#serious-discussion
1 messages · Page 259 of 1
Noo not really but I like few songs of him!
or ig advisor chan
Wow
Good luck, how much longer?
Algebraic geometry?
yeah
Its a band but i love you
Oh well
Hope it’s going to be a bit more interesting than it sounds 🫶
till 9
till 9 is okay-ish
Thank you! Me too then🤭🤭
I guess it depends in part on whether it's "constantly doing things until 9" vs "on the clock until 9 but not necessarily super busy"
Good luck 💀
Either way yeah you're at the home stretch Ally!
And what are you guys working on?
Okay that's a plus
Hope you can get some rest and respite soon
Good night 😄
ahh... I hope you feel better in the morning
🤍
have a wonderful night! 
Thank you guys!! You too! 😭🫶
Good night!🪷💁
Omg u math ppl r stinky
im doing my presentation in the math building actually cuz i was assivned and
wHY THIS smell LKKe 19 cans of ASS
Ur one of the exceptions dw
oops idk why it didnt reply
Have you considered heeding the advice in your PFP?
Advertising of this form isn't allowed
What a description

Who knows Gilbert strang?
Like personally?
my friend took his class and met him at MIT
wtf
My highest was a Rem 😭
lmfao
what is trignometry
google it
trigonometry is the study of triangles, except it's secretly the study of circles
No comment
Wow,really cool
Im jkk
i thought it is the study of ratios of side lengths of triangles
Its shy
Hi omg im free
Im so tired but
Idk im feeling kinda down
Might take a walk in the city
a walk couldn't hurt c:
I love night walks to clear my mind 
but u get eaten alive by the mosquitoes
yea ikr
you already know where im enrolled in anyway so i might send night walk pics to u
I'd love that 
night walk photos are always so beautiful 
wtf why only 14k i usually get like 10k a day but this is barely above normal for a work day
I need to get it to 20k
Keep it up! You're on the way!
you can do it! 
ughhhh i just came back from vacation and now i’m trapped in the suburbs again
hopefully no one attacks me in the streets
so jealous
Ha ha ha
Ok
We'll put out a notice that we'll beat anyone up who tries
we dont talk about that one day i was in a town at 9 pm in the winter
Uh oh 
i knew someone was gonna clip that smh
unrelated but i would like to know if anyone plays deltarune in this server
15.6k
Time to recircle even though my phone is nearly dead
I played chapter 1 loooong ago when it first came out, didn't try chapter 2 or now the full version
BASED
Nah, It's all about Sigma Sigil around here
Good work! Keep going!
I dont really need this but thank you
Fair enough lmao, you're welcome
it was funny
he meant well but thats not an area where i need encouragement because im good with it
but i could use it in areas like math
Guess that's just my default when I see people getting good work done :P but yeah you def got it when the math comes back around
(general statement)
Some people really want to dominate everyone or be dominated by anyone (this is for my ex)
I feel like the last parenthetical contradicts the first
Also
What
Its general statement for some people
I think "for some people" makes it fairly non-general
And my ex come in some people
"For all people except the ones for whom for this is false"
Anyway more crucially than the nature of generality... I just wonder if this really needed to be shared with the class
She really thinks she can dominate everyone
But reality is something else she don't know about this
My bad 😂
I imagine these results are likely not applicable to all
Sure they aren't applicable to anyone
But if they r applicable to someone that means someone isnt right in mind
Anyway like I said, the generality of this is secondary
The really important thing is that I question whether mathcord was the ideal place to talk about who dominates whom
🥀
Yeah, not the most ideal place
Im talking this in discussion
And I see everyday people talk some shit
Can't i do today?
People talk about a lot of things here, obv this channel isn't math centric
Lol just chill
And I haven't yet said anything about rules because domination is a bit vague
I'm just confused
I'm too
I’m three
I thought it is good place to talk about this other than other channelz
Four
A Rem? That's a level? I haven't played it yet...
It can go to negative infinite too
And there is lot of possibilities
What can happen
something something i genuinely need to start buffing up my analysis now that I think about it
meh, that’s an August problem lolll
Any interesting sequences you’ve come by?
I don't like analysis btw
I can think of 1 I saw recently
that’s chill, I like algebraic /topology more anyways
Hey chat?
daily reminder
(for now)
Ur summer might be full of problems mines isnt
this makes me slightly uncomfortable
-1/12 moment
Lol
meh i like it, adds a reason to get out of bed and do stuff
with minimal effort required other than ur brain
/pencil
/paper
/computer for occasional coding
I really thinks they have some misstep doing that series
But idk 😐
no and thats kind of you
What the helly??
Yes it is
everyone has different areas where they need or want encouragement
i love that of you
and as for math you help me so much with it
-1/12 bell curve meme
so while i may have said that about walking you make me a happier person when you encourage me in math
I meant you got the encouragement when it comes to the math! But also I think you'll do well!
Okay wait maybe I'm blanking out but where is this coming from?
Yeah for sure, and I'm glad! Will def keep doing that then!
derivative of zeta at s=0
That's good
Oh oh
this is the sort of thing that shows up when computing zeta regularized determinants of Laplace operators or whatever
oooh sounds interesting
My mind defaulted to thinking Gamma function when I saw a factorial
But then thought wait that's not...
my phone is really dying anyway so ill leave it at 18.8k that makes me mad but wtv dumb phone
tfw computing the determinant of an operator on a Hilbert space
Phone battery running low is a pain
@verbal gazelle welcome to the mathcord! 
my phone battery is always running low fr
Look we still haven't proven the zeta function and ∞! = √2π looks so wrong. I see a pattern 
even my vision is stronger than my phone battery
sometimes it does last a little longer if brightness is basically nonexistent
A bot?
I'm not a bot 
Once I'm working, I'll have a much smaller case I bring around with me which will hold a laptop, laptop charger, phone charger, and that external battery for a phone

I thought you were
but my iphone has lately had a really shitty battery life—as the Big Apple intended 😔🥀
Just so that I have fewer incidents when I'm at < 10% and panicking
Accept my sincerest apology for that
LMAOOOOOOOOOOOOOOOOOO this made my day
dw, I'm not offended at all 
plenty of people have mistook me for a bot by now
I get a new one each day
Haha...good, my lady
Daily? Dang
often more than daily
I guess people are just used to welcome messages from bots
sometimes several times a day
IM DYING IM SRY
I'm glad c:
Me too, the way I talk, they somewhat contemplate me as a "Bot"
no! it's good that you got a good laugh out of it 
your profile star animations are too good to be a bot, i fear
my backpack had a pouch for a portable battery and a usb cable that goes from inside the pouch to the front strap
my days are better when you're smiling 🤍
Excatly, what happened...
And I apologize
no seriously, you're fine 
ur so endearing higher u make my day
Cool, I was searching for some good mentors this side
But got ignored by most of em
Which is blue
if I had seen you sooner, I'd have greeted you 
I just looked at your message history and realized you got ignored for days 
r these kids gonna get out the shower or what
well theyre college freshman i shouldnt call them kids but
nah
I do anyway w my co workers
they r kids
in any case though, we're happy to have you here, and I hope you enjoy your stay
@verbal gazelle
true
what if one of them is 27
It's aight
like do u admit ur a kid
I run my own server...
And was planning to add some real good professor in it
ur a kid to me then
maybe not past the age of 40
amukh is an adult
if I’m expecting a freshman to walk out, but they’re 30, I’d be a lil spooked loll
and ur a kid
or 60
Eh some people start college late. Maybe they work a while, had circumstances that meant they had to delay, etc
True true
they act like kids most of the time anyway
With the way prices are going I could def see working for a while before college so as to minimize loans catching on
As suremark would say, they're kids almost surely
oh yeah im so excited to see what trump has done to my financial aid package
there was someone in my english class in college who seemed like 40s
may dropout frfr
frfr
It’s actually a nightmare paying for college without any form of external help
be it financial aid or family support
Or both at this point...
I’m literally expecting to have to drop out anytime now 😭🙏
Hopefully the aid will stay intact
maybe trump will make it so financial aid only goes to rich people
more money for the rich
I would actually be shocked and intrigued
{insert suremark sticker}
ah? is the admin currently attacking financial aid for students? 
at that point he’s doing it for the plot
hes been cutting a lot of things
What plot
He won't directly cut much except for perhaps Pell Grants and federal aid, but because his shenanigans have hurt university funding at large
die
trump won i might as well dropout and work unpaid for him
The worry would be that they will meet a lesser percentage of students' demonstrated needs in order to offset the losses from the federal research cuts
I will at least try to enjoy the spot I’ve got now while I still have it
I think that's less likely, their inclination seems to be to cut the funding on the level of research (accept fewer grad students, hiring freezes, cancelling research projects)
it was nice knowing you all before trump makes me work all day for free and have no breaks
See you again maybe never
His future memoir where he claims to save education via some backwards play
maybe even making “trump scholarships” or smth
wait is that a thing?
probably
Which makes sense on some level: if they lose the money for a given research grant, they just cut the project that grant funded but keep the educational mission going
Still obviously sucks but yeah hopefully financial aid won't be affected?
Sure with his nephew elon
At that point we just deport ourselves fr
I’d lowkey be down for that
i agree i want that free 1000
especially cause my bf is an international
just deport me out of this shithole
so there’s concern there already
this could be taken two ways…
I'll deport myself into the US and then deport myself out for a free $1000 
but make sure my Springer books make it with me!!
Like okay we claim the $1000, go to a different country where we're gonna thrive more, but still have citizenship to come back if things improve
Win win
That's brilliant plan
deporting urself into the us costs $5m (if trump’s golden visa becomes a thing)

but would we get the choice to pick said country?
What country u want
/which countries would be open to taking a (potentially) huge influx of Americas
Germany?
Well, in my case the default is Morocco, but we're not broadcasting this strategy too much
So we won't be part of a package
also im not comfortable with elon musk and trump’s relationship i saw them two making out in public and being so cuddly
pls save that for private like look one of u owns teslas go to ur office or smth and stop burning our eyes
oh nvm i forgot theyre fighting rn
yeah didn’t you see elon had a black eye
Against whom
Americans!
Meanwhile his wife smiles at, barely kisses him, and otherwise seems like she despises being in the vicinity of him
Wahaah
messiest break up of 2025 so far
culture at its finest
21thh century's most epic relationship

oh wait the name sorry dami
In my case my inclinations are gonna be Germany/Austria/Switzerland, Morocco, Malaysia/Singapore, Australia/NZ, Gulf Arabia, or East Asia
So many choices
Huh
Wait I was typing and thinking about my thing and fully missed what you sent LMAO so I guess all is well
they could also make applications based off of if the country is seeking special skillsets (like math ppl)
i just sent the trump and elon musk couple meme but i deleted it bc the tiktok username (not the image of trump and elon musk) is a bit 18+ inappropriate
didnt realize
but at that point it would be like a preferential-accelerated visa app anyway lol
apologize to the rest of us too smh
noooo not the culture
i dint need to apologize to u suremark
i what
ur next child will be flux
idk ab the other 2
unhinged
Yeah I mean, in my mind
- Germany is good from the pov of jobs/economics (Austria seems very nice and is right next door so throwing that in, Switzerland if I have a ton of money)
- Morocco is my family's original country and is a good place to do business. Maybe not ideal to be an employee
- Australia/NZ are anglosphere and seem very pleasant
- Gulf Arabia might be good economically. If I work there it has to be as an American, as a Moroccan I'd get a lot less money (they pay based on nationality) and I think the treatment can be a bit rough, but I'm close enough to Arab that maybe there's some momentum in navigating culture
- East Asia seems great economically and quality of life, might be a bit of a departure from what I'm used to
- Malaysia/Singapore seem to have great quality of life (weather might be a bit rough and Singapore is kinda expensive, Malaysia being Muslim might give some inertia with navigating cultural matters)
its ok uncle daminark can support u if needed
LMAO
Bro is ready
do NOT lmao at that
what if any of these require giving up ur us citizenship
be nice
Trump doesnt know
hell support ur birthing
no
pregnancy is hard suremark
Thing is, when the US is good and if you have money, imo it is the place to be. So I'm kinda hesitant to give up American citizenship
So probably if I move to a place where citizenship requires giving up American, I'll just go for residency
I feel like some northern european countries could also be nice
I think quality of life is good in eg Scandinavia but they might not be as good as somewhere like Germany for career advancement
u seem so excited ab pregnancy
hm
lets add 2 more children
sharp and deltoid
Like there's the question of where I'd wanna work and build myself vs where I'd wanna retire
Germany is good choice
ally this is the 21st century ppl don’t have kids anymore. sorry to burst ur bubble
be proud of ur children
kawaii~chan will be upset to hear this
unc daminark wish he had kids
so be grateful
he hasnt been able to found his furry partner yet
Unlike u
Bc ur very furrizzmatic
My WHAT
that is a word
I disavow this notion of a furry partner
triple combo goes crazy
be proud
ur furry wife will treat u well
her name is um
idk what u like i forgot
manifolds
I pre-emptively divorce
Absolutely not
Right and dami is a fluffy meowing cat for his discord mod team
he meows at them
and purrs when he doesnt wanna do work
if that’s not marriage material, I don’t know what is
My strongest motivation to work is you saying this
oh a furry girl will find him one day dw
hes already laid his eyes on some furries
omg we love a hyper-productive furrball!💃💃
hopefully they dont find him too furry
Pre-emptive divorce
can always take him to the groomers
your local Petco does a decent job usually
to clean his fur yeah
damn none of these words are in the bible
fr
dami u dont even read the bible
If the alternative is being called a furry...
omg yes!! suremark and his senpai kawaii chan from mystreet had the most adorbs slow dance ever~! ✨💕 they were moving suuuper gently, like fluffy clouds floating on a soft spring breeze ☁️🌸. suremark held kawaii chan close, all careful and sweet, making sure not to step too hard so their dance felt like a soft, cozy hug 🥰🤗. kawaii chan’s cheeks turned the cutest shade of pink—like a little sakura blossom blooming just for him 🌸💖—and she smiled all shy and sparkly, resting her tiny head on his shoulder like the sweetest little kitten 🐱💞. the whole room felt like it turned into a magical fairy tale, with sparkles ✨ and soft pastel lights all around them, making their moment feel like a dreamy bubble of happiness 🎀🌈. every beat of the music was like a heartbeat just for them, making the world slow down so they could stay wrapped up in their soft little love story forever~! it was seriously too cute, like straight out of a magical anime episode~! 🥺💫💕

how did you even get the llm to do this
Genesis 27:44
“And tarry with him a few days, until thy brother's fury turn away”
it was in ur journal
they just spelled things differently in the old testament
ok shower time brb 🧼 🚿
ew whats that smell
i showered 2 times today
my appearance reminded you to shower
good thing i did
math ppl r stinky
erm hello??
that one room in the math building smells
So i dont redact my statement anymore bc of u
wtf
idk wtf youre talking about
yeah bc i wanna be better than u
u smell awful
Imma be honest ryc you did kinda walk into that one
even my armpits smell like petunias
the alternative is you being related to chip
flux*
youll get there some day
dami dislikes chip more 100%
Do I?
would you rather ban flux or chip
@coarse bridge water beam ur romantic partner is missing u
I might just ban you in my indecision
pls stop projecting ur fantasies
Not mine
You mentioned chip in your last couple of messages tho bro, projection from who?
Pf// Suppose for the sake of a contradiction that what you claim is true.
Recall that lying is a sin.
Hence, a contradiction.
▪️
Ok brb
speaking of fantasy ig ur shower was fictional
last couple of messages who? brother this the first time im speaking in discussion 2 dont play around lil vro
lil vro
Okay, more than a couple, three times even. I guess the obsession lies deeper than I thought
3 messages ah yes -> a lot the fact u care so much makes me think u want in
dw i can dm him for u
tell him that u have a fat crush
💔
How does he have this much a stranglehold on your minds LMFAO
who even uses this channel cuh
He's a chill friend of mine but you guys are gonna be writing fanfiction at this rate
ally*

LMAO
You basically already did the other day 
literally
LMFAOOO
WHAT
hi
hi
paw
Yoooo, wait wait wait 🫸-----🫷
Flux with a bunny tag?
bugs bunny flux bunny!!???
no
i talk about all my friends like this
i mention them
including u
hes not special dude
☠️
this reminds me of "discord has chipotle in a chokehold"
iykyk
I was talking more about water beam in that moment LMFAO but yeah fair
I do not know. Might keep it that way tbh
yes

Gn
Good night ally
Good night!
Goddammit
GOODNIGHT ALLY
cya
Go to sleeeeep
Rent free in their heads
have a wonderful night ally 
@coarse bridge Lmfao dumb fuckin muted ass
Now
Sounds good, gn fr this time
I feel like using calculus to generate pi is cheating somehow
how else are you supposed to get Pi?
Hm, I was gonna say summations or something, but I guess that's still more or less calculus
Perhaps I shoulda said, using integration to generate pi feels like cheating
why
¯_(ツ)_/¯
Nothing wrong with integration, there's a lotta neat stuff there
✅
was it not the ratio of the circumference to the diameter of a circle?
and it's how you calculate it is it not?
oh well without a circle
😭
hi
Hey there!
i know
use picks theorem
take a square of side length 2
plot a circle in it
of radius 1
find the distance of all points in the square from the origin by using cond r>1 or r<1
divide no of points in circle/no of point outside + in circle
look we know area of circle with radius 1 is pi and of a square with side lentgth 2 is 4
so like Nin/(Nin+Nout)=pi/4 for so like you can approximate pi using this
@jade kindle
✅
or just integrate a circle with a given radius and divide it by the radius squared
the only issue is
The Antiderivative is weird
the antiderivative of the half circle function looks like the integral symbol
Coincidence?
I dont think so
2arcsin(1) in radians
,w 2 arcsin(1)
hello
Sup
can u do middle terms in scientific calculator
no
you would need a calculator with a CAS
computer algebra system
nah i need to use it in exams to save time
then just learn it properly then
if you can't be bothered to factor just use the quadratic formula
you don't know what the quadratic formula is??
search it up dude
it's actually pretty dumb that they don't teach it this year
they should teach that and factoring and completing the square together
then your unit would be complete
ok thanks
not knowing transformations or vertex form isn't the end of the world but those are also important
np!
omg jealous
i can sleep in more today however i get bad headaches from naps most
of the time
or if i go over 8
Ahhh that sounds very tiring, not getting enough sleep ☹️
When do you get vacation?
From work
i do get enough
8 hours
i just cant rly get myself to go beyond it
im off like 3 days a week and last week i had like 5 days off
but im fully done once mid july hits
Jealous 😭
its not a longterm job
what year r u if u dm asking
hi ally
hi flux the furry
Oh I’m still in high school, have two more years left, then it’s uni time
what time do u start and leave
Like in a day?
Yea
for hs
btw nat meet flux the furry and flux the furry meet nat
@gentle birch
8-16 then I have work then I have job from 17-19
Hello flux the furry 🤩
8 am TILL 4 pm?
Yeah 😭😭😭
i request you to not
oh getting very formal today okay
learning from other people aside from that stupid brainrotted older brother u have
I’m sorry 😭
Sounds good
What should I address you then :D?
u did nothin
im just pulling his leg
hes done worse stuff to me and everyone in this server
he can handle it for one last time
Call him flux
Is that alright? That I call you flux? 😄
Btw how long is ur day? If you don’t mind me asking 😅
WHAT
🤧
shut up ryc
I see it now 💀
Oii no bullyin 😂
can somebody help in bio (dna replication drawbacks)
i need to improve my sleep
delete mathcord
good night
work or university
it's ok
its not bc of mathcord
just a normal day of both ig 😄
maybe try getting off from social media?
ye i can try that ig
6 hours of sleep a day
I get 12 hours of sleeping 
Has anyone tried both fxcg50 and fxcg100 if so which one is better?
only happens if I'm lucky or I have nothing to do
I stay up all night
Help us, the People, Boycott The Hatch. Roblox is turning dark with child predators in their events. Join us in this revolution and stop Roblox. https://x.com/hashtag/BoycottTheHatch?src=hashtag_click
i remember this
there was a (roblox) sex game creator involved in their latest event
wtf
uni varies , work can be 10 or 6 hours
but im only working in the summer
LMAO
Idk if itll help but try blinking rapidly
it makes ur eyes tired
I feel bad for flux sometimes, but then again... I don't
LMFAO
Hi, shy girl

does this actually work
Hi joe mama
for me yea
erm
“Turning”

Hi little kid
Ouch
where do u live
Wet shoes give blisters
Not enough people sung the rain rain go away song
Or some people did and it went away to here
Literally been raining all week in finland
hi :3
<@&268886789983436800>
Wait I did not see what happened
probably for the best
you didn't miss anything good
I thought I was going to get banned lmao
just ascii "art" spam
pfft
hi :3
calls mods
"mods, kill this man"
I'm gonna guess some sussy ascii art
@novel shell welcome to the mathcord! 
Thanks😊
Math is fun
Very logical
And there’s (usually) always a reason for something
Ill think ab sharing
its smth i prefer keeping confidential now
ok
STOP
🤣
Dami can you property (T) pill me sometime
omoi wa rain rain
Yess! I gave a short talk earlier this year about property (T) and it might've been in my top 3 talks ever
swag yeah I hear about it in the context of lattices in Lie groups but idrk how to think about it or why it's interesting
Okay now I'm here for a bit. So what do you know about it so far?
I will say most of what I know of it is fairly impressionistic
So I'm not sure eg of the proof that lattices in higher rank groups satisfy (T) or anything
But I can tell you the story
you realize im taller but sure i will be a shortie
Ur not taller
Ur 4’8
Brutal
btw dami atp i dont even wanna go to the dining hall for dinner im just gonna skip it and stay in this building
i dont wanna walk in the rain
My shoes have never been this wet before
Ah yeah valid. Esp given that dining hall food... May not spark joy

dami do u wanna see a vid of how wet my shows are
The only place on this campus thats open for food has 10 dollar breakfast sandwiches
Pretty #epic
Do show!
Actually i forgot im wearing black converses
oops i disappeared for a bit. But yeah, just the story would be appreciated. Maybe like some of the motivation and some corollaries of satisfying property (T) that I might care about (I think you have an idea of what kinds of results I like)
so u cant rly tell
:3
U rly are a furry
stop it get some help
U LITERALLY say :3 :3 :3
:3
Automorphic forms r banned then
:3
Deltoid short? Lol I hope it was true
1 more year and I'm out of waterloo
u after loo
Yup!
True
U so tall
:3
im the shortest in mathcord
there are 13 year olds in mathcord
Okay so, I think the original motivation for Kazhdan was understanding finite generation of lattices in higher rank groups?
Recap of the definition, you put this "fell topology" on the space of unitary irreps of a group, and you wanna say that trivial rep is isolated. This is where the notation comes from, parentheses mean there's a ball around trivial rep (T)
Thank you!!!
are we talking about kazhdan lusztig polynomials?
If you translate what that means in practice using the definition of Fell topology, this means that if π:G->U(H) has an "almost invariant vector", ie a sequence v_n of unit vectors such that π(g)v_n - v_n goes to 0 uniformly on compact subsets of G, then there's an actual invariant vector
I think there's a bit of a dynamical/"spectral gap" interpretation in play, but I'm not recalling the details offhand. That said, I can give a graph theory mini story that might have some bearing on things
How tall r u
Yeah, I guess I'm trying to learn some of the dynamical side of things too. This feels reminiscent of like ergodic actions vs mixing actions and realizing these things via the representation of G into L2 and characterizing invariant vectors there
Fell topology?
i never realized how tall traffic lights were until i saw the pic of a person standing next to one
Namely, you can define the notion of a Cheeger constant of a graph, which measures connectedness. Cheeger = 0 iff disconnected
Given a family of graphs (say k-regular for some k), you say they form an expander family if their Cheeger constants are bounded uniformly away from 0
There's an inequality relating the Cheeger constant to the spectral gap (smallest non-zero Laplace eigenvalue), so really an expander family can be thought of as a family of k-regular graphs which have a minimum spectral gap
If you study random walks on regular graphs, that spectral gap being larger implies a faster mixing rate for the walk
Almosg as tall as me
smh dami graph theory pilled
Now, you can show probabilistically that most large k-reg graphs are good expanders. But constructions of explicit families usually used property (T)
The theorem here is, if Gamma is a finitely generated group with (T), say S is a symmetric generating set, then the family of Cayley graphs Cay(Gamma/N, S) are expanders
Dami im gonna take notes of ur math
Ur rly knowledgeable
In this
Smartie very impressive
it's like he nearly has a phd or something
graph theory mentioned
So this at least gives a plausible connection between property (T) and dynamics. I think there are some notes of this one guy whose name I keep forgetting where he unpacks a bit more what the connection is between a general group theoretical spectral gap and almost invariant vectors
Ok let me credit
Thanks Ally! :)
But yeah so, Walter you mentioned L^2 and I'm glad you did
Because this lets me set up (T) vs amenability
How much do you know about amenability? The answer is probably more than me because I only know one and a half things
im 4 11
But yeah there are a few characterizations of amenability.
One is that the group carries an invariant mean, i.e. an invariant linear functional on L^{infty} which is non-negative and has norm 1
One is something about fixed points if it acts in a nice way on an appropriately adjectivized subset of a TVS (maybe it was affine action on a compact convex subset?)
One is that the regular representation of G on L^2 has almost invariant vectors
Ur fine
The last thing is what I wanna focus on. Okay let's say G is both amenable and has (T).
Because G is amenable, L^2 has almost invariant vectors
Because G has (T), L^2 must have an invariant vector. That has to be the constant function
If the constant function is in L^2, that means G is compact
,calc 12+5*(4-2)+8
Result:
30
isn't it wonderful what modern technology can accomplish
why does the invariant vector have to be constant?
The action is translation
So mean
But yeah so amenable + (T) => compact already has some nice consequences
For one, you can show that Z is amenable. Because it's not compact, it doesn't satisfy (T)
But then it's not too hard to show that a quotient of a property (T) group has (T). If π:G/N -> U(H) has almost invariant vectors, okay pull that back to G, tada.
In particular, any group with (T) can't have Z as a quotient. This is a cohomological property, since H_1(G,Z) is its abelianization
You can phrase this as H_1(G,Q) = 0, or H^1(G) = 0
Is property (T) preserved upon passing to finite-index subgroups?
Yes
In fact finite covolume
I see. I like this a lot. I didn't know why people cared if mapping class groups satisfy property (T) but I guess it's evidence for/against Ivanov's conjecture on H_1 of finite index subgroups
That's exactly where I was going with this
I'm pretty sure this inspired Ivanov
Im jk vereerrrrity
At least is post-hoc motivation
i read this as post-ho
Since okay we think cohomological stuff for lattices in higher rank groups probably should mirror stuff in MCG for higher genus surfaces
LMAO
I still haven't actually taken a GGT course so I'm missing a lot of motivation for questions people care about in this area
Right yeah, I'd like to fully understand this analogy
Neither have I, this is all hearsay/some frantic Googling for talks I gave 
lmao
Group homology 
Likewise, if I get to stay in math I really wanna learn this story
Even if I don't, I still wanna learn this recreationally
It's so surprising how deep that goes
Group cohomology is cool
Sheaf cohomology scary
But yeah my first application was that (T) was (possibly time traveling) inspiration for Ivanov because I know you like MCG
Yeah, it is
It also sounds scary to me tho
Like, what does group homology/cohomology truly measure
replace Group with Galois and it won't be as scary trust
That's why you're so good at Lie theory
Other than yap like "homology of BG"
Lie algebra cohomology is goated
Wait, is Lie theory connected to this somehow
from my experience if you tell me H^43(X,Z) = Z^44 or something that doesn't tell me anything
Damnit, I got whooshed
What the hell does this MEAN
Scary math people talking math language
but like if you explain connections with other stuff then yeah it gets a meaning
If X is a space, I think it does
yeah you added more context now it makes sense
It has 44 43-dimensional holes (or whatever yapping one can think of)
I meant for general cohomology theories
Ah, yeah
But yeah here's another application. Turns out, if Gamma is a lattice in a higher rank semisimple group with... Idk finite center? No center? Some applicable adjectives
And N is a normal subgroup
Then Gamma/N will have both (T) and amenability. So it's compact. But it's also discrete. So it's finite
This is called the Margulis normal subgroup theorem
A take: Homology should not have been generalized beyond bounds to give birth to motives and the like
Can this be renamed to math-discussion-2
in rough language it means "some space X has 44 43-dimensional holes"
Yeah, I'm trying to learn this + superrigidity to get a decent idea of why Margulis was goated
Oh sounds like smth only smart math ppl can know
Well tbf, if you think about it, who IRL cares about holes anyway
It's very intuitive lol. Take for example a circle (here we are not taking it to be filled. Only boundary). This has 1-dimensional hole
Damnit, true
But according to this reasoning, a straw has 1 hole
Showing amenability is quite hard as far as I know. But (T) is easier. I would explain it but I don't know exactly why
Some general theorem which is true up to maybe being a bit false: if G is a higher rank semisimple algebraic group over a local field, then it has (T)
Also, if H is a closed subgroup of G and G/H has finite volume, then G and H have (T) together
Okay so that means lattices in higher rank groups satisfy (T).
And it's preserved under quotients
Me dont know topology but carry on Yes
I may be saying wrong things, add adjectives as needed
But if nothing else, SL(n,R) and thus SL(n,Z) has (T) :P
sure
Good enough for me
Dami do u think u could win a topology bee
Why is this property named (T)?
It sounds important
Thanks for explaining all of this, helps a lot to appreciate it more when I see it randomly pop up
tbh idk how to explain H_1 and H_2 with english terms other than 1 or 2 dimensional holes
lmfao
Also interesting that lattices are everywhere
I remember there was a specific term for these but i forgor
Oh, wow
Quite simple
Its fine im just not full of the knowledge u got u tried ur best
Reality bc i dont study this stuff
So yeah Margulis normal subgroup theorem, cohomology, expanders, vague idea of spectral gap are the main kinda nice facts associated with (T) in my mind. Also something about rotation invariant measures on the sphere, though I know that story much less. Look up Banach-Ruckiewicz (modulo spelling) if you're curious about that
Too many theorem
If only I knew what T is, but anyway
Basically it means the trivial representation is an isolated point in the space of unitary irreducible reps of a group
You need to ask what the topology is on that space, it's called Fell topology
Any direction you go, there are so many facts and perspectives you have to take in, so overwhelming tbh
too too many topology
okay so if you have a sphere (it's hollow). The hollow in that sphere is 2-dimensional as opposed to the hollow in a circle for example (it's not the same as the hollow of a sphere)
you see where im going with this
So irl it means that if you can find unit vectors v_n in a unitary irrep such that gv_n - v_n goes to 0 (say uniformly on compact sets), then you can find a single vector for which gv = v
Space of irreducible reps, now that is scary I guess 
I could be studying cohomology of arithmetic groups but instead I'm required to compute character formulas for unitary representations
Lol wait this is the thing that was solved for n=2, 3 by Drinfeld using like automorphic reps and Jacquet-Langlands
rotation-invariant measures on the sphere
I fear this might be a bit complicated at this stage?
Not just your explanation but the topic at large
it's hard to explain in text
the only time i got it working is when i was using a blackboard explaining
True but also it's the kinda thing where... Idk if you're not used to topology already it might be a bit tricky to imagine "data in a space" or to think about spaces combinatorially
I think I will check out this book
hi
So thing inside sphere is 3d but surface 2d
Does solids of revolution help me with this XD
Time for you to explain then
ig Margulis'/Sullivan's proof only works for n >= 4
Does my knowledge gaps make this too impossible
Well it okay u r teaching me discrete math soon So :3 maybe me learn bit of pure math for once
LMAO I mean Deltoid is giving a fairly good explanation, and probably comes as close as possible to explaining it understandably to a very early student.
I just think it's one of those things where it's hard to appreciate algebraic topology at all if you're not as used to certain ideas that typically come later
(Except me still not doing it)
give me a hard math equation please
Just giving interest to U pure math ppl
Hmmm, interesting, I see.
Anyway I hope there would be a way to more simply present these all
I have a plan coming together and I think you'll enjoy it
Or maybe there lies the innate complexity in the property of (topological?) groups and stuff
Not necessarily impossible but a bit tricky. I will say many students see some of these things for the first time in grad school
It's so funny when ally is typing, because discord gets very confused by the unicode directional marks
TRU I do notice that haha
Oh I was talking about the parallel convo
Like Ally is in calculus, so homology and such is tricky to explain at this stage
Yeah I was just thinking and talking about the T property and amenability
Tbh if she does calc 3 I can start to explain a tiny bit of algebraic topology. Just talk about winding numbers off curves
Maybe there is a way to present it to highschoolers or something
There might be a way? Like tbh I don't really know a ton about the Fell topology myself
Who u how u know my name ally
My background is hypersketch
Ally not listed on me profile
But yeah I know the interpretation as just almost invariant vectors vs invariant vectors
Ok time to learn about stationary measures 
Lol, I'm not stalking you, I promise. I just noticed people calling you that
Oh oh
I need to learn those one day
perhaps
what that mean
Honestly I should get an industry job with excellent work life balance and just learn a lot of math on weekends with no research pressure or smth
industry job
work life balance
hmmmmmmmmmmmmmmmmmmmmmmmmmm
it's probably possible with some shenanigans
Who did good in diff equations
me :3
I mean... Arguably any job is tricky to have work life balance for. Industry might be slightly better in some ways because for many types of jobs you can just kinda clock out and that's it
Invariant vectors are still quite involved if you think about it, like how topology is complicated as well
Idk, I am just daydreaming about how stuffs might be explained without deep intensive dive to subjects
I cannot
maybe if I can use Von Neumann algebras to prove certain cohomology groups vanish or stabilize
I am too busy
how
Oh I mean high schoolers learning (T) is hopeless generically
I can at most answer one off questions or concerns in the relevant topic channel from time to time
well I am wondering if I can do that. Idk anything about Von Neumann algebras
:3
Yeah, I wonder what is the essence of the reasons
I never got a bad grade in an ODEs class?
That is because I never took it but I think some of the basics I can kinda guess because of linear algebra
My superpower in math is guessing tbh
Maybe each of these are complicated concepts which somehow is gathering interests.
Suremark was right about GPT brain
a regular ODE class is just some tricks getting thrown at u
hard to appreciate DEs at an undergrad level idk
Yeah that's the vibe I get tbh
This recent class I took we got justifications for the tricks and some cool theory thrown in :3
Wow
I only had a couple weeks of ODEs in my analysis class
meee except when the work life balance isn’t so good sometimes 
But not quite yet full fledged proofy ODE
Ok u tutor me
oke :3
tbh I still get amazed when I learn new stuff about DEs
Isn't ODE just like, Pattern match using the tricks given to you
Ur alreasy. my tutot anywY
First ODEs class so it seems. Some ideas are hidden
Like the whole oh particular and general solutions is just first isomorphism theorem
yeah undergrad ODE be like
Yeah, never got further than first ODE class
analysis be pattern matching which inequalities to use
Idk I dont like Analysis
SAME
Dami r u gonna give up calc 3 tutoring position to someone else so i dont botber u too much with it
I still get exposed to DEs in number theory. The ideas make my mind go crazy
Isn’t everything pattern matching to some extent
Maybe suremark