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.

Spindle_diagram

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.

octant

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!

I am a minority in academia

5 Jul

I started trying to write this post and ended up looking at SO MANY articles and thinkpieces related to academia, minorities, affirmative action, high school, independent/charter schools, microaggressions, and interventions.  This topic is way too complicated for my humble little corner of the internet to take on to any kind of depth, so I’ll just talk about my experiences instead and maybe put in a few links.

I’ve written a lot about being a woman in mathematics (see first post, second post, nth post), and a little bit about race and a bit about the intersection (but really that link is nationality and gender).  But I haven’t written too much about being a Vietnamese person in mathematics.  Part of it is that Vietnam is in Asia, so I’m Asian-American, and the stereotype is that we’re good at math and doing great in academia since you can see a bunch of Asian professors in math.  Most of those Asian professors are Asians-from-Asia which is different than Asian-Americans.  Asians-from-Asia face a whole different experience and set of difficulties than Asian-Americans.  A quote from Americanah, by Chimamanda Ngozi Adichie, somewhat hits this:

The only reason you say that race was not an issue is because you wish it was not. We all wish it was not. But it’s a lie. I came from a country where race was not an issue; I did not think of myself as black and I only became black when I came to America.

So that’s point 1: Asians-from-Asia and Asian-Americans are different.  For instance, Asian-Americans are part of a system of structural racism, while Asians-from-Asia can encounter this system but not have the same understanding and possibilities for complicity/empowerment/action as Asian-Americans.  There’s even a term in Vietnamese for us, Viet Kieu: “Foreign but not foreign, Vietnamese but not Vietnamese.” College friend’s blog post on being Viet Kieu.

Point 2: “Asian-American” is also an unnecessarily general term and erases the difficulties that communities of people from very different nations and backgrounds have with their relationships with the US.  I saw a link recently that I can’t find saying that “Asian-American” as a term is going out with the next census and it’ll actually break us down into parts of Asia instead of, you know, people with ancestry from the world’s largest continent.  Anyway, because of a variety of historical factors (stuff like the Chinese Exclusion Act and racism and xenophobia), lots of [East and South] Asians who have immigrated to the US are highly skilled H1-B visa holders.

Heard of the “model minority” thing?  Maybe it’s because children of doctors, lawyers, and engineers are more likely to know about how to become doctors, lawyers, and engineers.  Versus, say, if you take a swath of the general population of a country and plop them in a new country, you’ll probably get the same percentage of highly skilled workers in that swath as you do in the new country.  Probably less because the certifications of the old country aren’t valid in the new country.  You guessed it, I’m talking about Vietnamese-Americans!  There are many successful Southeast Asian-Americans, but numbers wise, we’re worse off than lots of other ethnic groups.  For instance, we drop out of high school!  Even within Asian-Americans you can compare numbers: Vietnamese poverty rate around 15%, and Filipinos around 6% (national average about 14%).

Point 3: Data time!  Especially in light of the Yale thing last year and all the stuff about diversity in academia, let’s just look at some research about diversity in academia.  There’s a Consortium on Race, Gender, and Ethnicity collecting and organizing great work on how to diversify academia.  This table is from their FAQ page:

Table 1. Distribution of Full Time U.S. Faculty, by Race/Ethnicity (1988-2010)

I’m not a big data head, but we can compare the numbers above with the census numbers in 2010:  5% Asian, 13% black, 1% Native American/Alaskan Native, 16% Hispanic, 72% white.  It doesn’t add up right because the census counts Hispanic as ethnicity, not race, but the table above doesn’t or something (also I rounded).  It looks like Asians are doing okay!  And then you remember my point 1, and the data doesn’t differentiate between Asia-Asians and Asian-Americans.  So it’s unclear what’s happening, but it’s pretty clear that programs like SACNAS are necessary (and other programs that target, for instance, African Americans and not-science).  I’ll mention here that I’m a Mellon Fellow and so have benefited from a program that specifically targets these issues.  And that leads us to…

Point 4: what to do.  The American Psychological Association put together a HUGE pamphlet on “Surviving and Thriving in Academia.”  That’s great!  There was an article in Science in 2011 about a simple intervention that helped minority students succeed in college: basically it fought off stereotype threat by saying that everyone struggles with feeling like they belong in the first year of college, and then it fought off victim-ing (I don’t know the word for telling people that they are victims and need help) by having the students make videos for future students telling them that everyone struggles with feeling like they belong in the first year of college.  It’s pretty cool!  I loved this blog post about a psych Ph.D’s experiences with racism here in Austin.  Quote from it:

Being black isn’t hard; being black is awesome. It’s being the subject of discrimination that is hard, and that is a fight we can all fight together.

Okay I lied and only wrote commentary on a whole bunch of links instead of a memoir of my own experiences.  I feel much more engaged with the issue of being a woman in math than with the issue of being a racial minority in math, but I also think both of these things are very important to my identity.  So there’s that!  I’ll maybe say more about intersectionality in another blog post.

Barely related to the content of this post: here’s a video that I watched a few months ago and LOVED.  Get past the cheesy weird preview screen and production by MTV and there’s a surprising about of history, data, and analysis in this.

Minimum rank of graphs with loops

28 Jun

A few months ago I was one of two Distinguished Graduate Speakers at this awesome conference, Underrepresented Students in Topology and Algebra Research Symposium (USTARS).  The conference started in 2011, when a bunch of advanced grad students and postdocs (I don’t think anyone on the committee was a professor yet) decided that there needed to be a place for underrepresented students to see other people like them doing the math they did.  And now they’re all professors and still organizing this traveling conference!  So many people of color!  So many women!  So many homosexual people!  (Not so many trans people…) So many first-generation college students!  So great!  Too many slides!  (I pretty actively hate slide talks unless there are pictures that are impossible to make on a chalk board.)

YenTalk

Credit: Erik Insko for this picture of me giving a talk at USTARS 2016!

Anyway, I wanted to blog about some math I saw there, based on a talk by Chassidy Bozeman, a grad student at Iowa State and a fellow EDGE-r.  The talk is from a paper that resulted from her graduate research course, which Iowa offers to intermediate grad students and gets them started on research (hence the eight authors on that paper).  I thought it was fun and approachable!  And I have one crumpled page of two-month old notes, so we’ll see how this goes.

First, let’s remember what a graph is: a collection of vertices (dots) and edges (lines between dots).  A graph is simple if there’s at most one edge between any two vertices, and if no vertex has an edge to itself (aka a loop).  A loop graph allows loops, but still no multiple edges.

graphs

Left is a simple graph, right is a loop graph

You can associate an infinite family of symmetric matrices to a loop graph.  These square matrices will have the number of vertices of the graph as the number of columns and rows, and the entry a_{ij} will be 0 if and only if there is no edge between the corresponding vertices and  in the graph.  The rest of the entries will just be real nonzero numbers.  This infinite family is useful for defining the minimum rank of a loop graph: it’s the minimum rank over all of this infinite family of matrices.  The rank of a matrix is a measure of how large it is.  For definition by example, \left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \end{array} \right) has rank 1, and \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) has rank 2.

So the goal of Chassidy’s talk was to characterize all loop graphs whose minimum rank is the full size of the graph.  A couple more definitions before the theorem: a generalized cycle is a subgraph whose connected components are either a loop, a single edge, or a cycle.  It’s spanning if it touches every vertex in the graph.  Spanning generalized cycles don’t always exist!

generalizecycle

Components of a generalized cycle: loop, edge, cycle

nonexist

No spanning generalized cycle, womp womp

Theorem: A loop graph has minimum rank equal to its size if and only if there exists a unique spanning generalized cycle.

Quick nonexample: here’s a picture of a loop graph that has more than one spanning generalized cycle, so from the theorem we know it doesn’t have minimum rank four.

nonunique

This is theorem 3.1 in their paper.  It’s also comically the end of my notes on this talk.  Here are a few facts they use to prove the theorem:

  • If is an induced subgraph of G (that means the vertices of H are a subset of the vertices of G, and its edges are all the edges that appear in G within that subset), then it minimum rank is bounded above by the minimum rank of G.  Succinctly, mr(H)\leq mr(G).
  • If is a disjoint union of a bunch of connected components, the its minimum rank is the sum of the minimum ranks of those components.  mr(G)=\sum mr(G_i).

Here’s a picture of my notes!  Read the actual paper if you want to know more!

20160628_221615

Subgroup separability problem set session(non-elementary)

22 Jun

Update #1, five hours later: Zach Himes posted a great comment fixing my iffy part, and I also video chatted with Nick Cahill about it and we came up with an alternate solution which is still maybe iffy.  Adding both below.  Nick also added a comment about the second exercise, using group actions.  What do you think?

I’ve read Sageev’s PCMI lecture notes several times by this point; it’s the basis of my and many other people’s work (in particular, my friend Kasia has an impressive number of publications related to this stuff).  And every single time I get stumped on the same exercise near the end, so I thought I’d try to write up a solution, crowd-source it among my blog readers, and figure out something correct.  For reference, these are exercise 4.27 and 4.28 in his notes, but I’ll lay out the problems so you don’t need to look them up if you don’t want to.  Please comment with corrections/ideas!

A thing that mathematicians care about is the structure of a group.  We say that a particular subgroup H<G is separable if for any group element that’s not in H, we can find a finite index subgroup K that contains H but doesn’t contain g.  Intuitively, we can separate from H using a finite index subgroup.  Here’s the cartoon:

output_37OpDk

If H is separable, then given any g not in it, we can find a finite index subgroup that separates H from g.

The first exercise is to show that this definition is implied by another one: that for any group element that’s not in H, we can find a homomorphism f: G\to F where F is a finite group, so that the image of under the map doesn’t contain f(g).

So let’s say we start with such a homomorphism, and our goal is to find a finite index subgroup that contains but not g.  Since we’ve got a homomorphism, let’s use it and try K:=f^{-1}(f(H)).  Since f(g)\not\in f(H), we know this definition of excludes g, as desired.  Then we need to show that K is finite index in G and we’ll be done.

What about the first isomorphism theorem?  We have a map G\to F, and we know f(H)<F, and is a proper subgroup since f(g) isn’t in f(H).  This next bit is iffy and I could use help!  

  1. (Original) Then we have a map G\to F/f(H) induced by the map f, and the kernel of this map is K.  By the first isomorphism theorem, the index of in is the size of the image of this map.  Since F/f(H) is finite, the image of the map is finite.  So has finite index in G, as desired.  [What’s iffy here?  You can’t take quotients with random subgroups, just with normal subgroups, and I don’t see why f(H) would be normal in F unless there’s something I don’t know about finite groups.]
  2. (based on Zach Himes’ comment) By the first isomorphism theorem, ker has finite index in F.  We know ker is contained in K, since 1 is contained in f(H) [since 1 is contained in H, and f(1)=1, where 1 indicates the identity elements of G and F].  It’s a fact that if \ker f \leq K \leq G, then [G: \ker f] = [G:K][K: \ker f].  Since the left hand side is finite, the right hand side is also finite, which means that K has finite index in G, as desired.
  3. (Based on conversation with Nick Cahill) We can think of F/f(H) as a set which is not necessarily a group, and say that G acts on this set by (g, x) \mapsto f(g)x.  Then K=Stab(1):=Stab(f(H)).  By the orbit-stabilizer theorem, [G:K] = |Orb(1)|.  Since F is finite, the size of the orbit must be finite, so K has finite index in G, as desired.

The second exercise has to do with the profinite topology.  Basic open sets in the profinite topology of a group are finite index subgroups and their cosets.  For instance, in the integers, 2\mathbb{Z}, 2\mathbb{Z}+1 are both open sets in the profinite topology.  Being closed in the profinite topology is equivalent to being a separable subgroup (this is the second exercise).

So we have to do both directions.  First, assume we have a separable subgroup H.  We want to show that the complement of is open in the profinite topology.  Choose a in the complement of H.  By separability, there exists a finite index subgroup that contains and not g.  Then there’s a coset tK of which contains g.  This coset is a basic open set, so is contained in a basic open set and the complement of is open.

Next, assume that is closed in the profinite topology, so we want to show that is separable.  Again, choose some in the complement of H. Since the complement of is open, is contained in a coset of a finite index subgroup, so that is not in this coset.  Let’s call this coset K, and call its finite index n.  We can form a map f: G\to S_n, to the symmetric group on letters, which tells us which coset each group element gets mapped to.  Then is in the kernel of this map, since is contained in K, but is not in the kernel of f since it is not in that coset.  In fact no element of H is in the kernel.  So we’ve made a homomorphism to a finite group so that and have disjoint images, which we said implies separability by the previous exercise.

Okay math friends, help me out so I can help out my summernar!  So far in summernar we’ve read these lectures by Sageev and some parts of the Primer on Mapping Class Groups, and I’m curious what people will bring out next.

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