With that title and reception I can imagine people bookmarking this for „later“ and feeling good about it. But who reads that stuff really?
To each their own, but 700+ pages for material that is done in my experience in the first 2-3 weeks of undergraduate math is more disheartening than empowering for a student, in my opinion.
If you can open a math book anywhere in the last 20% of pages and just start reading, you are looking at pop science and not lecture notes.
If you are interested in this kind of role, it is usually called Executive Assistant in the corporate world. Had the fortune to support my CEO and attend e.g. all board meetings (because I did the minutes). This is at a >10$ bn revenue established company. Very rewarding, but experience depends very much on the style of your boss.
A map between algebraic curves is defined by polynomials. That the map is defined over K means you can find a coordinate system such that the equations of the curves and the equations of the morphisms have coefficients in K and not some larger ring, eg the complex numbers or a large extension field of F_p(field of p elements)
Good question, no there is no difference for elliptic curves, which you can think of as 1-dimensional geometric objects (curves) which posses group structure. A good map in this category should respect the geometry (be a so called rational map, ie defined by polynomials in a suitable coordinate system) and the group structure. Interestingly all these maps are either constant (map everything to 0) or surjective.
For higher dimensional geometric groups (abelian surfaces etc) one usually wants to make a distinction and calls the surjective homomorphisms with finite preimage isogenies.
Actually, a Banach-Tarski-like result is impossible in 2D space, since there is a Banach measure (= volume definition to all subsets of the plane) that extends the usual volume definition (e.g. for circles).
The crucial idea that makes Banach-Tarski work in 3D is the insight that the set of rotations around an axis through the origin in 3-space has a free subgroup F on 2 generators (finite strings of A's, B's and their inverses). From this fact the proof is quite easy, but this comment is too small for it.
Sorry, but the breathless way, that maths is often discussed on HN, makes me feel uneasy.
It feels strange to see adults that opine on every subject, from nuclear fusion energy, to virology and financial markets, like they know it all, to suddenly "I was never good at math", like a clichee party conversation.
I mean, I get it: It first feels strange and magical, since even the explanations of some of the vocabulary take more time than we are willing to devote to a single thought. But instead of digging in and looking up what "Borel measurable" might mean, the HN crowd rather watches the x-th numberphile video/emotionalized Quanta blurb.
/rant
More to your points:
> Your kid has a greater chance of being a multi-millionaire NBA player than being smart enough to understand this stuff
There are >5000 math phds each year, so no, getting into the NBA is harder.
> Even 18th century math would be a challenge for many math grad students. Just crazy
Not sure, what this is supposed to mean. Certainly as a math grad you should be able to _understand_ 18th century math. Now, to _come up_ with the stuff is something else entirely. But I'm not sure how many engineers would claim they had discovered the telegraph, were they be born instead of Gauss.
Regarding life insurance product offerings: disability insurance is often advised by consumer protection groups for young professionals, who crucially depend on future income.
Regarding savings products, nothing wrong IMO, with tax subsidizing a "consume later" mentality, if the products are cost-effective.
Ok, so this article is not about going meat free. Just about factory farming. If you "crave meat" maybe you could consider non-factory farmed meat, for all the reasons mentioned.
Regarding your 3 points:
'Availability' is a problem with non-meat? I would think in most neighbourhoods that have a butcher, you should find a supermarket.
'Satiability', really? You don't feel full after baked potatos with sour cream? Mac and Cheese? A pizza quattro formaghi? Noodle soup? Indian curry? All of the above and an apple pie? You need that patty, right?
As a life-long meateater, I can very well understand your last point on taste. Hard to argue with taste. Since I don't want to miss it either, I will have meat from time to time as a special treat. Usually when eating out fancy. So about 1-2 times a month. Makes me looking forward to those dinners even more and I don't have to go the hardest 10% of the meat-free way.
Why would the robot cost the same? If it costs the same, then there is no incentive to introduce robots in the first place. I think the wide-spread automation is a testament to the much lower price of robots compared to human labour, no?
Maybe it really is a cultural difference. I think there is a continuum of valuations at work here:
If a contractor gets his materials cheaper or does have cheaper labor available, I don't see any obligation or market process to pass these savings onto clients, when the work is of the same quality.
On the other hand, when you delegate work to an expert, and he doesn't tell you about the 99% savings you could employ with existing technologies, I feel this is bad conduct, regardless of output. It just amounts to a 9900% markup on his value. I don't think this kind of markup is ethical, but probably the market will catch up to this expert and revalue him.
You say you would feel "sad". To be honest, I would feel cheated. And to set up this smokescreen with pseudo-gardening people (or prerecorded data entry screencaptures to stay with the article) is outright fraught, IMHO.
Yes, to make the analogy better, say the gardener invented the robot while on your "gardening time".
Edit:
Writing a script might be expensive, but is the market price of this script really a lifetime of data-entry salary? Seems unlikely, or we would not see as much automation.
A question for the people that have a positive attitude towards keeping time savings for yourself instead of passing them on to your employer.
Would you be also okay with the following situation: You have someone remodell your garden. He comes up with an estimate of 1 week worth of full-time work for two people. Based on your experience this seems reasonable and you agree on a fixed price of 10000$.
Two scenarios of what happens next:
1) He shows up the next day with a gardening robot you never knew existed and he never told you he would use. The work is done in 30 minutes. He goes home to spend time with his kids. (Or maybe contract more "2 week projects"?)
2) Two gardening people show up every morning to greet you when you drive to work. They leave when you come back and make steady progress every day. On the last day you come home early and witness that actually the daily work is done by a gardening robot in 5 minutes and the two people actually drive home during the day to spend time with their kids.
You're right, I was thinking about univariate polynomials. The "preprint" is only concerned with functions of one complex argument, so this should suffice.
To each their own, but 700+ pages for material that is done in my experience in the first 2-3 weeks of undergraduate math is more disheartening than empowering for a student, in my opinion.
If you can open a math book anywhere in the last 20% of pages and just start reading, you are looking at pop science and not lecture notes.