If you were interested in joining a group that encourages you to actually read and discuss that ever-growing collection of pdfs sitting in your downloads folder...
On the topic of teaching kids math: I have a vivid memory from kindergarten when the teacher was showing us shapes, and asked us to name them (we already knew basic names, and we knew that we were supposed to say the most specific ones, e.g. square instead of rectangle etc). Now she showed us a square, so all the kids yelled "square". But then she rotated that square 45 degrees, and the kids yelled "rhombus", because we knew that "rhombus" was "a skew square". And the teacher said "no, it's a square, it's the same square, don't you see?", and just rotated it back and forth. I remember we were baffled and lowkey amazed.
In retrospect, this lesson taught me not only about group actions (Klein's Erlangen program vibes), but also about semantics - that the words are just tools to carry ideas, that one is not smart by knowing definitions ("a skew square is called a rhombus") but by identifying patterns and their flexibility.
I guess this was one particular lesson that just clicked for me, but sometimes I wish more lessons were like that instead of memorizing another definition and feeling smart for sounding latin.
Cannot sleep so: spren on Roshar (minor stormlight spoilers ahead). We know there are various humanoid spren, in prnciple able to bond with humans. There are also horselike musicspren, and I don't know if it is canon or not, but it makes sense to speculate that they somehow bond with horses. This way Rhysadium horses are like human Radiants. So in general, animals not native to Roshar should be able to bond spren. Now other than humans and horses, there are two other widespread non-native species: rats and chickens. I dare not speculate about rats, they must have their ways. But I am really curious about Radiant chickens.
We know from WaT that Mraize has never left Roshar, so his Aviar is suspicious at least..
I don't even think that is really true. So many papers are just referencing theorems from other papers and taking them as a blackbox. Otherwise it is infeasible, we don't have years and years and years of life to spend on each paper in the modern publish or perish paradigm.
No. You are not doing it for some moral high ground. You are doing it because it is the far west out there. Everyone makes mistakes.
The author, the reviewer, the editor. You don't want to be adding to the mistakes just because you don't believe reading 10 pages of proof is unfeasible.
I have found mistakes in papers and in books. It happens.
Yes there are people who invent theorems because they misunderstand them, or for what other reasons, but I don't see why you need to follow their footsteps.
An annoying risk is to have your papers sent back because you used a false result and thought that it wasn't your job to check it when submitting a paper. Which would really be wasting a lot of time.
Moreover these days there are some awful people who believe they can just feed it to ai and be done with it. Without checking anything. Because they believe it isn't their job as scientists, to be sure that what they write is actually correct. Are you sure you want to go and trust whatever they spat out?
It is and always has been a jungle out there.
I am sorry, but we all have been doing the unfeasible. Don't go believing that understating a proof is a waste of time. Because if you do you will never be happy here. That would be really sad.
I am not advocating for never understanding the theorems you use. But, for instance, should every graph theorist personally go through each sentence of the proof of the graph minor theorem to be willing to invoke it?
I mean, this just conflicts with the advice I've gotten from multiple professors and a postdocs that I need to learn to "blackbox things" or I will never get work done.
I thought for a moment there was some misunderstanding but I believed it couldn't be true since the conversation started talking specifically about learning.
There is a difference between browsing or learning a paper or a book or a subject and adding to the literature thanks to it.
There is no point in going through hundreds of proofs on your first or even any reading of a subject. It's literally what started this thread!
It is different the moment you want to cite those results in a paper. That's all. Black boxing is fine and all when you go through exercises or just new literature. It's not fine when your own results hinge on them.
If you have professors who said to you that you don't need to check the proofs of theorems from papers or books you are literally using for your results, I'm sorry, but they have given you terrible advice.
I disagree. As an algebraic geometer, I trust EGA, GAGA, BCHM, big papers like that. And in general, if I can prove a very special case of a theorem, and get the vibe why it should be true (sometimes very vaguely) + if the theorem has already been used by respected people, I feel free to use it. I mean, even if there are mistakes somewhere, then either the whole field is wrong, or it will be fixed someday anyway. This is also why I trust StacksProject (not blindly, but like, if I see the reason in case of varieties, I trust that they worked out the details for stacks or whatnot). Otherwise there wouldn't be much progress, and personally, I would be very much behind.
On the other hand, if I plan to use something from outside my field (topology, for instance), I either have to reprove it myself, or ask people. If a topologist can convince me that the result is true, I trust them. I guess the key point is to "trust the community". There is certainly some psychology/sociology involved which I cannot make precise now, but to follow your metaphor, I feel that at least algebraic geometry is no longer wild west, it has become a safe space (at least away from the borderlands)
I just gave a horrible class, sent homework with screwed up formulas, and two emails correcting it. To all the students of mathblr: if your teacher does shit like that, or is shitty in any other way, please skip the classes. Mathematics is pure beauty and you do not have to suffer having it obscured. You can learn everything on your own; just don't waste your time and energy.
Below a cut: a vent about how I wish I didn't screw up do badly.
Well now I know that it should be a reddit question or sth, but since this is a safe space and reddit seems not, I decided to vent here.
And I know that I am privileged, even more than average rich white cishet etc, so the best I can do is to shut up, but well this is why this text is under the cut and #the reader is encouraged to skip this chapter.
So the upshot is I suck at teaching. Which is kinda non-issue when it comes to basic undergad where the students just want to pass and do not pay attention anyway, but when it comes to more advanced classes, the students are actually smart (I am talking olympiad/putnam level, smarter than me anyway), they actually want to learn, and I am wasting their time, discouraging them. I would love to just sit down with them and say hey, I find this subject difficult, too, but ultimately beautiful, let's learn together - but there is a distance of age, academic title and simply the fact that I am being paid for knowing that stuff already.
Generic classes mean: prepare homework, and then let them present the solutions, or give hints whenever they're stuck (there is also the lecture, but this is a separate thing). So preparing homework freaks me out: I know I have to give some trivial exercises, but in order not to insult them, I must give more challenging ones, too. And I know that a rule of thumb is that if the teacher can't solve the exercises on the spot, they are too hard. But I can't solve anything on the spot (really, I wiggled my way through but I never did a solid computation and it shows). So I can open the book, rewrite a problem, check a solution somewhere (including chatgpt out of despair) and try to assess how hard would be to come up with it (which is stupid because following a written solution and coming up with it is two different things). And I have to somehow sync with the lecture, which is easier if I am the lecturer, but generically I am not, and again, the lecturers are smart professors who can advise me like "well give them the standard obvious exercises like this and this", and well first of all I have no idea if I understand you correctly, let alone how to solve it, but also these are two exercises, and I get the impression that the students expect like 20 every week. Which I lowkey know is an overestimation but it gives me hope that I won't screw up all of them.
Because here it comes - I have to prepare those exercises every week, so reasonably fast, and I freak out as described above, which takes most of the time, and best I can do is to throw some exercises which give solvable vibes. Which means that I will screw up writing them (last time I gave an exercise with typos in *every* formula, and of course I joked that there is always a bonus exercise to correct my typos - but this can be done for obvious typos like a sign, but if the four variables are randomly permuted, this just means that the students will waste their time in computing nonsense). And the only way I see is to give them many exercises - maybe it is just because I want to prove to them that I am not that dumb. Which is stupid, I know, but I can always say that it is to appeal to different interests or whatnot. Probably a reasonable way out would be to give them, like, four, and have written down solutions. But I am too stressed or tired to do it in advance, and then it's too late, and here we are. And the professional way would be to have a list of 100 exercises before the semester, with full solutions, but see above.
And then comes the day before the class, when I do write down the solutions, which means I stay up late, which means that I give the class powered by adrenaline and lack of sleep instead of genuine knowledge, so the amount of insight I share is strictly negative.
And the worst part is that my main job should be research. And if somebody asks what do I do for a living, I cite my research from years ago, because since then I just postpone everything because of teaching. Well, because of freaking out. And of course I get responses like "bah, teaching, wait for the administrative tasks, then you will see". And there are joint projects I participate in, which have deadlines, so basically I am failing the coauthors as well as students.
But the people are supportive! The people in charge told me that they will not give me any administrative responsibilities unless I ask for it (which I won't); and I could ask for reduced teaching load (which I did not, and it's too late now). And like emotionally I have full support too. Which only adds to the guilt of screwing up those wonderful people, and those extremely smart students.
Anyway, vent's over, thanks for reading. Again, if you are a student and you end up in awful class, skip it. If you need a grade, just ask for a way to get it without attending, there is no reason why attendance should be obligatory. Every area of research, no matter how exciting, can be presented in an awful way, and conversely, no matter how shitty a university course is, the underlying science is beautiful.
So I just had a dream that Brandon Sanderson published a textbook in algebraic geometry. It was ridiculously thick, typeset and illustrated like a ttrpg rulebook, and narrated by Hoid, of course.
Here's something I think is fun - what do you most associate with all of the letters of the alphabet in math? It doesn't really matter if you think of the letters as being uppercase or lowercase. Here's what I think:
a, b - the Pythagorean Theorem
c - a scalar
d - differentiation
e - Euler's number
f, g, h - functions
i - the imaginary number
j - a quaternion
k - a scalar, but specifically when it's an integer
L - the Laplace transformation of a function
m, n - variables, but specifically when they're integers
A, B, C - objects in a category
D - derivation of a ring
e - an idempotent
f, g, h - morphisms
i, j, k, \ell - numerical indexes
m, n - integers used for dimensions
O - avoid it myself
p, q, r - other numerical indexes
s - variable for a little perturbation
t - variable for time
u, v - morphisms
w, x, y, z - variables
A - a specific type of curve which often appears in my stuff
B - base of a fibration
C - some curve
D - divisor
E - exceptional divisor
F - fiber of a fibration
G - a group, but more frequently "I ran out of notation"
H - a (hyperplane) section, something horizontal
I - identity? avoid
J - almost complex structure
K - canonical class
L - line (or something that resembles a line), or line bundle
M - manifold
N - normal bundle
O - script for structure sheaf, otherwise avoid
P, Q - points
R, T - something big or invariant (don't ask)
S - surface
U, V, W - open sets or vector spaces
X, Y - varieties
Z - closed sets
Lowercase are mostly variables (a,b,x,y) numbers z - complex, p,q - prime (or points), small parameters (s,t) or functions (f,g and u,v for nonvanishing). There are also Greek ones, but I really should go to work now...
Also I think that it is kinda cool that, within a community, one can see a symbol and have immediate intuition what type of object it refers too, despite the fact that it cannot be rigorously defined (otherwise we would run out of symbols). It should be an easy exercise in linguistics to study this phenomenon, maybe its even done? (Also you can recognize the community by looking at such intuitions, another feature of language/psychology)
Here's something I think is fun - what do you most associate with all of the letters of the alphabet in math? It doesn't really matter if you think of the letters as being uppercase or lowercase. Here's what I think:
a, b - the Pythagorean Theorem
c - a scalar
d - differentiation
e - Euler's number
f, g, h - functions
i - the imaginary number
j - a quaternion
k - a scalar, but specifically when it's an integer
L - the Laplace transformation of a function
m, n - variables, but specifically when they're integers
A, B, C - objects in a category
D - derivation of a ring
e - an idempotent
f, g, h - morphisms
i, j, k, \ell - numerical indexes
m, n - integers used for dimensions
O - avoid it myself
p, q, r - other numerical indexes
s - variable for a little perturbation
t - variable for time
u, v - morphisms
w, x, y, z - variables
A - a specific type of curve which often appears in my stuff
B - base of a fibration
C - some curve
D - divisor
E - exceptional divisor
F - fiber of a fibration
G - a group, but more frequently "I ran out of notation"
H - a (hyperplane) section, something horizontal
I - identity? avoid
J - almost complex structure
K - canonical class
L - line (or something that resembles a line), or line bundle
M - manifold
N - normal bundle
O - script for structure sheaf, otherwise avoid
P, Q - points
R, T - something big or invariant (don't ask)
S - surface
U, V, W - open sets or vector spaces
X, Y - varieties
Z - closed sets
Lowercase are mostly variables (a,b,x,y) numbers z - complex, p,q - prime (or points), small parameters (s,t) or functions (f,g and u,v for nonvanishing). There are also Greek ones, but I really should go to work now...
So I am probably, like, a month late, but what is the deal with arxiv requiring an English version of each submission? Is there any legitimate reason for this discrimination? Does this work retroactively, so the free and accessible copies of SGA or whatnot will be taken down?
I totally understand boycotting arxiv now. I can't afford that, but great names, like Olivier Debarre, can... and once they start doing it, arxiv will scatter across HAL, personal webpages, researchgate, and ultimately the idea of free access to most up-to-date research will die? I was aware that existence of arxiv was essentially hacking capitalism (there was an xkcd for that) and a testament to how uniquely chill and no-nonsense mathematics is... But being born and raised in the world with arxiv, I cannot imagine a world without it.
There is some community, especially in French or German speaking countries, pushing for publication in these languages, I guess to preserve original terminology and ideas it carries. And of course, enforcing English *as the first language* is a kind of gatekeeping, and any gatekeeping is bad in principle (of course, I guess nobody has any problem with having their work translated afterwards).
Now I have no problem with meeting this requirement, I would've met it anyway, but apparently there are people who do, for whatever reason: patriotism, accessibility, convenience, I don't know. My point is that I cannot blame them, as there is apparently no reason to impose such a rule. So I am really scared that people start boycotting arxiv (you see in comments some people which declare they will). And if this happens, arxiv will no longer be a primary source of new research, which of course will create unfair advantage for people with access to specific conferences or journals.
The great struggle for me in algebraic geometry is that once again people are ALWAYS WRITING EQUALITIES WHERE THEY REALLY ONLY HAVE ISOMORPHISMS!
Please, as a beginner I do not yet have the confidence to hand wave over these things. I must experience that keeping track of the isomorphisms is always trivial and can be safely hand waved away before you ask me to just do that.
This is why I have to tend to my perversions (desire for proofs to be clear and complete and fully elaborated) in private before I can be ready to head out into public and socialise with the other algebraic geometers.
There is a somewhat aggressive response to this on J. Milne website. While I do not agree with the tone, I agree with the standard abuse of notation, where "=" stands for "there is a UNIQUE FUNCTORIAL iso". Of course, in the lecture this should be always spelled out. And it is a good exercise to write down the iso, and even better to spot such abuse. So the student should not be yelled at, they should get their final grade increased
So I am probably, like, a month late, but what is the deal with arxiv requiring an English version of each submission? Is there any legitimate reason for this discrimination? Does this work retroactively, so the free and accessible copies of SGA or whatnot will be taken down?
I totally understand boycotting arxiv now. I can't afford that, but great names, like Olivier Debarre, can... and once they start doing it, arxiv will scatter across HAL, personal webpages, researchgate, and ultimately the idea of free access to most up-to-date research will die? I was aware that existence of arxiv was essentially hacking capitalism (there was an xkcd for that) and a testament to how uniquely chill and no-nonsense mathematics is... But being born and raised in the world with arxiv, I cannot imagine a world without it.
vaporwave
s-u-f-j-a-n-s-t-e-v-e-n-s
pixelskylines
turnnoffyourmind
snowghoul
blakekathryn
gamechangershow
ellievsbear
