Pumpkin gnocchi with sage brown butter

26 Oct

In my last two weeks of pregnancy I really, really wanted cinnamon rolls.  Turns out cinnamon rolls are one of those things that are everywhere when you aren’t looking for them, and then mysteriously disappear as soon as you want them/are super gigantic and can’t travel very far.  I ended up getting some from the grocery store (the Mariano’s ones were way better than the Whole Foods ones), and an un-iced one from the bakery down the street (why would you have an un-iced cinnamon roll?!)  These were not satisfying, so the last thing I baked before baby came was those awesome vegan pumpkin cinnamon rolls that I always make when I want cinnamon rolls.

Cinn-fully delicious.  This is apparently a standard cinnamon roll pun.

Cinn-fully delicious. This is apparently a standard cinnamon roll pun.

They were packed with butter because I’m not actually vegan.  I should’ve made cream cheese frosting for them instead of the vegan simple white icing (powdered sugar + milk + vanilla).

Anyways, almost every time I make those cinnamon rolls I end up making pumpkin gnocchi, because they don’t use an entire can of pumpkin.  They leave just enough leftover pumpkin to make a recipe of gnocchi.  Pumpkin gnocchi = way easier/faster than potato, though probably the same if you start from raw pumpkin.  It’s taken me three hours to make potato gnocchi, while pumpkin come together in about half an hour if you don’t bother to do the dimpling (which I don’t because I’m lazy).

Cinnamon rolls are not a necessary ingreedyent, but make them if you're feeling greedy.

Cinnamon rolls are not a necessary ingreedyent, but make them if you’re feeling greedy.

So few ingredients!  Flour, pumpkin, eggs, salt, pepper, nutmeg.  Also, in our defense I made the gnocchi the day after the cinnamon rolls.  So that’s why we’ve already eaten four of them.

Orange you glad I make so many puns for you?

Orange you glad I make so many puns for you?

The key to gnocchi is to not mix it too much- a very, very light touch is needed, or else the dumplings get gluey/chewy instead of pillowy.  Start your pot of water boiling now.  Once your ingredients are barely mixed, you need to lightly roll the dough into four ropes (about half an inch in diameter) and cut the dumplings with a fork.

Let me show you the ropes, kid.  These are the ropes.  No literally.

Let me show you the ropes, kid. These are the ropes. No literally.

This would be a terrible pillow fight

This would be a terrible pillow fight.

I cut one rope at a time, then toss that batch of gnocchi into the pot.  It takes about the same amount of time to cut a rope as it does for one batch to cook.  The gnocchi are done when they rise to the top of the water.  Use a slotted spoon to pick them up.

I wonder if the author of Tortilla Curtain makes pasta a lot- you know, b/c he's Boyle all the time

I wonder if the author of Tortilla Curtain makes pasta a lot- you know, b/c he’s Boyle all the time.

Leave them in a colander and toss them with a little olive oil so they don’t stick to each other.  IMG_20140926_120122557

 

I have a windowsill herb garden so I threw tons of sage leaves into a most of a stick of butter.

When horses take photos, I wonder if they go "Say Gee!" (sounds like sage)

When horses take photos, I wonder if they go “Say Gee!” (sounds like sage)

Sage brown butter, if you haven’t had it before, is amazing.  Just put butter into a pan until it sizzles, then stops sizzling (that’s the browning part), and then put in a ton of sage leaves.  Recipes on the internet call for eight leaves, which is crazy.  Put in as many as you can.  The crispy sage leaves are delicious!  And they impart such a rich flavor to the butter.

Once the sauce is done, throw the gnocchi in the pan, toss, and eat with some grated parmesan on top.

IMG_20140926_121350251_HDR

 

Pumpkin gnocchi with sage brown butter sauce

Gnocchi recipe is unadapted from The Skinny Fork, so I won’t write it here.

Sauce:

Melt 6-8 TB of butter in a pan, until it sizzles, foams, and subsides.  Butter will be nut brown.  Add 8-20 sage leaves and let them fry for a few minutes.  Don’t let them burn!  Eat with anything.

I have a baby! Also people are being mean to me (breastfeeding)

16 Oct

This is clearly not a usual Baking and Math post (neither will show up here) but I owe a post and I’m not operating on a ton of sleep right now so here’s a real fast update.  This baby was born last Tuesday, 10/7, at 7:11 a.m. and he’s great.

 

justbabySome not so great things: he was a bit jaundiced-looking so we had to keep taking him back to get blood drawn to find out if we had to stick him under UV lights.  It’s difficult to transport newborn babies, especially if you don’t own a car.  But he’s fine now so that’s great.  Other not so great thing: breastfeeding him hurt like hell, to the point where I’d be crying in pain.  On Sunday, after doing our blood test, I thought I’d feed him before we got in the car and headed home.  So I tried, and then I started screaming in pain (this pain was a slow ramp up over the course of the first week of his life).  Luckily we were already in the hospital, so we thought the nurses/lactation consultant could help.

Some background if you have not had a child lately: something we call the Breastfeeding Propaganda Machine has been in full force for several years, with slogans like “Breast is best” and “A mother’s breast is where a baby belongs” printed on posters, and unhelpful breastfeeding classes where they say things like “breastfeeding shouldn’t hurt if you’re doing it right.”  Also doctors, nurses, lactation consultants (that’s a full time job!), and other parents who are ready to pile on the guilt with said slogans and facts: mother’s milk gives baby necessary antibodies, promotes closeness, helps mom lose weight, is the correct choice.

I find the BPM incredibly insensitive to the many mothers out there who a) can’t for some physical/emotional reason breastfeed, b) don’t have the time to breastfeed because they need to work, or c) have their own reasons for not breastfeeding.  Theoretically I have bodily autonomy and don’t need to justify my personal decisions as to what I do with my body to other people, but I guess that’s not true.

Things that were said during our conversation (keep in mind I am sobbing in pain at this point) (Sarcastic responses in parentheses):

“Hmmm this shouldn’t be hurting you this much.” (Thanks.)

“Breastmilk is the perfect mix for your baby.” (Great.)

After telling her we’d been supplementing because his weight had decreased too much the first week: “Oh, well, if it’s medically necessary then that’s a good reason to supplement.”  (Oh, thanks for your judgment I really needed that.  Because it’s your right to judge whether my reasons for supplementing are legitimate or not.)

My husband was boiling mad at this point.  We left and the lactation consultant lady literally followed us down the hallway and kept saying unhelpful things to us.  We wish that she and the nurses had been more patient/mother-friendly.

For the record, despite my acknowledgment of the BPM and us joking for weeks that if we didn’t breastfeed our baby would grow up stupid and ugly, I still protested when my husband very reasonably suggested supplementing with formula after I’d had another hour-long feeding session of our constantly screaming, not-sleeping, weight-losing baby.  Because I’d internalized the BPM messages, I felt frustrated, guilty, and ashamed that I couldn’t provide my baby with thebest.”  I wanted to keep trying to exclusively breastfeed him even though all signs pointed to it not working.  Once we started supplementing, he slept!  And didn’t constantly scream!  And everything got better and was amazing!  But it still took me days to get over my hang-ups (I’m still getting over them, honestly) that I wasn’t good enough for my baby.  I wanted to push through the pain and do everything for my baby, feeling that if I didn’t breastfeed, I was a bad mother.

If you know me personally, that last paragraph should sound pretty out of character.  I pride myself on being confident, not giving a damn what other people think of me, and also being quite difficult to offend/affect emotionally (unless you’re my husband or mom).  But I’d listened to the BPM for too long, and it hurt my family.  Thank goodness I have a reasonable husband who talked me down to supplementing and pumping until I was in less pain.

Anyways, these were some disjointed thoughts on what’s happened to me over the past week and a half or so.  If I had more sleep/was operating at more of my usual intellectual level I’d add commentary to the following links, but I’ll just leave them here:

A woman who can’t physically breastfeed and the absurdity of the BPM trying to get her to

Fantastic well-researched article saying that medical literature isn’t as strong as the BPM says

Everything you do to baby affects the family- breastfeeding might not be the best choice for the family

Pro breastmilk- basically we don’t understand it yet but it’s still good

 

 

What is hyperbolic space? (updated)

2 Oct

Thanks very much to reader “ilikemathyoudont” for pointing out that I, yet again, messed up the triangle inequality.  Corrected it below.  If you are like me and always forget it, you should draw triangles like I do below, and probably you’ll get it right.

I spend a lot of my time (most/all of my time) thinking about shapes/structures/arguments in hyperbolic space, so I thought I’d take a post and explain what it is.  Maybe I’ll be able to put up some research-y posts one day about it.  NOTE: in this post, we’re only going to talk about two dimensions.  In the future we might talk about higher dimensions (like three, the one we live in).

First, let’s intuitively figure out what I mean by “space.”  For us, a space is somewhere that a point can walk around and measure how far it’s moved.  The way we measure distance is called a metric (I’ve written about metrics before)- here are a few examples.  Let’s take as our space the plane a.k.a. a flat land.

The orange path (taxicab metric) takes 14 steps between the two dots.  The green only takes 10 (Euclidean metric).  I just made up the purple one but it's a weird length.

The orange path (taxicab metric) takes 14 steps between the two dots. The green only takes 10 (Euclidean metric). I just made up the purple one but it’s a weird length.

Each of these metrics measures a different distance between where a point starts and where it ends.  There aren’t too many rules to be a metric:

  1. you need to have distance zero if and only if start point = end point,
  2. distances need to be positive (or zero, see 1)
  3. we need to satisfy the triangle inequality: for any three points x,y,z in your space, this should be true: d(x,y)+d(y,z)\geq d(x,z)

I actually use the triangle inequality ALL THE TIME and I always forget what it is and need to draw a triangle.  Essentially, you need to be able to draw a triangle with the distances, so the sum of the two sides can’t   must be longer than the third side.

The orange one is fine.  The yellow guy on the left isn't quite a triangle, but his sides are long enough that we can swing around and make a triangle.  But poor blue- there's no way to use those little stumps to close up into a triangle.

The orange one is fine. The yellow guy on the left isn’t quite a triangle, but his sides are long enough that we can swing around and make a triangle. But poor blue- there’s no way to use those little stumps to close up into a triangle.

So one way you can have a different metric space is by taking your space (like the plane) and putting a different metric on it.  But what if we change the space?  Like, what if instead of walking around the plane, we walk around a sphere (like the Earth)?  Our space will have different properties.  For instance, if you walk in a straight line on the plane, you’ll never get back to yourself.  But you can walk around the equator (with, yknow, walk-on-water shoes and infinite endurance) and end up right back where you started.  Somehow the sphere is fundamentally different from the plane.

Let’s have a super short history lesson.  Once upon a time (around 300 B.C.), a Greek named Euclid wrote a super cool book called the Elements.  In it he wrote a bunch of definitions and five axioms (things that we assume are true without proof), which laid the groundwork for the study of geometry.  These are the axioms:

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. Given a line and a point not on it, there exists exactly one line that passes through that point and never intersects the line.

And here are the pictures that illustrate them:

postulatesFor a really long time, people were happy with the first four and unhappy with the fifth one.  Euclid’s fifth postulate really irked mathematicians for a millenium or so, and in the mid-1800s  we finally got non-Euclidean geometry, which accepts the first four postulates and rejects the fifth one.

There are a couple of non-Euclidean geometries (hyperbolic geometry is one of them), but I think spherical geometry is a little bit easier to get your head around first, because the Earth is a sphere (thank you, history).  Let’s get back to the metric on the sphere.  We want distance to measure the shortest path between any two points, so instead of drawing a straight line on a map we do those funny arcs that they have in airline maps.

Flight map for Air Canada

Flight map for Air Canada

These arcs look funny, but if you had Venice and Toronto on a globe and taped a piece of floss to the globe between the two, it would map out the arc exactly as it looks on the map above.  Turns out all lines in spherical geometry can be extended to great circles on the sphere, a.k.a. the longest possible circle you can draw.  The equator is an example of a great circle.  Or any circle that includes the north and south poles.  And that brings us to Euclid’s postulate no. 5- given a line on the sphere, and a point not on that line, there’s no way to draw a line through the point which is parallel to the first.

No matter how we try to draw a line through the mole, it'll hit the equator

No matter how we try to draw a line through the mole, it’ll hit the equator.

Another fun thing about spherical geometry: triangles don’t add up to 180 degrees like they do in Euclidean geometry.  This picture from Wikipedia proves it better than I can:

Triangles_(spherical_geometry)

On the sphere, you can have a triangle with way more than 180 degrees: this one has 230 degrees! But in flat space, all triangles add up to 180.

So a few differences we’ve noticed between flat and spherical geometry so far:

  • In Euclidean geometry, there’s exactly one line parallel to an original line that goes through some other point.  In spherical geometry, there are none.
  • In Euclidean geometry, all triangles add up to 180 degrees.  In spherical geometry, they add up to more than 180 degrees.

(Side note: this number-of-degrees-in-a-triangle fact is equivalent to the parallel postulate, so these two facts are basically the same).

There’s a natural question that comes up from these two differences:

  • Is there a geometry with more than one line parallel to an original that goes through some other point?
  • Is there a geometry where all triangles add up to less than 180 degrees?

The answer is yes!  This is called hyperbolic geometry and is where lots of research lives nowadays.  In this land, if you draw a line there are infinitely many lines parallel to it that go through some other point.  And all triangles add up to less than 180 degrees.  There are many models of hyperbolic space, but we’ll just look at two.  The first one is the Poincare disk model.  Escher does a really good picture for this:

All the fish are the same size!

All the fish are the same size!

In this model, the outside circle represents the end of space a.k.a. infinity (we call it the circle at infinity or the boundary of hyperbolic space).  One way I’ve explained this picture is imagine that there’s an infinitely large bowl printed with all these fish, which are all the same size.  If you stick your head into the bowl, the fish at the “bottom” of the bowl will be pretty big, while the fish in your peripheral vision will get smaller and smaller the further away they are from the bottom of the bowl.  This is a nice way to start to wrap your head around hyperbolic space, which is fundamentally different from flat space in the opposite way that spherical space is.  We say that spheres are positively curved, while hyperbolic space is negatively curved (and flat space isn’t curved or has curvature 0).

The metric is a little harder to see in this model, so mathematicians often use the upper half-space model instead.  It’s sort of like using a map to think about the Earth instead of a globe.  When we use maps, they’re finite, because the Earth has finite surface area.  But in hyperbolic space, since we go off to infinity, we’re going to have to use something that is infinite.  So we use the top half of the Euclidean flat grid.  While straight lines on a map of Earth are arcs, as we saw above, straight lines on this model of hyperbolic space look a little different.

Straight lines are ones that go straight up to infinity, and segments of half-circles whose diameters lie on the bottom line

Straight lines are ones that go straight up to infinity, and segments of half-circles whose diameters lie on the bottom line

This model includes the boundary at infinity too, but it’s infinitely far away up (just like infinity in Euclidean space is infinitely far out).  If you have two points (x,y) and (a,b) in the Euclidean plane/flat space, the distance formula (which measures the metric) is \sqrt{(x-a)^2 + (y-b)^2}.  To write this in terms of differentials (nope, not defining that now), we can say ds^2 = dx^2 + dy^2 for the Euclidean plane.  In the upper half plane model of hyperbolic space, the metric is ds^2 = \frac{ dx^2 + dy^2}{y^2}.  Roughly, this means that the further up you go, the shorter horizontal distances are.  That’s why the fastest way between two points on the bottom line is using those half circles we drew above.

OK that’s our introduction to hyperbolic geometry.  I really wanted to put in a math post before my life derails for a bit.  So I apologize if we don’t have a post for awhile after this one- I’ll be dealing with a newborn.  Here’s a picture of me now:

IMG_20141001_223003556

 

 

(almost Paleo) gluten-free Ice cream cake construction

23 Sep

For my husband’s birthday last month I decided to make him an almost-Paleo ice cream cake.  This is easily made fully paleo (by which I mean no refined sugar, no flour, no dairy).  I’d never made an ice cream cake before- I think next time I’ll go more classic and just do a cookie crust with ice cream on it and frosting, instead of an actual cake which is then frozen.

I've been sorta blah lately, but I'm doing o-ca(k)e

I’ve been sorta blah lately, but I’m doing o-ca(k)e

For the record, two years ago when I did this for his birthday I made this incredible paleo birthday cake with coffee in it.  Fun fact: I had to fly it down to Austin (where he lived) for the party, and went through TSA with a tupperware of definitely-liquid-ish way-more-than-three-ounces homemade coconut frosting nestled into the cake caddy I’d bought for the occasion.  When they asked what it was, I said “frosting for this surprise birthday cake!”  I think I looked super cute that day AND I had a cake as my personal item (+backpack for carry-on) so they let me through.

Anyway, this year was more low key cake-wise.  Also I somehow didn’t have cocoa in my house so I cheated and used some fancy-pants hot chocolate mix we had instead, and cut down on sugar.  You are most definitely not supposed to do this, per the internet and baking mavens, but whatever.  I live on the wild side.

I think the pregnancy is driving me a little COCOa.  In fact, I'm sure it is because the word I'm looking for is "loco," right?

I think the pregnancy is driving me a little COCOa. In fact, I’m sure it is because the word I’m looking for is “loco,” right?

This cake is super easy to put together.  Whisk together the eggs, maple syrup, and vanilla.  Recipe calls for agave, I’m also pretty sure you could use sugar if you aren’t picky.  I cut down on my syrup because of the sugar in the hot cocoa mix.

They should warn you that the hormones maple you in all different directions at once

They should warn you that the hormones maple you in all different directions at once.

Then add the almond flour and sprinkle with salt, baking soda, and cocoa.  If you aren’t lazy you could mix the dry ingredients separately, but why do that?  Because it’s more uniform and better generally and what’s the problem with making one more bowl dirty, Yen?

Anyways.  Mix until mixed.  I thought this batter looked a little suspect and reviews said that it was a bit dry, so I pulled a Hershey’s and added 3/4 cup of boiling water to moisten the cake up.  Totally works.  Butter and flour a springform pan (this part is important if you want to make an ice cream cake!!!) and toss your batter in.

Everyone dislikes taxes, but what about the paper that we use?  Are any of those sheets like one day, I want to be used for taxes?  I'm an aspiring form?  (springform pan...)

Everyone dislikes taxes, but what about the paper that we use? Are any of those sheets like one day, I want to be used for taxes? I’m an aspiring form? (springform pan…)

Bake until done, about half an hour.  Then let cool completely.

While the cake is cooling, take out your dairy free homemade possibly paleo avocado coconut chocolate chip ice cream (or any flavor of ice cream) and let it soften.  Once it’s soft (maybe half an hour depending on your freezer), spread it as evenly as you can on your cake.

This recipe is far, far from canon.  How could we imbue it with an air of dignity/conservativism?  Just Pat it more and it'll Bu-canon.  (I really felt like making a Pat Buchanan pun)

This recipe is far, far from canon. How could we imbue it with an air of dignity/conservativism? Just Pat it more and it’ll Bu-canon. (I really felt like making a Pat Buchanan pun)

Then let it freeze again.  For frosting I went with whipped cream, which you can also do with coconut cream.  Since you’re freezing it, you’ll want to add a tablespoon of unflavored gelatin or agar-agar (which I for some reason have, despite not having cocoa powder) so that the frosting doesn’t just melt when you take out the cake.

Un-springform the cake and frost it with the whipped cream, then freeze it again.

Next on Discovery/Food Net-channel: Naked and Afraid- A Story of Frosting

Next on Discovery/Food Net-channel: Naked and Afraid- A Story of Frosting

Next up on ESPFood: Interviewing the NY Knicks on Frost(ing)- an epic battle for the truth (actual tagline for Frost/Nixon)

Next up on ESPFood: Interviewing the NY Knicks on Frost(ing)- an epic battle for the truth (actual tagline for Frost/Nixon)

To decorate, I melted some chocolate chips in the microwave and stuck them in a plastic bag.  Snipped off the corner and wrote on the cake.  It was great because the cake was so frozen it immediately made a Magic Shell effect.

IMG_20140825_195111963

 

Paleo ice cream Cake:

Chocolate cake, adapted from epicurious:

2 c almond flour

1/2 c cocoa

1/2 tsp salt

1/2 tsp baking soda

3/4 c maple syrup

2 eggs

1 TB vanilla

3/4 c boiling water

Prep a springform pan: butter and flour it (or oil and almond flour it).  Mix the maple syrup, eggs, and vanilla together.  Add the remaining ingredients and mix well.  Add the boiling water and mix.  Pour into the pan and bake at 350 for 25-35 minutes or until a toothpick comes out clean.  Let cool completely

Ice cream, whatever flavor you want that’ll go well with chocolate.  Let soften while the cake cools.  Spread on the cake (do not un-springform the pan yet!).  Refreeze for an hour.

Whipped cream for frosting:

1 c of whipping cream, or 1 c of the thicker stuff from the top of a can of coconut milk

2 TB of powdered sugar, to taste (up to 1/4 cup)

1 TB of gelatin or agar-agar

1/2 tsp of vanilla

Whip all ingredients together until it looks and tastes good.  Spread on your frozen cake (unspringforming it now seems good so you can get the sides).  Refreeze for an hour.  Decorate!

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.

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