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.
This post is
- a list of “latex on the internet” links that I organized recently, and
- a bit of a love letter to mathapedia.com
Most important is the following: go to the mathapedia.com sandbox here, click on “load random sketch” until you get the following
- 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
- 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
- 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
- a stackoverflow post about alternatives
- Mathjax. Peter Krautzberger, who runs mathblogging.org (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.)