## Unorganized Common core thoughts, math anxiety, also help me?

27 Aug

Disclaimer: I hate high-stakes testing.  Hate it.  This post is not about high-stakes testing; it is about content that happens to be used in high-stakes testing.  Imagine I’m writing about Holden Caulfield, and try to banish all memories of Catcher in the Rye plot pop quizzes from your mind.

My cousin asked me about my thoughts on common core.  Here’s an actual paragraph from the common core website:

These standards define what students should understand and be able to do in their study of mathematics. But asking a student to understand something also means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

This sounds awesome!  I often think I understand something well, and then try to write it down only to realize that there are more subtleties than I thought, or that I had a fundamental misunderstanding, and in general that I am wrong.

On March 16th of this year I thought I had an answer to something I’m working on.  Aaaand… today is August 27th and I’m still working on it.  But I didn’t realize my answer wasn’t complete until I started writing, and I wrote “, because” and had nothing to follow it.  This is frustrating and ridiculous, and I don’t expect every kid in America to become a mathematician and work on a problem for six months.  But I do believe that trying to do something, thinking you’re right, and then trying to explain it and realize you’re wrong after six minutes is a good experience and makes you a better critical thinker.  Question your beliefs, analyze your reasoning, explain yourself- all good things.

The above is all theoretical (I was a math and philosophy major in undergrad).  So I looked up some common core examples.

Here’s a comment from this article, as reported by a Washington Post blog:

Try doing multiplication in long drawn out word form like this one: 3, 6, 9 what is the 12th number in this sequence? My son can’t just read that and think the 12th number automatically. He has to write them out. He also cries over this type of math.

The problem itself is great (I’m also not opposed to crying).  Most parents can help their kids with 12 times 3 equals 36 “automatically,” but figuring out why 12 times 3 is 36 is hard.  This question offers justification first, and encourages the student to explore and discover multiplication for himself.  I was quite bad at my times tables as a kid (I still remember nonsense like “Six and eight went out on a date.  When they came back, they were 48!”) because it didn’t make any sense.  But writing out a sequence 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 is just like practicing spelling or lay-ups, and it makes sense.

I started finding more examples of common core type problems, but I think talking about one of my own past experiences is more helpful.

During my first semester of graduate school, I sat down to my first analysis exam.  I’d never been a big analysis fan (though now that I’ve taken it umpteen times I actually enjoy it!), and I felt a bit shaky about the material before I walked in the room.  The professor handed out the exams and said “you have one hour.”  If this isn’t clear to you by now, I’m good at math.  But as I leafed through the problems I started silently FREAKING OUT.  I felt something like vertigo, and also felt very very cold as I thought about trying to finish this exam in the allotted time.  My hands were shaking and I was a little sweaty.  This was my first experience at math anxiety and it was TERRIFYING.  My skin is prickling as I type this, remembering the experience five years later.

I can only imagine what it must be like to feel that with math for years in elementary-high school.  Math anxiety is real, and unfortunately contagious.  There’s been some amount of backlash to common core/shaking up math education, and I’m certain that some of it is rooted in past traumatic memories of math anxiety.  All I can say is, I support students, I believe in them, and that “the most important thing is to stay calm.”

If you google “common core math” you’ll get lots and lots of mild vitriol, but you also get some good stuff.  Here’s a side-by-side comparison of some math problems and explanations of why they switched.  Here’s a cool explanation of the new subtraction.

Here’s an article called “The Ten Dumbest Common Core Problems” that I’ve run into a lot [I dislike it because it doesn’t give credit to where any of the pictures/examples came from], and my thoughts on 1-10.  Again, not an elementary-high school educator, just a math person:

1. 7 + 7 = 10 + 4 = 14.  Maybe adding a word like “number bonds” is strange and new, but memorizing that 7+7 = 14 can be hard (I’m a terrible memorizer, but a good reasoner).  Breaking 7 into 3 and 4, and then pairing up to get a 10?  That’s how I add large numbers quickly, and I had to figure that out myself.  Helping kids learn the trick is great.
2. Awesome, visualizing subtraction.  See above explanation on the new subtraction.
3. This is a worksheet with a typo.  So… unclear why it’s included.  If it didn’t have typo, it’d be a good sheet.
4. This worksheet’s picture doesn’t make sense.  But the math does.  (Figure out what’s unknown if you have parts of a whole)
5. More visualization of addition and subtraction.  I literally did this with 12 year olds when I was 15, using quarters on a table.
6. I like this one too.
7. I always hated “carrying the one” so I’m all for the new addition.
8. I don’t understand this.  This doesn’t mean I hate it, it means I want to know what it says.
9. This isn’t new; I did this worksheet as a child.
10. Another typo

Final anecdote in this jumble of a blog post: how I learned to divide fractions.  They tell you to just multiply by the reciprocal.  Like I said, I’m bad at memorization, so I’d often do random nonsense that seemed sensible (divide the tops, multiply the bottoms).  No matter what I tried on my own, I felt like I just couldn’t get this dividing fractions thing. One day, my dad drove me to Baker’s Square to buy a pie.  He asked how school was going and I told him that I would never get how to divide fractions and I’d always get it wrong and math is dumb and I was just a sad little 8 year old sitting in that passenger seat.  He said, okay, well, if we split this pie among our family, how much pie does each of us get?  And I said “one fifth.”  What about if just you and I take the pie?  “One half.”  What if I give you the pie?  “One.”  What if half a person gets a pie, how much pie does one person get? “Tw-OHHHHHHHHHH.”  Crystal clear, this concept that had been frustrating me for weeks.  Why do we use the reciprocal?  Because if half a person eats a pie, that whole person eats TWO PIES.

That’s my happy story of understanding the madness behind the method.  I’ve seen dozens of students who know to multiply by the reciprocal, but who don’t understand why, which makes word problems quite difficult.  Conclusion: so far, I’m all for common core math.  The important thing is to stay calm.  And ask questions.  And maybe not help your kids with homework?

Oh one more thing: I’m interested in giving back to the community/doing some volunteer work and putting my skills to use.  I told my advisor yesterday that I don’t have any skills, and he responded that I have math skills and I should go find something like the free clinic for math.  Do you have any recommendations for me to help people with my math skills?

## Cinnabon mots

20 Aug

I made these fantastic and very large cinnamon rolls last week.  They are fluffy and light and soft and airy and buttery and pretty delicious and remarkably un-vegan.  Personally I prefer my pumpkin vegan cinnamon rolls, which are a bit heavier, smaller, and denser with a stronger cinnamon/ginger/nutmeg flavor and filling, and they don’t really require frosting.  These white cinnamon rolls definitely require frosting- the cream cheese/butter combo makes them super delicious and ties together what’s basically a soft white roll + brown sugar and cinnamon.  It’s just down to your preferences- are you a sugar cookie or a spicy ginger cookie kind of person?    This was from a recipe called “Clone of a Cinnabon” and they are quite similar to that mall staple.

If you don’t have three hours to spare/you want cinnamon rolls RIGHT NOW, check out these super fast banana ones that I’ve made over and over again.

I feel a lot of pressure to come up with butter and butter puns… maybe I should start doing repeats and milk them for all they’re worth

I microwaved my milk for thirty seconds and put it in the mixer bowl, then added the yeast- you want it to be warm but not too hot.  Tickle the yeast, don’t kill them.  Also, turn on your oven to 350 while you prepare the dough, so you can have a warm place for the dough to rise.

Another way the kitchen aid stand mixer has changed my life- no more ten minutes of kneading required!  Just use the dough hook and the mixer does it for you!  I’ve never been a good kneader so this is great.  I did use a whisk to mix the liquid stuff first though (the yeast with the milk, then the egg and melted butter).

LOOK AT THIS ONE HANDED ACTION SHOT! Do you know how hard it is to crack and open an egg with one hand? The eggstremes I go to for you.

Now that my measuring cup was empty (because the milk was in the mixing bowl), I could use it to melt six tablespoons of butter- another 30 second zap.

Ghee whiz it’s easy to melt butter

Add that in to your mixing bowl, then measure out your dry ingredients: flour, sugar, salt.  Toss them in, and use that incredible dough hook to knead the dough.  Or knead it yourself on a floured surface for ten minutes, until it makes a nice soft ball.

It’s so great that after you bake something, you get to eat a baked good! If you try sometimes, you just might find you get what you knead.

If kneading yourself, turn off your oven beforehand so it’s the right not-killing temperature for the dough to rest.  Otherwise, turn it off, then put the dough back in a bowl, cover it with a kitchen towel, and stick it in the off-oven for an hour.  Go do something else, like attend to the little monster who’s been crying at your feet this whole time:

Worst sous chef ever

Anyways, when you come back to it the ball should be twice as big as before and so fluffy.  Plop it onto a clean floured counter, leave the kitchen towel on it, and make your filling: this is just a mixture of brown sugar and cinnamon, but I like adding white sugar to cut the molasses-ness of it.   This time I used the Cinnabon method and spread butter, then topped with sugar/cinnamon, but I actually think it’s better to mix the butter with the sugar/cinnamon ahead of time and use that as a filling, like we did with the pumpkin rolls.

Roll out your dough into a nice thin rectangle and fill it.  I used to use wine bottles or tomato sauce jars for this, but we got this awesome fancy rolling pin as a wedding pressent and I love it.  It’s too large for a drawer and sits on our counter looking mildly rustic/pretentious and I love it.  It’s like you’ve been eating jelly beans your whole life of various quality, and then someone gives you some Jelly Bellys.  That’s how I feel about this rolling pin.  (I love Jelly Belly jelly beans a LOT.  If you’re ever near Fairfield, CA I highly recommend the free Jelly Belly factory tour.)

Swing your partner round and round… Now dough

See…

Dough!

Notice that I don’t put the cinnamon sugar all the way to the edge- this helps keep the filling from leaking out.

Now roll that log up (long side rolls up) and pinch the ends and the seam.

Don’t be fooled- this would make a terrible rolling pin, even though it looks about the right size and shape.

Cut it in half, then cut each half in thirds, and cut each third in half.  If that sentence was hard to read, just cut it into 12 pieces in a reasonable way.  Put them into a buttered 9×13 (I used a pyrex), toss it back into that warm oven for half an hour.

Notice how far apart they all are- feeling a little bit uncomfortable, a little bit like middle schoolers at their first dance.

They grow up so fast. This is more like the club when they’re leaving college.

Take out the big cinnamon rolls, then heat the oven to 350.  Bake for 20 minutes, until very lightly browned on the top.  Meanwhile, switch to the regular paddle (not the dough hook) for your mixer and beat some butter with some cream cheese until homogenous, then add a bunch of confectioners sugar, a dash of vanilla, and a bit of salt.

The proeblem with studying for speelling tests at the last minutee is you might end up addding extra letteers to words. Creamming is not the answer.

Let the rolls cool a bit before your frost them, so the frosting doesn’t totally melt everywhere.  Like 5-10 minutes.  Then frost and eat these beauties.  We actually split them usually because a single one is so huge.

Fake cinnabons, adapted from “Clone of a Cinnabon” on allrecipes

1 c milk

1 packet of yeast

2 eggs

6 TB melted butter

4 1/2 c flour

1 tsp salt

1/2 c sugar

Filling:

3/4 c brown sugar

1/4 c white sugar

3 TB cinnamon (+ginger, nutmeg if you like that)

6 TB softened butter

Frosting:

1 package cream cheese (8 oz)

1/2 c butter (1 stick)

3 c powdered sugar

1 tsp vanilla

1/4 tsp salt

1. Heat up the milk until warm, then mix in the yeast.  Turn on oven (to whatever temperature).  Mix the egg and melted butter with the yeast-milk, then add the flour, sugar, and salt.  Mix until it turns dough-like, then knead until soft (10 minutes by hand, 3-4 minutes by stand mixer).  Turn off oven, and cover the kneaded dough and put into oven for an hour.
2. Filling: mix all the filling ingredients together: you’ll actually want to cut the butter into the sugar so you’ll end up with a crumbly delicious mess.
3. Take out your dough from the oven once it’s doubled in size, and plop it onto a floured surface.  Let rest for ten minutes, then roll it out into a big rectangle (12″ by 8″ or so), and spread the filling on it.  Roll up from the long side, tightly, and pinch the seams and end.  Cut into 12 pieces, and place into a buttered 13″x9″ pan.  Cover and put back into that warm oven for half an hour
4. Take out the rolls, heat oven to 350, and bake the rolls for 20 minutes until very lightly golden on top.
5. Meanwhile, beat the cream cheese and butter together, then add the sugar, vanilla, and salt and beat until smooth.  Frost the rolls about 5 minutes after they’re out of the oven.

## Quick post: research updates of friends

18 Aug

I noticed a few papers up on arXiv last week that correspond to some old posts, so I thought I’d make a quick note that these people are still doing math research and maybe you are curious about it!

We last saw Federica Fanoni and Hugo Parlier when they explored kissing numbers, and they gave an upper bound on the number of systoles (shortest closed curves) that a surface with cusps can have.  This time they give a lower bound on the number of curves that fill such a surface.  Remember, filling means that if you cut up all the curves, you end up with a pile of disks (and disks with holes in them).  So you can check out that paper here.

Last time we saw Bill Menasco, he was working with Joan Birman and Dan Margalit to show that efficient geodesics exist in the curve complex.  This new paper up on arxiv was actually cited in that previous paper- it explains the software that a bunch of now-grad students put together with Menasco when they were undergrads in Buffalo, NY (UB and Buffalo State) during this incredible sounding undergrad research opportunity– looks like the grant is over, but how amazing was that- years of undergrads working for an entire year on real research with a seminar and a semester of preparation, and then getting to TA a differential equations class at the end of your undergraduate career.  Wow.  I’m so impressed.  I got sidetracked: the software they made calculates distances in the curve complex and the paper explains the math behind it and includes lots of pretty pictures.

My friend Jeremy did a guest post about baklava and torus knots a long time ago, and of course he’s got his own wildly popular blog.  He also has a bunch of publications up on arXiv, including one from this summer.  They’re all listed in computer science but have a bunch of (not-pure) math in them.

The paper I worked on over that summer at Tufts with Moon Duchin, her student Andrew Sánchez (note to self: I need a good looking website I should text Andrew), my old friend Matt Cordes, and graduate student superstar Turbo Ho is up on arXiv and has been submitted: it’s on random nilpotent quotients.

Moon and Andrew and others from that summer have another paper which has been accepted to a journal, it’s also about random groups and is here.  It was super cool, I saw a talk at MSRI during my graduate summer school there and John Mackay (also a coauthor on that paper) was in the audience and this result came up organically during the talk.  Pretty great!

There’s another secret project from that summer which isn’t out yet, but I just checked two of the three co-authors webpages and they had three and four papers out in 2015 (!!!)  That’s so many papers!  So I don’t know when secret project will be out but I’ll post about it when it is.

## A Linear Algebra Exercise

13 Aug

Back when I went to that excellent MSRI graduate summer school (video of lectures available at the link), one professor had us try a few seemingly-random exercises in order to motivate some definitions.  The exercises by themselves were pretty fun, so I thought I’d share them and maybe get to the definitions (which I have since used in my research).

Linear Algebra is really hard, and it’s one of the first courses you take after calculus (if you take courses after calculus).  I’m very often surprised by how often I use it and how applicable it is (I’m pretty sure my husband also uses it often).  A few months after I first saw an eigenvalue, I ran into an old friend who told me “I still don’t know what eigenvalues are useful for.”  Fast forward five or six years, and now I see eigenvalues EVERYWHERE!  My senior seminar of undergraduate was basically a semester all about eigenvalues (spectral analysis).  Anyways, this is all to say I still don’t feel very proficient at linear algebra, but I am very appreciative of it.  I hope you feel that way about lots of math/baking too!

In these exercises, we’ll live in vector spaces.  We’ll do a “definition” by example here: Euclidean space, or real space, is an example of a vector space.  Instead of thinking of coordinates as points in space, we think of them as little arrows that point to a point and have a specific length/magnitude (the distance of the point to the origin).

The purple point is (2,3). The orange vector has magnitude (square root of 13) and is pointing at the point (2,3).

Just as groups are the fundamental object of study in algebra, vectors are the fundamental object of study in linear algebra. Vectors are abstract sets with two operations (instead of one, which groups have): you can add vectors and you can scalar multiply them (which means make them longer or shorter).  There are a bunch of axioms that vector spaces have to satisfy, but you can just think of arrows in Euclidean space.

Next we need to understand the concept of an inner product.  This is an operation where you put in two vectors and get out a number (in our case, real, but generally complex or in any “field” in quotes because I haven’t ever defined what a field is).  In our example, if you have one vector that points to (2,3) and another that points to (9,11), you could do 2*9 + 3*11 = 18+ 33 = 51 and get 51 as the inner product of the two.  This is written $\langle (2,3), (9,11) \rangle =51$.  So we multiply all the matching coordinates, and then we add up all those products.

There’s a few axioms that inner products satisfy:

1. Conjugate symmetry: $\langle b, a \rangle = \bar{\langle a, b \rangle}$  In our case, since we’re in the reals and not the complex numbers, we can ignore the bar and read this as symmetry.  From our definition this is true.
2. Positive-definiteness: $\langle a, a \rangle \geq 0$ for any a, and equals 0 only if a is 0.  In our case, $\langle a, a \rangle$ will be a sum of squares, which is only 0 if every single coordinate is 0.
3. Linearity in the first argument: $\langle ax+c, b\rangle = x\langle a, b \rangle + \langle c, b \rangle$.  You can/should check this is true for our example.

From our definition of inner product over Euclidean space, we can come up with a formula for the magnitude of a vector (remember, that’s the distance from the origin to the point that we’re pointing at).   We have $\langle (a_1, a_2, \ldots, a_n), (a_1,\ldots, a_n) \rangle = a_1^2+a_2^2+\ldots+a_n^2$.  By the Pythagorean theorem, we already know the distance of a point to the origin: it’s the square root of this quantity!  So we have that the magnitude of a vector is the square root of its inner product with itself.  In symbols, we write $\| a \| = \sqrt{\langle a, a \rangle}$.  Any time you have an inner product, you can define a norm $\| \cdot \|$ in this way.

Here’s the exercise: let’s say you have a bunch of unit vectors (vectors with magnitude 1) and you take all of the pairwise inner products.  So if you have four unit vectors a, b, c, d, you’ll have all these inner products, which we’ll arrange in a matrix:

$\left( \begin{array}{cccc} \langle a, a \rangle & \langle a, b \rangle & \langle a, c \rangle & \langle a, d \rangle \\ \langle b, a \rangle & \langle b, b\rangle & \langle b, c \rangle & \langle b, d \rangle \\ \langle c, a \rangle & \langle c, b\rangle & \langle c, c \rangle & \langle c, d \rangle \\ \langle d, a \rangle & \langle d, b\rangle & \langle d, c \rangle & \langle d, d \rangle \end{array} \right)$

That’s all fine and dandy and well and reasonable.  Here’s the exercise: if I give you a matrix A, can you tell me if you can make A out of some unit vectors in this way with some inner product?

Notice that in the question, we didn’t say a particular inner product (we’re using one example, remember?)  This just says if you have a matrix, can you find an inner product and some unit vectors that make it.  So to take a few stabs at this problem, we’ll have to use all those axioms of inner products?  Definitely we need A to be symmetric (that means that the entry in the 2nd row, 3rd column is equal to the entry in the 3rd row, 2nd column etc. etc.) by property 1.

Property 2 and our discussion above say that all the diagonal entries have to be 1 (because we started with unit vectors).

Those first two conditions are necessary for A to even be a candidate, but just because A satisfies those conditions doesn’t mean that there’s a collection of unit vectors that form it.  (These are “necessary” but not “sufficient” conditions.)

The actual answer is that A must be non-negative definite: this means that if A is rows by columns and you pick a collection of complex numbers, this sum $\sum_{i,j} a_{ij}c_i\bar{c_j}$ is greater than or equal to zero (it’s not negative).  Parsing the sum expression: take each entry of A and multiply it by the complex number corresponding to its row, and by the conjugate of the complex number corresponding to its column.  Add up all of these products.  If this number is non-negative, then yes, your matrix can arise from a collection of unit vectors.

Done!

Just kidding!  Did you think I’d just throw an answer at you and not explain it?  That’s not how we do!

As always, the great question in math is why? Just getting a random condition thrown at you shouldn’t feel satisfying (well, it might, but too bad I’m taking away your satisfaction).

Let’s find out why the condition is necessary:

Squaring any number makes a positive number, so if you add up all your unit vectors and take the norm of that and then square it, you should get a positive number.  Even if you throw random complex scalar multiplication on to those vectors, the norm squared will still be positive.  So in equations, we have $0 \leq \| \sum_{i=1}^n c_iv_i \|^2$.  If we unpack this definition, we get

$0 \leq \| \sum_{i=1}^n c_iv_i \|^2$

$=\langle \sum c_iv_i, \sum c_jv_j \rangle$

$=\sum \langle v_i,v_j \rangle c_i \bar{c_j}$

$= \sum a_{ij} c_i \bar{c_j}$

So we get the condition is necessary (didn’t even use the unit-ness of the vectors here).

How about sufficiency?  Well, we have to come up with an inner product and some unit vectors.  So take some unit vectors $u_i$.  Any vector will be written $\sum t_i u_i$.  We’ll define an inner product on this space by $\langle \sum t_iu_i, \sum t_ju_j \rangle_A = \sum a_{ij} t_i \bar{t_j}$.  Go check that this inner product satisfies the three axioms above.  Great!  It’s an inner product (this relies on the fact that A is non-negative definite).  If you let all the $t_i=1$, then you can form A from the inner product of the $u_i$s.  That’s it!

Congratulations!  That was a hard exercise session!  We did it!

## Blueberry cheesecake ice cream

6 Aug

You might eat dinner after you make this ice cream. Make sure you leave rum for dessert!

It’s summer!  Time to bust out the ice cream maker!  This ice cream is super fancy compared to the other ice creams/sorbets I’ve made (you can also just blend a melon and put it in the ice cream maker and it’ll be great).  To make any ice cream cheese cake ice cream, just toss in some softened and blended cream cheese and a sleeve of graham crackers.  That’s pretty exciting.

This recipe has two parts: first you make the cheese cake ice cream base, then add a blueberry sauce.  This is IMPORTANT: don’t just put blueberries or blended blueberries in your ice cream without cooking them fully, because then you will have little ice chunks of frozen fruit in your final product.  Not great.  With almost no tweaking, you could use this exact recipe to make any fruit here cheesecake ice cream: strawberry, raspberry, peach…

If naming conventions were more standard, a shredder would be called a word processor.  Also how great would this food processor be with a cartoon wooden spoon giving me tips about how to use it?!

The cheesecake ice cream base is easy: blend cream cheese, half and half, milk, sugar, salt, and rum (yum).  I used a food processor for this.  You want chunks of cream cheese so don’t worry about mixing it really well.  Also, cream cheese is very rich, which is why we’re using half and half here instead of cream.  You can use cream if you want.  Then toss that in the ice cream maker while you do the rest.

To everything, churn churn churn

There is a season, churn churn churn. That season is summer. Making ice cream in winter is not something I think of doing

To light crush the graham crackers, I stuck my sleeve (unopened) into a bigger bag, and then started smashing it with the end of my French rolling pin.  Smashing it with a pot would also work.  I liked leaving it in the sleeve to minimize graham cracker crumbs flying everywhere.

I filled out a form about home ownership the other day. Asked if I would’ve bought a house without granite countertops, my first thought: “that would be crushing!”

The blueberry sauce takes a few minutes but is worth it for the fanciness factor.  Heat up berries with water and sugar…

Man, I lowthe unripe barrys

And add corn starch, lemon juice and zest, and salt.  Stir and stir and heat, and just like cranberries in cranberry sauce, eventually the blueberries will burst their blueberry-y goodness out into a tasty sweet and thick sauce.

Surely they made lots of versions of the robot in 2001: Space Odyssey. Maybe he was even the fifth version: HAL-E. If there were a botanist obsessed with that movie who discovered a new species of single-ovary fleshy fruit, he could name it HAL-E berry. I bet you know whose favorite fruit that’d be.

Sometimes I name inanimate objects and talk to them. If I had used a pan here instead of a pot, I would’ve named him Peter. The brand would’ve been Peter’s creator, so I could call that jammy manufacturer J.M. Berry.

Cool down your berry sauce as much as your patience allows (I threw it in a tupperware and stuck it in the freezer while doing other stuff), then swirl it with the churned cream cheese ice cream and add in the graham cracker pieces.

If workers at my fridge manufacturer took bets on who could mix this ice cream the best, it’d be a sw(h)irl pool

Stick the whole thing into a tupperware into the freezer.  It’s so good.

I didn’t take a picture of the final product so here’s a picture of my baby reacting the way I bet you are to this news:

Blueberry cheesecake ice cream, adapted from Joy the Baker:

1 package cream cheese, softened

1 c milk

1 c half and half

1 c brown sugar

1/2 tsp salt

1 tsp rum/bourbon/whatever you want

1 sleeve of graham crackers (not crumbs)

at least 1 c blueberries, more if you want

1/4 c sugar

1/4 c water

2 tsp corn starch

1 lemon

First, use a food processor or blender to mix the first six ingredients until mixed (it’s fine/expected that it’ll be lumpy).  Throw that in your ice cream maker.

Smash the sleeve of graham crackers into small pieces (bite size or smaller).  Chunks are fine here too.

Heat up blueberries with sugar, water, corn starch, and the juice and zest from the lemon over medium heat.  Stir occasionally, until berries have burst and sauce is thick (5-10 minutes).  Cool the sauce (put into freezer for speediness).

Carefully swirl cooled sauce into churned ice cream, then mix in graham cracker pieces.  Freeze for several hours.

30 Jul

This talk happened in March and I still remember it (and I was super sleep deprived at the time too).  Immediately after the talk, another grad student and I were chatting in the hallway and marveling at how good it was.  He said something like “I feel like a better person for having gone to that talk.”

A few days later, I ran into the speaker and told her that I had loved her talk, and she said “I’m super unintimidating so feel free to email me or ask me if you have any questions.”

During the talk, at one point she said (again, up to sleep-deprived memory coarseness)

“It’s more important that you learn something than that I get through my talk.  There’s no point in rushing through the material if you don’t take something away from this.”

All of these quotes are to say that this was probably the best talk I’ve seen (and I’ve seen lots of talks).  Particularly because of that last quote above.  Speaker put audience before ego, and that is a rare and beautiful thing (the other contender for best talk I’ve ever seen was by someone who recently won a big award for giving incredible talks).

The good news is that this is something anyone can do – mathematics at this level is a matter of practice and good habits, and not “talent” or “genius”.

OK, done fangirling!  On to the math!

We’ll be talking about a property of groups, so brush up from a previous blog post or wikipedia.  First we need to define a  (total) ordering on a group: a binary relation ≤ that satisfies three properties (which you’d expect them to satisfy):

1. Transitivity: if a≤b and b≤c, then a≤c
2. Totality: for any a, b in the group, a≤b and/or b≤a
3. Antisymmetry: if a≤b and b≤a, then a=b.

A few examples and nonexamples:

• The usual ≤ (less than or equal to) on the real numbers is an ordering.  For the rest of this post, I’ll freely switch between using ≤ to denote being in the group, and being in the real numbers (it should be clear when we’re talking about real numbers).
• Comparing heights of people is not an ordering: it’s not antisymmetric (see picture)
• Ordering words in the dictionary is an ordering: if you’re both before/at the same place and after/at the same place as me, then we must be the same word.
• Consider the group $\mathbb{Z}_2\times\mathbb{Z}_2$, which you can think of as a collection of ordered pairs $\latex \{ (0,0), (0,1), (1,0), (1,1)\}$.  If we define an ordering by (x,y)≤(a,b) if x+a≤y+b, then we’d break antisymmetry.  If we defined it by (x,y)≤(a,b) if x<a and y≤b, then we’d break totality (couldn’t compare (0,1) and (1,0)).

Top: reals are good to go. Middle: just because we’re the same height doesn’t mean we’re the same person! Bottom: (0,1) and (1,0) don’t know what to do

• Can you come up with a relation that breaks transitivity but follows totality and antisymmetry?

Notational bit: we say that a<b if a≤b and a does not equal b.

Now we say a group is left orderable if it has a total order which is invariant under left multiplication: this means that a<b implies ga<gb for every g in the group.

Let’s go back to the reals.  If you use multiplication (like 3*2=6) as the group operation, then the usual ordering is not a left(-invariant) order: 2<3, but if you multiply both sides by -2 you get -4<-6, which isn’t true.  However, if you use addition (like 3+2=5) as the group operation, then you see that the reals are left orderable: 2<3 implies 2+x<3+x for every real x.  This is a good example of the fact that a group is a set and a binary operation.  It doesn’t make sense to say the real numbers are left orderable; you need to include what the group operation is.

Here’s an interesting example of a left orderable group: the group of (orientation-preserving) homeomorphisms on the real numbers. (Orientation preserving means that a<b means that f(a)<f(b), all in the reals sense).  If you don’t feel like clicking the link to prev. post, just think of functions.  To prove that the group is left orderable, we just have to come up with a left-invariant order.  Suppose you have two homeomorphisms g and defined on the reals.  If g(0)<f(0), then say g<f.  If f(0)<g(0), then say f<g.  If g(0)=f(0), then don’t define your order yet.  If g(1)<f(1), then say g<f.  And so on, using 2, 3, 4…  Looks like a good left order, right?  WRONG!

Pink and blue agree on all the integer points, but not in between

If g and f agree on all the integers, they could still be different functions.  So we haven’t defined our order.  We need a better left order.  What can we do?  I know, let’s use a fact!

FACT: the rationals (numbers that can be written as fractions) are countable and dense (roughly, wherever you look in the reals, you’re either looking at a rational or there are a bunch in your peripheral vision).

So now we do the same thing, but using the rationals.  Enumerate them (remember, they’re countable) so use $q_1,q_2,q_3\ldots$ in place of 1,2,3… above.  It’s another fact that if g and f agree on all rationals, then they’re equal to each other.  Let’s make sure we have an ordering:

1. Transitivity: If f≤g and g≤h, then that means there’s some numbers (call them 2 and 3) so that f(2)<g(2) and g(3)<h(3).  But since we had to go to 3 to compare g and h, that means g(2)=h(2).  So f(2)<h(2), so f≤h.
2. Totality: if I have two different homeomorphisms, then there has to be a rational somewhere where they don’t agree, by the second fact.
3. Antisymmetry: We sidestepped this by defining < instead of ≤.  But it works.

Here’s a “classical” THEOREM: If G is a countable group, then it is left orderable if and only if it has an injective homomorphism to $\text{Homeo}_+(\mathbb{R})$.

Remember, injective means that each output matches to exactly one input.  Since we showed that there’s a left order on the group of orientation preserving homeomorphisms on the reals, we’ve already proven one direction: if you have an injection, then take your order on G from the order of the homeomorphisms that you inject onto.  So if h is your injection and g, k are your group elements, say that g<k if h(g)<h(k) in $\text{Homeo}_+(\mathbb{R})$.

One thing Mann does in her paper is come up with an example of an uncountable group that doesn’t do what the theorem says (she also does other stuff).  Pretty cool, huh?  Remarkably, the paper seems pretty self-contained.  If you can read this blog, you could probably do good work getting through that paper (with lots of time), which is more than I can say for most papers (which require lots of background knowledge).

That brings me to the “also”: I’ve been quite tickled to be asked about applying to grad school/what grad school entails a handful of times, some of those times by people who found me via this blog.  So please email me if you’re interested in whatever I have to say on the subject!  I’ve applied to grad school twice and have friends in many different departments and areas.  I hear I can be helpful.