “spoon” in Vietnamese BOOM BILINGUAL WORD PLAY)

You can also get this dish in Chinese restaurants, where they often put oyster sauce on it.  But we’re cooking Vietnamese today, so fish sauce all the way!  Also, I haven’t seen this dish a ton in Vietnamese restaurants, but it’s in most homes- we compared it to how most American restaurants don’t have peanut butter and jelly sandwiches, but that doesn’t mean they aren’t ubiquitous.

My great-aunt has a nifty tool for splitting the woody stems so you can eat them.  But I’m lazy/don’t like stems so I just cut them off and discarded them.  Make sure you wash the veggies really well- just like spinach it’s easy to get dirt in the leaves.  Then chop them into two-inch segments.

Y'all have too many expectations of me and my puns- why can't you leaf me alone?

Y’all have too many expectations of me and my puns- why can’t you leaf me alone?

Heat up a few tablespoons of oil in a big skillet over medium-high heat.  Then roughly chop up some garlic cloves and put into the oil.

Garlic can be so sixy (I split one of the cloves in half when peeling it)

Garlic can be so sixy (I split one of the cloves in half when peeling it)

I mean, it's in the name: garlic cLOVE.

I mean, it’s in the name: garlic cLOVE.

Let that cook for a few minutes until very lightly brown, then dump in all of the veggies.

After the chief of police in Houston went on a radical diet, people started calling him light Lee Brown

After the chief of police in Houston went on a radical diet, people started calling him light Lee Brown

If you’ve washed them thoroughly and not dried them, the water sticking to them should be enough.  But if it’s not (like if you start seeing dry looking leaves around), add a handful of water (a couple tablespoons).

This cooks pretty fast- 5-6 minutes fast.  Just like spinach!  Give it a good stir every minute or two; I’m not a constant-stirring kind of person (so I’ve never made risotto).

Cheerleaders eat a lot of salad, right?  They're always cheering RAW-RAW!

Cheerleaders eat a lot of salad, right? They’re always cheering RAW-RAW!

I wonder if some of them like to mix it up and chant PARTIALLY COOKED PARTIALLY COOKED!

I wonder if some of them like to mix it up and chant PARTIALLY COOKED PARTIALLY COOKED!

Just kidding, I know no cheer routine would be so ridiculous.  That just isn't DONE.

Just kidding, I know no cheer routine would be so ridiculous. That just isn’t DONE.

While that’s cooking, make your nuoc cham-dipping sauce.  This is a lot of sugar, some lime juice, some fish sauce, minced garlic, and water.  My mom always says to do it to taste, but it’s roughly equal amounts of sugar, lime juice, and fish sauce as a base, then add an equal amount of water (so double the volume by using water).  Then add a tiny pinch more sugar, and whisk it all together.  Taste it and see if it’s too limey or too fishy, and add water/sugar/lime until it tastes good.  You can mince a few garlic cloves and/or a few hot peppers and add them too- I went with garlic this time.

You could call my mom Ursa Major- she's a big dipper

You could call my mom Ursa Major- she’s a big dipper

My mom likes to dip the cooked veggies in the sauce, but I was feeling lazy so I just poured a bunch of it over the vegetables.  We don’t like adding fish sauce to dishes that are cooking because then the whole house smells like fish sauce- just add it afterwards.  Then I took the leftover garlicky sauce and poured it over some steamed salmon, and we had a meal with white rice.

IMG_20140901_125100421

 

Rau muong xao toi (from my mom)

A bunch of ong choy/water spinach/morning glory

Six-eight cloves of garlic

1-2 TB Fish sauce

Half a lime

1-2 TB Sugar

1. Wash the greens WELL.

2. Heat up some oil in a big pan.  Cook rice/protein now if you want.

3. Roughly chop five or six cloves of garlic and add to the oil.

4. Chop up the greens into two inch pieces.  Discard woody stems.

5. When garlic is lightly brown, add the greens to the pan.  Stir.  Cook for five-six minutes, until everything is wilted and cooked-looking.

6. Mix juice of half a lime with 1 TB of sugar.  Add 1 TB of fish sauce and 3 TB of water, stir until sugar is totally dissolved.  Add a pinch more sugar.  Taste.  Add more fish sauce, lime juice, or sugar to taste.  Optional: mince two more cloves of garlic and add to dipping sauce.

7. Either serve warm greens with dipping sauce, or pour sauce over greens.  Eat!

Stop what you’re doing and make avocado coconut chocolate chip ice cream

22 Aug

I was in the middle of writing a blog post about the apple-blueberry-yogurt pie I made for the fourth of July, but then I tasted this ice cream that I just finished and changed my mind.  It’s SO GOOD.  The light avocado flavor plus the coconut milk and a teensy bit of lime and salt just make the creamiest indulgence.  This still has cream in it, but could be done vegan (just use more coconut milk).  I’m in love with this ice cream.  And also, I’m in love with ice cream (I didn’t bring the ice cream maker with me to Boston, but since coming back I’ve made strawberry ice cream and strawberry-lime-mint sorbet which were both also awesome and I’ll put the recipes at the bottom of this post).

If I used maple syrup or agave or something instead of sugar, this would be paleo, right?  It’s my husband’s birthday on Sunday and I want to make him something as good as the cake I made two years ago.  I’ve been going through a big carb phase for the past seven months, so this would be a thank-you to him for putting up with all the pasta/bread/not-paleo-at-all stuff we’ve been eating.  And what would be better than an avocado coconut ice cream cake with an almond-chocolate base?  I’ll update if I do that!

Incidentally, let me brag about him for a second: he’s currently in the woods in Oregon, running in one of the world’s largest relay races from Mount Hood to the coast.  I’m currently sitting at home with a bowl of ice cream and a laptop.  This is exactly what I imagine we look like right now:

I've never been in an eating contest.  I'm more of an amateur-portion person- you can tell from this picture (because it's not in PRO-portion)

I’ve never been in a competitive eating contest. I’m more of an amateur-portion person- you can tell from this picture (both because of the size of the bowl, and because it’s not in PRO-portion)

Anyways.  I picked up five small avocados for a dollar at the amazing produce shop down the street earlier this week, and I set them in a basket with some bananas waiting for them to ripen.  The problem with buying five avocados at once when it’s just you at home is you have to use them all at once, or you’ll have overripe ferment-y ones (remember the raw avocado pie from last summer?).  So I thought of ice cream or massive smoothies- avocado is definitely for sweets in Vietnam, with avocado smoothies being a huge thing (blend an avocado with ice and sweetened condensed milk.  Drink.)  I also didn’t have any milk at home, but I did have two limes and a can of coconut milk.  Also some friends brought me chocolate with coconut in it.  So that’s what happened!

Lately I've been very clumsy and stubbing my toe a lot.  This makes me mix up things- am I an avocado for this ice cream, or an AdVOCAte?  Stubbing my toe made me lose the extra 'o' (and also made me anagram)

Lately I’ve been very clumsy and stubbing my toe a lot. This makes me mix up things- am I an avocado for this ice cream, or an AdVOCAte? Stubbing my toe made me lose the extra ‘o’ (and also made me anagram)

Non-custard ice creams are so easy- just put everything in a blender.  Plop your avocado pieces in (I cut avocados in half, then use a spoon to scoop out the flesh), some sugar, some cream, the juice of two limes, and a can of coconut milk.  Blend.

Ah, the admirable avocado (certainly not PIT(i)ful in this picture)

Ah, the admirable avocado (certainly not PIT(i)ful in this picture)

She's been in some complicated relationships-it's still hard for the coconut milk to open up.

She’s been in some complicated relationships-it’s still hard for the coconut milk to open up.

She's just sick of all the li(m)es, but realizes that we all li(m)e sometimes to make life a little smoother (or perhaps more zesty- have i mentioned you look REALLY good in that?)

She’s just sick of all the li(m)es, but realizes that we all li(m)e sometimes to make life a little smoother (or perhaps more zesty- have i mentioned you look REALLY good in that?)

Blend that up til smooth, then toss it in the fridge while you go do other things.  Also, cut up a chocolate bar into little chunks.

We could try to be honest all the time, but everyone has a DARK side- and BARing all might not be a great idea

We could try to be honest all the time, but everyone has a DARK side- and BARing all might not be a great idea

People will CHOC up outcomes to what yOu say- so there's no need to explain you're LATE because you want to spend less time with them.  Just blame traffic- a little li(m)e won't hurt anyone.

People will CHOC up outcomes to what yOu say- so there’s no need to explain you’re LATE because you want to spend less time with them. Just blame traffic- a little li(m)e won’t hurt anyone.

This was my first time leaving the mixture in the blender instead of transferring to a bowl.  It makes SO MUCH MORE SENSE because then you don’t splash all over when you pour the cream into the ice cream maker.

I'm not advocating lying all the time, but hey, any PO(u)Rt in a storm!

I’m not advocating lying all the time, but hey, any PO(u)Rt in a storm!

Note: DON’T put the chocolate chips, nuts, mix-ins, whatever in when you first pour in the mix.  Wait until it’s about the thickness of greek yogurt so that the chips get evenly distributed.

IMG_20140822_175735793

Avocado coconut chocolate-chip ice cream (vegan, paleo options available)- adapted from David Lebovitz, who adapted it from this awesome looking cookbook Absolutely Avocados.  Makes about 1 quart.

1 can coconut milk

3-4 small avocados or 2 big avocados

3/4 c sugar (or agave syrup/other sweetener for paleo)

2 limes

1/2 c cream (or more coconut milk)

pinch of salt

chocolate (I used a 3 oz bar of dark chocolate with coconut in it)

Blend all ingredients except chocolate together.  Chill in a fridge for at least 20 minutes.  Start your ice cream maker and throw in the blender-ful of stuff.  Chop chocolate into small chips, and add in about 10 minutes into churning process.  Enjoy!

IMG_20140822_165348162

Bonus recipe: Strawberry-lime-mint sorbet, adapted from the Cuisinart recipe book that came with my ice cream maker

Note: While for ice cream I just throw the sugar in, by now I’ve decided that simple syrup is the way to go with sorbets.  Otherwise you’ll get grains of sugar ruining the smoothness of the sorbet

1 c water

1 c sugar

1 bunch mint leaves (I took them from my garden so I don’t know how many there were.  Maybe 15)

3 limes

2 c strawberries (one pint according to this helpful website)

pinch of salt

Bring the sugar and water to a boil to make a simple syrup.  Let it boil while stirring for a minute, then turn off the heat.  Throw in the mint.

Meanwhile, cut up and measure your strawberries.  Blend them with the juice and zest of the limes and the salt.  Discard the mint.  Add about 1/4 c of the simple syrup to your berry blend, and blend.  Taste.  This is probably too sour.  Add more syrup 1/4 c at a time until the mixture is a little bit sweeter than you’d like.  Chill in the fridge, clean up, and throw in the ice cream maker.  Yum!

I used my leftover mint simple syrup (I had about 1/4 c) in a strawberry iced green tea.  Delicious.

How to cut a watermelon (for science!)

13 Aug

It’s summertime, which means watermelon!  There are actually two parts to this post: first, the smarter way to cut watermelon (so you don’t end up with watermelon juice all over your face/hands/clothes), and second, a goofy little experiment I did the other day to figure out the best way to buy watermelon.

For my entire life I assumed eating watermelon, like eating a ripe juicy peach, was just a naturally messy ordeal best done outside or over a sink.  Then this summer a friend showed me the right way to cut watermelon- watermelon fries!  The traditional wedge shape means every bite releases more juices to go everywhere else, while the fry fits entirely in your mouth and is a lot like eating a fry.  See schematic of happiness:

On the left I'm positively melon-choly.  On the right I don't understand w(h)at 'er problem is.

On the left I’m positively melon-choly. On the right I don’t understand w(h)at ‘er problem is.

You start the same, cutting a watermelon in half and laying it flat side down for safe cutting.  Instead of making on vertical slice and lots of horizontal slices (as you would for wedges), you make equal numbers of vertical and horizontal cuts.  This forms a lattice- it’s best if each square is about an inch to an inch and a half long.

 

PRO TIP: cutting board that fits inside a baking sheet = no mess to clean up after cutting

PRO TIP: cutting board that fits inside a baking sheet = no mess to clean up after cutting

Wedges were very in fashion a few years ago, but now they're passe

Wedges were very in fashion a few years ago, but now they’re passe

It's all about the stilettos now (I'm not sure what stilettos are but they're definitely skinnier than wedges)

It’s all about the stilettos now (I’m not sure what stilettos are but they’re definitely skinnier than wedges)

Discard the corner/end pieces that have no watermelon, and voila!  Watermelon fries!

Now you don't need to go to a generic American chain restaurant to say TGIF! (Because everyday can be Fry-day)

Now you don’t need to go to a generic American chain restaurant to say TGIF! (Because everyday can be Fry-day)

Here’s a real life version of the right side of the cartoon above:

IMG_20140713_140203173

 

Now for the second part of the post!  What’s the best way to buy watermelon?  That is, if you can’t carry a 15-20 pound watermelon yourself (for whatever reason).  Sometimes stores carry big slices of watermelon, so I’ve bought plenty of 5 pound slices this summer, which I highly recommend.  But yesterday the store only had a box of precut watermelon for $3.90 or a personal sized watermelon for $4.99.  So of course I bought both, for you blog readers (or because I wanted watermelon).

We wait & watch... which watermelon will win the war?

We wait & watch… which watermelon will win the war?

Any guesses?  Because of the labor involved we guessed that the mini watermelon would win, but then again, mini seedless watermelons seem like a luxury item, like a labradoodle or a pluot, so they might charge a premium for them.  Volume wise they look similar, but humans are pretty bad at estimating volumes- we’re good at linear estimation but bad once we get to higher dimensions (how many acres is the lot where you live?) [For that matter, what is an acre?]

I decided to go by weight, but also tried a rough volume calculation ahead of time.  Clearly there’s space in the box, but I didn’t want to measure each piece and calculate volume so I just did the volume of the box.  Similarly, there’s rind in the watermelon and it’s not a perfect sphere, but hey, this is why I’m a mathematician and not an experimental scientist.

Box: 7″ x 6″ x 3″ = about 126 cubic inches of watermelon

Mini watermelon: 6″ diameter = 4/3* pi * 3^3 = about 113 cubic inches of watermelon

Again this volume calculation is pretty bad: you can see how much space is in the box above, and once we cut open the watermelon, how much rind there actually is.  After measuring the box/melon, the first thing I did was weigh the precut melon using my husband’s fancy scale and a bowl.  Then I ate a lot of that melon, and started cutting the mini watermelon (not the smart way).  I wanted to get as much watermelon as possible out of the mini melon, but still cut off all the white parts since the precut didn’t have any white parts.

Cutting a melon this way is not very appeeling- I just did it for science

Cutting a melon this way is not very appeeling- I just did it for science

Mini-melon fries!  I guess fries are already mini potato fries (holy crap regular-sized potato fries are often called 'wedge fries' all the puns don't work!)

Mini-melon fries! I guess fries are already mini potato fries (holy crap regular-sized potato fries are often called ‘wedge fries’ all the puns don’t work!)

One thing that I thought was crazy was the next picture: I’ve unwound the measuring tape to 6″, the diameter of the mini melon, but you can see how much more melon there is.  Then again, the bowl isn’t so deep so this isn’t that crazy.

So that's how long six inches is!

So that’s how long six inches is!

Also, I needed to do three weighings for the mini melon because the scale couldn’t hold that much.

Box: 542.0 grams

Mini melon: 1162.0 grams

The mini melon held twice as much melon as the box did!  It was really crammed in there vs. all the space in the box, I suppose.

So the final tally:

Box: 138.97 g/$

Mini melon: 232.87 g/$

So the mini melon is clearly the better deal, right?  You get almost twice as much melon per dollar spent, even though you spend an extra dollar.

HOWEVER.

The precut watermelon was perfect- sweet, juicy, with just the right amount of bite.  I ate almost the entire box while doing this “experiment,” which means I ate an entire pound of watermelon in half an hour.  It was like every piece was from that magical inner zone of watermelon which isn’t too mushy (the exact center) but is still sweet (not touching the rind).

Meanwhile, just about every piece in the personal watermelon was a rind-piece.  It’s just not big enough to get to that magic zone.  Even the store clerk thought I’d picked out a good one!  I’m going to make gazpacho with those two pounds of watermelon because I want to hide the lack of sweetness.

Next time I want watermelon (like tomorrow), I’m going to pay the premium and buy the precut.  I would encourage you, however, to buy a whole watermelon and cut it into fries as above.  That is, if you can carry a watermelon.

The Apology and why it bugs me

3 Aug

I want to remark at the beginning of this post that I love math people.  We’re a little weird, very friendly, and generally quite open-minded and supportive (at least, this is true of the math people I know, a.k.a. geometric group theorists and friend fields).  There’s one thing that really, really bugs me that many (definitely not all) math people do when talking math with each other.

Also, I’m really into lists right now.

As I’ve mentioned earlier, I’m doing an exciting research program this summer involving four faculty, five graduate students, and three undergraduates doing at least five research projects.  With so many different experiences and different personalities interacting, there are lots of times when apologies are required:

  • Interrupting someone in the middle of a productive thought (actually people don’t apologize for this enough.  Reminds me of this post from the What is it like to be a woman in philosophy? blog)
  • Stealing someone’s notes/pen/paper/seat
  • Talking over someone (similar to the first thing here)
  • Probably more things I can’t think of right now

And I’m totally down with all of those.  They make complete sense- apologies are a nice lubricant for social and professional interactions.  But there’s one apology that really bothers me, which comes up in these situations:

  • Not knowing something that you’ve never been exposed to/had a reason to explore
  • Not being able to read the mind of someone who isn’t communicating clearly (related: this old post on teaching)
  • Having a different background than someone else, mathematical or otherwise
  • Being better at processing things in a visual rather than audial way, or vice-versa

These all come down to one thing: you’re a different mathematician than whoever you’re talking to.  And this is the thing that you might say in this situation:

Sorry, I’m slow.

I dislike this so much!  I’ve heard very many mathematicians say this over the past few weeks, whom no one would call “slow.”  One reason for my distaste ties in with the whole “women apologize more” bit, explored in a Pantene ad, dissected by Time, and perhaps most effectively explained in this spoken word video.

To be clear, this is not a women-only problem (while I’ve noticed more women do so than men, men also do this).  I dislike the phrase “sorry, I’m slow” because

  1. I’m apologizing for an adjective that I’m applying to myself- ->I’m apologizing for who I am.  [I am not a person who likes doing this.  I certainly apologize when I make mistakes/do bad actions, but to judge myself on my character, and invite you to pass that same judgment?  Not fun.]
  2. I’m devaluing my contributions to this conversation.  If I don’t take myself seriously, how can I expect you to?
  3. By saying these words aloud, whether I believe them now or not, I convince myself and you that I am, in fact, slow.  Just like if I looked in a mirror everyday and said “I’m ugly” I would eventually believe it.
  4. I’m perpetuating a system of these apologies- now whenever you’re in a conversation and struggling to understand what’s going on, you’ll be tempted to say “sorry, I’m slow” and cause 1-3 to happen to you.

Maybe the worst part of “sorry, I’m slow” is that there are good reasons to say it: when faculty/those further along say it, it encourages undergrads/younger folks that they aren’t the only ones who feel this way.  Similarly, if you say it in a group of peers, it builds camaraderie (in the way that teenage girls insult themselves in order to get compliments from each other).  When younger people say it to older people, mentorship instincts kick in and older people often share personal stories of some other time they felt slow.

Really what I’m saying is that “sorry I’m slow” is bad because it makes you believe that you’re slow, and it’s good because it tells everyone else that you also think you’re slow.  I just wish people didn’t pass these value judgments on themselves.  =(  I suppose this post is why I’m a mathematician, not a psychologist or sociologist.

From here: http://cheezburger.com/5218979584

From here: http://cheezburger.com/5218979584.  Also, I’m the puppy and the cat.

 

Pineapple upside down cake

24 Jul

Last month I thought about going to a party for a friend’s birthday, but then sat inside and played board games instead.  To make up for it, I baked him a cake!  This actually happens fairly often, but without the cake part.

Anyways, pineapple upside down is my second favorite cake (it’s hard to beat carrot cake).  Also, I’ll take this inopportune time to give a shout out to the incredible Alliance Bakery down the street from my old apartment, who made our hilarious and wonderful wedding cake.  It was a delicious light lemon cake with a mango mousse filling and a simple buttercream frosting.  Also, because our initials spell out YAM, we asked for it to be made in the shape of a sweet potato, but to try very hard to make it not look like poop.  They did a fantastic job and I wish I had more pictures of the cake… unfortunately I only have one and it’s in B&W

You should try to get there early; there are often hordes of people at Alliance (WOW reference...)

You should try to get there early; there are often hordes of people at Alliance (WOW reference…)

Also that photo (and all my wedding photos) were taken by my best friend who happens to be a professional photographer, so if you’re in Singapore and need photos taken I recommend her.

ON TO CAKE I PROMISE IT’S NOT A LIE.

One, I already have a spouse.  Two, why would you wed pants?  Marry-chinos, yeah right.

One, I already have a spouse. Two, why would you wed pants? Marry-chinos, yeah right.

All I had to buy for this cake was the pineapple and the cherries.  I’m curious what I’ll do with maraschino cherries (any ideas?  Shirley Temples, I guess?).  Also, this is a one-bowl cake, which is so great! (Two if you use a separate bowl to microwave butter)

First, melt half a stick of butter and pour it in the bottom of your cake pan, then sprinkle brown sugar over it.  Takes about 30 seconds in the microwave to melt half a stick of butter.

This is getting me so many brown ie points with my friend

This is getting me so many brown ie points with my friend

Then cram as many pineapple rings as you can into the pan-I got in six.  If you’re using a square pan you might have better luck.  Also throw in as many cherries as will fit, because when else will you use them?  I realize that I did not need to put maraschino cherries in the cake but they’re literally the cherries on top so it’s hard to say no.

If you like it you should put at least six rings on it

If you like it you should put at least six rings on it

Fun fact: pineapples don’t come from trees.  They’re actually the only edible species of the bromeliad family of plants- this is the only fact I remember from signs at the University of Washington greenhouse in 2011, when I visited to see a corpse flower.  Those are flowers that smell like rotting carcasses, hence the name.  I recommend checking it out if you’re ever near one (they don’t bloom often).

Back to cake.  Melt a stick of butter in the bowl (again I did 30 seconds and it gets almost all the way melted, and if you stir the blobby butter it’s basically melted), and mix in the sugars.

Everyone was bowled over by this cake

Everyone was bowled over by this cake

Does not melting the butter all the way really work?  You butter believe it!

Does not melting the butter all the way really work? You butter believe it!

Then add the rest of your wet ingredients.  It really does help to mix butter and sugar together at the beginning of your recipes when baking, then adding wet, then dry.  Otherwise the sugar doesn’t incorporate as evenly/well.

Yes, I milk my jokes as far as I can- please laugh to protect my eg(g)o

Yes, I milk my jokes as far as I can- please laugh to protect my eg(g)o

I sort of gave away what happens next.  Here’s another chance to use more bowls, if you mix your dry ingredients first.  Or you could sort of sprinkle the other dry ingredients over the flour and whisk it all together (one less bowl to clean!)

Whenever someone says they like my blog, I'm floo(u)red

Whenever someone says they like my blog, I’m floo(u)red

Also have you used rubber spatulas (not an Amazon affiliate link I’m not that fancy I just googled ‘rubber spatula’)?  They’re amazing for scraping!  Fun fact I was first introduced to these when I was about seven and baking a cake with my Swedish grandmother (long story not actually related to her but did go to her house every month and called her “Grandma”), and she showed it to me and called it something like a ‘fun ruiner’ or ‘child spoiler’ because it doesn’t let you lick the bowl.  Cool story Yen.

If you couldn’t tell the program I’m at right now is sort of intense and I’m a little unfocused because I’ve been mathing all day.  Yay math!  It’s SO MUCH FUN I’ll have to blog about this summer’s math sometime too.  Random groups!  Nilpotence!  I like that word because it’s like you’re so helpless you aren’t even impotent, you’re nilpotent (this has nothing to do with the definition of nilpotent).

BUT SERIOUSLY LOOK HOW AMAZING THIS SPATULA IS

BUT SERIOUSLY LOOK HOW AMAZING THIS SPATULA IS

Right, cake.  Pour the batter over the pineapple topping, pop it in the oven for 40 minutes, let it cool for a bit, and invert it onto a plate.

Yes I love this spatula.

Yes I love this spatula.

Also new cake pans, thanks wedding registry!

Also new cake pans, thanks wedding registry!

IMG_20140622_202511843

Pineapple upside down cake- recipe lightly adapted from Sally’s Baking Addiction- honestly I suspect hers might be better than this one so you should check that out.  But if you want the cake I made see below

3/4 c butter (1 and a half sticks)

1 1/4 c brown sugar

1/4 c white sugar

can of sliced pineapple

maraschino cherries

1 egg

1 1/2 c flour

1/2 tsp baking powder

1/4 tsp baking soda

1/4 c oil

1/2 c milk

1 TB vanilla

 

First, melt half a stick of butter in the microwave, then spread out into a cake pan.  Sprinkle 1/2 c brown sugar over, then top with pineapple slices and cherries to your liking.  Don’t throw out the can of pineapple!  Measure out 1/4 c of the juice for later.  Turn on oven to 350.

Next, melt the other stick of butter, and mix in remaining brown sugar and the white sugar until smooth.  Add in the egg, pineapple juice, oil, milk, and vanilla, and mix until smooth again.  Then add flour and sprinkle the baking powder/baking soda over.  Mix well.

Spatula that batter into your pan, and bake for 40 minutes or so, until a toothpick in the middle comes out clean.  Let cool.

Run a knife around the edges of your cake, then put a plate on top of it.  Flip the whole thing upside down and lightly bang on the bottom of the cake pan to drop the cake onto the plate.  Yum!

Chris’s super-easy and super-impressive French toast souffle

15 Jul

I’m a huge sucker for deals.  The other day I wanted ice cream and was going to buy a pint, but then saw that for just $1 more I could get TWO half-gallons (a two for the price of one deal).  So I ended up getting those, only to realize when I got home that I’d accidentally bought LOW FAT ice cream (egads!)  So I feel like I got suckered into buying a product that I didn’t want.  That said, we’re still eating the ice cream; it’s just not as good.

That story was just an intro to say that this dish is a really good deal.  It’s very, very little work for a lot of return-literal “ooh”s and “aah”s.  My friend Chris (who, incidentally, made one of my favorite sites involving Rodents Of Unusual Size)  sent this to me via one of those chain-email recipe exchange things that never work.  Lucky for me I got this recipe, which I’ve made again and again since 2011.

You'll maybe not want to whisk life and limb for this souffle, but definitely try it to your fullest CAPacity

You’ll maybe not want to whisk life and limb for this souffle, but definitely try it to your fullest CAPacity

I realize I took the above photo a little bit late- the first thing you want to do is put a stick of butter in your pan of choice (Chris likes deep and narrow; I like whatever I have on hand), and stick it in your oven, which you’ll heat to 425.

Not a pun: I think I'll start calling grilled cheese sandwiches "butter melts".  Sounds so good right now.

Not a pun: I think I’ll start calling grilled cheese sandwiches “butter melts”. Sounds so good right now.

While the butter is melting, put all your other ingredients in a bowl and whisk, whisk away.  I do the eggs first, then other stuff.

I hate it when people sugar-coat bad news

I hate it when people sugar-coat bad news

Just give it to me and maybe I"ll figure out if it's a mixed blessing or not

Just give it to me and maybe I”ll figure out if it’s a mixed blessing or not

By the time your oven is preheated, your butter should be melty and bubbly.  Pull out the pan, pour your bowl of delicious stuff right on top of the butter, and toss it back in the oven.
If someone with ombre blonde-brunette hair hangs from her knees on monkey bars, would you call her upside-Brown?

If someone with ombre blonde-brunette hair hangs from her knees on monkey bars, would you call her upside-Brown?

This POURcelain dish is perfect for what's happening here.

This POURcelain dish is perfect for what’s happening here.

Who needs root beer floats when you can have a butter float?

Who needs root beer floats when you can have a butter float?

After thirty minutes, you’ll have a gorgeous, poofy souffle that took none of the work compared to usual souffle recipes.  I’ve had it with cinnamon sugar on top, maple syrup, or my favorite, toss on cut up berries (any kind) and a spoonful of sugar- the heat will slightly cook the berries and sugar.  Serve IMMEDIATELY (or the poof will go down) with a side of berries.  I love the almost crispy chewy crust in contrast with the creamy center pieces.  It’s all delicious.
2013-08-31_12-08-03_154
Chris’s French Toast Souffle
1/4 cup (1 stick) butter
3 eggs
1.5 cups milk
6 TB sugar
3/4 cup flour
1/4 teaspoon salt
Optional toppings: more sugar, cinnamon, maple syrup, berries…
Put the stick of butter in a pan, place in oven, and preheat oven to 425.
Whisk eggs in a bowl.  Add milk, sugar, flour, and salt and mix until smooth.
If the butter isn’t bubbling yet, wait.  When it is, pour batter into the pan over butter.
Bake for 30 minutes.
Serve immediately with desired toppings.
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