wow moddy u made wew sad
#serious-discussion
ancient flame
LOL
wispy pike
Why is this even useful
I think it's called "reverse triangle inequality"
In Swedish, it's called "omvänd triangelolikheten"
More precisely, why is this useful if one needs x to evaluate it accordingly to the equality?
pure sun
it's typically written as |x-y| >= |x| - |y|
and that's the form that people would typically apply it in
summer sandal
One thing that comes in mind is proving every convergent sequence in R is bounded.
For every e > 0, pick N and we have |x_n - L| < e for all n >= N. That is |x_n| < e + lL| for all n >= N. Let M = max(|x_1|,…,|x_(N-1)|, e + |L|).
I think I have it right.
|x_n| - |L| <= |x_n - L| < e. Yeah I have it right.
brazen badger
wwa
lunar spear
I saw a wild aussie last weekend
ancient flame
LOL
tepid token
does someone here know how to write an x without lifting their pencil
i saw a tiktok on it a few weeks ago and ever since watching it i always notice when i write a x lol
fathom swallowBOT
strad
tepid token
YEah
honest veldt
Looks kinda like this
tepid token
Do you mind writing one out in steps and taking a picture?
honest veldt
I'll try, give me a second
honest veldt
The person grading your final will be so impressed by your x that they'll just unconsciously give you that A+ without thinking
tepid token
thats the important thing that matters
hexed hound
Ok now that's a cool x
sleek wing
but not quotient monoids
they absolutely do
how could quotient rings exist if quotient monoids didn't
bright hill
Why is this even useful
this gets abused af in real analysis lol
storm sage
^^
cold needle
adding and subtracting something is a good trick to know
tender tulip
this gets abused af in real analysis lol
I literally used it 2 days ago
Specifically proving two functions from N to an ordered field are cauchy, then so is their product under the field
neat lintel
"cauchy sequence"
cold needle
lmao
cold needle
oho okay
very nice
did u register for classes or
i forgot what happens in preview
lol
mine was virtual so its a fever dream
ancient flame
I'll do that on friday
cold needle
ah ok
dang
so sad im actually going to be back in gnv saturday
lol
just brushed by each other ig
ancient flame
LMAO
tender tulip
"cauchy sequence"
"the product of two cauchy sequences" seems less intuitive then " product of the functions"
ancient flame
rip
tender tulip
at least for me at first
ancient flame
the worst timing lmao
cold needle
fr
ancient flame
I'm sure we will have plenty of opportunities during the semester
bright hill
"the product of two cauchy sequences" seems less intuitive then " product of the...
I literally have no idea what you're talking about
tender tulip
I literally have no idea what you're talking about
let f(n) be a function from N to R, and g(n) be a function from N to R
if both are cauchy, ie, for all epsilon greater than 0, there exists an N such that for all n, m > N
|f(n) - f(m)| < epsilon
then f(n)g(n) iscauchy
bright hill
so a cauchy sequence?
tender tulip
y e s
bright hill
neat lintel
lmao
tender tulip
b ut product of cauchy sequences sounds weird as shit because you don't know what the product is
neat lintel
tender tulip
it could be convolution (?) or some shit idk
cold needle
bright hill
ah yes, refusing to use established jargon because it "sounds weird" is such a good idea
no one would be confused at all /s
bright hill
it could be convolution (?) or some shit idk
this also makes no sense btw
tender tulip
fair
grizzled grove
Is class field theory necessary to learn automorphic forms?
Class field theory is so hard
vivid halo
Is class field theory necessary to learn automorphic forms?
sort of
it is possible to stay strictly on the automorphic side and never study anything related to the Galois side, in which case you maybe don't need class field theory
as soon as you care at all about the actual correspondence between automorphic and Galois then yes you need class field theory, since it's the simplest possible case of this correspondence
for what it's worth you do not need to learn all the proofs of class field theory, especially in the global setting. Also understanding the proofs of class field theory is pretty separate from being able to do explicit computations with it
pure sun
I second what ng said. I never bothered to learn most of the proofs of global class field theory but I did successfully use global class field theory frequently
grizzled grove
for what it's worth you do not need to learn all the proofs of class field theor...
Yeah but why do we need to know the proof of the local class field theory more than global class field theory. I am still absorbing that upper numbering and lower numbering thing introduced in inertia groups. These all concepts are insane and highly non trivial. For example coming up with tamely ramified and wildly ramified stuff is not easy to understand because of weak algebraic thought process of mine. The thing is langland programme could be appreciated best if one knows both the world. Like you said Automorphic side and Galois side. Maybe I am more into analytic stuff that is why I feel automorphic side maybe easier for me to understand first and then I come back to Galois side or motivic side
vivid halo
the proof in the global setting is harder than in the local setting
there's a proof in the local setting that is extremely explicit and lets you do computations with extensions of p-adic local fields, so it's worth learning
there's a more complicated proof using group cohomology that is maybe less worth learning
it's certainly less explicit
grizzled grove
Kronecker Weber theorem uses local class field theory. In global class field theory what is more to be done after we already have kronecker Weber theorem
vivid halo
global class field theory is a lot more than just Kronecker-Weber
I guess maybe if you only care about Q
then Kronecker-Weber recovers a pretty substantial part of the story sure
grizzled grove
there's a more complicated proof using group cohomology that is maybe less worth...
Yeah but like I can see that math involves art of packaging. And till now i have understood only inverse limits. But there is also higher form of packaging information and that is cohomology. And maybe scheme theoretic language is also packaging
vivid halo
yes, but that's just something you have to deal with
but okay sure if you only care about Q then you can avoid a lot of additional complication
it's just you'll end up with a description that doesn't generalize to other number fields
grizzled grove
It is impossible to understand automorphic forms from books or lectures. So I am watching lectures of Kevin buzzard and they aren't that easy too...especially the local class field theory. So I will watch a course on class field theory. None of the books have detailed explanations about why we are doing stuff that way.
Like Ravi vakil has algebraic geometry book written in informal way
I wish to see informal books on every topic
vivid halo
Milne's book explains the "why" of class field theory fairly well I think
but yeah imo it's very hard to get into automorphic forms stuff without first knowing algebraic number theory and class field theory
since a large goal of Langlands is to generalize class field theory, you have to understand that first
grizzled grove
I know algebraic number theory till chebotarev density theorem. But now class field theory is dangerous. But as far as I can see in iwaniec book there is no class field theory at all
It's just automorphic forms on GLn
vivid halo
yeah just read Milne if you want to learn class field theory
grizzled grove
And the definitions aren't that complicated
vivid halo
Milne also has an algebraic number theory textbook
grizzled grove
Okay I'll look at it. I have Kiran kedlaya notes also
vivid halo
and the class field theory textbook cites it sometimes
so that's nice
Cassels-Frohlich is also a standard text on this
vivid halo
lol
Milne's book is Tex'd if that matters so much
certainly you'll have to get used to reading old typesetting at some point lol
maybe even learn French
grizzled grove
Yeah I think french people have some advantage that math is natural to them because language helps in how you think
Like in some language humor is natural
vivid halo
also yeah this stuff is quite hard but it's also quite doable, you just have to buckle down and learn some hard stuff
cohomology is unfamiliar but it's just a tool you have to use sometimes
grizzled grove
And I think I have to learn representation theory of both finite and infinite dimension of finite and infinite groups?
Milne covers that?
Btw till now i thought mordel Weil group was hardest...but now I think Weil deligne group is hardest I know of, which I still not understand. Maybe groups which I have no idea about like selmer and Picard maybe more harder than this. By hard I mean the definitions. Not how mysterious.
vivid halo
Milne covers that?
not really
and yeah by far the biggest part of Langlands is representation theory
I wouldn't really call the infinite dimensional representation theory a prereq to Langlands so much as it is Langlands
so usually people will like
learn some rep theory of finite groups and then learn Lie theory
grizzled grove
I still thank to God that Weyl group didn't come in this langland business. Because Weyl group is too much algebraic
vivid halo
meaning finite-dimensional representations of Lie algebras
and then learn the infinite dimensional stuff by example
e.g. learning the relevant infinite dimensional representation theory for GL_2
vivid halo
I still thank to God that Weyl group didn't come in this langland business. Beca...
lmao
Weyl groups show up all over the place in Langlands
grizzled grove
Why do the Lie group come in picture...there was no manifolds to start with ...
Weyl group? But till now only Weil deligne group has come in local class field theory
vivid halo
if you have an automorphic representation of GL_n(A_Q) say
then this decomposes as a tensor product of representations of GL_n(Q_p) over all primes p, and a representation of GL_n(R)
GL_n(R) is a Lie group, and we usually study representations of this in terms of certain representations of its Lie algebra. The relevant definition is something called (g,K)-modules.
so we're studying infinite dimensional representations of a Lie group, and of p-adic Lie groups
the local Langlands correspondence says that these infinite dimensional representations correspond to certain representation of the Weil-Deligne group
grizzled grove
This tensor product breaking into archimedean and non archimedean places sounds familiar . Like product of absolute values is 1.
vivid halo
so that's how these things show up
the tensor product over places of Q is because the adele ring A_Q is being used here
grizzled grove
Yeah like the gamma factors corresponding to infinite and finite places
In tate functional equation
vivid halo
right
but okay so now the problem is about studying certain infinite dimensional representations of G(Q_p) and G(R) where G is some algebraic group
there are people in the Langlands program who just like
study the rep theory of G(Q_p) and that's it
they just study that part of the program
I think by now the rep theory of G(R) is more or less understood to the extent that Langlands wants you to understand it
grizzled grove
Btw the field are of two type. Local and global. But global has example extension of Laurent series over finite field.
Local has finite extension of rational function field over finite field.
How these two are different
vivid halo
global means either number fields or function fields over a finite field
i.e. finite extensions of Q, or finite extensions of some F_p(t)
grizzled grove
Sorry I wrote wrongly
vivid halo
local fields are either like
R, C, finite extensions of Q_p, finite extensions of F_p((t))
grizzled grove
But what about just the simple finite field.
vivid halo
those are sort of like
"local fields of dimension 0"
where local fields in the usual sense are "local fields of dimension 1"
there is some higher dimensional notion of local field that includes examples that you would expect
e.g. F_p((t_1,...,t_n)) is a "local field of dimension n"
grizzled grove
I see ...Wikipedia doesn't talk like that. So finite field is in both local and global because in both case dimension 0 will give finite field
vivid halo
yeah
there's also a sense in which Deligne-Lusztig theory is "Langlands for finite fields"
Deligne-Lusztig theory is some way to completely classify the representations of algebraic groups like G(F_p)
more generally finite groups of Lie type
grizzled grove
Like Function field and Laurent series of dimension 0 would not have that indeterminate T involved? That's why we are left with just finite field
vivid halo
yeah exactly
you can mix these two notions as well
for instance Q_p((t)) should be a local field of dimension 2
grizzled grove
This Deligne is coming everywhere ...makes me jealous
vivid halo
yeah Deligne is the best
other than Grothendieck maybe the best algebraic geometer who ever lived
grizzled grove
Yeah I see many fans of Grothendick but till now i haven't come across his work to appreciate. For me Dirichlet is best and that's why my profile pic is of him.
vivid halo
but yeah literally the entire automorphic side of Langlands is rep theory especially of the infinite dimensional type
which is great if you like rep theory and harmonic analysis
grizzled grove
I don't like any of this stuff till now unless I come across its arithmetic applications. I am more from the background where I wish to understand asymptotic behaviour or arithmetic functions or to look for zero free regions of derivatives of L functions. Or converse theorems for L functions.
vivid halo
probably the automorphic side will be where you want to be then, since that's the side where we can say the most about analytic properties of L-functions
this stuff does have arithmetic applications (think arithmetic applications of modular forms and things like this)
grizzled grove
Thanks for the discussion.
Yes my mentor does subconvexity stuff on them and told me to read that but i usually read first whatever I am not told to read to understand why I should I read what I was told to read.
reef carbon
alright so this is more of a soft question but like. probably complex-analysis related if anything
idrk where to put it cause im not like, specifically looking for help with anything
but anyway
i was fucking around with a complex function plotter and i decided to plot the function $f(z) = \sum_{n=1}^{\infty} nz^{2^{n-1}}$ (or rather its 20th partial sum but it looks close enough to the real thing)
fathom swallowBOT
Ann
reef carbon
and what i got was... this
and this gives me such hyperbolic vibes but i have no idea if that even remotely makes sense
reef carbon
(for comparison heres the identity function plotted in the same thing)
neat lintel
what do the colors even mean
reef carbon
hyperbolic vibes
i mean that the picture looks like it's a poincaré disk projection of something in the hyperbolic plane
hue denotes argument, saturation denotes magnitude
neat lintel
anything that is a disk looks like the poincare disk
ok ig there are circles around that seem to have boundaries similar to geodesics on H2
so I'll give you that
reef carbon
it looks like a disk with things getting exponentially smaller as you get closer to the boundary
neat lintel
Ok
bright hill
I was watching a 3b1b video and this literally sounds like some classic lowmath shitposting
fervent pebble
all math is lowmath
sacred forge
einstein notation is interesting
neat lintel
no
arctic grove
neat lintel
this is barely a shitpost
junior stump
hmm
the mathematic hidden behind sudoku is actually pretty interesting
i really wonder how did they get this formula
neat lintel
by counting in smart ways
wispy pike
hey quick question
does the factor theorem hold true for all polynomials? Including irreducible polynomials?
unborn trellis
I assume you're talking about polynomials over C, which all split
Irreducibles in that case are just (x-a), so yes
wispy pike
How about over R?
unborn trellis
Then you have irreducibles of degree 2, right
wispy pike
yep
unborn trellis
take x² - 1
wispy pike
yep tohught about (x-1)(x+1)
unborn trellis
yeah, you are correct about that
naive chasm
Sorry meant to write x² + 1
not over F2
unborn trellis
But (x+i) is not an element of R[x]
So factoring doesn't work unless the field is algebraically closed
pure sun
it does work if the root is in the field you're considering. e.g. if c is a real root of a real poly f(x), then (x-c) is a factor of f(x) in R[x]
wispy pike
Still new to term field, but from your answer, the answer is no
pure sun
in this case field just means the set of numbers you're considering
wispy pike
ohh
pure sun
i.e. are you considering real numbers or complex numbers
wispy pike
okay that makes more sense!
Thank you guys for the replies and insightful answers. I now have a better understanding of the theorem, I'm happy!
pure sun
np
junior stump
by counting in smart ways
that doesn't really answer my question
neat lintel
How do i learn more abt
Complexity
And computer science
Like should i get some textbooks
neat lintel
How do i learn more abt
As a complete beginner
Who heard some things about it
Like i heard about polynomial time and big O notation
Or smth
And the yt videos
naive chasm
Complexity
comp theory or comp of programs ?
neat lintel
Programs ig
dire spindle
How can I get my younger brother more into computer science
I tried getting him to do some CS50 Python, which starts off extremely basic
but he only wants to solve problems which he already knows how to do
sick burrow
does anyone know of any good windows applications for scratchwork?
using my computer as a tablet that is
OneNote is like a solid 90% of what I want but my two issues with it are (1) I keep accidentally bringing up text boxes which then brings up an on-screen keyboard and is really annoying and (2) I'd ideally want a few more features for doing geometric stuff like better shape tools and a compass or angle measure or something
a hypothetical perfect app would also have a drawing to TeX converter built in but that might be too much to ask
vivid halo
honestly onenote is still probably the best on windows
sick burrow
should I use "onenote" or "onenote for windows 10"
because apparently they're different applications
vivid halo
there's not much of a difference other than like different user interface
onenote does have some latex functionality
it's somewhat limited but it's better than nothing
sick burrow
I don't really care about TeX functionality beyond the dream of an app that would let me handwrite things and turn them into TeX (and even that wouldn't be that helpful)
vivid halo
yeah unfortunately this isn't really a thing
detexify is already hit or miss enough with single characters 
uncut socket
@chilly coral @lunar spear I think you guys would enjoy reading the first sentence based on your jokes in that channel
And the fact that's a lot to memorize while getting kidnapped
chilly coral
Dudes prolly senku or sumn
lunar spear
<@521464336284254228> <@852256521986768957> I think you guys would enjoy reading...
Im more confused with the speed limits
peak tide
i guess that knowing they are political science majors allows you to define a nonuniform prior distribution on destinations they might be taking you to, which should improve your answer
bayesian estimation ftw
uncut socket
Im more confused with the speed limits
Why?
uncut socket
@molten tartan ECE? Electrical Computer engineering?
kind horizon
I try but I just feel so unproductive
neat lintel
i get burned out extremely easily and this is a good cure
neat lintel
I try but I just feel so unproductive
you're going to be more productive later if you rest now and then work later, rather than try to force anything while burned out
you're not being unproductive by taking a break, you're just increasing productivity later
the mind, much like the body, must rest
storm sage
I try but I just feel so unproductive
happiness is productive :)
you can do more when you are happy
think about what you define as "productive" and why
molten tartan
<@615703104004030496> ECE? Electrical Computer engineering?
Yep!
carmine creek
happiness is productive :)
this is so wholesome
fading zealot
My mom asked me if I want an air conditioner in my room, I already had two electric fans and I'm fine, so I said that since the energy consumption of an air conditioner is equal to nine fans, unless I cannot stand the temperature even when there are eight fans operating in my room at once, I won't need an air conditioner.
My mom then said that my logic is ridiculous but cannot point out why.
Is my logic ridiculous?
neat lintel
energy consumption does not necessarily equal energy output
this doesn't sound like a real conversation that real people had
peak tide
another consideration, fans generally run nonstop unless you manually micromanage them, whereas air conditioners typically have thermostats
fading zealot
Hmm
pearl moth
fans and air conditioners perform different functions
fans actually make your room hotter
pearl moth
another consideration, fans generally run nonstop unless you manually micromanag...
they have to to avoid slugging
neat lintel
fans actually make your room hotter
not necessarily, it could be blowing air from in and out. not a closed system
lunar spear
mine be like 1,5m from the window
neat lintel
how do i get accees to ask questions in the algebraic topology thread
lunar spear
advanced role
tawdry falcon