Ok that makes more sense with my intuition actually
#serious-discussion
vivid halo
mhm
blazing pawn
And then Gal(bar{k} | k) measures the etale covers that do come from base changes
vivid halo
yup
blazing pawn
which is where exactness comes in basically
vivid halo
yes exactly
that part is clear I think, it's the splitting that's a little funny
I mean maybe think about it like this
vivid halo
how did we cook up this projection π^et_1(X)->Gal(\bar{k}/k)? We certainly didn't say what it did on elements, rather we specified it by looking at what it induces on all finite quotients
blazing pawn
where K_U is the composite of all the etale extensions over U?
blazing pawn
how did we cook up this projection π^et_1(X)->Gal(\bar{k}/k)? We certainly didn'...
Yeah this was the dumb diagram i made earlier 
vivid halo
something like this yea
well right so what we said was to specify π^et_1(X)->Gal(\bar{k}/k) we need to specify how to turn finite etale covers of Spec(k) into finite etale covers of X
so for the splitting, we need to specify how to turn finite etale covers of X into finite etale covers of Spec(k), which is where we use the k-rational point x
what's the dumbest way you can think of doing this
hint: use the fact that a k-rational point is a map x:Spec(k)->X
blazing pawn
compose...?
Like literally just compose with the map
Wait
I do not understand directions
vivid halo
no no we start with a finite etale cover Y->X
blazing pawn
blazing pawn
Lift?
vivid halo
pullback!
vivid halo
for this you need the fact that finite etale covers are preserved by pullbacks
but then yea that's your splitting
blazing pawn
I see
vivid halo
I like this better than what Szamuely wrote lmfao
vivid halo
this also gives you a way to, you know, actually compute things lmfao
okay so here's how you get splittings that don't come from k-rational points
you use something called k-rational tangential basepoints
suppose X has a puncture at some k-rational point x in a compactification \bar{X}
so like, we can't take x as a k-rational point of X since it's missing from X, but we would really like to remember it's there somehow
to remember it's there we need one extra piece of data, namely a nonzero tangent vector v at x in \bar{X}
blazing pawn
Hmm
vivid halo
we can't really look at fibers of x in finite etale covers of X, since these would just be branch points and that's not what we want
but we can look at the fibers of the tangent vector!
blazing pawn
I would like to make sure i understand the k-rational case better tbh
vivid halo
so e.g. here the fiber over v is {e_0,e_1} so this sees the fact that this is a degree 2 cover
blazing pawn
or at least szamuelys claim because i think in general i just need better intuition for pi_1(U_bar{k})
Hmm is there a way of thinking about what elements of the etale pi_1 correspond to
finite quotients are etale morphisms right
blazing pawn
Oh
vivid halo
vivid halo
not least since the groups themselves can be really complicated and it's like, not even clear how to write down what an element is in a concrete way
deep mango
lmao
blazing pawn
Bleghhhhh
vivid halo
this is, by the way, why I don't like Szamuely's proof: it's really requiring you to think about elements of this thing, and how it acts on points, which is weird
sly thistle
Hello n groupoid
vivid halo
occasionally you do have to contemplate these things but it's best not to
hi 
but yea moth I mentioned at some point that "fiber functors" FEt_X->Fin can be thought of as basepoints of X
certainly we get fiber functors from any point x of X, by just taking the fibers of finite etale covers over those points
but certainly this tangential basepoint business also gives us fiber functors, by taking the fibers of finite etale covers over a tangent vector
these give you more splittings than you expect for the etale homotopy exact sequence by the discussion above
vivid halo
what's also super cool about this is it remembers more information about your covers than ordinary points do
in the above picture, if you remember the branch point y and the fact that e_0 and e_1 are attached to it
blazing pawn
The idea is that at branch points our fibers are fucked cause non etale but because branch points and their fibers are discrete the tangent vectors will be pointing in the "fibers near x"
And we can study those, right?
vivid halo
this is encoding not only the degree of the cover but also the ramification indices!
blazing pawn
vivid halo
and yea that's the idea, the tangent vector gives you a direction where you can move infinitesimally away from where your cover is bad, and then take fibers once things are good
blazing pawn
Epic
vivid halo
this is also cool because like
blazing pawn
Now i want to learn about this
vivid halo
if you imagine some "moduli of splittings" it's sorta like
blazing pawn
RYC sully... Sad
vivid halo
there's some part of the moduli that literally just looks like X itself, and then you're gluing on collars to all the punctures of your curve
if you insisted on tangent vectors of norm 1 the moduli is a Riemann surface with boundary
vivid halo
where you've replaced each puncture with a circle
oh I just mean like it's a space where the points correspond to splittings of the etale homotopy sequence
certainly X should embed into this space, since any point of X yields a splitting
sly thistle
Imma be honest I have no idea what y’all talking about
blazing pawn
I do not know what the etale homotopy sequence is either
vivid halo
it's the short exact sequence we've been talking about
and we've been talking about constructing splittings of it, which is what Szamuely doesn't do a good job explaining
vivid halo
yea
blazing pawn
The pullback idea does seem a lot more intuitive
But he didnt really do the "k-rational points as morphisms"
vivid halo
yea, which is why I assume he took this approach instead
in the schemes chapter the approach I mentioned is taken
blazing pawn
He takes the much more based approach of simply not explaining anything
vivid halo
based
vivid halo
so like if we try to visualize this space of splittings of this short exact sequence for say X=P^1-{0,1,\infty}
and say we restrict to only unit tangent vectors
what space should we get?
we should get a pair of pants!
the boundary circles correspond to unit tangential basepoints at 0, 1, \infty respectively
blazing pawn
Its all ive ever wanted...
vivid halo
the open part is the thrice punctured sphere
if you don't insist on unit tangent vectors replace each red circle with C*
okay we might worry though like
did we miss any splittings? Do all of them come from k-rational points and k-rational tangential basepoints?
This is Grothendieck's section conjecture, it's very open
blazing pawn
oh yes this was in the book
we want a bijection from k rational points + tangential basepoints to sections
sly thistle
yknow i keep writign things then deleting them because i realize theyre not adding anything
vivid halo
that's right
blazing pawn
we have injectivity but not surjectivity
sly thistle
so if you see me typing a bunc that is probably why
blazing pawn
Wow I am so smart.
vivid halo
in the case where X is proper we don't need to care about tangential basepoints
yea so this conjecture is cool because it's saying like, the arithmetic of rational points is encoded group theoretically in this sequence
which is kinda wild
deep mango
vivid halo
wait I forgot the most important part lol
blazing pawn
Ok this was pretty cool even though idk how well i understood it
vivid halo
this is like, definitely not true in general lol
for curves you need genus g>1 in the proper case
in the non proper case you need negative Euler characteristic
sly thistle
for curves you need genus g>1 in the proper case
there are curves of genus>1
this wasnt covered in the manual
nvm thats a shitty joke sorry
vivid halo
lmfao
deep mango
blazing pawn
elliptic curves btfod
vivid halo
it's a nontrivial result that the conjecture in the proper case follows from the conjecture in the proper case with no rational points
deep mango
"""""curves""""""
blazing pawn
Yes 
vivid halo
which is nice because it's maybe easier to show that a set has to be empty (usually by cooking up some cohomological obstruction) than it is to show some set has like precisely 934583094 elements
sly thistle
bruh this conversation is making me feel like such an idiot lmao
blazing pawn
Why
vivid halo
that result is really nontrivial but there's a kinda banal observation that the conjecture in general follows from the conjecture for no rational points
deep mango
bruh this conversation is making me feel like such an idiot lmao
you should feel like the genius
sly thistle
because theres a lot of things i dont understand
deep mango
for not getting sucked into their crankery
sly thistle
and i dotn even know what to ask
blazing pawn
sly thistle
lol
vivid halo
that's fine, you're not expected to!
sly thistle
yeah but i wanna thats the whole problem
deep mango
if you even want to ask a question, you know more about this than i do
vivid halo
like the intended audience of the conversation is moth who is reading a book on this topic specifically
sly thistle
yes
vivid halo
idk do you know about topological fundamental groups?
sly thistle
no
vivid halo
rip
sly thistle
i mean i know the meme explaination that loops do something but that isnt really much lmao
vivid halo
There's a notion of fundamental group in topology, there's a more familiar definition in terms of loops up to deformation in your space and then you can prove that this group classifies covering spaces of your space
in algebraic geometry there's no good notion of loops, since spaces are no longer nice topological spaces
but there is a good notion of finite covering space in algebraic geometry
so it's possible to define an algebraic version of the fundamental group in this setting: instead of defining it in terms of loops and then proving this statement about covering spaces, we simply define this object as whatever object makes this theorem about covering spaces work
the reason why this is cool is things like field extensions in Galois theory are covering spaces in this sense
sly thistle
and thats equivalent in the nicer rcase?
vivid halo
it's not equivalent, but it's related!
There are some spaces in algebraic geometry, say varieties defined over C, where you can get a complex manifold out of this
and you could take the topological fundamental group of this complex manifold
and ask how this is related to the algebraic fundamental group of the variety
they aren't isomorphic, but they are up to something called profinite completion! Which sort of makes sense, because the algebraic fundamental group only classifies finite covers, whereas the topological fundamental group classifies possibly infinite covers (and the profinite completion is exactly the operation that like, removes all infinite quotients of your group in some minimal way)
sly thistle
i remember reading p-adics are profinite but i didnt really know what that meant
vivid halo
yea like the p-adic integers Z_p are a great example of a profinite group
this basically means that they are "built" out of finite groups
in this case, it's the finite groups Z/p^nZ for all n≥1
vivid halo
yea another example is the profinite integers \hat{Z}
which is equivalently \prod_p Z_p
sly thistle
profinite integers are profinite?
vivid halo
in this case it's built out of the finite groups Z/nZ for all n≥1
this is the algebraic fundamental group of say the punctured complex plane C-{0}
sly thistle
why
vivid halo
the topological fundamnetal group of C-{0} is Z
sly thistle
what
vivid halo
the generator of Z corresponds to a simple loop around 0 in C-{0}
vivid halo
so then there's a theorem that says if it makes sense to compute a topological fundamental group (for C-{0} this is fine since we can regard this as a manifold) then the algebraic fundamental group is the profinite completion of the topological fundamental group
hence the profinite completion of Z
sly thistle
and apparently the profinite integers are the profinite completeltion of Z
but like
what does this actually mean
vivid halo
mhm! Here's one way to see this
vivid halo
quotients of the fundamental group correspond to covering spaces. Quotients of Z are of the form Z/nZ for n≥1, or Z itself
(all but one of these quotients is finite)
sly thistle
yes
vivid halo
the covers corresponding to the finite quotients are like
vivid halo
sly thistle
pacman ghost
sly thistle
garfield
vivid halo
what the profinite completion (and the algebraic fundamental group) are seeing is only the finite covers
the profinite completion basically threw out this infinite cover
by producing some group that has all the same finite quotients, but no infinite quotients
sly thistle
so do all profinite groups correspond to the profinite completion
of some fundimetnal topological group
vivid halo
nope!
sly thistle
damn
vivid halo
have you seen much Galois theory before?
vivid halo
absolute Galois groups of fields are examples of algebraic fundamental groups
and these are rarely profinite completions of groups
they can get really wild
even like Gal(\bar{Q}/Q) is a mysterious object; it's a profinite group and we can only study its structure very indirectly
sly thistle
yes
vivid halo
so the thing that moth and I were talking about was an exact sequence of algebraic fundamental groups that blends these two extremes
the group on the right is some absolute Galois group, so very arithmetic. The group on the left is the profinite completion of some topological fundamental group, so very geometric
vivid halo
this is definitely cleanly in the realm of arithmetic geometry since you're like
taking ideas from Galois theory and forcing yourself to think about them geometrically
idk it was shocking to me that you could even do this when I first learned this stuff
sly thistle
wdym
arithmetic geometry tho, thats closer to home for what i haave some grasp on i think
vivid halo
I mean, this is effectively saying "you learned about covering spaces in algebraic topology, you learned about Galois theory, it turns out these are the exact same picture but in different contexts"
sly thistle
i mean that sounds pretty amazing but i dont really know enough about either
to appreiciate it
vivid halo
that's okay, but hopefully that gives you a slightly better idea of what's going on?
sly thistle
ig
vivid halo
idk I hope I haven't wasted your time rambling about this lol
vivid halo
me too thanks
I'm literally just vomiting words without thinking
I'm still so pissed that the GT conjecture was a complete mirage
yea lmfao rip
deep mango
should i read all of it?
vivid halo
actually who knows maybe the paper that Mochizuki wrote that disproved this is wrong like his other papers
deep mango
or should i ignore all of it... hmmm...
vivid halo
lmfao ultra did I tell you one of the early paper ideas that Daniel gave me to do?
deep mango
a good 6 or 7 minutes of my valuable time
sly thistle
read what
vivid halo
"reprove the main theorem of Mochizuki-Hoshi-Minamide but just write it like a normal person"
sly thistle
my time isnt valuable dw
sly thistle
why does it seem like any proof of Mochizuki is basically unreadable
vivid halo
and spread across like 6 papers
but in a way that's like
not at all essential
unfortunately the issue is like
you kinda have to read all of the papers first before you can do and simplify everything
and that's just like
not a good use of time
there's another kid that did this recently but rewrote most of the results of the Lafforgue brothers
3rd year grad student
sly thistle
wtf
vivid halo
let me find the paper I may be misremembering
at the very least it's one of the really hard Lafforgue papers
sly thistle
read what
sly thistle
can i see it
sly thistle
oh i thought they meant some paper
deep mango
i'm well versed in all your integrals nG. thanks for passing them along.
i really liked the one where you did a u-substitution.
deep mango
like operator valued integrands and stuff? or operator valued measures? what exactly do you mean?
maybe neither of those
then nothing
bronze pelican
deep mango
sly thistle
i feel like that was a reference
bronze pelican
What GT Conjecture
sly thistle
What GT Conjecture
groethendiek teichmullar maybe? idk just guessing by the letters
deep mango
geez tuts.
vivid halo
so you can prove that Gal(\bar{Q}/Q) is a subgroup of Out(\hat{F}_2) and you can prove a lot of different relations that the image of Gal(\bar{Q}/Q) has to satisfy, so it's very tempting to ask if you can obtain Gal(\bar{Q}/Q) as an explicit subgroup of Out(\hat{F}_2) defined by some relations
essentially what Mochizuki-Hoshi-Minamide show is that a large class of relations that everyone expected should be sufficient cannot possibly make this idea work
This was the Grothendieck-Teichmuller conjecture because people defined this Grothendieck-Teichmuller group GT as a subgroup of Out(\hat{F}_2) satisfying some relations, and then the conjecture is that GT=Gal(\bar{Q}/Q)
bronze pelican
vivid halo
what Mochizuki-Hoshi-Minamide show is that GT is essentially as close to being isomorphic to Out(\hat{F}_2) as possible, and the relations don't cut the group down in any meaningful way
sly thistle
wack
vivid halo
obviously it's still possible in principle to come up with totally new relations that do the job, but this requires a completely new approach
I suppose one can't fault everyone for getting things SO WRONG because it's not like Out(\hat{F}_2) is an easy group to work with, it's basically impossible to have any kind of feeling for what the relations are actually doing
also it's not like there is a very convincing philosophical reason why this conjecture has to be obviously false
Mochizuki has some philosophical explanation involving IUTT
sly thistle
how tf is the explaination involving IUTT supposed to be intuitive. isnt there a whole issue that noone can understand mochizuki wrt IUTT
sly thistle
fair i havent really kept up with the whole thing utlra
vivid halo
implicit in this setup is the identification of the algebraic fundamental group of P^1-{0,1,\infty} with \hat{F}_2, which is VERY noncanonical, it depends on a choice of embedding \bar{Q}_l->C
he does some song and dance about why this makes everything fall apart which I don't find 100% convincing
and then says something about like
IUTT giving some notion of "analytic continuation between different embeddings \bar{Q}_l->C"
which just sounds like fartsmelling
sly thistle
im scared cause that sounds like a real thing
vivid halo
like I see what he's trying to say
it's just he doesn't actually give any details on what he means by this lol
there is a way in which the conjecture is salvageable which I occasionally think about but haven't got anything paper worthy out of
there is a variant of GT that is known to be isomorphic to Gal(\bar{Q}/Q), it just diverges a bit from the original philosophy
sly thistle
ngroupoid proves abc conjecture eta?
vivid halo
it is still a subgroup of automorphisms of a profinite group satisfying some relations
I've tried writing down the relations and it's kind of a pain
but it should be doable in principle
essentially there's a definition of "GT_V" given a set V of varieties
the original GT conjecture claims that GT_V is isomorphic to Gal(\bar{Q}/Q) when V={M(0,n),n≥3} is the tower of genus 0 moduli of curves
and this is just false
there's a similar statement for when V={M(g,n)} and this is false too
vivid halo
but there are sets of varieties V where we actually know GT_V=Gal(\bar{Q}/Q)
it's just not clear (and probably false) that you get finitely many relations when doing this
for V={M(0,n),n≥3} you get only finitely many relations, two coming from M(0,4) and one coming from M(0,5); this is the "two level theorem" that says everything in the tower is determined by the first two levels
there's absolutely no reason to expect this to be true for other sets V
so it's possible that the conjecture is still salvageable but you just end up with something that is utterly useless
idk I'm just salty because I spend so much time thinking about this conjecture and wrote a whole undergrad thesis about it and it ended up being completely false lmfao
vivid halo
lmfao
vivid halo
it's fine lol
nimble shuttle
meant to post in iuvory
vivid halo
provided you have perfect knowledge of how to distribute electrical resources, then this is a good thing
however economies don't operate on perfect knowledge
vivid halo
oh well that's easy
solar panels during peak hours produce more electricity than people consume, provided enough people have them
errant rock
they pay people to take the excess ?
vivid halo
sure it would
sly thistle
wouldnt this simply just lead to excess power not being used?
sharp mulch
MIT Technology Review is old white conservative propaganda but repackaged to target young people
vivid halo
the price is negative insofar as the company would have to pay money to deliver this electricity
compared to the cost of just taking the solar output
sly thistle
sure but if there is more electricity available than there is needed
wouldnt the excess not be delivered at all
vivid halo
well no, it has to go somewhere
sly thistle
so then at what point is it negative
vivid halo
depends on the method of generation
but for instance this negative price thing isn't exclusive to solar
here it is happening to natural gas
Negative Waha prices, which happened several times in 2019 and 2020, occurred because energy firms were not able to build pipelines fast enough to keep up with growing gas output associated with record Permian oil production.
That forced some drillers to flare record amounts of gas or pay others to take it rather than shut oil wells because they could make enough money selling crude and other liquids to cover their gas losses.
this explains how this happens
nimble shuttle
this has been posted in ivory like 3 times already
oh really i thought it was just the first post of the thread
vivid halo
yea, you're losing money no matter how you operate
blazing pawn
I like the replies in the comments talking about how this is "free solar" as if we all pay factories directly for the power they generate
Twitter upon discovering what a utility company is:
vivid halo
obviously there will be some amount of money loss as (hopefully) people move towards renewable energy sources, as the non-renewables are priced out
but this is actually an economic problem that isn't as simple as "lol oil companies should just take the L"
the reason it's a problem is that solar's main drawback is that it can't deliver enough power during off-peak times
this has one of two solutions:
vivid halo
- have enough batteries to store energy produced during peak hours to sustain through non-peak hours
sly thistle
battery doesnt ahve to be a chemical battery tho
vivid halo
this is ideal, but it's like really cost prohibitive and admittedly not environmentally friendly at all given how much lithium you have to remove from the earth
sly thistle
dont in britain they use some water battery according to some tom scott video
vivid halo
sure there are other methods to this but it's not a very good short term solution obviously
like yea eventually this seems ideal
in the short term you're left with
- rely on non-renewables during off-peak hours
this is ideal especially as people are transitioning to fully renewable sources
even if we could offset like 50%+ of energy demands with solar that would be a gigantic win
sly thistle
yeah
i dont think anyone is asking for immediately 100% energy supplied by renewables
vivid halo
the issue now is that if this isn't organized carefully, you can have peak hour solar fucking up the prices and economics behind non-renewables even during off-peak hours
that's largely the issue
sly thistle
yea
vivid halo
yea, you can't price out things that you're still forced to rely on
if we get to a point where we no longer have to rely on non-renewables then yea fuck it price them out
but until then you have to be a little careful
velvet dagger
Look oil companies just direct a lot of profit into R&D for energy storage
They get rich when they figure out something huge that just solves the problem
neat lintel
😮
velvet dagger
ezpz why'd nobody think of this before
viral barn
hi, this is a random question, but could one use statistical mechanics on like, idk, ant colonies and traffic jams? i guess, cos one can consider ants and cars as microscopic entities, leading to large scale entities
supple torrent
How do I do abstract algebra in a week!?
torn willow
supple torrent
Groups, rings and fields, field extension and galois theory
;-;
And some introductory part which I have an idea of
It's a refresher course so covers a lot but in less depth
But then I wanna revise the ideas too, any resources for that would help
deep mango
But then I wanna revise the ideas too, any resources for that would help
You could use fraleigh's intro abstract algebra book, it should be quicker to read.
People here hate it but it's probably the right thing if you want a quick course with a little less depth
deep mango
Hello.
twilit cradle
Can someone answer my question in #book-recommendations
devout nacelle
vagrant kestrel
hi, this is a random question, but could one use statistical mechanics on like, ...
Yes both of those have had statmech applied to them
viral barn
chjeeee
thorn brook
narrow rock
Disliking Hatcher is like groveling over Rudin.
sick burrow
I thought people liked Hatcher
meager sonnet
Hatcher requires a lot of visualisation and intuition
leaden torrent
hatcher is a good text, people are just contrarians
leaden torrent
you just have to think about mathematics like an actual mathematician does
rather than like a 2nd year undergrad does
deep mango
modest rune
Disliking Hatcher is like groveling over Rudin.
Ive hated rudin since day one (for bad reasons)
sick burrow
What does it mean to "think about mathematics like a 2nd year undergrad"
Like what are you doing that you shouldn't be
Is it just everything?
I mean I just finished my second year of undergrad and I don't know shit so
leaden torrent
focusing on fine details of proofs rather than conceptualization/arguments
see that one Tao blogpost about "rigorous" vs "post-rigorous"
modest rune
Honestly I think it helps with hatcher to have someone who already knows the big ideas
Because hatcher doesn’t always express them well
Like he sort of just does the details without clearly describing the intuition
And if you’ve never seen how topologists think
Its not necessarily obvious
bronze pelican
But then you read Szamuely's book and a lot of the fine detail are just wrong and this fucks up your conceptual understanding
modest rune
Composition is a fine word for that
Insofar as it captures the idea of doing one thing and then another
leaden torrent
composition should only refer to writing fuchs-style counterpoint
bronze pelican
Composition of automorphisms of the universal cover
deep mango
Ultraproduct pre-post-rigorous.
Typical foundations. Clinging to the safety blanket of precise exposition.
latent forge
Typical foundations. Clinging to the safety blanket of precise exposition.
Is this a continuation of last time
deep mango
untold sapphire
hard agree lol. I have had people try to tell me the same line on Hatcher, that it's just meant for more mathematically mature people who are able to fill in the gaps themselves. I think that's nonsense.
odd narwhal
Hatcher is good if you already know the material
Or know someone who already knows the material
velvet dagger
I felt like Hatcher is good for people who already have it lol
But there are ways to communicate said visual intuition. Hatcher does a trash job at that
modest rune
i think hatcher is a good book if you are smart (already know AT) but not if you are dumb (have not learned AT)
velvet dagger
Also I think he overshafts algebra side lol
velvet dagger
He doesn't draw the pushout diagram in Van Kampen, takes way too long to do category theory, etc
velvet dagger
I remember 3 being kinda bad, 2 was the high point
modest rune
takes category theory as a prereq
thats my hot take for today
i wont take any questions
blazing pawn
I have a question
3.1 and 3.2 are ok 3.3 is kinda messy
I feel like "category theory as a prereq" could mean a lot of things
nimble shuttle
the szamuely errata probably has errata
bronze pelican
Szamuely fkin sucks
untold sapphire
i have one good friend who likes Hatcher a lot but she's an infty-cat gigabrain who works with these crazy formal models for homotopy theory so i think for her the geometric content of hatcher is a breath of fresh air
velvet dagger
One day I will probably procrastinate so hard on stuff I'll write an AT book and it'll make the rest obsolete
Tbh nah probably won't happen I'll procrastinate by playing video games
modest rune
We can write it together dami
sly thistle
i have one good friend who likes Hatcher a lot but she's an infty-cat gigabrain ...
For a second I thought you wrote thatcher and I had a massive recoil
untold sapphire
homotopy theorists are notorious Tories
velvet dagger
Also holy fuck this paper I'm trying to read is typo city
Fuckin annals paper too
How is that matrix in SL_2(R)?
It's fixable if you make either gamma u or alpha u a negative
And if instead of dividing the diagonal terms by 2 you multiply the other ones by 2
So unless the rest of the argument isn't even susceptible to that modification things are fine
But like why this is a 21 page Annals paper I feel like it could've been edited better
bronze pelican
smh
deep mango
Just gonna ignore whatever dami is going on about. Hatcher is good for a person who is more topologically minded and less algebraically minded, because it doesn't really do enough topological exposition and it does a little too much of the diagram chases and stuff.
So Hatcher is good for me
I could see it being very very bad for someone who is the opposite
sharp mulch
Lol
rancid meadow
well if hatcher sucks and ryc likes hatcher that must mean that ryc sucks
sharp mulch
Lol
sharp mulch
Downvotes to the left as well
deep mango
Ugh... Why am I cursed to be the only wolf in a flock of idle sheep...
rancid meadow
why is it called GL_n
isnt this just the group of invertible matricies
what does GL mean
modest rune
general linear
rancid meadow
what does that mean
modest rune
to distinguish from special linear
rancid meadow
deep mango
Keep in mind that if you want a group of linear operators
rancid meadow
The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
thx wikipedia
rancid meadow
im not actually sure what general linear position means
modest rune
i think thats completely post-hoc
deep mango
My thought was that it is "the most general group of linear operators"
modest rune
i think they called it general bc it was general
yeah i think that still ryc
i dont buy it
deep mango
General linear position of a group of vectors just means they're linearly independent I think
deep mango
Because this is what you get "generically"
If you pick vectors at random
Something like that
rancid meadow
modest rune
~~its not like the general in general position makes sense anyway
rancid meadow
if u pick random vectors with uniform probability what is the chance they are linearly independent
modest rune
i think least restrictive makes sense for GL
general makes sense for GL
especially since SL exists
deep mango
if u pick random vectors with uniform probability what is the chance they are li...
100% if by uniform you mean like, pick a unit vector
rancid meadow
unit vectors arent very interesting
deep mango
You cant pick numbers from R uniformly randomly
rancid meadow
deep mango
rancid meadow
ok i slept through all of intro probability
vagrant kestrel
this is a case for finitism
deep mango
What would be the probability of picking a point between 0 and 1?
rancid meadow
right yeah okay i figured the answer to my question was going to be P = 1
deep mango
Yes
rancid meadow
but why is this the case
rancid meadow
that you cant pick numbers from R uniformly randomly
modest rune
just use the measure on S^1
deep mango
Well the real answer is that probability spaces are finite, and there is only one translation invariant borel measure on R which assigns the unit interval a non infinite measure
(a multiple of) lebesgue measure
And lebesgue makes R infinite measure
There's some proof of this
It doesnt matter
modest rune
this is just the haar measure isnt it
deep mango
The point is "theres no sensible way to do it, because probabilities are countably additive"
deep mango
So if [0, 1) has prob 0, so do all the other intervals
And then the union of them has prob 0 too
Cause countable sum
Why are probabilities countably additive? Because it makes sense to say that the chance of any one of a sequence of things happening is bigger than the chance of some event in a finite subset of things in the sequence happening.
(this ends up implying that the chance of something in the sequence is actually the sup of the chances in finite subsets)
Maybe slim has a better explanation of morally why probabilities are countably additive
I dont really know why this is natural myself
I think it makes sense
I just don't have a good reason why
Like
I don't know how to say to someone who doesn't know measure theoretic probability that it doesnt make sense to pick numbers uniformly from R
Something about the uniformity spreading out over all of R means its zero everywhere
Like you're trying to take a constant function and say its integral over all of R is 1
summer nest
devout nacelle
Something about the uniformity spreading out over all of R means its zero everyw...
What does uniformity mean in this context?
forest jackal
yeah for similar reasons if you choose a random point in a high dimensional ball, it will with high probability be:
a) very close to the surface of the bounding sphere
b) very close to the equator
students often find b) especially counterintuitive, because by symmetry that is of course true for ANY equator!
devout nacelle
Oh, makes sense
leaden torrent
its a probabilistic argument
timid spindle
yeah for similar reasons if you choose a random point in a high dimensional ball...
I think it's b but the logic behind it is bad.
umm why can't I text in #numerical-analysis even tho I have the role
forest jackal
yes it is both.
last oxide
violet spade
Is there a proof for this
You pick a random unit vector by taking a gaussian vector X~N(0,I_n) and normalizing it X/||X||. If you have two random vectors, Choose a basis so the first random vector is the first basis vector. Then the dot product is just the first coordinate of the second vector in this basis, which should be small since there are n coordinates. Or use gaussian concentration and Central limit theorem for more formal
forest jackal
the way I got my students to do the equivalent problem of equatorial concentration was computing/bounding the hypervolume of an equatorial slice of the n-ball or the surface measure of the same slice of the (n-1)-sphere. orthogonality to a fixed vector (the pole) is of course equivalent to lying exactly in the equatorial slice.
mortal oasis
You pick a random unit vector by taking a gaussian vector X~N(0,I_n) and normali...
Ahhh I see so basically the dot product gets close to zero because there are just so many points
Wait that’s kinda odd to think about
Why can’t I just visualize N dimensions
is this similar to the curse of dimensionality specifically in knn
half meadow
Hello cab you guys help me with a simple question?
leaden torrent
reef carbon
summer nest
I'm nobody 
toxic schooner
neat lintel
outer lion
toxic schooner
pale orchid
languid root
⛓️ ✂️
toxic schooner
whole copper
faint loom
brave hollow
pure sun
@astral marsh how is your orientation at MIT going?
Or has it wrapped up? Last time we chatted I think you had just started it
modest rune
arch seems to be quite busy playing league
pure sun
I mean i spent one year of college just playing dota 2 so i get it hahaha
astral marsh
<@705929438814273608> how is your orientation at MIT going?
it's an academic summer program
lasts through mid august
it's going fine
pure sun
Ah ok, that’s good to hear
pure sun
Hahaha sounds about right
astral marsh
arch seems to be quite busy playing league
thats just because they actually legit let us leave as soon as we finished our calc 3 midterm
i got so much free time from that
pure sun
Instead of waiting until the exam time is over?
pure sun
Yeah that’s really nice (and also in my experience it’s the standard for all college exams)
astral marsh
i see
pure sun
Yeah i remember standardized tests in hs are the worst cuz its like, you have 4 hours and if you finish in 1 you just sit there
But in college, for every proctored exam i ever took, you just leave when youre done
The only requirement i ever put as a prof is that if you finish in the last 5 minutes you have to stay
Cuz otherwise it is super distracting dor those trying to finish
astral marsh
Ah yeah
pure sun
Which i thought was reasonable and never got complaints about
modest rune
my HS always let me leave
modest rune
i went to a private school and they probably just willingly broke the standardized tests rules
small school full of nerds
mortal oasis
Thanks!!
fallen delta
when are you considered a homie
ancient flame
when you get the honorable role
modern pawn
Something's mentioned about it in the #info channel, I assume you just get deemed honorable?
There's no set things that get you the role, I don't think
deep mango
basically it's usually a pattern of consistent help / contribution, often in more advanced channels than just the question ones, which is noticed by other honorables/mods enough that they nominate you for the role.
neat lintel
it's basically up to two things
- were you nominated by other honorables
- is there no moderator blocking the decision to make you honorable
cold needle
yes
narrow rock
trusted can also nominate
errant kiln
can they block as well 
narrow rock
probably ye
errant kiln
Imagine being disliked by one of the many honourables/trusted but liked by all the others
odd narwhal
On that note, what does it take to get active role
errant kiln
It gives you access to the well-renowned #feet as well, so it's really worth becoming honourable.
errant kiln
On that note, what does it take to get active role
being active
odd narwhal
Define active
odd narwhal
That's not well-defined, i'm reporting you to #foundations
narrow rock
you should be afraid of jesse.
neat lintel
You can't get honorable
blazing pawn
an elitist club
PTY dont call it that im trying to get you into it
bronze pelican
I rarely post outside of the general channels and #mathematics-voice
Only times I go into advanced mathematics channels is to ask a question
sharp mulch
deep mango
at very least trusted. i mean come on
sharp mulch
Lol Jesse in shambles
vivid halo
I do, which is why I'm not honorable
bronze pelican
This is a conspiracy against light mode users
velvet dagger
This is a conspiracy against light mode users
No it's not
Conspiracies are secret
I am telling it to your face
Ur bad
For using light mode
astral marsh
limber perch
wow geometric algebra really is the chad way of framing physics equations
swift crescent
Is anyone studying math while working in a less-related field?
quasi jettyBOT
edgy atlas
Curious, do you math majors here take paper notes or LaTeX notes?
I always take paper notes b/c it helps me remember better I think
modest rune
Most math majors take paper or like tablet notes
static loom
just drawing figures is too much of a hassle in latex for it to be viable imo
edgy atlas
I'm interested in tablet notes but with a caveat. I can never get as "thin lines" or as much precision on tablet notes as I seem to get on paper
hahah TikZ is a nightmare I agree
had to use it for my discrete math class 🤮
modest rune
i cant speak to other tablets
but you can both zoom in and do stupidly thin lines on an ipad
rancid meadow
i take ipad notes and then type them up in latex after
sharp mulch
I used to take latex notes but it was slowly giving me carpal tunnel and repetitive strain injuries
I do not recommend
sick burrow
I solve this problem by not taking notes
sharp mulch
I started taking notes because I kept falling asleep in class
Or
I started taking latex notes because I kept falling asleep in class
I used to take notes on paper
And would fall asleep
sick burrow
Lmao
cold needle
ballsy
sharp mulch
Lecture halls were simply too good for sleeping
static loom
I scribble a few things down, reference it maybe a few times before I throw it out at which point it's either in my memory or I roughly know what to google/ where to look up
sharp mulch
Someone droning on monotonously
cold needle
nice n cold
sharp mulch
Comfy-ish seats
They were either nice and cold and dark and breezy
Or hot and humid
Both of which are conducive to falling asleep
devout nacelle
now im sleepy
Try some 🍞 🧈 🍞 to dodge sleepiness
narrow rock
arent you in florida metal?
narrow rock
how is it 1
sharp mulch
Lol
narrow rock
you can go to sleep
sharp mulch
You can always sleep
narrow rock
You can always sleep
not at 1
cold needle
my body had to shut down
devout nacelle
Eh, power naps are fine imo
sharp mulch
Practice self care
devout nacelle
I often recharge myself mid-day, it works fine
devout nacelle
Ange Instagram self-care celeb arc
cold needle
someone tell me a story
im bored
and i don't want to watch my entomology lectures yet
blazing pawn
So meta!
blazing pawn
Ultra have you seen the grimes butter bread
cold needle
Writing it down for future reference
blazing pawn
is that what we've been referencing the whole time
blazing pawn
Grimes terrifies me
sharp mulch
Who is grimes
blazing pawn
Who is grimes
idk singer or something
She has really terrifying diets like only eating spaghetti for 2 years
Im like 99% sure someone else invented this first
sharp mulch
I see
blazing pawn
I think ive seen this in a TV show
Or something
Collective unconscious
Oh also grimes has this thing where she takes a stick of butter, like a full stick, melts it in a pan
and then takes a slice of bread
and saturates the bread in the melted butter until it absorbs it all
And then she spreads jam on top and eats that
rancid meadow
that sounds so much worse than blocks of butter
sharp mulch
Pop artists trying to keep up the trend of dying by 28 or something
blazing pawn
she also apparently like
When she wakes up she eats pills of caffeine
falls back asleep
blazing pawn
and then wakes back up when the caffeine pills have entered her system
rancid meadow
a lot of people do that. its a good way to work
cuz u get 20 mins of nap and wake up amped
blazing pawn
Azealia banks is a really bad person but shes funny
blazing pawn
Im not groggy for 20 minutes when i wake up
rancid meadow
thats cuz ur 12 moth
blazing pawn
Sorry ur weird
cold needle
lmao
sharp mulch
Are you groggy for 20 minutes when you wake up?
No groggy for me
Mornings are wonderful
Sunshine
The sun
blazing pawn
Wow
rancid meadow
yeah okay ange is perfect
sharp mulch
All praise the sun
blazing pawn
Sunshine and the sun
modest rune
If I sleep enough I wake up fine
blazing pawn
Hello maximilian J
modest rune
But I get a migraine if I don’t consume caffeine quickly
cold needle
maximillian J
sharp mulch
Wake up at sunrise
cold needle
lmao
blazing pawn
I think theres only supposed to be one l
sharp mulch
As the first rays of dawn peak over the mountains
blazing pawn
That sounds concerning
cold needle
How poetic, ange
blazing pawn
Caffeine dependency
devout nacelle
Wake up at sunrise
Vampire moment
blazing pawn
Scary
devout nacelle
headaches are a sign to me to take caffeine
Caffeine or nap for me
blazing pawn
I dont want to be caffeine dependent because what happens when i get too lazy to buy more
blazing pawn
Also its expensive
rancid meadow
u just order on amazon moth
rancid meadow
its not
sharp mulch
Ewww stop simping for bezos
rancid meadow
its like $16 for 400 pills lol
blazing pawn
I never get tension headaches
cold needle
reach out to a local premed
blazing pawn
Like maybe 1 a year
cold needle
they have extra
blazing pawn
So theyre weird to me
Virgin minor inconvenience tension headache vs chad debilitating pain cluster headache
cold needle
neat lintel
I'm the police, what do you need?
rancid meadow
i knew u had cop vibes
blazing pawn
I wonder if i should get medication for my headaches
neat lintel
i knew u had cop vibes
You will have to come with me
sharp mulch
Oh my
neat lintel
People online say to take LSD
LSD:
Let's
Swallow
D...
blazing pawn
Hilarious!
devout nacelle
blazing pawn
moth microdosing lsd at 17
Reality
sharp mulch
Lots of people (mainly silicon valley tech bro types) talk about microdosing on various drugs
blazing pawn
It was probably inevitable
neat lintel
LSD:
Let's
Swallow
D...
Finish the D word
devout nacelle
Do you have anything better to contribute?
modest rune
Its not microdosing
modest rune
Iirc full dosing lsd has indeed been shown to reduce migraines
sharp mulch
Oh you want moth to take full doses?
neat lintel
Do you have anything better to contribute?
Law
sharp mulch
Oh my
blazing pawn
Theyre these things called cluster headaches
sharp mulch
Have you considered meditation in the mornings
modest rune
I forget the difference
sharp mulch
Wake up at sunrise and meditate for 30 minutes outside
Face the sun
Let the solar energy enter you
viscid pecan
I hear cluster headaches are worse than migraines
sharp mulch
Be a plant
modest rune
Listen to “solar power” by lorde
devout nacelle
bronze pelican
I have a fkin headache
blazing pawn
Cluster headaches is like localized pain in ur eye
bronze pelican
😵
modest rune
Oh
sharp mulch
Have you considered yoga
viscid pecan
Microdose some mushrooms for cluster headaches
modest rune
I guess I get cluster headaches then
blazing pawn
Its fun because the veins start inflating so u can feel ur own skull crushing ur organs
viscid pecan
Works for me
modest rune
Oh no
sharp mulch
Also house plants work wonders
modest rune
Not that bad
cold needle
wtf moth that sounds horrifying
neat lintel
LSD:
Lucy in the
Sky with
Diamons
viscid pecan
Great for migraines
devout nacelle
Yeah Moth, you should ideally seek professional help for this.
blazing pawn
burning
neat lintel
LSD: Landing Ship Dock
blazing pawn
or stabbing
neat lintel
I'm a moth
viscid pecan
Go to neurologist
cold needle
...
blazing pawn
doesnt really feel like that
neat lintel
Go to neurologist
I already went there
blazing pawn
I phrase it that way because it sounds funny but its not that dramatic a fingernail pierced my cornea
So it just hurt a lot
modest rune
I think moth should flippantly use lsd to try to cure it
neat lintel
I may have a mechanical dysphagia so neurologist is the only one who can determine that
blazing pawn
I think moth should flippantly use lsd to try to cure it
Based(?)
neat lintel
@cold needle , now I'm trolling a little bit
modest rune
Don’t do it alone for sure