serious-discussion

[vague lion]

you can make it a contradiction proof pretty easily

and that's a common alternative formulation

is all

[leaden torrent]

Sure, but it doesn't have to be

Why make your proof less direct than necessary

When even Euclid didn't do that

[vague lion]

for fun

[leaden torrent]

(admittedly Euclid's original proof was insufficient, he only proved the n=3 case lmao)

(but it obviously generalized)

[dense belfry]

I think you can phrase cantor's proof a similar way though

[leaden torrent]

(mathematical language of that time just couldnt express the notion of "arbitrary list")

[vague lion]

like, it's common to the point that i think that's how i first saw the proof

[dense belfry]

you can say "If I'm given a countable list of real numbers I can find a real not in the list"

[leaden torrent]

Sure but that doesn't prove anything unless your list is allowed to be arbitrary

But you need the assumption that R is countable for the list to be arbitrary

[dense belfry]

what do you mean by that

[vague lion]

if you have a countable list then if you just find one more

it's still countable

so that sucks

[dense belfry]

well we are proving for each countable sequence of infinite binary strings that that sequence is not an enumeration of all infinite binary strings

We have a uniform process to find a new infinite binary string not in our list

sounds pretty good to me

[leaden torrent]

So? You've shown you can't find a list of all reals. So what?

[dense belfry]

so isn't that what the reals being countable means?

[leaden skiff]

that's what the reals being uncountable means

[dense belfry]

thanks ninja

[leaden torrent]

No, countability means in bijection with N

If a set is countable we can biject it with N with a list

[dense belfry]

I'm not sure what your point is here

[leaden torrent]

Why don't you just add the new element to your list?

[dense belfry]

well the point is that no lists work

[leaden torrent]

For finite lists that is convincing

[dense belfry]

I don't see why it's not convincing for countable lists also

[leaden torrent]

But for infinite lists, why can't we just repeat the process infinitely and add all the elements?

The answer is because that'd increase the cardinality

And we assumed that R was countable

[dense belfry]

I don't like that line of reasoning tbh

[leaden torrent]

Perhaps I'm not expressing it well

FYI you can phrase Cantor's proof in a direct way

You talk about functions instead of lists

And then the direct proof is very natural

[dense belfry]

lists are functions

[leaden torrent]

Well yes

But you can be more explicit

[dense belfry]

tbh I use lists as functions enough that I just use them interchangably

[leaden torrent]

Okay sure

Then the proof is direct and just an argument about directions

About surjections*

Autocorrect

[dense belfry]

sure

[leaden torrent]

Idk id be willing to buy an argument that Cantor's proof is basically a direct one as well

But he phrased it contradictively

Whereas Euclid phrased his directly

(in fairness, proof by contradiction wasn't even accepted in Euclid's time so he was kind of forced to)

(see sqrt(2) stuff)

(but still)

[dense belfry]

I think I'm happy from just seeing the parallels between the 2 proofs

they feel like they are trying to do the same thing to me now

[leaden torrent]

They basically are

It's just a phrasing thing

The fact that Euclid's proof is constructive is actually important to logicians

Whereas they don't care so much about whether Cantor's is since, well

If you're working with R

[dense belfry]

well certain logicians

[leaden torrent]

You don't care about constructiveness

[dense belfry]

idk tbh, it just doesn't seem to be something that I've heard many computability theorists talk about

[leaden torrent]

yeah obviously not all logicians care

[neat lintel]

Hi

[neat lintel]

Hello

How are you?

hello C A T

[fair mural]

hello cat

[untold sapphire]

?

[full storm]

The shuffle product is the dual of the exterior product and forms an alternating multilinear algebra. https://en.wikipedia.org/wiki/Exterior_algebra#Alternating_multilinear_forms

pages 112-113 touch on this algebra and its relation with products used in clifford algebra and grassmann-cayley algebra

This is just an algebraic approach to defining this product

there is also a combinatorial one that exists but i am not familiar with it

[neat lintel]

the big thing is decomposition of what this means

oh mb

this is really old

[full storm]

https://people.math.rochester.edu/faculty/doug/otherpapers/weibel-hom.pdf This book also goes much more in depth on the shuffle algebra algebraically

Also called the regressive algebra in different contexts

[neat lintel]

why did he take down his pure mathematics video? (pardon the ping)

[random kelp]

i saw a video of him

and his eyes were like

👁️ 👁️
👄

[burnt dune]

ahlfors is the best book to exist

[neat lintel]

Draw the projections of a hexagonal pyramid of base side 25 mm and axis height 60 mm resting on the edges of its
base on HP with its triangular face perpendicular to both HP and VP.

No

[neat lintel]

Hello everyone, I am gonna leave this server

for like a few months

and then come back

farewell

[neat lintel]

ok

Not a crime

[vivid halo]

they come up a lot in Lie theory since for example the graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra on that finite set.

The shuffle product also comes up a lot when you work with iterated integrals and combinatorics of simplices: the product of two simplices is a union of simplices and the combinatorics here is governed by the shuffle product; since iterated integrals are just ordinary integrals over simplices, this tells you that a product of iterated integrals is a sum of iterated integrals over shuffles of the integrands

[obtuse fjord]

👍
Thanks! I have seen the terms "free Lie algebras" thrown around quite a lot in this theory more recently. This latter description is where I'm encountering the shuffle product though, in signatures from rough path theory. I think Terry himself has sufficient description of the shuffle algebra in his Saint-Flour notes for what I need, but I will look into the Lie theory if I need more

[opal linden]

wow this new help system is cool

[vivid halo]

@narrow rock okay so let n≥1, let q be a power of a prime p, and let a be in F*_q. Then you can consider the Kloosterman sum:

$\mathrm{Kl}n(a,q)=(-1)^{n-1}\sum{\substack{x_1,\hdots,x_n\in\mathbb{F}q\ x_1\hdots x_n=a}}\exp(\frac{2\pi i}{p}\mathrm{Tr}{\mathbb{F}_q/\mathbb{F}_p}(x_1+\hdots+x_n))$

[fathom swallow]

nGroupoid

[vivid halo]

these are essentially trigonometric sums

[jovial ember]

[vivid halo]

they show up all over the place (e.g. in spectral theory of automorphic forms, in additive combinatorics, etc)

[narrow rock]

hmm

[vivid halo]

the thing that Katz did that takes up one of these monodromy books is like

[jovial ember]

Is that actually the product = a

Or should it be sum

[vivid halo]

yes this is the product = a

[jovial ember]

Weird

[narrow rock]

why would it be sum = a

[jovial ember]

Idk

[vivid halo]

it just is

[narrow rock]

then all the terms are the same

[jovial ember]

That’s what my combinatorics brain told me

Oh yeah

[narrow rock]

what

[jovial ember]

Why are you danning him,

He’s right

[vivid halo]

oh I see what you're saying yea

[jovial ember]

Swag

[vivid halo]

okay so like

one can show the following bound |Kl_n(a,q)|≤nq^{(n-1)/2}

moreover this bound is optimal

another thing you can show is that (the angles of) these Kloosterman sums are equidistributed as a ranges over \bar{F}*_p with respect to the Sato-Tate measure

how do you show both of these things

well what Deligne and Katz did was the following. One can define a certain complex of sheaves on G_m over F_p so that by the Grothendieck-Lefschetz trace formula the trace of Frobenius on the stalks of this sheaf are the above Kloosterman sums

[narrow rock]

[vivid halo]

one shows that these complexes are concentrated in degree 0 and are local systems of rank n, tamely ramified around 0 with unipotent monodromy with a single Jordan block, and totally wildly ramified around \infty with Swan conductor 1, and pure of weight n-1 (which implies this optimal bound)

Moreover Katz showed the converse: if you have a rank n local system on G_m with these ramification properties, then it is isomorphic to a Kloosterman sheaf up to the translation action on G_m

what Katz then did was the following: since you have a rank n local system on G_m, you get a geometric monodromy representation π_1(G_m/\bar{F}_p)->GL_n(Q_l(µ_p))

let G_geom be the Zariski closure of its image. Katz then shows that G_geom is Sp_n for n even, SL_n for n odd and p odd, SO_n for n\neq 7 odd and p=2, and G_2 for n=7 and p=2

Katz then has all these arguments that let you conclude like, if G_geom is suitably large (as is the case for these Kloosterman sheaves) you can deduce equidistribution theorems for Frobenius traces with respect to the Sato-Tate measure on G#_geom the set of conjugacy classes of G_geom

so the form of the geometric monodromy group constrains your equidistribution results and tells you what measure you're equidistributed with respect to

it's also just very surprising that G_2 comes up in this example like this

[narrow rock]

what information do these kloosterman sums encode

[fair mural]

@neat lintel since other people are using this channel now, and your question will get pushed up, it’s best to ask in #math-discussion

[vivid halo]

yea so these Kloosterman sums come up all over the place, so the general philosophy is you can get results in a lot of areas if you have good geometric control over these kinds of trigonometric sums, which is what these results are telling you

one situation where it comes up is like

if you have a classical cusp form of weight n≥2, how do you prove the Ramanujan conjecture for this? Well, the clean way to do it is you realize this modular form as occurring in the l-adic cohomology of a local system on a modular curve so that the Fourier coefficients match the traces of Frobenius on l-adic cohomology, and then the Weil conjectures tell you that this l-adic local system is pure of weight n-2 which gives you the bounds a(p)≤2p^{(n-1)/2} on the Fourier coefficients

what if you have modular forms of even lower weights than this, for example weight 0 (possibly non-holomorphic) automorphic forms

well, in this case you have the Kuznetsov trace formula which relates some god awful expression involving Kloosterman sums to an integral transform + spectral terms, where the spectral terms are sums of Fourier coefficients taken over spaces of holomorphic and non-holomorphic automorphic forms.

so if you have good bounds on Kloosterman sums, you can get good Ramanujan type bounds in the cases where Deligne's proof doesn't apply

[narrow rock]

cool

I need to read a bunch of things to understand it better tho

[vivid halo]

the other thing with Katz's monodromy stuff is like

you can use some of his results about moments and monodromy to prove a good chunk of Weil II

this is what he does in his AWS notes and it's kind of a cool (albeit somewhat unusual) strategy to the proof

[jovial ember]

#serious-discussion message

I made a 2-way teleporter

[fringe needle]

But how

Are you pointing to the channel instead of to a message

[jovial ember]

Nope

[fringe needle]

But it doesn’t highlight any messages when I transition 🤔

[jovial ember]

That’s cuz it’s not high up enough

If more messages populate it will

[fringe needle]

Lemme just drown this one

A

A

A

A

A

A

A

A

A

A

A

A

[jovial ember]

Don’t think it’s high enough

Wtf

[fringe needle]

Yeah lol

[jovial ember]

It should be linking to the message

:(

It should be to that message

#serious-discussion message

Okay I changed the teleporter

[fringe needle]

Ah someone has revealed your secret

[gentle bay]

Ha.

[jovial ember]

Yeah lol

[gentle bay]

Easy peasy.

[jovial ember]

It’s not that complicated

[fringe needle]

I’m just shtupid

[jovial ember]

It even said (edited)

[fringe needle]

Yeah I didn’t see it lol

[gentle bay]

Yeah exactly.

[fringe needle]

I be turning blind

[gentle bay]

Or are you just tired?

With studies?

[fringe needle]

In 20 years I’ll be blind from computer screens and dead from blasting metal

Deaf *

Hopefully not dead

[gentle bay]

Ah.

[fringe needle]

That too lol but not because of studies

Just going to sleep too late

[jovial ember]

[fringe needle]

It’s hollow knight’s fault

I cannot sleep until I beat the pantheon of hallownest

Even if it mean not thinking straight on my practice analysis exam and messing up

[gentle bay]

I could not defeat that first big-fat guy.

[toxic schooner]

[fringe needle]

Zuck

Git gud

[pale orchid]

git --force -gud

[primal crater]

[timid spindle]

gut push -f professor knife

[snow lintel]

*git

[timid spindle]

Nah it was meant to be gory

[pale orchid]

gut push sounds like bowel movement

[static loom]

💩

[pale orchid]

gut push --force

[static loom]

lol

don't forget to flush with sudo rm -f /bowl afterwards

[timid spindle]

Hahah
cat 💩 > /dev/null

[pale orchid]

i guess this strictly falls under "shitpost" category

[timid spindle]

That sounds like a horrible idea if there's no white noise

[jovial ember]

Hey edd

What rhymes with Edd

And is something I use everyday

[pale orchid]

bed

[jovial ember]

Yeah

[pale orchid]

[timid spindle]

sad

[pale orchid]

i am, but it doesn't rhyme

[timid spindle]

Big fluffy hug

[inner finch]

edd more like ded

[devout nacelle]

Would've said head but you don't really use it

[jovial ember]

Yeah

[pale orchid]

chmonkedd

[primal crater]

very enlightening convo

[toxic schooner]

indeed

[timid spindle]

points juicing

[strange goblet]

schwifty

[timid spindle]

Let's get scheifty

[stray kite]

when defining a function f:X->Y, is there any way to leave Y arbitrary so that it is the "largest set" possible here which would contain all codomains? like, basically so that it doesn't restrict the range of f in any way?

[leaden skiff]

wdym "all" codomains?

[neat lintel]

there is no "largest set". see Russel's paradox

but if you are working inside some "universe" set, you could have Y to be that one

[stray kite]

like say you have f(x) = x, f : R -> Y
Y should be the 'largest' set such that any value from the domain which is valid in the definition can output a value(i.e. here Y = R)

[neat lintel]

do you know what a function is?

[leaden skiff]

that's what the co-domain is?

like, the image of the function need not be the domain

like for e^x (from R to R)

the domain is R, the co domain is R

the image is (0,R)

[neat lintel]

and the codomain is R

they are talking about codomain

[leaden skiff]

yeah

yeah i typod

[stray kite]

ok i realise what i was saying doesn't make much sense now lol

there is no "largest" codomain because you can always make a superset of the codomain to get a larger codomain which is still valid

[leaden skiff]

yes

[neat lintel]

why would you want to have a largest codomain anyway

[leaden skiff]

actually, is e^x: R -> R
a different function than e^x: R -> (0,R) ?

[neat lintel]

yes

[leaden skiff]

or are they the same function

[neat lintel]

it's a different function

the latter is a corestriction of the former

[leaden skiff]

so even e^x: R-> R - {-5}
is a different function

[neat lintel]

yea

a function is determined by it's domain, it's codomain and it's graph

[leaden skiff]

wdym its graph?

not all functions can be graphed

[neat lintel]

the graph of a function is the set of all (x,y) such that f(x)=y

[leaden skiff]

oh you mean the image?

[neat lintel]

no

you need the information of about what elements map to which

image alone is not enough

[stray kite]

ig i was just looking for a way to say "some arbitrary codomain which is a superset of the range"

[neat lintel]

you can say that

[leaden skiff]

that's just the domain x image

[neat lintel]

no

[leaden skiff]

isn't it?

[neat lintel]

suppose domain is {1,2} and codomain is {1,2}

and function is identity

[leaden skiff]

1 -> 1
2 -> 2

[neat lintel]

forget the swap

[leaden skiff]

image is {1,2}

mmm i see

[neat lintel]

domain x image = {(1,1), (1,2), (2,1), (2,2)}

[leaden skiff]

wait...

that's the function???

[neat lintel]

graph = {(1,1), (2,2)}

[leaden skiff]

what you've described is the function no?

[neat lintel]

the graph part yes, the domain x image no

[leaden skiff]

so the graph is literally just the function?

[neat lintel]

along with the domain and codomain

a function is a triple (D, C, G) where D is the domain, C the codomain and G the graph

[leaden skiff]

mm i see

[stray kite]

whats a graph

[leaden skiff]

this

[stray kite]

i see

i guess G \subseteq domain \times codomain?

[neat lintel]

yes

not any subset works tho

[stray kite]

right

[neat lintel]

it needs to satisfy this property: for every x in domain there is exactly one (X,y) in the graph such that X=x

[leaden skiff]

yeah otherwise it's a relation

[neat lintel]

it is a relation either way

[leaden skiff]

otherwise it's just a relation

[neat lintel]

yea

[tribal dove]

prove the pythagorean theorem non-visually

[wild jay]

what

[tribal dove]

do it

[wild jay]

?????

[neat lintel]

with eyes closed?

i can try

[wild jay]

based

[tribal dove]

as in don't use pictures in your proof

[neat lintel]

Suppose that x and y are orthogonal. Then ||x+u||¹ = <s+y,x+y = ||x||² + <x.t+y,x+||y||² ? ||x||^2 '||y||^2

it didnt go very well it seems

i'll do it with eyes open now ig

[tribal dove]

ok

[wild jay]

you just needed to take the quadruple integral then rotate 270 degrees by the z-axis and finishing with converting to polar form and using arc length integral

or that

[last oxide]

<

[neat lintel]

Suppose x and y are orthogonal. Then ||x+y||^2 = <x+y, x+y> = ||x||^2 + <x,y> + <y,x> + ||y||^2$. Since $x$ and $y$ are orthogonal, <x,y>=<y,x>=0, therefore ||x+y||^2 = ||x||^2+||y||^2.

[fair mural]

is this an actual proof

[neat lintel]

yea, it's pretty standard

you will learn it when you learn linear algebra probably

[wild jay]

False

didn't see it

[tribal dove]

thank you

[neat lintel]

that's very weird

[neon tundra]

But how do you motivate 2-norm unless you already know the Pythagorean theorem holds?

[neat lintel]

good question

rotations

you have usual norm in R

and extend it to R² in a way to be invariant for rotations

[neon tundra]

Then you need to define "rotations" first, though.

There's probably some way to get through it purely symbolically if we can show that R² with the 2-norm is the only model (up to such-and-such) of Euclidean geometry with this-or-that nice properties.

[neat lintel]

yea, now that i think of it might not be so easy, there's a way to do it assuming some this-or-that nice properties tho

[neon tundra]

Getting all the way will surely take us places along the way, considering e.g. that we need to exclude models that turn out to be hyberbolic geometry instead.

[neat lintel]

parallel postulate

[neon tundra]

Yeah.

[honest veldt]

Is it normal for topology to feel like we're just defining random stuff at first? Does it start making more sense later? (munkres, by the way)

[neat lintel]

idk, honestly i didn't like munkres much

i prefer Willard, and i also think its more motivated

[blazing pawn]

It is kind of normal because 95% of people dont care about point set

[mild nebula]

It does start making more sense later

Like, we don't really care about open sets and closed sets, we care about the topological properties that they can define. At this point, you haven't been taught the "why" yet @honest veldt

[untold sapphire]

yeah

[tiny marten]

it is a very frontloaded field

for something that isn't like, clearly a graduate level topic

i took an elementary topology class, it was a 4th year and one of the most frontloaded undergrad classes i took

it's not really any more frontloaded than measure theory or anything like that tho

also point set topology is great, point set gang

also anything that depends on set operations is kinda flim flam for a while

[honest veldt]

Ahhh I see. Lecturer did keep bringing up how the first week or two would be really formal, basically just set theory, and that it gets more interesting when we get to the "properties", whatever those might be

Thanks a bunch, you all, I was getting worried I'd have to deal with just pure set theory for months on end

[honest veldt]

What is, generally, the difference between point set and algebraic topology? Is algebraic the one with all the funny looking shapes?

[tiny marten]

point set topology is a bunch of kajiggering of the definitions of the topology in various sets or spaces that setup what people usually expand on using more sophisticated theories

you study the quality of compactness and continuity by playing with points in the spaces

which is pretty fun imo

and algebraic topology is like, setting up correspondences between algebraic theorems and topological objects with respect to the foundations from the point set stuff

cause algebra is more clear and has all these nice structures

in general it becomes v abstract

[honest veldt]

Ah, so algebraic topology builds on point set?

[tiny marten]

yeah tho at times it isnt clear they are the same field tbh

so there is like, algebraic topology and general topology + whatever

it is like connecting the qualities of the space to structures or things that must maintain some specific value due to the topology, then supercharging it with algebra and crazier stuff

[honest veldt]

That does sound pretty interesting, I'm guessing that's where all the meat of the crop is

In any case, thanks a bunch, you've gotten me a little excited for it again

[tiny marten]

np also yeah algebraic topology is the real influential one now

btw the point set stuff is important for intuition

[shy pollen]

i hate my mindset in regards to work and fun

i have not had fun in years

god i am depressed, i just want to sell my company and not give a fuck about work anymore

[tiny marten]

yeah workaholism

i seem to always be working on shit and neglecting my social life

[shy pollen]

it’s that i feel bad abt myself and don’t believe i can be exceptional if i do the parties and all that

idk why i think like that

[tiny marten]

yeah it's probably not true if you can moderate yourself

[queen urchin]

how do you even learn advanced math

I have no idea where to start

I'm trying to learn group theory and i feel like i skipped a few steps

[bronze pelican]

What do u know so far

[queen urchin]

algebra

some linear algebra

calculus

i remember a tiny bit of geometry but not much

i forgot how to do proofs

[vague lion]

you gotta know how to do proofs

and basic set theory

and that's about it

linear algebra is nice sometimes bc it links back to it a lot

[frozen venture]

Given an arbitrary plane in R^3 which goes through the origin, can you find an orthonormal basis for it?

[neon tundra]

Yes. Gram-Schmidt.

[mellow moon]

apply three times the vector product, then divide by the norms

by construction they will be orthogonal to each others

then normed

hence this is an orthonormal basis

[frozen venture]

Thank you

[still verge]

<@&268886789983436800>

[devout nacelle]

Thanks, dealt with. In future, you may DM @polar panther .

[neat lintel]

Damn.

[still verge]

Got it thanks

[mint patio]

Is the Laplace Transform an operator?

Or I guess are integral transforms operators?

[pale orchid]

sure

differential operators too

[mint patio]

How is it a differential operator if it's defined by an integral?

[pale orchid]

i mean one can also define differential operators

or just "yes"

[mint patio]

Okay :P Does it show up in functional analysis?

It's obviously a linear operator

[pale orchid]

yep

[mint patio]

pogeurs

Seems like such a cool subject

I know Ryc really likes it

[neon tundra]

In functional analysis, "operator" is any linear transformation V->V.

[mint patio]

Oh

[mellow moon]

You can define many Differential operators as integral Operators with specific properties

especially their inverse

[mint patio]

That's a very broad definition
I haven't taken any functional analysis Anatole, in case that comes up

[pale orchid]

oh that's another interesting bit, yeah. you can usually convert integral expressions into differential ones

that's where "green's functions" pop up

[mint patio]

Is that like how Maxwell's Laws have an integral & differential form?

[mellow moon]

no

It is far different

It's more like how Biot-Savart law/Newtonian potential solve some equations

but in a far more general way

[mellow moon]

Just in case you will need way more than just Functional Analysis lecture, except if you look at very specific Integral Operator.

[mint patio]

What else could you need?

[mellow moon]

Operator Theory and Spectral Theory like Fredholm at least

[pale orchid]

fredholm

[mint patio]

Got it. There's so much aaaaa when am I ever gonna get time to learn it all

[pale orchid]

these are the topics that made me realize i should've studied math instead of engineering lol

no way to easily handle cool problems without spending months on learning basics

[mint patio]

Same lol, I wish I knew more about the math career/job scene so I could maybe convince my parents to let me switch]

[mellow moon]

You have the time, I personally learn about it during my Master Degree

[mint patio]

So your undergrad wasn't a math degree Anatole?

I'm second year

[mellow moon]

It was

Just there is too much technology to get there before the end of the 4th or 5th year

[mint patio]

Too much technology?

[mellow moon]

Lectures inbetween

this is a very long travel

[mint patio]

Oh okay

Problem is I'm too busy suffering through engineering (god I suck at physics 😭) to even get through the prerequisites/in between stuff

Semester's almost over though so I will definitely study what I can over break

I am going to review real analysis then move onto complex

[mellow moon]

I don't know where you live, since as you said "I wish I knew more about the math career/job scene"

[mint patio]

America 🤠 Texas to be specific

[mellow moon]

this could be difficult depending on your place

[mellow moon]

Austin ?

[mint patio]

Dallas, three hours away from Austin

[mellow moon]

I knew Dallas were in Texas, but not that far

[pale orchid]

texas is big

[mint patio]

What's in Austin? I heard a lot of companies were moving there but idk any specifics, especially not when it comes to the math scene

Texas is very big :P

[mellow moon]

Just an my friend's uncle works there, and there are few well known Mathematicians too.

[static loom]

🤠

[mint patio]

Hopefully I can do something there someday 😁 I don't know anything about what I'm going to do with my life, not hopeful for engineering. I am about to fail Statics so turbohard

Granted I'm doing a lot better at the end of the semester, I was missing a lot of physics fundamentals when I first started

So maybe when I retake I will have hope

[mellow moon]

I wish for you

[mint patio]

Thank you

[mellow moon]

Maybe hope is more convenient ?

Which one is more natural ?

hope/wish ?

[mint patio]

I think hope is technically right here, but it's all the same lol! You got the meaning across

[mellow moon]

Okay, thank you !

[neat lintel]

you know what else is big?

[fair mural]

no

stop now

[neat lintel]

tensor bundles

[wild jay]

[mellow moon]

[tribal dove]

sigh my math class is far too easy 😒

[tiny marten]

which class?

[untold sapphire]

Uh oh.

[tribal dove]

Advanced topology

[tiny marten]

like a second course after an initial topology class?

[tribal dove]

Nah it’s actually “advanced functions”

Grade 11

[tiny marten]

oh

is this like pre or post calc?

[tribal dove]

Uh idk maybe pre calc but def not post calc

[neat lintel]

bruh math is silu

[ancient flame]

silu

[neat lintel]

why dont they call topology space math

reduce jargon

geometry is study of shapes

topology is study of topologies which are visualizablr

[ancient flame]

math is just "number theory"

[deep mango]

I think topology literally means something along the lines of space math

Why dont you want fancy names for things

[jovial ember]

Space math is less descriptive than topology

Space math could mean anything

[ancient flame]

bc simple = better

always

no exceptions

algebra should just be "equation math"

calculus should be "fancy squiggle math"

ez

[fair mural]

space math

sounds amazing

[leaden torrent]

Topology = topos + ology = "the study of place"

So basically space math, just in Greek

"topography" has a similar etymology, "writing/drawing places"

[vapid phoenix]

https://tenor.com/view/uhh-cheesed-to-meet-you-gif-22801166

hello everyone

[mortal igloo]

Someone once said topology is fundamentally the study of connectedness

[jovial ember]

Hehe

Disagree

[fair mural]

as someone who knows nothing about topology, is that really wrong

[mortal igloo]

Obviously that’s gonna need a disclaimer that “connectedness” isn’t meant to refer to the technical term because obviously that’s a definition within topology

[vagrant kestrel]

topology is the study of surface-level understanding

[mortal igloo]

https://en.m.wikipedia.org/wiki/File:Spot_the_cow.gif

An actual spherical cow in the topology Wikipedia page

This is gold

Maybe one can say topology is the “graph theory” of continuous spaces

[mint patio]

any geniuses want to explain how the fuck it became a negative

it's either a typo or I'm a dolt

[fair mural]

definitely a typo

L{f(x)+g(x)} = L{f(x)}+L{g(x)}

[mint patio]

right ok

so ig my final answer should only be off from theirs by a negative

[jovial ember]

Maybe that term is 0

So they’re the same

😨

[fair mural]

chmonkey uses his chmonkey brain

[nimble shuttle]

hm

sin(-x)=-sin(x)

[fair mural]

1 = -1

[nimble shuttle]

yea

problem?

[mint patio]

I figured it out

Very weird by the prof

Apparently the - and + are equal there

[fair mural]

feather is so smart and cute and awesome

[mint patio]

no u

[fair mural]

good job feather

[mint patio]

also I love linear algebra

this shit be easy

[fair mural]

i’m trying to start linear algebra

[mint patio]

spectral theorem who ?

have fun it's like the most useful thing ever LMAO

how are you learning ode's without LA?!

[fair mural]

i didn’t need any LA until systems of odes

even then i didn’t need to actually learn LA

just know a few things

but i wanna be completely ready for pdes

[mint patio]

oKKKK moving straight into pdes 😎

[fair mural]

i haven’t even done odes in like a week lol

i’m slacking on math

[mint patio]

me2 bud

me2

😔

[thorn brook]

yo what can be done to ensure that mathematics leads societal development in a positive direction?

I'm supposed to give a small presentation about this and the topic is malware

I have no idea what to say

[static loom]

what do you mean the topic is malware?

[thorn brook]

tbh I don't even know

I'm supposed to talk about malware

[thorn brook]

and answer this question

yo that was good tho wasn't it

at least it's something

doesn't matter for me lmao, I just want this done

ye okay I will take a note of that

ye but how can I prove that a system is secure? Is that what you mean?

[static loom]

you could argue for the creation of malware that uses AI to give math problems of an appropriate level for people to solve before they can access their phones or computers to better ensure society is put towards better mathematical literacy

[thorn brook]

oh shit lmao

okay so maybe I can say that by investing more in this CS stuff, society would be more secure or something

oh that's a good example

[static loom]

yeah, I think anything else would not really be bringing a positive societal impact

making better or more secure software doesn't really reach the aim of 'societal development in a positive direction', as it doesn't serve society writ large, but rather maintaining the bourgeoisie's status quo

didn't you ever hear the tale of Robin Hood?

[thorn brook]

ye true but I just want to have something to talk about

no I don't think so

okay this will get me started. Thank you both so much!

[river linden]

chaotic good. based

[leaden torrent]

this identity only holds for the real logarithm

ie a > 0 and a ≠ 1

[vast surge]

a\in(0,1)\cup(1,\infty)

[real sigil]

log = ln?

[jovial ember]

Wait sorry won’t it hold for a = 1 as well?

Or am I being massively dumb, you should end up with 0 either way?

[wise rose]

Any grad students in theoretical CS or pure Math? I'm finishing a BS in EE/CompE, which is a bit more pragmatic. But I'm looking at grad school. What made you decide on choosing your MS or Ph.D.?

I was one of those who was "always good at math" and two grades ahead all through grade school and finished first-year Calculus before I graduated high school (I attended college instead of the second half of high school). But then I stopped doing all math and I worked, where I really used very little math.

But then I fell down the Haskell rabbit hole. It started with "I wonder what this programming language is about" and ended up reading abstract algebra and type theory (TAPL and HoTT) books.

[surreal sapphire]

mniip 2.0

[wise rose]

I went back to university and am finishing an EE/CompE degree, which I considered to be fairly useful. So I had to take a second year of maths (multivariable calc, ODEs, intro linear algebra). My courses are primarily math-based but they're not higher maths.

[surreal sapphire]

(on a serious note, you might be interested in asking mniip, iirc they also got into math via haskell or functional programming and are currently looking into pure math phds with a degree in physics i think?)

[wise rose]

interesting. I am new here. I just joined last night as part of my denial about finals this week.

[surreal sapphire]

mood

i think mniip (the server owner) is fine with being dm'd or you can hope to catch them some time

[wise rose]

But I'm not surprised. The HoTT book came out and there's been so much discussion about dependent types with Haskell, Agda, Idris, and then Rust which is mainstreaming some type theory stuff into more practical languages.

[surreal sapphire]

other than that my reason for wanting to do a phd is it being the only way to do more math and not wanting to get a real job

[wise rose]

My goal is probably formal verification. I want to make formal verification more accessible.

[surreal sapphire]

i mean its probably achievable

[wise rose]

I need to take some more undergrad classes. Unfortunately, you can't go from a BS to MS at my satellite campus. Not only do we not have graduate-level classes, we are missing at least one of the undergrad prereqs for the MS (in any of CS, CompE, EE, or SoftE). So I will transfer to the main campus and talk to professors there.

None of them are really working on the stuff that I want to work on. But everyone says to do a Ph.D. at a different university than your BS anyway. However, there are some that have published some interesting papers on model-checking. I can also speak to the Math professors, as I might take the math proofs class (we do logic and proof-writing in intro to philosophy, discrete math, and intro linear algebra but the math department has their own proofs class) and then some algebra.

My satellite doesn't have a math department. The couple of math courses for engineers are taught remote.

[untold sapphire]

chef's kiss

[untold sapphire]

Recently i've been realizing that the zullip groups for some of these subjects are pretty active. i'm in the HoTT zullip, the categorytheory zullip, the Coq zulip and the agda zullip and they all have lots of people talking about types all day long

good community, and much easier to get involved in than transferring universities

[untold sapphire]

i went to undergrad for EE and then later on decided to get into category theory and type theory and went for a phd in pure math. interesting switch. i'd be happy to discuss it more but i'm heading out for the moment

[wise rose]

Is that this? https://zulip.com/

[untold sapphire]

yeah

coq.zulipchat.com

[wise rose]

I haven't used it before but I will look into it.

I am also supposed to be finishing a radio assignment.

[untold sapphire]

agda.zulipchat.com

etc.

[wise rose]

OFDM encoding in Matlab.

and my first final (in Control Theory) is at 7am tomorrow.

[untold sapphire]

wow i don't remember what this is. it has been a while since i earned my degree.

and oof good luck control theory is beautiful but uhhh no subject is beautiful when the final is bearing down on you

[wise rose]

OFDM (Orthogonal Frequency-Division Multiplexing) is a telecommunications protocol used in 4G and 5G

[untold sapphire]

orthogonal frequency-division multiplexing? sounds like some wild shit. my textbook for networking was this wild book that went on random tangents about communism

[wise rose]

it's radio telecommunications, so its rather different than networking

https://www.5gtechnologyworld.com/the-basics-of-5gs-modulation-ofdm/

[untold sapphire]

ok, yeah i just mentioned it because my networking class went a little bit into stuff like that, i guess because they wanted us to get some minimal exposure to it even if we didn't take a whole class

QAM and stuff

[wise rose]

we did that.

yeah, lots of protocols. I have no idea when I will be able to use it because MATLAB simulations don't seem very pragmatic and I just use a library, not implement a communications protocol on my own.

My electives could have been better this semester -- Controls, Radio, and DSP (plus Ethics and Senior Project)

[untold sapphire]

Damn i don't remember any of this stuff. guess staying out till 2am and then getting up early for class in college didn't do wonders for my long term retention. I had to learn a little control theory for a power system i was roped into working on

wind turbine generator

[wise rose]

next semester it will probably be Discrete Math, Automata/Formal Languages, Programming Language theory, and then I have to retake the Assembly programming class because I didn't finish it last year

and senior project -- it works pretty well already. I'm hoping to learn to print PCBs so we don't have a breadboard and do some stuff with outputting data over Bluetooth via an STM32 chip instead of to the Arduino console.

[untold sapphire]

that stuff sounds like a lot of fun. sounds like you have a lot to look forward to, have you ever flipped through the book by hopcroft and ullman on automata

or sipser's book on the theory of computation

[wise rose]

Yeah, I'm working on it now.

I also have that.

I haven't heard whether I will be allowed to take the Automata class because Discrete Math is supposed to be a prereq and I filed a petition with the department to take it concurrently.

But it should be fine.

[untold sapphire]

hopefully

[wise rose]

I'm doing the 1979 book and watching Ullman's lectures on edX

[untold sapphire]

oh i didn't realize he had lectures, neato.

[wise rose]

https://www.edx.org/course/automata-theory

It's running from whenever. So it gives you due dates but unless you pay, nobody grades your homework and you don't take exams. The lectures were recorded a few years ago (he's getting up there) and they're the same lectures from class but done separately in front of a camera, so they're easy to hear.

Watching some lectures online, the camera is kind of far back and you can't hear well and it's hard to make out what is on the board. But this is different.

The course is a sophomore CS course, so I'm sure it's a bit basic but because I am interested in theoretical CS stuff, I want to learn it for real and also chat with a different professor who wrote a lot of papers on automata theory and model-checking.

[devout nacelle]

Recorded video lectures on Theory of Computation by Sipser are available on the MIT OCW channel btw, he's following his own textbook.

[wild lantern]

That's useful.

I'm taking a course following that book next semester.

[wise rose]

I haven't asked the professor which book we're using. I have him for another class this semester (although our lectures have ended, we just have a paper due tomorrow). Syllabus for another professor at the same school says they're using the Cinderella Book but the new 3rd edition (which is what Ullman's lectures also use)

Some people complain that the more advanced stuff isn't in the new edition but I don't know enough about the topic to say. However, it's a classic and it's not exactly something that changes much (unlike, say, compilers)

Another professor's syllabus says they use Sipser with Ullman as additional reference.

[pale orchid]

you're doing cs?

or electrical eng? kinda hard to tell with the courses you mentioned. the first few things you mentioned were part of some communications courses i took

[neat lintel]

???

[frank orchid]

is the server’s minecraft server active?

[deep mango]

yeah

[neat lintel]

ryc

what do you study in math

or plan on studying research professionally

[wise rose]

I'm EE, CompE minor; Master's will probably be CompE although I'm back and forth with CS. It's the same department but I might have to apply to one or the other.

[pale orchid]

icic, EE was the impression i got

[wise rose]

My electives so far have been Instrumentation Design, Control Theory, Radio Telecommunications, and DSP. I try to keep it broad.

I'd basically take the same courses whether my Master's degree is labeled "CS", "CompE", or "EE"

[pale orchid]

strong telecomm / signal processing vibes :x

i do very similar stuff

[neat lintel]

lowkey need some help fr

[wise rose]

well, depends if I finish this telecom lab report tonight 😅

[neat lintel]

this shit hard for me

☠️ ☠️

[wise rose]

If you need help with a question, read #❓how-to-get-help

[neat lintel]

she assigned me 5th grade work

okay

thank

[wise rose]

I don't know what I want to do exactly. What I'm interested in is more theoretical stuff.

but interests and getting a job are not always the same thing

I'm going to transfer to the main campus and take another year of undergrad classes where there are a lot more options. I have to take undergrad classes anyway even if I start a Master's, so I might as well do it while financial aid is paying my tuition. I will take more CS (system programming and OS), digital design (a second Verilog course, VLSIs and ASICs), more computer architecture (I took one course but there's like three available) and some more math (more proofs, abstract algebra, linear algebra)

we'll see if I go the hardware (HDL) or software route for job but it'll be embedded systems either way

I want to add more of the theory to embedded, doing formal verification to prove that critical systems work.

[pale orchid]

aha, embedded systems is hot stuff rn. in that regard though, you'll probably end up doing very little telecomm/sigproc. it can still be useful so that you are aware of the types of problems people run into out there, but that's only like a contextual frame

[wise rose]

I just pretty much took anything that was available that wasn't power.

although in studying for the FE exam, I probably should have taken more power

[mint patio]

When sketching phase portraits for linear systems

with node points

How do I know how to draw the eigenvectors?

Like how tf were these drawn

[deep mango]

Do you know how to plot points

They're just the lines through those two vectors

Like

The points (t, 4t) for the (1, 4) vector

And the points (-t, t) for the (-1, 1) vector

They're marking down all the possible eigenvectors (remember that eigenvectors are still eigenvectors when you scale them)

[mint patio]

I almost typed "oh so you just plot the vector" then realized how 4head that sounded but I think it's right LOL

So you just plot one eigenvector (just taking t = 1), then draw a line through the origin to get all the others?

[deep mango]

Yes

[mint patio]

pog that's easy

ty

[mint patio]

Can you take complex ordered derivatives?

So the ith derivative of a function

Could be real valued or complex or whatever else makes it work

[ancient flame]

yes you can

[fair mural]

how would you do that

cauchy formula for repeated integration?

[ancient flame]

idk

probably

but I know you can

not much more than that lol

they can also be fractional or irrational or really whatever you want

[vivid halo]

Yes, one way to do this is the Fourier transform formula for diffintegrals

You can’t really use the Cauchy formula/Riemann-Liouville because it will spit out 0^i for constant functions

[mint patio]

Why is 0^i undefined?

[mint patio]

I will search this eventually thank you

Is there any physical or geometretical interpretation of a complex-order derivative?

Holy fuck I butchered that word

Sorry just woke up 💀

[deep mango]

you define powers by complex numbers using the exponential. z^w = e^(w log z) and pick a branch of log. but log 0 is always undefined.

and in fact there's no way to make sense of it at all, even trying to call it -infinity. it's a nonisolated singularity, if you approach from different directions the real part will approach -infinity while the imaginary part can approch anything.

[mint patio]

Ohhh

so 0^z is entirely undefined

[deep mango]

well

if w is real

then it's ok

or

if z is real if you are saying 0^z

[mint patio]

I never learned about the different types of singularities :sadge:

[deep mango]

since then it's e^(real * log z) as z approaches 0, but now this limit is 0 from every direction since the real part of the exponent approaches -infinity.

ok this is for positive reals

[mint patio]

Interesting, I see it now

Ok so if z has nonzero imaginary part then

[mint patio]

Why is it fair to call it 0 since the limit is 0 but we can’t do that with the complex case?

[deep mango]

it's probably not fair

idk

i'm not sure what you mean

when we're working with natural numbers, 0^n = 0*0*...*0

we extend that to rational numbers

and it's still 0

and then we take limits to extend it to real numbers and it's still 0

[mint patio]

oh is this cause reals are defined as limits (using cauchy sequences)

[deep mango]

this just doesn't make sense with the way we define powers of complex numbers

it's mostly just a residue of that

[mint patio]

reeee how is this supposed to be easier has its own set of new rules

although a lot of things do get easier when it comes to calculus

thonkers

Thank you Ryc

[mint patio]

If you have two equal eigenvalues how are you supposed to get two eigenvectors

[tall badge]

whats ur matrix

[mint patio]

uhhh I don't have a specific example in mind

[tall badge]

take the 2x2 identity

whatre the eigenvalues?

[mint patio]

just 1 right?

[tall badge]

whats the 1-eigenspace?

[mint patio]

good question 🥴
so we have Ix = x
wait so is it just every vector lmao

[tall badge]

yes, R^2

whats a basis of it?

[mint patio]

well just {(1,0),(0,1)} works right

[tall badge]

yes

so its an eigenbasis

[mint patio]

Yup, I see it now

Well that's cool

[tall badge]

in general u find as many (linearly independent) eigenvectors as u can by finding a basis of each eigenspace

note i say "as many as u can" bc some nxn matrices dont have eigenbases, ie less than n linearly independent eigenvectors

take $\m{2&1\0&2}$

[fathom swallow]

RokettoJanpu

[pale orchid]

you can alternatively reformulate this as finding the general solution of a problem with infinitely many solutions. the general form of these infinitely many solutions has you find basis vectors of the null space, and these are precisely the eigenvectors

whenever these so-called "free parameters" pop up, you're finding a basis for the null space

[cold needle]

jcf

[pale orchid]

jentucky cried ficken

[mint patio]

Say we're considering a linear system of ODEs (consider only the 2x2 case) where the two eigenvalues are equal but there are two distinct eigenvectors

Then the origin is a proper node, and stability depends on sign of course

But how do you draw it? Do you plot both eigenvectors and then just draw straight lines intersecting the origin between them?

So it looks like a star

It's not clear from the drawing the professor provided

If we have eigenvectors v1, v2, would it be like

I do not like this stuff

😭

All the material on the final takes so long to do

I wish it was 2 hrs 45 mins instead of just 2 hrs

Laplace Transforms of discontinuous function (by far the most time-consuming), power series solutions, linear & nonlinear systems

And the Laplace problems they give are always so much freaking work

[mint patio]

How would I write a product as a "sequence" of sorts? Or with Pi instead of Sigma? Idk cause it's not got a perfect pattern

The denominator here

Instead of writing that entire thing I'm wondering if there's a way to like write it as a_k or some other compact notation

Then define a_k or whatever off to the side

Or is it not worth it

[prisma burrow]

$$\frac {x^3^k} {\Pi (3i-1)(3i)}$$

Is this what you are asking for?

[fathom swallow]

✿ Ƹ̵̡Ӝ̵̨̄Ʒ Lepidopterian (◕‿◕✿)
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[mint patio]

Something like that yes! But not sure if that expression exactly, I haven't expanded it
Let's see

What's the starting index?

Wait i = 1

Okay

(3-1)(3) * (6-1)(6) * (9-1)(9) ...
= 23 * 56 * 8 * 9 ...

Yes!!!!!!

Thank you that's exactly what I wanted

[prisma burrow]

Hint your answer is given right in the question.

Look at the last terms.

[mint patio]

Ahah yeah but I've never worked with the product symbol before, I guess it's pretty obvious though

Wait

Oh

lol

I thought that was just for the leading coefficient

I'm an idiot

That's perfect

Since the product in the solution uses the same indexing variable as the x terms

Could I not just write it as Pi (3k)(3k-1) within the sum

So $\sum_{k=1} \frac{x^{3k}}{\Pi_{k=1}(3k)(3k-1)}$

[fathom swallow]

kanga gang bimbo lover feather

[mint patio]

Let's see

[prisma burrow]

Umm there is a mistake with your notation.

[mint patio]

What's wrong?

[prisma burrow]

The iteration variable should not be k again.

[mint patio]

Why not? In the picture it's defined with k

Oh

Wait is it this

[prisma burrow]

This is different. This is a product of terms say (3i)(3i-1) with i going from 1 to k.

[mint patio]

So $\sum_{k=1} \frac{x^{3k}}{\Pi_{i=1}^k(3i)(3i-1)}$

[fathom swallow]

kanga gang bimbo lover feather

[mint patio]

OMG YES

[prisma burrow]

Yes.

[mint patio]

I figured it out right before you typed it poggies

Alright got it 😁 Thank you!!

[prisma burrow]

Good for you.

[mint patio]

I'm confused, how can we start the series at 0? Doesn't that result in the first term being division by 0?

[ancient flame]

product inside sum

damn

[prisma burrow]

probably a mistake, the denom clearly starts at 3.

[mint patio]

Yeah but the thing is

This is the expanded form

Oh

[prisma burrow]

nevertheless there is probably a mistake.

[mint patio]

Maybe I messed up wait

ok I think they just like

did something weird

For context this is the general solution to a second order ODE

Why can they just pick a_0 = 1 and a_1 = 0

when there were no initial conditions?

I just left them as a_0 and a_1

I got the same y_1 as them in the end

but my y_2 had an x as the first term instead of 1

So I wrote my series as x + ...

and I was just gonna leave the x there then write everything with sigma from x^4 onwards

I think that was their bad

As far as writing y_2 goes

$\sum_{k=1}^{\infty} \frac{x^{3k+1}}{\Pi_{i=1}^k(3i)(3i+1)}$

[fathom swallow]

kanga gang bimbo lover feather

[mint patio]

Yeah?

This should have gone in #❓how-to-get-help sorry

Got carried away 🥴

[fair mural]

@mint patio seems correct

[mint patio]

vro

how do you live without commutativity

shit goes crazy

[jovial ember]

You choose to reject non commutatige rings

They do not exist to me

They cannot hurt me

🙉

I do not hear them

🙈

I do not see them

🙊

I do not speak them

[mint patio]

do you chair them.

btw

I love linear algebra

[jovial ember]

Linear algebruh

[jovial ember]

What are you referring to

I do not know what you are asking if I chose

So I cannot answer this question

[fair mural]

do you monkey them

[mint patio]

Do you chair non-commutative rings?

[jovial ember]

What is “them”

[jovial ember]

All I see you say is

[mint patio]

OH

[jovial ember]

“Do you chair ?”

[mint patio]

HAHA

Clever

[fair mural]

does chair monkey chair

[jovial ember]

[deep mango]

[mint patio]

what

[deep mango]

how do you live with it...

without noncommutativity you have to construct pathological rings like Q to get any interesting behavior

[mint patio]

how is Q pathological

[deep mango]

idk ask chmonkey

[mint patio]

I thought pathological meant an example that exists to be counterintuitive

or like

bad behaved

[jovial ember]

Q isn’t a pathological ring

Lol

[mint patio]

that kinda stuff

[deep mango]

yes it is

[jovial ember]

As an abelian group it’s kinda whacky

[deep mango]

it's commutative but it's not a product of Z's

[jovial ember]

Q is literally a prime field

That’s its group structure…

[deep mango]

how can something be a pathological group but not a pathological ring

rings are supposed to be even better than groups

[jovial ember]

Because as a ring it’s nice

It’s a fucking field

As a field it’s not too nice because most fields are nicer than it

But it is way better than almost every ring because it’s a field

It’s like literally any group is very nice considered as a monoid because they actually have inverses

But as a group it can start to seem pretty shit because compared to other groups it’s badly behaved

[deep mango]

wat

but

wat

wouldn't that make Q a really nice group

[jovial ember]

No Q is weird as a group

Okay

The real thing here is

[deep mango]

😵‍💫

[jovial ember]

As a ring the additive structure is way less important

It doesn’t matter if Q looks kinda weird as a Z-module

Because it’s literally a field

[deep mango]

hmm

alright

i'm convinced

[mint patio]

yeah

me too

me reading every sentence chmonki says when he's talking math

[deep mango]

none of this fits into my perception of math that everything is either trivial or pathological

[jovial ember]

Lmao

It’s weird because as a field Q starts to seem like shit

Because fields are soooooo fucking nice

But rings are scary beings

They are so fucked up

[bronze pelican]

Q is the most basic field there ever was

[jovial ember]

So even being a shitty field is great as a ring

Yeah but PTY there’s a lot of reasons Q is weird as a field i think

[fair mural]

@mint patio me reading literally everything ryc says

[jovial ember]

Like from a model theory perspective or something

[bronze pelican]

wfym

[mint patio]

ryc makes sense I’m just too braindead to follow

[bronze pelican]

do you even know any model theory chomkey

[mint patio]

chmonkey is like ng

[jovial ember]

No but I know model theorists who say Q is fucked up

[mint patio]

saying random words and hoping people believe it’s real math

[deep mango]

see chmonkey? i do make sense after all.

[mint patio]

algebra is fake.

[bronze pelican]

okay then your judgement is invalid

[jovial ember]

Maybe this is my geometer brain vs your number theory brain

[deep mango]

yep chmonk is definitely the one making things up. not me

[jovial ember]

I don’t like that Q isn’t algebraically closed

[tiny marten]

as an uncountability denialist Q is my favorite set for doing math

rational analysis

[jovial ember]

Mf is willing to accept countable infinite

But uncountable is too far

Smfh

[tiny marten]

that's right, they are all one

[jovial ember]

Actually maybe PTY is right

[deep mango]

i'm willing to accept countable infinity and the first uncountable cardinal, but not the cardinality of the continuum

[jovial ember]

Q has a lot of nice things going for it

But also it’s kinda bad

[tiny marten]

Q is very rich in certain computational methods

[fair mural]

ryc always makes sense what the heck

[bronze pelican]

Q is the most basic field you could ever think of

[deep mango]

Z_2 in shambles

[jovial ember]

Yeah but is basic = good?

[tiny marten]

outside of that it is mostly R without its evolution stone

[bronze pelican]

i dont know what good means

[jovial ember]

Lmao

[bronze pelican]

if anything, C is weird as a field because its isomorphic to proper subfields

[jovial ember]

I think that isn’t too surprising like

Big things tend to be iso to its proper sub things

C is amazing since it’s universal in a sense for almost all char 0 field stuff

[deep mango]

if you're not isomorphic to a proper subobject then you're totally fucked up IMO

[jovial ember]

Using some fucking magic over Q

Like if your statements only use like

Countable (or maybe |R|) amount of stuff

[bronze pelican]

wdym

[jovial ember]

So like for statements like that

You can consider the subfield generated by that

Or like

The point is you do stuff with transcendence degree over Q

And you can reduce to proving a statement for C

And then you can use GAGA and stuff

It’s kind of like how you can Freyd-Mitchell almost anything

By taking a subcategory with all the objects you need

Then reduce to R-mod

I don’t remember the specific details but this makes char 0 algebraically closed alg geometry often reduce to complex geometry

[bronze pelican]

i dont realy follow

[jovial ember]

It’s because I don’t remember how it works very well

[bronze pelican]

but imo i think C is only good it you consider it's topology/geometry

[jovial ember]

But that’s super powerful

I think that’s intimately related to its algebra in a way

Its geometry explicitly

Via alg geo stuff

Then to its topology and analytic stuff via GAGA

Maybe your view on what “as a field” is different than mine

But in my head C exists foremost as a field and also as like the nicest object in the entire world

[bronze pelican]

as a field, C is isomorphic to C_p = completion of algebraic closure of Q_p, even though they have very different topologies