m00n

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.

It is also true that this a 500fold increase in prevalence when compared to the general population.

The 5-year survival rate of lung cancer noticed due to e.g a sore throat is below 10%.

A course in Arithmetic by Serre is for me THE book to get into Number Theory

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.

The statistics worked for me too, just the streak was reset.

And they broke everybody’s streak in the process while reusing today‘s word…

Way to destroy a good brand.

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.

The actuarial profession is quite broad, in my experience.

For the teams I worked with, memorization of formulas was to their daily work as whiteboard coding would be to the daily work of a sw-eng.

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.