Carnival of Mathematics 138

19 Sep

I am super pumped to be hosting the 138th Carnival of Mathematics, a monthly round-up of great stuff on the internet that has to do with math.  Last month it was hosted over on the AMS Blog on Math Blogs by Anna Haensch (who also has a lovely podcast called The Other Half ) and Evelyn Lamb (my fairy blogmother who also writes for Scientific American, Slate, etc.).  Curiously, the first thing that came up when I looked up “138” was a 1978 song by the horror punk band The Misfits: https://www.youtube.com/watch?v=SOqVs-K1eEo

A little bit catchy, but not quite my cup of tea.  What is my cup of tea?  An integer sequence that does not appear in the amazing OEIS!  138 is the smallest product of three primes such that the third is the concatenation of the first two: so 138 is 2\cdot 3\cdot 23, 777 is 3\cdot 7 \cdot 37, and 4642 and 10263 are the next in this sequence (I applied for an account to OEIS to submit this, so if you find a smaller one or the next several in the sequence let me know).  So off we go, to venture into the great unknown (aka the internet)!  This month we have a fun mix of grammar, history/sociology, math, and games!

  • First, grammar: Manun Shah over at Math Misery [which hosted CoM#136] posted Does 11=8 + 3? to chat about linguistics and mathematics.  If you love those memes about the Oxford comma, this post might be right up your alley.  My favorite sentence: “I bought orange juice and dishwasher detergent and drank it for breakfast.”  He argues we should think more about linguistics when teaching math, and the implicit biases that language might have on students as they learn mathematical reasoning.
  • Also in grammar, Thomas Oléron Evans of Mathistophiles posted Understanding an Abused Unit: The Kilometre-Hour, which delves deeper into a specific example of how common language can hurt mathematical understanding, and he uses a DELIGHTFUL example of turtles.  Favorite sentence here: “Unless they sidestep across the beach, of course, like some sort of synchronised reptilian dressage.”  Here’s an amazing graph from his example:
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A delightful graph from Mathistophiles

  • Keeping up with his fun theme, Thomas submitted a SUPER FUN game by David Peter called cube composer which is all about functions.  Do please do go play it!  It’s so fun!  It’s an intuitive individual puzzle game, and you can fly through the levels.  It also remembers your progress, but you can skip levels if you want too.  Here’s a screenshot of me picking random functions and not solving the puzzle:
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A screenshot of me poorly playing cube composer

  • A high school math teacher friend texted me about another game called Square It on a website for math educators supported by the University of Cambridge.  We proceeded to lose, lose, lose, and continue to lose against the computer.  I played against my spouse a few times, and he won his first time.  Those darn programmers and their algorithm-minds!  For the record I did eventually win.  This one is faster and lighter than cube composer, and it offers different mathematical questions to think about in different size grids.
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Here I am losing in Square It to a computer

Square It! is also great to play with kids.  Two more submissions are also kid-friendly.  

  • Brent Yorgey of Math Less Traveled submitted his post about Making Tesselations, which delves into the math behind the delightful new children’s book Tessalation! that I will definitely be reading to my toddler.  In his post he also talks about ants on a doughnut, and y’all know how we mathematicians love our analogies to lean toward the absurd.  Here’s a wonderful joke Brent nonchalantly sneaks into his post:  “The ant is so small that it can’t tell that the surface it lives on is curved. To the ant, it just looks flat. (You may know some tiny creatures in a similar situation who live on a sphere.)”  So fun.
  • Matthew Scroggs submitted a post dear to my heart, written by Belgin Seymenoglu over at Chalkdust Magazine: an ode to the delightful half hour film Donald in Mathemagic Land.  The post includes a link to the 1959 cartoon, which I’ve watched many times since I first saw it when I was 15.  I guess this is a fun month, because I really really recommend watching this cute and funny cartoon that includes actual math in it.

Now for a sort of random assortment of history/sociology/whatever posts that have to do with math:

  • Paul Plowman wrote a 2 minute 20 second guitar song using the number e, and posted about it over on his blog as Transcendental Music.  Sort of a silly but fun little exercise like memorizing digits of \pi, and the riff sounds nice.
  •  Poppy Brady submitted a story she wrote for The Voice about Nira Chamberlain, titled Teachers said I could be a boxer, but Never a Mathematician, one of those feel-good stories about mathematicians.  One fun quote by Chamberlain: “I also like what British mathematician Sir John Kingman once said that ‘mathematicians are better if they stay a bit childish and play the game as a game.’”  I think that keeps in line with our theme this month!
  • Over in Nature, Davide Castelvecchi wrote a news story about how the Majority of mathematicians hail from just 24 scientific ‘families’, a result by Floriana Gargiulo who analyzed the Mathematics Genealogy Project.  Every grad student has used the MGP to stalk their advisor/potential advisor to see their lineage, and you can print out a tree and give it to your advisor as a defense present!  So it was fun to read about someone actually analyzing this trove of data.
  • Finally, saving the best for last, the brilliant Evelyn Lamb, explainer extraordinaire, wrote a post on Roots of Unity at Scientific American about How Converting Between Addition and Multiplication makes Math Easier.  If you don’t follow/read everything that Evelyn writes, you really should.  So approachable and so lucid.  She also wrote a fun piece about using different bases for your birthday candles, so you should read these two articles, follow her on Twitter, and tell her happy birthday!

If you run into anything interesting on the math-internet from now until October 15th, submit them to the Carnival of Mathematics site; the Carnival will be hosted next month at the online magazine Gonit Sora.  Hope you had as much fun as I did with the submissions this month!

My mom’s Banh Xeo (Vietnamese turmeric crepes)

7 Sep

Over the past twenty years or so, bánh xèo has been a mainstay of family and friend gatherings if my mom is around.  Yen’s visiting home from college?  Better make bánh xèo!  You want to have some friends over for dinner?  Time for bánh xèo!  It’s a Saturday in October and the cousins are coming around?  Let’s do some bánh xèo, baby!

These are a super fun party food (where the party is the food): everyone gets a plate and a little saucer for dipping sauce, and the giant fresh pancakes go on platters in the middle.  You take a small amount with your chopsticks, trying to get the perfect mix of batter, onions, bean sprouts, and meat/seafood, and wrap it in a little taco of lettuce, mint, apple/cucumber, and whatever other herbs you have on hand.  Dip the whole taco in your sauce and eat it in two or three bites.  Then do it again.  And everyone does it at once!

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I don’t know about your storage capacity, but for me this is a mega-bite.

Note: these can definitely be made vegetarian or vegan.  The batter is vegan, and for filling we’ve had great luck with mushrooms, sprouts, and mung beans.  For the dipping sauce, substitute soy sauce for the fish sauce and add a little bit more lime and a little less sugar to balance out the saltiness.

I’m not usually a huge mise en place fan; I like chopping while things are cooking, but bánh xèo happen so fast that it’s worth it to put in extra prep.  For eating, you need to wash all your vegetables and put them out on a platter.  You’ll probably have time to make the dipping sauce while a pancake is cooking, but it takes a lot of time to pluck Vietnamese perilla (shiso would work in a pinch) and mint leaves off their stems, leaf lettuce, thinly slice cucumbers and apples, etc.  So lay those out.  Now the batter has to rest for ten minutes or so after mixing it, so do that before you start chopping your raw ingredients.

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I want to put a pun here but I can’t because the perspectives on this photo are too confusing.  The ingredients are on a counter; my feet are several feet below; the bowl on the right is inside a sink that the cutting board is hovering over.  

Growing up we always had squid in our bánh xèo, but we stopped doing that when we moved away from easily accessible Asian markets.  So now it’s mostly pork (a lean, unfatty cut like tenderloin is good here) and shrimp, mainstays of Vietnamese cooking.  My mom prefers buying the unpeeled shrimp because she says they’re sweeter and more flavorful (I agree), and we slice them in half down the spine so they cook super fast.  You want very thinly sliced pork; it helps if it’s chilled.  Also thinly sliced onions are key, as well as a ton of green onion (the more the better).  My mom puts in two bunches of sliced up green onion per packet of batter, which makes around 6-8 very large servings (depending on the size of your pan, that’s 10-20 pancakes).  Hence the party food-nature of this dish.

The batter is important- if you’re super hardcore I guess you could make your own, but we always buy from a bag.  Our favorite is the one with elephants on it, below–certain other brands are a little too gluey or not crispy enough, or include coconut flour and pretend that coconut flour+water=coconut milk.  If you can’t find Bon Con Voi Banh Xeo batter, just try to buy one that requires adding coconut milk.  Then follow the instructions on the bag: mix in the turmeric, a can of coconut milk, and a can or two of water (read the label).  Let sit while you chop the other stuff.

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Photo from this mysteriously empty website, Nguyen Eternal.  You don’t need to get on a boat to buy this bot, which is made here in the USA!  (Bot means flour in Vietnamese).

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In the shadow universe, the copy of me is so jealous of this batter.  She’s a green un-Yen. (Yes I have made this pun before no it does not get old.)

Now it’s time to cook!  Mix those prechopped green onions into the batter.  Heat up your nonstick skillet (or two if you’re ambitious).  Add in a bit of oil and a handful of thinly sliced onions, then after 30 seconds or so put in your thinly sliced pork and shrimp and cook them for a minute or two, until they’re done.

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Stop hogging all the space!  There’s MUCH ROOM in the pan; even the shrimps deserve some breathing room.

Now ladle on just enough of the batter to cover the pan if you swirl it.  Think the thinness of crepes.  I’d lean toward ladling on not enough, swirl, and then add in a bit more to fill the pan rather than having a thick gluey pancake.

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If the band Rudimental were making banh xeo, I bet they’d invite Ed Sheeran over for this step, so they could lay-it-all (LADLE) on him.

Now you’ll have a pale daffodil soft thing with some cooked meat inside it, and a clearly slightly liquid middle.  Fill it with a generous handful of bean sprouts (mung beans if you’re into that sort of thing too), and cover it with a lid to lightly steam the sprouts while cooking the pancake.  Give it about a minute (so this whole process is taking you about 5 minutes per pancake), until the pancake is no longer pale but a golden turmeric color, and its visibly cooked.

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I should invite my cousin Scott over and ask him to help me cook this, and then try to gently prod him into calling the middle uncooked portion liquid gold.  Take that, furniture polish!  

Gently fold it in half like an omelet, and slide it out of the pan onto a waiting plate.  Serve immediately with dipping sauce and that premade plate of veggies!

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My mom’s banh xeo.  Makes so much (serves 8 hungry people)

1 package of banh xeo flour

1 can coconut milk

2-3 bunches of green onion

2 lb pork tenderloin or butt or any non-fatty cut

2 lb shrimp

1 big onion (yellow or white is fine)

2 packages of bean sprouts

2 heads of lettuce (I prefer red leaf, my mom likes Romaine but she is wrong here)

2 cucumbers, 2 Fuji apples

2 bunches of Vietnamese perilla or shiso, 1-2 bunches of cilantro, 1-2 bunches of mint (the regular grocery store mint is fine)

2 limes

2 TB Fish sauce

2 TB Sugar

1 carrot and a carrot-sized piece of daikon if you’re feeling ambitious (we did not)

  1. Batter: Finely chop up all your green onion, then follow package instructions.
  2. Veggies: Wash everything.  Leaf the lettuce and herbs, cut the apples and cucumbers into thin slices (cut the cucumber in thirds lengthwise, then cut into slices).
  3. Dipping sauce: Mix two tablespoons of sugar with the juice of the limes and 2 TB of water until dissolved.  Add in two tablespoons of fish sauce.  Taste.  Adjust levels of everything until it’s not too sweet, not too salty, and not too sour.  If it’s overpowering, add more water.  If you’re ambitious, finely grate the carrot and daikon into the sauce; they’ll slightly pickle while it’s sitting and offer some textural contrast when you eat.
  4. Pancakes: Thinly slice pork (about 1/8 of an inch thick if you can, 1 inch by 2 inch rectangles), peel shrimp and slice in half, thinly slice onion. Heat large skillet over medium high.  Add about 1 TB of neutral oil and half a handful of onion, stir.  Then add 4-8 pieces each of pork and shrimp and gently saute until just cooked.  Ladle only a little bit of batter over the cooked ingredients, and swirl to cover pan.  Edges should cook very quickly.
  5. Cover pancake with a handful of bean sprouts, then cover with a lid.  Leave for 1-2 minutes, until bean sprouts are slightly steamed and center of pancake is cooked and edges are lightly browned.  Fold in half with a spatula, slide onto a plate, and serve immediately with veggies and dipping sauce.

Note: you’ll be in the kitchen for a while with this; we usually have two or three pans going at once to feed a big crowd.

What is a “trigger warning”? What is a “safe space”?

26 Aug

This week a professor friend of mine posted and lauded the letter that University of Chicago president sent out to all incoming freshmen which said a bunch of reasonable and universal/noncontroversial stuff, and also this paragraph:

Our commitment to academic freedom means that we do not support so called ‘trigger warnings,’ we do not cancel invited speakers because their topics might prove controversial, and we do not condone the creation of intellectual ‘safe spaces’ where individuals can retreat from ideas and perspectives at odds with their own.

They, being a professor, then did some research into what exactly trigger warnings and safe spaces are, and then posted that this letter was “an embarrassment.”  So why did my friend have such a change of heart?  For that, let’s learn what these phrases mean.

trigger warning is a statement given before someone interacts with new material (e.g. a book, an article, a video, etc.) which alerts them of potentially disturbing content (e.g. graphic description of sexual violence, war scenes, etc.).  There’s a great op-ed piece in the NYT from last year that delves deeply into this:

Triggered reactions can be intense and unpleasant, and may even overtake our consciousness, as with a flashback experienced by a war veteran. But even more common conditions can have this effect. Think, for example, about the experience of intense nausea. It comes upon a person unbidden, without rational reflection. And you can no more reason your way out of it than you reasoned your way into it. It’s also hard, if not impossible, to engage productively with other matters while you are in the grip of it. You might say that such states temporarily eclipse our rational capacities.

The idea behind trigger warnings is that they give time for the reader/viewer/listener to prepare themselves or brace themselves for what’s coming, in hopes that they can rationally deal with the material.  In practice, they’re basically the same as the little box that says “this movie rated PG-13 for containing partial nudity” and most people can ignore them (idea from that op-ed).  There are many arguments out there against trigger warnings; one of the most cited is the “Coddling of the American Mind” article, where they analogize avoiding difficult subjects with phobias:

A person who is trapped in an elevator during a power outage may panic and think she is going to die. That frightening experience can change neural connections in her amygdala, leading to an elevator phobia. If you want this woman to retain her fear for life, you should help her avoid elevators.

There are great points in that article; namely, that turning disclaimers of difficult readings into optional readings does a disservice to students (they offer the example of professors not wanting to teach rape law).  I think the authors of this article and the professor of the first op-ed can agree that the point is for students to engage with difficult material.  However, the Atlantic piece authors believe that not including trigger warnings is the way to do so, while the NYT piece author believes the exact opposite.  So which is it?  Why the different views?

This is why mathematicians love definitions.  The Atlantic piece authors start with the same definition as I did, but then add another consequence:

Trigger warnings are alerts that professors are expected to issue if something in a course might cause a strong emotional response. For example, some students have called for warnings that Chinua Achebe’s Things Fall Apart describes racial violence and that F. Scott Fitzgerald’s The Great Gatsby portrays misogyny and physical abuse, so that students who have been previously victimized by racism or domestic violence can choose to avoid these works, which they believe might “trigger” a recurrence of past trauma.

And therein lies the problem!  Trigger warnings exist so that students can engage with a work, but the authors say they exist so students can choose to not do so.  This jumbled definition is why there seems to be a fundamental disconnect between some news sources which laud “not supporting trigger warnings” (The Federalist (conservative/libetartarian), Reason magazine (libertarian), Intellectual Takeout (conservative/liberal)) and those that do not (Slate (liberal), Vox (this is a great piece), New Republic (this is not as good as the Vox piece)).

Of course, nothing happens in a vacuum (besides math?).  This whole controversy is part of a larger trend as demographics change, political landscapes change, and universities change.  There are many examples in practice of students actively refusing to engage with material that might question their assumptions (first example that comes to my mind was the Duke freshmen not wanting to read the non-required recommendation “Fun Home”), and this is something that all these public intellectuals of various political backgrounds want to avoid.

There’s a lot to say about this, but I want to move on to the next definition, so I’ll leave you with this article by NYMagazine, which happily acknowledges that the U of C letter totally messes up on ‘trigger warnings’ and ‘safe spaces’ but also says that it has some important things to say.  Here’s a quote:

First, in addition to the absence of evidence that trigger warnings have any impact on the average college campuses, there’s also solid evidence that many of the tropes that have taken hold about “coddled” or“microaggressed” or “oversensitive” or anti-free-speech college students are seriously overblown. In many cases, these ideas have been bandied about so gleefully and frequently and uncritically by conservatives that the terms themselves have lost all meaning….But: There have absolutely been recent instances in which campus outrage has snowballed out of hand, in which protesters have actually impinged on the ability for real debate to take place, and these episodes matter.

Funnily enough the article refers to the controversy at Yale as also being disproportional, which I also wrote about in a similar vein as this post.

Next, a safe space is a physical location for marginalized people (historically LGBT folks) to exist with allies without fear of marginalization/hate speech.  Examples are church basements for Christian youth groups, gay bars, and the U Chicago LGBTQ Safe Space program.  The U of C letter refers to “intellectual safe spaces” as what it’s against, and includes a semi-definition as spaces where “individuals can retreat from ideas and perspectives at odds with their own.”  This is not a normal use of the term, and there again seems to be some confusion about what the phrase “safe space” means.  There’s a fantastic article on Vox that goes into what a safe space is and why it matters in depth:

You don’t have to explain to other black women why your hair is the way it is, she said, or what a certain word means, or countless other little cultural signifiers. “Everybody has a need to just be able to be themselves somewhere, without having to do that translation and without having to always be on guard to justify yourself.”

I liked this Vox Explainer article so much because it really delves into the reasons that people are against the idea of safe spaces.  To close this SUPER LONG blog post, here are the last three paragraphs from it:

Some people get upset because they don’t understand why they can’t be included in a certain group, or why their input on certain issues might not be welcome. A man might ask in good faith whether catcalls are really just “compliments” when women are trying to discuss their own experiences with street harassment, and he might be taken aback when those women get immediately upset or exasperated with him. To him, perhaps he was just asking an innocent question and trying to have an intellectual debate. But to the women, it’s pretty insulting to suggest that their life experiences are up for “debate” — plus they’ve heard remarks like these a hundred times, and nine times out of 10 it just derails the conversation, so they’re just sick of dealing with it.

The question of who belongs and who doesn’t, who is excluded and who isn’t, is a constant worry for most of us. But on top of the personal rejections that everyone faces in life, people in marginalized groups also have to face the feeling that society wasn’t really designed for them; that it considers them an afterthought at best. People in dominant cultural groups are used to rejection, but they’re probably not used to that kind of rejection. And they’re probably not used to being forced to pay attention to all the little social cues and codes that others pick up when trying to navigate a society that isn’t inherently made to fit them.

It’s not easy to deal with shame, hurt feelings, or fear during these kinds of cultural clashes. But particular spaces or identities are rarely the most productive things to blame for the strife. Inside or outside of safe spaces, the real problem is usually a failure of empathy, and the real solution is treating others with humility, respect, and compassion and being willing to learn from our own mistakes.

 

The non-academic job search (Part 1)

10 Aug

This may come as a surprise to you, but I’ve decided after much soul searching this year that I will not be applying for postdocs this fall and following the steps to success in mathademia.  Please do not take this as an indictment of academia (though I also feel that;  each of those words is a separate link about academia & motherhood), and I highly recommend postgraduate study in math if you’re interested in it (so does Evelyn Lamb on Slate!)

I have loved my time doing something I love on a flexible schedule which gave me lots of time to spend with my son, organize conferences, travel, blog, bake, exercise, and have the life I wanted.  It’s also been very difficult to have the highly unstructured environment, little oversight, and lack of regular collaboration.  Also very little money, but I married a person with an actual job so he can support the kid, and as a young 20-something I didn’t need much money (especially with math conferences covering travel and accommodation costs!)  So I love math, I love mathademia, but I don’t love teaching enough to do it full-time yet, and I don’t love research enough to want to move my family only to move them again 3 years later, and possibly 3 years after that again.  Hence I am starting my non-academic job search, and I thought y’all could join me on my journey.

When I first started toying with leaving academia, a friend of mine who also has a Ph.D. in math told me: the hardest thing about leaving academia is deciding to leave academia.  It’s been several months of pro/con lists, discussions with friends and family, and days of feeling sad and hopeless vs. days of feeling inspired.  You can’t help but feel like a failure when you make this decision, because all the exemplars of success that surround you are academics.  And that’s not even true for me; I try to know a range of people who do different things, but still in my day to day life and work it’s all professors etc.  Anyways, I got through this stage but it was rough.

Next I got some books!  Specifically, In Transition and What Color is Your Parachute? + Workbook.  I’d heard of Parachute, and I talked to a friend on the phone who used to be in consulting who said that everyone who left his company was given a copy of In Transition.  So I spent a few weeks working through these, which was mostly about soul-searching and there’s some practical advice in there about informational interviews.  In those weeks I also contacted career services at Yale (they have someone dedicated to alumni) and UT Austin and got some short and helpful advice.

I rejoined Amazon affiliates so I could put these pictures in this post.  Buy the books from links above, I get money!


After using the books and career people and narrowing down to a few fields that might interest me, I used LinkedIn and the internet to find companies in those fields in the locations I’m interested in (Austin, where we are now, and where our families are).  After a few days of searching, that got me a list of 50 or so companies in a file I called “first impressions”, which I then went through again and checked out all of their websites which took another several days.  I deleted all the ones I couldn’t imagine working for or which didn’t exist anymore, which brought me to a list of about 20 companies.

I figured once I started contacting people they’d google me, so I updated my website and made it pretty and fancy!  Then, using my CV as a starter and a template that one of the career counselors sent me, I wrote a resume targeting these fields.  Based on that, I updated my linkedin.  The website took me an afternoon several months ago, the resume took me two weeks.

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How many pictures of myself did I put on my website?  A lot!

In the meantime several more companies got added to my list: I’m on several email lists and one alum sent out a note that his company is hiring, I looked on vault.com which ranks consulting companies on lots of metrics, including “work-life balance” and “least amount of travel” so that added a few companies, and I added a few dream companies (AAAS) which are not where I want to live but why not explore them and figure out what makes them dreams, and what things I want in the job I end up getting?

So about two weeks ago I added another sheet for contacts to this growing excel file which still has the name “first impressions”, and I used LinkedIn to find people who work at and people who used to work at each of those companies who are in my 2nd degree networks, and I wrote down their names and the name of my connection to them who can introduce us.  I also used LinkedIn to find people who were alums of any of the schools I’ve been affiliated with.  Now I’m starting to do informational interviews with those people- I asked for half hour phone conversations, but after the first one last week I think 15 minutes would suffice.  To get introduced, I write an email to our mutual connection and include at the bottom of the email an introductory note to the person I want to connect to, so our mutual friend can just forward it instead of having to write a whole long thing.

What’s great about informational interviewing people who used to work somewhere is they have no skin in the game if I end up at that company.  They can tell me why they left!

If you couldn’t tell this has been a summer-long project that I started at the end of May (at least, that’s when I redid my website).  I didn’t start out the summer knowing any of these things, neither what the tasks were nor how to do them, but I talked to that former consulting friend for an hour in May and he made all of these helpful recommendations, and I talked to the Yale alum counselor for a half hour in June, and I met with the UT counselor in July who gave me concrete advice as well.

I’ll keep you updated every few months with the progress of this!

Phylogenetic trees

2 Aug

I just listened to a two hour talk on phylogenetic trees, and they seem fun enough that I thought I’d share them with you!  Sorry I literally forgot to post last week, and then I realized I did not take notes on the stuff I wanted to post about (more pictures by Aaron Fenyes)- here’s a photo that I took and wanted to explain:

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Something twisted representation something skein algebra something unit tangent bundle

Instead, you’ll read about the basics behind the research of my friend Gillian (no website), which is being supported somehow by people who want to know about cancer.  First, some background: the field of study of phylogenetic trees is inspired by and informs applications in evolutionary biology.  All of the life-forms today (or at some slice of time) all trace back to a root life-form at time 0.  Each current life-form follows a series of branching away from the root to get to where it is now

Then we can represent this evolution via a labeled rooted binary tree: the root represents the root life-form at time 0, and each branching of the tree represents a different evolution.  The labels mark which life-form is which.  Of course this model isn’t perfect (I can’t find the word for it but it’s a thing where two different species evolve separately from the same ancestor, then meet again and make one species.  If we were to represent this information in a graph, it’d make a cycle and not be a tree), but it’s been fruitful.

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The rooted binary tree of the wikipedia picture: node 0 is the root life-form, then 1-7 are the life-forms at our current time.

Now let’s mathify this.  We’d like to encode the evolutionary information into our tree.  We’ve already decided that all life-forms will end at the same time (now), so if we just assign lengths to each of the non-leaf edges this will automatically determine the lengths of the leaf edges.  A leaf in a tree is a vertex with only one neighbor, and we call the edge leading to that vertex a leaf-edge.  Let’s call the non-leaf edges interior edges.  In the picture above, we have 5 non-leaf edges, which determine a tree with 7 leaves.  Using this exact configuration of labels and edges, we have five degrees of freedom: we can make those interior edges whatever lengths we want, as long as they are positive numbers.  So in math-terms, the set of phylogenetic trees (aka rooted, binary, labeled trees) in this configuration forms a positive orthant of \mathbb{R}^5.  You can smoothly change any one of the edges to a slightly longer or shorter length, and still have a phylogenetic tree with the same combinatorial data.

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This is from the paper I’m writing, but it does show that in 3D, there are 8 orthants cut by the three axes (red dot is the origin).  The pink box represents a single orthant.

What about phylogenetic trees with different combinatorial data?  Say, with different labels or different branching, but the same number of leaves and the same number of interior edges?  First we need to figure out what we mean by ‘different’.  For instance, the following picture from the seminal paper in this field shows three trees that don’t immediately look the same, but we don’t count as different:

Why aren’t they different?  Because they encode the same data for each life-form: reading from node 0 we see that first 1 branches off, then 2, then 3 and 4 in all three cases.  There’s some combinatorics here with partitions that you can do (one can label a tree with a set of partitions).  However, changing the labels so that first 2 branches off, then 1, then 3 and 4 will be a different phylogenetic tree.  In fact I can smoothly go from one to the other in the space that we’re creating: first I shrink the length of the green edge below to zero, which takes us to the middle tree (not binary!), and then extend the blue edge.

axis

Shrink the colored edges to get the same tree in the middle (not a binary tree)

We’re going to add these non-binary trees with one less edge length to our space.  Remember the tree on the left has an entire positive orthant, and the tree on the right has an entire positive orthant.  Shrinking the green length to zero means that we’re moving to one side of the left orthant: so we add this axis to our space (we have \{x,y\in \mathbb{R}^2: \ x,y\geq 0\} instead of strictly greater than 0).  We can glue the green and blue orthants together along this axis.  Here’s a picture from the paper:

spacetrees

Notice that they also have the origin filled in, with a tree with no interior edges.  This is the cone point of this space.  Now we’re finally ready to describe the space of phylogenetic trees: within each combinatorial structure/labeling, we have a Euclidean orthant in (n-2) dimensions.  Then these orthants are glued together along their axes in a specific way, and all of them are connected to the cone point.  This is called BHV(n), short for Billera-Holmes-Vogtmann space (in the paper they call it T(n) but that’s confusing to everyone else).  Here’s the picture of T(4):

t4

Each triangle represents an infinite orthant

There are 15 different orthants glued together in this picture, because the number of labelled rooted binary trees on vertices is (2n-3)!!.  The double !! means you only multiply the odds, a.k.a. (2n-3)(2n-5)(2n-7)… This is also known as Schroeder’s fourth problem , which as far as I can tell was open for 100 years.  Pretty cool!

If you truncate BHV(n) so it’s not infinite (just pick some compact bound), then it forms a nonpositively curved cube complex, and we love those!  CAT(0) cube complexes are great.  I haven’t blogged too much about them (first terrible post and then those truncated Haglund notes) but they are the basis of all that I do and the number one thing I talk about when I give math talks.  Whoops!  The gist is that you glue cubes together in not-terrible ways, and then the resulting complex has great and fun properties (like you can cut it in half the way you want to).

That’s about all I have to say about this!  Gillian is working on some stuff about putting a probability measure on BHV(n) [you can’t do it with certain conditions], embedding it into a small enough Euclidean space that still preserves some of its features, and finding an isometrically embedded copy of the phylogenetic tree inside BHV(n) instead of just the coordinate point.  Also, fun fact to prove to yourself (actually please don’t scoop my friend), find the automorphism group of BHV(n)!  It’s just the symmetric group on some number that has to do with n (n+1 or something like that; I can’t remember and didn’t take notes).

Again, the main reference for this is the seminal paper that should also be accessible as it’s meant for biologists and statisticians.

Chocolate orange almond cake (gluten-free)

19 Jul

A few weeks ago I met a friend at one of his favorite new coffee shops in Philadelphia, Frieda’s, which is maybe called FRIEDA for generations and has a very cool mission of essentially being a cool hip young coffee shop that welcomes old people.  I spent three hours there working on math and eating breakfast and having an incredible chocolate cake that had this beautiful aromatic orange flavor and big chunks of orange in it.

We spent a long time raving over this amazing cake, and the chef/co-owner walked over and chatted with us (he’s on a first-name basis with my friend, who goes there all the time) and told me the recipe orally.  Oral recipes include things like “add some cocoa powder” and “top with chocolate ganache,” so this was an adventure!  I also busted out the digital scale for this, but then measured stuff out for all y’all in the recipe at the bottom.

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Spouse bought me the new bag of flour. Why settle for just dia-monds when you can have ALL-monds?  

The cake has very few ingredients, though I forgot to add the cocoa powder above.  First you boil the oranges for 2 hours as you’re a very patient person.  Just kidding!  I covered these in water, considered putting a plate on top but didn’t, and microwaved them for 15 minutes.

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When it comes to American playwrights, do you prefer Williams, or Inge?

Boiling the oranges pulls out the bitterness from the pith (the white part), which is great for what happens next.  But while the oranges are microwaving, you might as well measure out your ingredients and mix them: sift together the almond flour and baking powder and cocoa powder.  I don’t have a sifter so I use a whisk, but sifter would be better for end-result cake texture.  Whisk up the eggs to get some air in there, then whisk in the sugar.

My action shot was too exciting for this!

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I was too eggs-ited!

You can also, if you want, skip all of those steps and dump everything in the food processor after you blend up the oranges!  CAREFULLY pull the oranges out of the hot microwave water and CAREFULLY cut them in quarters.  They’ll be soft but not falling apart, but the juice inside might be HOT.  It helps to buy seedless oranges for this part, because you don’t have to fish them out.  Then throw ’em in the food processor or the blender!

 

Orange puree!  If you’ve got a big food processor you can throw in all the other ingredients now and pulse it all together.  If you don’t, mix the puree with the eggs and sugar, then mix in the dry ingredients.  It’ll be lumpy because of the almond meal.

We’ve been really into The Great British Bake-Off lately, and my spouse thought he’d try to make a Mokatine despite the fact that he never bakes… in fact this was the first thing I’ve EVER seen him bake.  Anyway, we ran out of parchment paper so I told him to “butter and flour” the pan.  He did not know what I was talking about, so here are pictures:

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He’s a stick-ler for details and wanted to know “how much butter”.  I said “enough?”

Using your fingers or a paper towel or a stick like I did, rub butter all over a pan until the whole pane has a thin layer of butter on it.  Then spoon a few (2-3) tablespoons of flour into the pan and shake it around, rolling on each edge, until it’s evenly covered in flour.  Dump out any remaining flour.

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Who needs flowers when you can have flour? (but darling if you’re reading this I still like flowers)

Note that I didn’t do a great job above: you can see where the flour didn’t stick to a part I didn’t butter enough.  That’s exactly where the cake stuck to the pan later.  SO BE THOROUGH with your buttering and flouring.  Or, yknow, keep parchment paper in stock so there’s no flour on your gluten free cake…

Bake!  Let cool completely before frosting (but don’t leave it out too long for fear of losing moisture).  Frosting is MAGICAL GANACHE.

Did you know about ganache??? Somehow I had not made ganache before, despite having a baking blog for almost four years… it’s SO EASY and SO MAGICAL.  I’m into caps today.  You just pour hot cream onto chopped chocolate, and stir it until it’s frosting!  I am lazy so I used chocolate chips, which have extra stuff on them to keep them in their shape, so my ganache wasn’t perfectly smooth.  But still, it’s so delicious and wonderful (ganache is the center of truffles!  I didn’t know!)

 

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All of the photos were in different shots so I couldn’t gif it for you.  It starts out looking like failed hot cocoa when you pour the hot cream on the chocolate and wait for a minute, then like good hot cocoa as you stir it, and then shiny dark melted goodness, and before you know it (after a few minutes of cooling) you have frosting!  Then you can just spread that thick delicious stuff all over the cooled cake, and serve!  We actually don’t like chocolate very much and I made this for a friend’s chocolate-themed birthday party.  I will definitely make this cake again, sans cocoa powder (it’s SO orange-y and SO almond-y and SO easy).

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Chocolate orange almond cake (adapted from David Hong at Frieda’s)

For cake:

2 oranges

300 g (3 c) almond flour

200 g (1 c) sugar

6 eggs

1 TB baking powder

1/3 c-1/2 c cocoa powder (I did 1 c and it was too cocoa powdery; 3/4 c is the Hershey’s chocolate cake recipe, I think 1/2 c would be great)

For ganache:

1 c heavy whipping cream

1 c good chocolate, chopped, or fancy chocolate chips

  1. Put oranges in a bowl, fill with water so oranges float, and microwave for 15 minutes.  Every five minutes, rotate the oranges so that a different side is floating out of the water.  Or put a small plate on the oranges to keep them submerged.
  2. Meanwhile, sift together the almond flour, baking powder, and cocoa powder, or whisk them well.
  3. Vigorously whisk the eggs until frothy, then whisk in the sugar until light and fluffy and pale.
  4. Carefully chop the hot, soft oranges into quarters, then puree in a food processor or blender.  Preheat oven to 375.
  5. Whisk orange puree with the eggs and sugar.
  6. Mix the dry ingredients with the wet and mix well.
  7. Butter and flour a springform pan or line with parchment paper.  Should work on any pan; I just used a springform.
  8. Fill the pan, bake for 40 minutes or until a knife inserted in the center comes out clean (up to an hour).  Let cool completely before frosting.
  9. Heat up the cream over medium heat in a small saucepan.  It doesn’t need to boil, but should be pretty hot (if you aren’t sure, take it almost to boiling).
  10. Put the chocolate chips in a bowl, and pour the hot cream over.  Let sit for a minute, then start stirring with a wooden spoon or whisk.  Keep stirring until it turns into ganache.
  11. Frost your beautiful cake!

Productivity tips for solo workers

12 Jul

I just got back from a Dissertation Writing Retreat, put on by my undergraduate fellowship, Mellon Mays .  Twelve of us planned our days, talked goals and schedules, and tried out techniques for staying productive and keeping up our morale.  The first two days we were essentially locked in a room for four hours (two sessions of two hours each) and worked on our computers, using social pressure and a shared timer.  Then we weaned off to one session and then no sessions, with the expectation that we’d figure out how to use the time schedule ourselves.  The end of each day we had check-ins and discussed what worked and what didn’t.  So I thought I’d share with you some of the stuff I learned.

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#IlookLikeAProfessor #squadgoals (Faculty panel)

First, a few things I already knew:

  • Figure out where you can work.  When my partner worked from home we turned the guest bedroom into an office for him.  I’ve done parks, coffee shops, the office, and our home office and all were much more productive than the kitchen table, where I can see the dishes, the fridge (what’s for dinner tonight?!), the living room…
  • Write a task lisk for each day and focus only on those tasks.
  • Make sure to “have a life,” which in my case meant starting a baking and math blog.
  • Exercise!  Figure out some way to move your body.
  • Use SMART goals.  Specific, measurable, attainable, realistic, and timely.  I don’t really know what “attainable” means vs. “realistic” but maybe it balances out people with too low self confidence (at least it can be attainable) vs. people with too high self-confidence (remain realistic!).

years of grad school and those are the things I knew.  I’ve been in a rough space for the past few months, math and life-wise, so this writing retreat was the perfect detox/jump start for me.  Here are some things I learned!

  1. Break up goals into specific, manageable tasks.  I used to look at my planner post-it each day and see things like “work on paper X” and “read paper Y.”  On our first evening we listed our main goals for the week, and then took the top goal and split it into at least three specific tasks.  So my “organize paper” became 1) copy topic sentences (lemma/theorem statements), 2) skim paper and name techniques, 3) figure out which theorems use which techniques, 4) form flowchart, and 5) rearrange paragraphs so flowchart makes sense.  Then when I sat down the next morning I didn’t have “organize paper” to look at, but a really easy softball of a task to start my day and feel productive.
  2. Set out your tasks and goals the day before.  This has helped me SO MUCH.  I used to spend half an hour each morning reviewing the previous day and setting up what to do that day.  Here’s a picture of the Emergent Task Planner pages we were using. etp
  3. Try the POMODORO TECHNIQUE.    The idea is that you break up goals into tasks, and then set a timer and FOCUS on each task for 25 minutes (=one pomodoro), then take a five minute break.  You fill a little box for each pomodoro (=”pom”) next to your task that you took, and then you cross off the task when it’s done.  If your tasks are taking 4 or more poms, you’re not breaking up the tasks enough in step 1.  After four poms, you take a longer break (15 minutes).  DO NOT SKIP BREAKS.  The breaks let you work longer and feel more refreshed and ready- in my experience, if I skipped breaks then I’d do a pom or two less that day.  The timer is great!  I use a free app on my phone as the timer.  I also go one step further and put my computer on airplane mode for the poms when I don’t need the internet.  Speaking of which…
  4. Consider turning off the internet.  I was always getting stuck on a thing, and then getting frustrated, and then checking slate or gawker or national review or reason or twitter or facebook and reading an article or five before going back to the task at hand.  I’m pretty distraction-prone, so turning my computer to airplane mode and putting my phone away helped a lot during the retreat.  In regular life I’ve been setting computer to airplane mode and putting my phone on a shelf after setting the timer (I still need to pick up if daycare or nanny calls).
  5. Keep a master task list for the project/week/month.  We made an “activity inventory” of the larger goals we wanted to accomplish over the week.  Then at the end of each day once I had finished my tasks for the day and was looking to the next day, I’d refer to the activity inventory and cross off the major goals and see what was coming up to break into tasks for the next day.

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    Goals to finish this project, crossed off as I accomplished them.  Or, funnily, — if they stopped being necessary.

  6. Keep track of distractions.  The Pomodoro technique recommends putting an apostrophe in your list to show when distractions happen.  I did not do that, but I do make liberal use of the “notes” section at the bottom of the ETP sheet above, or a side notebook, just to sketch a few notes about ideas that cropped up.  Also, if I got hit by the math muse, I’d run with it and write it down as a new task with little time bubbles (I believe in staying flexible!)
  7. Various one-time techniques: pre-hindsight: think about a time you didn’t achieve a goal, and try to figure out what would have helped you achieve it.  Then try to implement those tools for success in future goals.  Put yourself first: spend the best part of the day (the time you’re most awake and aware) on the work that matters to you, and then deal with other peoples’ needs.  Take breaks: “You don’t realize you need a break until you’re fatigued, and by the time you’re fatigued it’s too late (to do more good work”-Shanna Benjamin, our amazing facilitator.

Good luck with all your work, blog readers!  I think this is also useful for non-solo workers, but it’s harder to keep track of because there will be other people and other schedules involved.  I did meet with a fellow grad student and we pom’d together, including a 25 minute conversation we had trying to figure something out.  Good luck to all of us!

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