Bott is often quoted as saying "Evrey mathematician has a secret weapon. Mine is Morse theory." Every time I see his name I've tried to look up a source. Finally found one:
Slightly off topic (since there's already been so many better answers than I can write), but our knowledge of the dino-slaying asteroid came much later: https://en.wikipedia.org/wiki/Alvarez_hypothesis
I really do think the incompleteness theorems deserve the attention they get, not just because of what they say about efforts to formalize mathematics and because of the historical context -- remember Gödel numbers came (just) before Turing and the first recognizably modern electronic computers. That numbers can represent things that are not numbers was (IMO) a revolutionary idea.
Having said all that, I'd taken mathematical logic in college to learn about incompletenss, but the most interesting things I got out of it were completeness and compactness. Non-standard models really can be quite interesting.
In addition to the above links on quasicrystals etc, it may help to have a bit of context on periodic tilings, which have very precise mathematical properties: https://en.wikipedia.org/wiki/Lattice_(group).
I had an even older iPad I was happily using for similar use cases. Until one day a family member bricked it and I needed to factory reset. No big deal, I thought -- nothing important on it. Turns out it needed to phone home to do the factory reset, and since the server it wanted to talk to was no longer up (or perhaps the address changed?) I couldn't factory reset the iPad.
If someone has a work-around I'd love to hear it. Until then, or until Apple changes this design, I think I'm done with iPads. I don't want to pay that much to "own" something that Apple can simply make obsolete by reconfiguring or turning off a server somewhere.
In my couple decades as an academic mathematician I've only ever met one. He was a strong advocate, and got me to install & try it, but I could never convert to using it fulltime.
Yup. Back in my day there was 1.00, a Civil Engineering course, a pretty standard intro to programming in plain old C. I don't know if it still exists. There was nothing of that sort in EECS, though there are lots of IAP courses (which take place in January, before spring semester starts). IMO a month is about right to spend on (leisurely) picking up a programming language for fun. A friend and I learned APL that way.
I have been using mu4e for years, and am generally happy with it, and yet... I've never recommended it to anyone else. Unlike, say, org-mode or magit, which I'd happily evangelize.
The pain points are what other commenters have said:
- I don't find the default config a good fit for me, and run it heavily customized. As someone said everything in Emacs turns into a project...
- Performance can be an issue, especially indexing new mail (and especially if you like to lug around a copy of most of your emails locally as I do). On a laptop while traveling this used to be more of a problem, but newer versions are notieably quicker and newer laptops have better battery life.
- HTML rendering isn't great. Thankfully I don't get too many important messages that isn't just plain text. This might be a reasonable use case for xwidget-webkit though I'd imagine there are security/privacy issues to work out. (Another Emacs project -- yay!)
When I started I thought it would be an efficient way to get through lots of emails, and it has been for the most part. I'm just not sure I've saved time overall unless one counts the hours configuring it as "entertainment / hobby" rather than "work".
I don't question this decision is sometimes (often) driven by the need to increase publication count. (Which, in turn, happens because people find it esaier to count papers than read them.) But there is a counterpoint here, which is that if you write say a 50-pager (not super common but also not unusual in my area, applied math) and spread several interesting results throughout, odds are good many things in the middle will never see the light of day. Of course one can organize the paper in a way to try to mitigate the effects of this, but sometimes it is better and cleaner to break a long paper into shorter pieces that people can actually digest.
- Differenting a function composed of simpler pieces always "converges" (the process terminates). One just applies the chain rule. Among other things, this is why automatic differentiation is a thing.
- If you have an analytic function (a function expressible locally as a power series), a surprisingly useful trick is to turn differentiation into integration via the Cauchy integral formula. Provided a good contour can be found, this gives a nice way to evaluate derivatives numerically.
Thank you so much for all this detail. This is very interesting & quite helpful, and it's great you were able to communicate all this with your friend.
For anyone else interested: I wanted to be able to typeset mathematics (actual formulas) for the students that's as automated as possible. There are 1 or 2 commercial products that can typeset math in Braille (I can't remember the names but can look them up) but not priced for individual use. My university had a license to one of them but only for their own use (duh) and they did not have the staff to dedicate to my students (double duh).
My eventual solution was to compile latex to html, which the students could use with a screen reader. But screen readers were not fully reliable, and very, very slow to use (compared to Braille), making homework and exams take much longer than they need to. I also couldn't include figures this way. I looked around but did not find an easy open source solution for converting documents to Braille. It would be fantastic to be able to do this, formulas and figures included, but I would've been very happy with just the formulas. (This was single variable calculus; I shudder to think what teaching vector calc would have been like.)
FYI Our external vendor was able to convert figures to printed Braille, but I imagine that's a labor intensive process.
Partway through the term we found funding for dedicated "learning assistants" (an undergraduate student who came to class and helped explain what's going on, and also met with the students outside of class). This, as much or more than any tech, was probably the single most imapctful thing.