Isomorphism Pedagogy


From this post of Ravi Vakil’s blog:


I’ve gradually come around to the idea that when learning about some category for the first time, the notion of isomorphism is pedagogically prior to the notion of morphism. I won’t argue about whether it is logically prior; that’s not my point. When learning groups, students first propose the notion of isomorphism (as they figure out what they mean by the intuition of two groups being the same) before the notion of how you map from one group to another. With schemes too isomorphisms come first.


In mathematics notation, we have a symbol for “is isomorphic to” (\cong). But this is the wrong notion in general: we need a symbol for “a map that is an isomorphism”. We already have a reasonable answer: a right arrow with a \sim on top. But might it be nicer to make it look slightly different, and have a symbol that looks like a \cong, where the bottom is a rightarrow? I like this idea because although it is (slightly) new notation, it is patently clear what it means, and also useful.

Love letter to Mathapedia

This post is

  • a list of “latex on the internet” links that I organized recently, and 
  • a bit of a love letter to

Most important is the following: go to the sandbox here, click on “load random sketch” until you get the following


Then play.

Misc awesomeness:

  • you can click (or touch on mobile) and slide the top graphic horizontally. The bottom one you can even slide vertically. 
  • This would be super useful for teaching the 2nd fundamental theorem of calculus. I am going to play around with not just having a slider, but having it add up the area as one slides.
  • this is device independent, in that it works on phones and tablets
  • you can feed this latex and pstricks
  • watch the short demo video on their frontpage

More thoughts:

  • I use latex, and in particular plain text, more than anything. I find it frustrating when people develop teaching tools (e.g., cool 3d images for multivariable calculus) in some other language that I don’t know and hence can’t modify, or have to pay money to use some teaching tool before I know that it is useful and easy to use. I have certainly found many instances where there is a visualization that I like but isn’t quite right for what I’m doing. (Mathematica and its vector fields/flow implementations [while awesome] is one example.)
  • It is actually kind of a pain in the ass to use the projector mid lecture in a classroom. And my department is a joint math and computer science department. 4 minutes to warm up the projector! I fantasize about my students having demo’s pre-loaded on their phones. 
  • would mathoverflow users find this useful? We already run mathjax, and this runs on top of that. Joseph O’Rourke famously includes lots of amazing visualizations (e.g) in his questions and answers. 

List of “latex on the internet” links

  • Writelatex 
  • Sharelatex (From last summer’s REU students, rapid preview. They used it for a few weeks, but at the time was too buggy and they went back to just using dropbox and resolving conflicts when they came up. Haven’t tried it since.)
  • mathim — math chat. I used to use this while skyping with my advisor Bjorn Poonen
  • mathML 
  • a stackoverflow post about alternatives 
  • jqmath 
  • The original mathoverflow latex hack is pretty great – we had a Javascript header that would send text between $$ to Scott Morrison’s server, compile it, and replace the text between the $$ with a link to the gif on his server. 
  • Mathjax. Peter Krautzberger, who runs (which is even better than my elaborate system of folders of RSS feeds), works on this too. Mathoverflow (and stackexchange in general) use mathjax.
  • A recent AMS Notices article about latex on iPad. I haven’t tried their setup. I’ve been using TexTouch, usually via a bluetooth keyboard at coffee shops and airports. It has a nice discussion about syncing. I use dropbox which does 85% of what I want it to do.
  • There are a bunch of profhacker articles about latex. Here’s one.
  • Prezi, a ridiculous and awesome alternative to giving beamer talks. This talk by Christopher Schommer-Pries has fueled plenty of my alt-beamer fantasies.

I compiled this list while thinking about mathapedia and a conversation with Cyrus Radfar, founder of  Kapuno. (I met Cyrus at ScienceOnline, kept in touch through twitter, and when he remembered that I was a mathematician he told me about mathapedia.)

Pink eye and automation


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My wife and I have a new tradition in Atlanta — every March, there is a marathon on our street. We wake up early (for us) and, rather than compete, we make mimosas, a sugary breakfast, watch the marathon from our southern porch, and do our taxes.


This year though I have pink eye. So no Mimosas. My left eye basically constantly feels like I’ve had the same contact in for several days. This is the 3rd time I’ve had pink eye. The other two times were more epic:

  1. woke up with pink eye on sun, sat, had 3 math finals on mon
  2. red eye back from our honeymoon, woke up with pink eye, spent the next week moving into our new place in Atlanta.

I’ve always liked how Anton geeked out when he had a blood clot and made a detailed web page about it. So I’m going to do the same. (Also, at this rate I’ll probably get pink eye another 6 times in my life, I may as well learn everything about it.)

Pink eye (conjunctivitis):

  1. Wikipedia 
  2. Webmd 
  3. CDC 
  4. Pubmed

Actually, there’s not that much to say about pink eye:

  1. Usually heals naturally in 7-10 days.
  2. When you wake up your eyes are glued shut
  3. Try not to infect the other eye.

One good tip for 3 — wipe from your nose to the outside of your face.  And in general, there’s something very satisfying about a midday, 3-hour, feverish nap.

The downside is that I’m often not quite lucid enough to do math. (And for instance, this blog post was written in a non-lucid state and edited later. Its like Hemingway said “Write drunk, edit sober.”) When I’m sick I usually pick up some free or cheap flash or iPhone game and play it. (Usually from one of toucharcade’s best of lists.)

Coincidentally, this weekend, Nimblebit was demoing their game NimbleQuest,

This was fun, but to really rock it you need to spend a lot of time leveling up. Which I don’t really have. There is *some* time lucid enough to do math, but I didn’t want that time to go to waste game-wise.


So I cheated — I wrote an apple script that would repeatedly walk my characters in a circle, collecting gems and whatnot, using applescript.

What is applescript
Basically its a high level language for Macs that lets you automate things. Usually I use Bash for this, but if I need to do something like click a particular spot on my screen, or resize windows, I use applescript.

Here’s a good description of applescript from Profhacker. Also, Profhacker is pretty cool, check that out too.

The following script just makes my game character walk in a rectangle (he attacks on his own), and at the end of every walk hits “return” three times, so that, if the character dies he can restart.

activate application "Google Chrome"
delay 0.1

tell application "System Events"
repeat 3 times
keystroke return
delay 0.1
end repeat
end tell
delay 0.575

tell application "System Events"
keystroke "w"
end tell
delay 3
tell application "System Events"
keystroke "d"
end tell
delay 1
tell application "System Events"
keystroke "s"
end tell
delay 3
tell application "System Events"
keystroke "a"
end tell

end repeat

To tell the truth this is almost as much fun as actually playing the game… So far the game seems to make about 250 diamonds an hour.

Arizona winter school 2013

I’m a day late with my weekly blog post because I’m at the 2013 Arizona winter school! I go every year (only missed one since 2005) since I’m from Tucson, and it’s as awesome as ever, with a record 232 participants (and only 234 seats in their auditorium).

So, short post today though — inspired by the twitter-addicted-throng of science-y types at ScienceOnline, I’ve started tweeting again. (I used to just tweet awesome extreme metal videos, but that got old.) I haven’t quite found my twitter groove, but have had quite a bit of fun livetweeting the #AWS (see also #AWS2013).

I haven’t been able to get too many others to tweet with this hashtag (I wasn’t quite ballsy enough to write the hashtag on their whiteboard) but its already been quite rewarding. The biggest payoff is that my tweets revealed to my postdoc mentor JSE that the fun he was missing and he impulsively (on the first day of the conference!) bout a ticket to Tucson.

Also, I thought it would be a bit awkward tweeting a math conference, but it turns out that it feels like an extension (or maybe subset) of my usual notetaking. I tend to not write down everything the speaker writes, but rather write down new insights, new project ideas, or funny quotes, a style which ports well to twitter.

Other fun —

  1. my cousin thought my twitter account was hacked (then remembered that I do math), and 
  2. my new friend Kris thought that the following picture was an “infinity omelet”


I’m so used to this picture being a “multi-holed donut” that I never made the connection to breakfasty awesomeness.

Overall opinion of tweeting a conference: 5/5, will tweet again.

Prym Varieties

(This post is incomplete and will later be proofread, filled in, etc.)
Today, new sunday routine, I search my “” file for lines containing the word “blog” and found this:

“brain dumps, of things I’ve learned recently and lots of fun facts (e.g., Pryms)”

So who is this blog’s target audience? Probably my own grad students (and, well, me at a later date). But non math people read this, and some senior math people too. What to do? Today for instance I’m going to write about Prym varieties. What can a non-math audience get out of a post like this? I’ll at least say that this encourages me to try to find pictures, facts, ect that I would not usually bother with.

Cool degenerations of curves

Another line says “blog post, cool degenerations of curves and more”, with a bunch of examples. Here’s a degenerate prym. (The top right curve is two copies of the left curve, but with “P of the first curve glued to Q of the second”.)
Two neat things about this picture
1) it is an etale double cover
2) The associated Prym variety is the Jacobian of the base curve

This second fact is key; it says that the Jacobian locus is in the closure of the Prym locus.

So what is a Prym variety?

The short answer is — it is (more or less) the only way that I know of to write down abelian varieties which are not Jacobians. I asked about this on Mathoverflow ages ago (examples of rational families of abelian varieties) and Prym varieties were (for me) the most managable example.
So how does one write down an abelian variety A that isn’t a Jacobian J? Well,

– A is dominated by a Jacobian (in particular, the Jacobian of some curve lying on A) so it makes sense to look for A’s in J’s.
– A map of curve C –> C’ induces a map of Jacobians J_C’ –> J_C, and the quotient A is an abelian variety that is often not a Jacobian
– This construction might not give you a principally polarized abelian variety.
– Geometric class field theory tells that two torsion points on J_C give etale double covers of C. This is (almost) the only case where one gets a PPVA.

Numerology of Prym’s
OK, the real reason I wanted to write this post is to collect all of the cool stuff I’ve learned about Pryms.

– Hyperelliptic begets hyperelliptic. In fact, this is an easy, explicit exercise (say, blay, in blay’s book):
– Trigonal begets tetragonal. Lies deeper. Dogani’s trigonal construction.
– Dimension counting. Call R_g the space of prym’s of dimension g. Then there is a map M_{g+1} –> R_g. It is a classical fact that this map is generically finite, with non-finite fibers coming from Dogani’s construction.
– Tetragonal usually begets a non-Jacobian.
– Tetragonal construction (characterize the tetragonal which do give Jacobians).
– Nodal degeneration trick. Keyword — Beauville admissible cover?

* Questions:

– Is there a good notion of Prym’s for graphs, or in tropical geometry? My first thought is no, but after Farbod S’s talk last week (reminding me of the emphasis on discrete graphs as models for metric graphs) maybe?

Beer, economies of scale, and emacs macros


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This is really a post about how awesome emacs is. To start, though, something sexier than emacs:


Belgian IPA 02.13.2013

Beer! I like to brew. My wife and I brewed 250 bottles (an amber and a Belgian witbier) for our wedding. Now that I have a house, I can do things like buy bulk grains and hops. But which hops to buy? What grain do I want 50lbs of?

So, I went to this awesome database of brewing recipes. I want to learn to make a solid Belgian Dubbel, and would like to see what the most common recipes are. I want to read all of the recipes labeled eg (extract-grain) and see what they do. I also don’t want to spend 10 minutes finding and opening the 48 eg links on this page (out of hundreds of links).

Now, I know this sounds trivial, but listen — this can either be a tedious exercise, full of right clicking, eyeballing, and RSI escalation. Or it can be a fun logic puzzle.


Macros. In my emacs setup, if I hit F3, it records my keystrokes. Hitting F4 stops the recording, and hitting F4 again repeats the recorded sequence of keystrokes.

Applications abound.

1) Line breaks. One of my pet peeves is the fact that gmail (and also most of my collaborator’s text editors) take normal text (one continuous line, formatted for your particular page)


and deform it into something like this (artificial line breaks; looks terrible if you scale)


I’m nerotic about these things, but don’t expect my collaborators to be. (But this blog’s audience? Compatible neuroses?)

Also — not something that a find-and-replace can fix. (Though query-replace with emacs, noting that C-q C-j inserts a linebreak, works well).

Basically what I did was the following sequence of keystrokes (on a mac):

– go to end of line (Cmd-right)
– go to next line (right)
– hit backspace (or delete)
– hit space.

One nice feature of this is that having it as a hotkey lets you decide when to use it. I basically want to use it for each paragraph, but not for, e.g., code or displayed math.

2) Beer! Back to beer, and a second screencast.

— view source (right click-select view source)
— copy to emacs
— start macro (f3)
– search for eg (C-s return; eg)
– move to front of line (Cmd-left)
– go up two lines
– highlight everything above this () or ()
– delete
– search for eg again
– go to end of line (Cmd-right)
– highlight (Cmd-Shift-left)
– cut (Cmd-x)
– paste at end of file (Cmd-down, paste)
– move to top of file (Cmd-up)
— end macro (f4)

Now I just hold F4 for ten seconds and I’m done.

Once I get a list of urls, I put “open” in front of each line (via macros or query-replace w/ C-q C-j), copy, and paste into a terminal and they all open as separate tabs in a browser. I can quickly browse them now.

(If I want to automate more, instead of opening, I can curl the url, write to a file, and do the same kind of macro as before to get the piece of the table with grain, hops, etc and get a big list of data to study.)

Epilogue. I’m not a trained programmer. MAGMA taught me most of what I `know’. I like coding bash scripts to do things and generally finding better ways to automate repetitive tasks in my life. Programmers might cringe at my use of macros (“Seriously? Just use regular expressions…”) But for a non-programmer, recording a series of keystrokes is a fun logic problem, while regular expressions requires reading a manual.

As a mathematician, in my idle time, I almost always want to do the first, and almost never want to do the second.

The LMFDB (L-functions, modular forms, and related objects database)

John Voight visited Emory last week and introduced me to the crazy LMBDB (L-functions, modular forms, and related objects database). (He also introduced me to the awesomeness of Fucsian groups.)


Contrast with this quote by Borcherds:

If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.

(From this great post by Scott Carnahan about Lafforgue’s work.)

Technically, these aren’t arrows between areas of math, but between mathematical objects, so its legit. Fun features:

1) If you hover over an arrow it tells you what the connection is.
2) The biggest selling point for me (and this came up in conversation with John, he wasn’t just gloating about his involvement in this awesome project) is the computations over number fields other than Q.
3) If you go to the page of a particular elliptic curve, it returns the isogeny graph. (Featuring conductors up to 299998!)
4) On a page for an L-function, it will even verify the Riemann Hypothesis. (Scoll to bottom.) It computes this in real time too. You can see a little, almost invisible progress bar in the top right corner.
5) I generally really like super hyperlinked mathematical things. (E.g. I LOVE the stacks project and its recent interface update.)