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)
A map of algebraic curves is not a special case of a map between two sets. There aren't really source and destination sets to speak of. The fact that it is given by polynomials really is a definition.
The moral "justification" of the definition is something like "algebraic geometry is precisely the study of such objects" or "we want definitions that are stable under ring base change, and this implies polynomials", etc., but we don't formally need to justify definitions.
[One could take a different route to defining those things, in which this becomes a theorem instead of a definition. For example one can define algebraic curves over a field k as contravariant functors from k-algebras to sets satisfying certain additional properties, and then maps of algebraic curves are natural transformations between those functors. The fact that they are given by polynomial equations is then a theorem. Just stating the "additional properties" for a curve is a rather daunting task, though, unfortunately.]
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.
Apparently the old Wordle URL redirects to the new NYT page, and it includes the statistics as a JSON in the URL so the new page know what's the old statistics is.
Ah! Thank you for this, I was wondering how they preserved the statistics through a move to a whole different site (where the old local storage of course can’t be accessed)
I find the idea amusing that exposing people's wordle scores would result in billion dollar damages. My life was ruined because someone was able to find out 60% of my wins are on 6 guesses!
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.
I was good enough in maths to get a PhD from Commutative Algebra, but the really good ones were on another level, where you could barely follow their thoughts (especially real-time; anything can be attacked with enough patience, but it was precisely the speed of their train of thought that humiliated you the worst).
People like Erdös were gods in the mathematical universe.
This is exactly right. I could also get a PhD degree in math myself (I dropped out after obtaining Master during which I obtained novel results in algebraic geometry), but after meeting and interacting with actually smart people, it became clear to me that I’m just not nearly on the same level. Research level mathematics requires completely another level of sheer brainpower that most people don’t even imagine exists.
Adding my voice to this too. I have a PhD in Differential Geometry and would consider myself to have been a decent student and researcher. The "good" people in my field were more than a head and shoulders above me, and the "great" people were somewhere off in the stratosphere.
The nature of Mathematics is that the potential depth of understanding and progress is essentially infinite, which frees truly spectacular minds from the constraints they would experience in other fields.
Is that because the "good" people in your field were just way more obsessive about the topic?
I feel like there are some topics that I'm obsessed with that I'm so much more informed on than most people in my field that I can run circles around them. They would call me super smart if the things I'm obsessive about mattered. Sometimes they have mattered. But I know better than to talk about them at length because people get bored.
I don't think you need that super fast brain to be good at math. I'm sure it helps, and I know some people who have it and I felt I could never be like them. But I've also known some really top mathematicians (one Fields medalist in geometry and one of the biggest cheeses in mathematical logic) who weren't like that, and I felt like I could keep up with them, at least on a conversational basis (I don't claim that I would have been much of a researcher if I had stayed with it, much less at their level). Pure brainpower goes a long way, but personality and commitment count for a lot all by themselves.
I 100% agree that some people are innately superior at math, the mental arithmetic abilities (at a very young age) of human calculators like Von Neumann are proof enough of that.
But I also agree with the other poster that it's kind of dangerous/distasteful to imply that mathematical ability is something that is not necessary to cultivate, or at least not worthwhile unless you're the next Galois.
A lot of students are already lacking in grit and give up on difficult subjects, not realizing that areas like math require a lot of discipline, struggle, and engagement to cultivate. This hierarchical nonsense about it only being worthwhile for the "chosen few" NBA superstars is not productive, especially with Ameria trailing most developed nations in mathematical and scientific literacy (which has real societal consequences, IMO).
I saw a study from long ago, maybe the 1980s, which researched US and Chinese high school education. As I recall, people in the US high schools mostly thought that success in education was due to natural talent, while people in the high schools in China thought it was overwhelmingly due to hard work. The kids in the Chinese schools did much better on the tests.
> This hierarchical nonsense about it only being worthwhile for the "chosen few" NBA superstars is not productive
Agreed. It also takes away the dreams of and opportunities from a lot of people.
The point here is that while it is perfectly reasonable and probably desirable to encourage people to become proficient in mathematics, there is an enormous chasm from there to most modern research mathematics. The sports analogy is that while it is good idea to encourage people to be physically active, for example by playing recreational basketball, it would be completely ridiculous to expect even a strong amateur basketball player to be able to hold his own in the actual NBA.
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.