Tweendom approaches

Spent 20 minutes today arguing with CJ over which is better, the Black Eyed Peas “I Gotta Feeling” or Carly Rae Jepsen’s “Good Time” (feat. Owl City.)  Caleb favors Jepsen, arguing that songs are made of “music, singing, and words,” and that “I Gotta Feeling” wins on words but loses on music and singing.  His judgment of “I Gotta Feeling” as a piece of music is that “the music doesn’t match the singing and 3/4 of the music is copied and the 1/4 of the music that isn’t copied is boring.”

I asked CJ what “Good Time” is about and he said “it’s about people who overestimate their life and think bad things never happen in it.”

 

What I looked like when I was 17 and talking about math

My mom just sent me this, from the 1989 Westinghouse (now Intel) science fair. I don’t usually think CJ looks very much like me, but in this picture I can kind of see it.

Persiflage on Scholze

Like everyone else I am wildly cheering Peter Scholze’s new preprint constructing Galois representations attached to torsion classes — torsion classes! — in the cohomology of locally symmetric spaces for GL_n.  I had been aspiring, and still do aspire, to develop enough of a global picture of how this works to write about it on the blog.  But I’m happy to report that it looks like Persiflage, who’s somewhat closer to the subject than I am, is going to do it at his place.  In his words:

This is mathematics which will, no question, have more impact in number theory than any recent paper I can think of. The basic intent of this post is to commit to future posts in which I will discuss the details.

At the risk of talking about stuff I dont understand yet, I’ll make one comment.  It seems that a key technical development is Scholze’s ability to use the language of perfectoid spaces to talk about things like modular curves and modular varieties “at infinite level.”  See how I reflexively put scare quotes there?  It’s because, when I learned this stuff, it was customary to pretend to talk about infinite level,  but really this was used as more of a metaphor; every actual argument I knew how to make took place in the pedestrian context of schemes of finite type over local and global fields.  (Others may have been more daring, I don’t know.)  Anyway, Scholze’s techniques seem to allow him to work fearlessly at the top of the tower, no scare quotes necessary, at which point new phenomena appear, phenomena which have implications even back at finite level.

(I am eager for this preliminary stuff to be corrected, refined, rebuked, and improved on in comments….!)

 

 

Interview with DeMarco and Wilkinson

Nice joint interview with Laura DeMarco and Amie Wilkerson at Scientific American.

I didn’t know this about Amie:

 I went to college, and I was feeling very insecure about my abilities in mathematics, and I hadn’t gotten a lot of encouragement, and I wasn’t really sure this was what I wanted to do, so I didn’t apply to grad school. I came back home to Chicago, and I got a job as an actuary. I enjoyed my work, but I started to feel like there was a hole in my existence. There was something missing. I realized that suddenly my universe had become finite. Anything I had to learn for this job, I could learn eventually. I could easily see the limits of this job, and I realized that with math there were so many things I could imagine that I would never know. That’s why I wanted to go back and do math.

This was basically me, too.  After college I got into the fiction writing program at Johns Hopkins, which made me think maybe I could really make it as a writer, and I deferred grad school and moved to Baltimore and wrote fiction all day every day for a year, and while I valued that experience a lot, there was not a single day of it where I didn’t kind of wish I were doing math.  Having had that experience — not just suspecting but knowing how annoying it is not to be doing math — took the edge off the pain of the painful parts of grad school.

AB notes June 2013

• Last night AB requested for her bedtime songs “America The Beautiful” (“the mountain song”) and “Sit Down, You’re Rocking The Boat” (“the boat song.”)  This makes me feel her musical taste is developing appropriately.  If I remember correctly, CJ’s go-to bedtime songs at the same age were “Here Comes Your Man” and “New York, New York.”
• For a while, AB has been using “they” in the objective, as in “I want to see they.”  Now this has spread to “he” and “she” as well.  Also, she substitutes “to” for “for” in constructions like “save it to me” and “wait to me.”  That makes sense, actually — what’s the explanation, if any, for “give it to me” vs. “save it for me”?
• After we went to the Brewers game she told me she wants to be a baseball player when she grows up. 

Is philosophy worse for women than math is?

My philosopher friends today are all talking about the resignation/firing of Colin McGinn, a pretty well-known philosopher as I understand it, who as it turns out has been sending e-mails to his graduate students describing…. well, there’s no real reason for me to describe it, I leave that kind of filth for the Chronicle of Higher Education.  

Philosophy and math have roughly the same male-female ratio, but philosophy has blogs like What Is It Like To Be A Woman In Philosophy? and math, as far as I know, does not.  Is that because math has actually created a culture friendlier to women than philosophy has?  Or is it because philosophy is closer to the social criticism tradition and philosophers are more likely to want to talk about these things openly?

I have one small data point.  I once heard a philosopher give a talk in which there was a weird joke about you have to be careful not to sleep with your graduate students because [some philosophy joke I didn't get and don't remember.]

Or rather, it read as weird to me, because I think it’s highly unlikely that someone would say something like that in front of a roomful of mathematicians under any circumstances.  Or if they did, there would be a burst of murmurs and everyone would be looking back and forth with the “Did he say that?” look.  On this occasion, only I was looking back and forth.  Nobody seemed to think it was weird, not the women, not the men.  It was an informal, jokey kind of talk.  But still.

 

 

In which I have a quarter-million friends of friends on Facebook

One of the privacy options Facebook allows is “restrict to friends of friends.”  I was discussing with Tom Scocca the question of how many people this actually amounts to.  FB doesn’t seem to offer an easy way to get a definitive accounting, so I decided to use the new Facebook Graph Search to make a quick and dirty estimate.  If you ask it to show you all the friends of your friends, it just tells you that there are more than 1000, but doesn’t supply an exact number.  If you want a count, you have to ask it something more specific, like “How many friends of my friends are named Constance?”

In my case, the answer is 25.

So what does that mean?  Well, according to the amazing NameVoyager, between 100 and 300 babies per million are named Constance, at least in the birthdate range that contains most of Facebook’s user base and, I expect, most of my friends-of-friends (herafter, FoFs) as well.  So under the assumption that my FoFs are as likely as the average American to be named Constance, there should be between 85,000 and 250,000 FoFs.

That assumption is massively unlikely, of course; name choices have strong correlations with geography, ethnicity, and socioeconomic thingamabobs.  But you can just do this redundantly to get a sense of what’s going on.  59 of my FoFs are named Marianne, a name whose frequency ranges from 150-300 parts per million; that suggests a FoF range of about 200-400K.

I did this for a few names (50 Geralds, 18 Charitys (Charities??)) and the overlaps of the ranges seemed to hump at around 250,000, so that’s my vague estimate for the number.

Bu then I remembered that there was actually a paper about this on the arXiv, “The Anatomy of the Facebook Graph,” by Ugander, Karrer, Backstrom, and Marlow, which studies exactly this question.  They found something which is, to me, rather surprising; that the number of FoFs grows approximately linearly in the number of friends.  The appropriate coefficients have surely changed since 2011, but they get a good fit with

#FoF = 355(#friends) – 15057.

For me, with 680 friends, that’s 226,343.  Good fit!

This 2012 study from Pew (on which Marlow is also an author) studies a sample in which the respondents had a mean 245 Facebook friends, and finds that the mean number of FoFs was 156,569.  Interestingly, the linear model from the earlier paper gives only 72,000, though to my eye it looks like 245 is well within the range where the fit to the line is very good.

The math question this suggests:  in the various random-graph models that people like to use to study social networks, what is the mean size of the 2-neighborhood of x (i.e. the number of FoFs) conditional on x having degree k?  Is it ever linear in k, or approximately linear over some large range of k?

Natural logs and products of no primes

The e-mail you get after you write an article about number theory is very interesting.  For one thing, you’re reminded of phrasings which have one meaning among mathematicians, but a slightly different one outside the tribe.

The majority of the e-mail I’ve gotten about the bounded gaps piece concerns two questions of this kind:  I’ll answer them both here, in case other readers are following the link from Slate to the blog.

Q:  You say that the number of primes less than X is about X/log(X), but don’t you mean X/ln(X)?

A:  When mathematicians say “log” we mean the natural log, the thing which in some other contexts (e.g. Google’s search bar calculator) is denoted “ln.”  But mathematicians never say “ln.”  (To be honest, we kind of think the base-10 logarithm should be called “lu.”)

Q:  You say that every positive number is the product of primes, but this is not true for prime numbers themselves, which can’t be expressed as products.

A:  A prime number is indeed the product of prime numbers!  It is the product of just one prime number, itself.

What about 1?  It’s the product of zero prime numbers.

 

Yitang Zhang, bounded gaps, primes as random numbers

In Slate today, I have a piece about Yitang Zhang’s amazing proof of the bounded gaps conjecture.  Actually, very little of the article is about Zhang himself or his proof; I wanted instead to explain why mathematicians believed that bounded gaps (or twin primes) was true in the first place, via Cramér’s heuristic that primes behave like random numbers.

And a lot of twin primes is exactly what number theorists expect to find no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt. If the twin primes conjecture were false, that would be a miracle, requiring that some hitherto unknown force be pushing the primes apart.

10,000 baby names of Harvard

My 20th Harvard reunion book is in hand, offering a social snapshot of a certain educationally (and mostly financially) elite slice of the US population.

Here is what Harvard alums name their kids.  These are chosen by alphabetical order of surname from one segment of the book.  Most of these children are born between 2003 and the present.  They are grouped by family.

Molly, Danielle

Zachary, Zoe, Alex

Elias, Ella, Irena

Sawyer, Luke

Peyton, Aiden

Richard, Sonya

Grayson, Parker, Saya

Yoomi, Dae-il

Io, Pico, Daphne

Lucine, Mayri

Matthew, Christopher

Richard, Annalise, Ryan

Jackson

Christopher, Sarah, Zachary, Claire

Shaiann, Zaccary

Alexandra, Victoria, Arianna, Madeline

Samara

Grace, Luke, Anna

William, Cecilia, Maya

Bode, Tyler

Daniel, Catherine

Alex, Gretchen

Nathan, Spencer, Benjamin

Ezekiel, Jesse

Matthew, Lauren, Ava, Nathan

Samuel, Katherine, Peter, Sophia

Ameri, Charles

Sebastian

Andrew, Zachary, Nathan

Alexander, Gabriella

Liam

Andrew, Nadia

Caroline, Elizabeth

Paul, Andrew

Shania, Tell, Delia

Saxon, Beatrix

Benjamin

Nathan, Lukas, Jacob

Noah, Haydn, Ellyson

Freddie

Leonidas, Cyrus

Isabelle, Emma

Joseph, Theodore

Asha, Sophie, Tejas

Gabriela, Carlos, Sebastian

Brendan, Katherine

Rayne

James, Seeger, Arden

Helena, Freya

Alexandra, Matthew

George

If you saw these names, would you be able to guess roughly what part of the culture they were drawn from?  Are there ways in which the distribution is plainly different from “standard” US naming practice?