serious-discussion

[neat lintel]

Sure

Where is it?

@true zinc you have an answer ready?

[true zinc]

No

I don't understand why you are listing various classical mechanics subtopics

[neat lintel]

It was diving

I forgot boyle’s law

[neat lintel]

Combine them and think what activity needs all of this

[summer crypt]

I dm'd you

[true zinc]

Ah

I was thinking of rowing or some other water-based activity, so I guess I was close

[ember lance]

Cool! I also like diving, but haven't had much chance

[neat lintel]

The sampling method COULD be accurate depending on how many people you take and who you take

[neat lintel]

I went for about 3 weeks

In december

[summer crypt]

it was 925 people from the ads and 185 from the open ended college surveys, so about, 1110 people in total

[ember lance]

What did you see?

[neat lintel]

It could be about accurate

[neat lintel]

Chinese nemo (yellow nemo)

[summer crypt]

do you know what the sampling method exactly is?

[neat lintel]

Hold on let me use the sampling method

[summer crypt]

ok!

[neat lintel]

There are more sampling methods

There’s this one:
$s^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2$

[fathom swallow]

TabMineCrafter

[neat lintel]

There is also n=N/1+N•e²

[summer crypt]

I used SurveyMonkey's margin of error calculator, put the total population and the sample size with a confidence score of 80 and it said 3% and with 99% it was 6% but they don't really mention the confidence score in the paper

[neat lintel]

Ok no
I used the wrong one

[summer crypt]

they are using what's called cluster sampling, right?

[fathom swallow]

TabMineCrafter

that is  $n_c = \frac{N_c}{N} \cdot n$ right?

[neat lintel]

As i remember

,ti @summer crypt

[fathom swallow]

This user hasn't set their timezone! Ask them to set it using ,ti --set.

[neat lintel]

,ti @true zinc

[fathom swallow]

The current time for thecatcollective is 03:41 PM (EST) on Thu, 02/01/2025.
minecrafter96024 is 7 hours ahead, at 10:41 PM (EET) on Thu, 02/01/2025.

[summer crypt]

can you explain this?

[neat lintel]

$n_c$ is the number of clusters to sample

[fathom swallow]

TabMineCrafter

[neat lintel]

$N_c$ is the number of clusters in the population

[fathom swallow]

TabMineCrafter

[neat lintel]

N is total population size

n is total sample size

[neat lintel]

What time is it for you?

[summer crypt]

wait... so how do I use this? Because there are multiple sub clusters as well, they divided the US into 9 regions, and those 9 regions into most populated, and then the remaining into more than 25,000 inhabitants and those with less, and then chose the highest populated regions and those with >25,000 inhabitants

[summer crypt]

past midnight

[neat lintel]

Where do u live?

@summer crypt

[fresh comet]

[neat lintel]

,ti @fresh comet

[fathom swallow]

The current time for ourfallenstars. is 01:47 AM (EST) on Fri, 03/01/2025.
minecrafter96024 is 7 hours ahead, at 08:47 AM (EET) on Fri, 03/01/2025.

[neat lintel]

Bros on discord at 1:47 am

[fresh comet]

you can sometimes find me here at 6am

[north topaz]

abhi to party shuru hui hai

[ancient lance]

tumara hindi kafi badya hai , bhai

[neat lintel]

,ti @ancient lance

[fathom swallow]

The current time for math_rocks is 12:52 PM (IST) on Fri, 03/01/2025.
minecrafter96024 is 3 hours and 30 minutes behind, at 09:22 AM (EET) on Fri, 03/01/2025.

[neat lintel]

,ti @north topaz

[fathom swallow]

This user hasn't set their timezone! Ask them to set it using ,ti --set.

[neat lintel]

@ancient lance wanna do math

[ancient lance]

Not now, I'm tired

Not well

[north topaz]

mazaak kar raha hai

[ancient lance]

nahi

[north topaz]

that's literally the title of a song

it's like if you copied and pasted xue hua piao piao (in characters) into chat

[ancient lance]

I see

okay

sorry

[neat lintel]

At, almost 1 pm?

[ancient lance]

well, i'm better now

[neat lintel]

G-ometry or algebruh?

[ancient lance]

wdym

I'm doing calculus rn

[neat lintel]

Oh ok

[neat lintel]

Where you at?

[ancient lance]

series

tests for convergence

[north topaz]

I thought you would know that anyways

that Indian party song is so famous

it has a six in the morning line

due to Indian pronunciation it really sounds like sex in the morning to me

which is weird and funny

cause those videos always have girls dressed up loosely and stuff, All India Bakchod made a vid on every Bollywood video ever

[ancient lance]

I don't follow mordernt bollywood tbh

I listen to songs form the 70s and 80s mostly

[neat lintel]

Like this?
$S = \sum_{n=0}^\infty a r^n$

[fathom swallow]

TabMineCrafter

[ancient lance]

like this

[neat lintel]

Same shit

S = sin(π/n)/1-(-1)

So sin(π)/2

Which is about 0,027

[ancient lance]

fair, yeah

[neat lintel]

S = a/1-r

[neat lintel]

When you buy an iPad to write and to do math on it just to realize paper writing is 1000x better but you have to stick with your goal of writing on ipad

,ti @ancient lance

[fathom swallow]

The current time for math_rocks is 02:48 PM (IST) on Fri, 03/01/2025.
minecrafter96024 is 3 hours and 30 minutes behind, at 11:18 AM (EET) on Fri, 03/01/2025.

[fresh comet]

,ti

[fathom swallow]

The current time for ourfallenstars. is 04:19 AM (EST) on Fri, 03/01/2025.

[neat lintel]

Is this real?

[solid snow]

that's up to you

[neat lintel]

So you cannot relate to it?

[neat lintel]

It’s up to me if you can relate to it?

[solid snow]

i'm saying whichever medium you prefer is up to you

[neat lintel]

But can you relate to it?

[modest anchor]

Yoo

[neat lintel]

Yoooo

,ti @modest anchor

[fathom swallow]

This user hasn't set their timezone! Ask them to set it using ,ti --set.

[neat lintel]

Guys

Let’s name cases where soh cah toa is useful

[icy locust]

We could, but then our list would circle back

[tame egret]

Estimating the magnitude of 3 forces in equilibrium

[neat lintel]

Calculating the initial velocity when launching, a bird for example

[true zinc]

aap donon yahaan hindi kyon bol rahe ho?

[ancient lance]

kyu nahi

in all seriousness, just for fun

[real rivet]

Hey everyone! I'm Gaspar. I'm teaching myself some maths, and I hope to learn with all of you ☺️

[neat lintel]

And i do maths with u coz i am bored

[neat lintel]

So where are you rn?

[real rivet]

Basic math, just started.

[true zinc]

I've got no issues with it tbh lol, just don't want y'all to get whacked by mods

[stuck tendon]

good morning folks

[neat lintel]

More specific

[north topaz]

wait Ryan that reminds me

can you speak Indonesian (Bahasa)?

[true zinc]

yea but we don't speak it much so we're very very rusty, especially compared to our hindi or spanish

[daring parrot]

How long do i read and write everyday , im a high school dropout and i wanna go to college for stem degree in my 20s

my vocabulary very bad and im incapable of writing essays

[true zinc]

1-2hrs, just start slow, read a text and copy it if it's short enough, eventually start summarizing and writing about the texts

[daring parrot]

thank u ryan

[daring parrot]

hey wdym by read a text and copy it

[true zinc]

Literally write it word for word

and by short I really do mean short, 3-5 pages long texts

[pure obsidian]

yes i recommend writing by hand if i were in ur shoes bc i remember stuff better when i write it by hand

[true zinc]

This helps immensely

[hidden oyster]

I never understood typing out notes it just doesn't feel the same imo

[pure obsidian]

real. altho i have to admit if ur a gamer or a programmer typing might just feel better

[hidden oyster]

I play an ok amount of video games but typing just don't feel right for taking notes

For programming it's all in my head no notes (probably not good but it do be like that sometimes)

[amber berry]

hey peg

[narrow carbon]

Weak English is an obstacle to learn maths

[amber berry]

then learn english intead of urdu.

[upbeat kraken]

I find listening to politicians very helpful

Because they talk around the bush and talk airy fairy they have to come up with flowery and complicated words so they sound smart so they talk a lot and they talk flowery

You can learn those words

And also works for interpreting practice

[real rivet]

Hmm, I'm about to dive in to "rules for multiplication"

[pure obsidian]

loll

[pure obsidian]

yeah ive tried typing stuff for notes and it goes in one ear and out the other when i do that

but it's good for when i want to make summaries of stuff

or even mind maps

i think i prefer mind maps that are typed over ones that are written by hand. they're just neater

[dry narwhal]

if u dont mind, what software do u use to type ur mind maps/make them digitally?

i'm curious as to what other people use for that stuff because i keep on hearing on those productivity youtube things that obsidian is really good for that, like they say it's a lot better than notion, but i have tried obsidian and it is definitely very comprehensive in terms of features but sometimes i find that it is too much for me

[dry narwhal]

like obsidian is so complex for me that i have difficulty using it if that makes sense 😭

[pure obsidian]

i wish i could remember but i dont. tbh i have not made a lot of mind maps, maybe like 10 at most in my lifetime lol

https://miro.com/mind-map/?irclickid=W00QfFQMQxyKRbnSyL1EEyV6UkCS5VRpuQ2Gxc0&utm_source=TechnologyAdvice LLC &utm_medium=cpa&utm_campaign=&utm_affiliate_network=impact&irgwc=1&rel="nofollow"# this one looks good tho

[dry narwhal]

it's okay!!

[pure obsidian]

haven't kept up with productivity apps.. is it new?

[pure obsidian]

ok cool 🙂

[dry narwhal]

i think it's a few years old

let me checkk

whoops i dont have image perms haha

Obsidian was initially released on 30 March 2020. Version 1.0. 0 was released in October 2022.

from google

[pure obsidian]

gotcha, i'll take a look at it

no offense but it just looks like evernotes to me 😆

[dry narwhal]

LMAOO i actually never used evernotes before

[lusty atlas]

is evernotes good

i can say obsidian is good in its use for me

screenshots + math mode in markdown + markdown tutorials

[dry narwhal]

oh yea omg i remember it had latex

and i think it had code support too

like it had lots of support for many different types of formatting

[pure obsidian]

i mean i used it for high school

like one high school course but i forgot lel

[dry narwhal]

i should probably get the hang of using productivity software more

especially since i am planning on taking computer science for college 🥲

i'm on my last year of hs

[pure obsidian]

that's good. no harm in using something different

[pure obsidian]

nice

[lusty atlas]

markdown has it so it does too

it also has RTL support for hebrew and arabic

[dry narwhal]

ooo okay

[lusty atlas]

yeah thats the courses i use it for rn

for math courses i use regular notebooks because latex is not doable

[dry narwhal]

what type of math appears the most in comp sci?

im gonna guess discrete math

[violet compass]

[violet compass]

You need inverse of trigonometry+ calculus and some Quantum mechanics only

[dry narwhal]

oooo okay

[violet compass]

ok 😌

[dry narwhal]

yay we're gonna discuss quantum mechanics in a few months

[violet compass]

I'm a bio student btw

duh

[lusty atlas]

im in my 2nd semester rn, and it really depends on the courses you have

like i have in my freshman year 2 real analysis courses, 2 linear algebra courses, and 2 discrete math courses

and each have different topics like graph theory and combinatorics which require different symbols and drawings

this should be the standard freshman year math tho

[dry narwhal]

ohhhh okk

omg good thing i took that discrete math elective last year 😭 🙏

[lusty atlas]

can you send the syllabus?

[dry narwhal]

okayy

wait i fear i might doxx myself if i do actually

because our syllabus is very different from other schools in my country

but for first quarter, we had modular arithmetic

[lusty atlas]

send a screenshot cropped or something?

[dry narwhal]

euler's totient function, fermat's little theorem

also chinese remainder theorem

[lusty atlas]

whats modular arithmatic?

[dry narwhal]

i won't have the best explanation ever, but it's like dealing with equations/congruences as how my teacher put it, "like how a number line is wrapped around a clock"

so you can have stuff like x is congruent to 1 (mod 5)

[lusty atlas]

do you still remember the proofs for those theorems? i mean theyre very hard to me

[dry narwhal]

"is congruent to" is gonna be like that three lined equal sign

[dry narwhal]

they didn't really discuss the proofs that muchh 😭 im sorry

[violet compass]

[lusty atlas]

so it's just normal arithmatic with mod?

[dry narwhal]

yes

[lusty atlas]

i see

[dry narwhal]

and also things like 7^2023 is congruent to x (mod 7)

[lusty atlas]

yes i understand

[dry narwhal]

i'm still in hs so it's nowhere near university level math

[lusty atlas]

did you have set theory also?

[dry narwhal]

yes

only basic tho

[lusty atlas]

up to what topic?

[dry narwhal]

like straight up venn diagrams

[lusty atlas]

very nice

[dry narwhal]

for graph theory we had like

[lusty atlas]

did u have graph theory?

[dry narwhal]

yepp

we had like

the different types of graphs

how to determine if it's an eulerian graph

or hamiltonian cycle

hakimi havel algorithm

idk if i spelled that right

[lusty atlas]

eulerian graph? u mean eulerian path or eulerian cycle?

[dry narwhal]

OH thosee

srry

like u can cross every edge exactly once

[lusty atlas]

yes we just learned it this week

did you do proofs with it?

[dry narwhal]

nope

[lusty atlas]

i still think it's a hard topic

[dry narwhal]

yess

especially in your case where u have to prove it

i'm scared for proving math LMAOO

[lusty atlas]

did you also learn ramsey graph? this is our next week's topic

[dry narwhal]

nope 😢

[lusty atlas]

me too 😥

is that elective an "subject" class? like how physics and cs are electives in my country?

[dry narwhal]

no it's actually completely optiional

[lusty atlas]

we had those electives for 3 years for example

[dry narwhal]

that specific one

[lusty atlas]

i dont understand

[dry narwhal]

it's like an optional class u can enroll

for ours

[lusty atlas]

when u say "class" what does it mean?

[dry narwhal]

like a subject

[lusty atlas]

schedule-wise

[dry narwhal]

ohh

basically 2 hour discussion

and homework

like the schedule wise u go to a classroom for it

[lusty atlas]

is it an after-school thing or inside the regular schedule like math?

[dry narwhal]

after school

[lusty atlas]

i think i get it

[dry narwhal]

yeppp

[lusty atlas]

we dont have those in my country

it's very cool, your syllabus seems pretty good as well

were there other electives?

[dry narwhal]

yess

there was nuclear physics

[lusty atlas]

like AI or other math ones?

[lusty atlas]

elaborate

[dry narwhal]

and i think robotics

that's it

[lusty atlas]

we had only basic physics and that was an elective that is a "subject"

[dry narwhal]

they don't have those anymore tho in our previous grade level

they cahnged the electives

idk what they are now tho!

[dry narwhal]

ohhh

[lusty atlas]

do you know this one?

[dry narwhal]

nope

i didn't enroll in it :')

[lusty atlas]

do you have a syllabus for either 1 of the 2?

[dry narwhal]

so in your country's vocabulary, "elective" means "class/subject" for mine?

[dry narwhal]

i don't have one sorry 😭

[lusty atlas]

yes exactly, we only have to cover 5 points with elective(s), so u can choose just 1 as well

[dry narwhal]

we had physics since 9th grade

it is a subject

[lusty atlas]

mandatory up till 12th grade?

[dry narwhal]

yess

[dry narwhal]

ohhhh okay

[lusty atlas]

u have a syllabus?

[dry narwhal]

what lessons did you have in physics until 12th grade?

like up to what topic did u study physics

in hs

[lusty atlas]

we had kinematics, dynamics, circular motion, gravity (newton's formula and stuff), electricity (electric circuits and stuff) + experiments in class

thats the broad stuff of the topics i remember

[dry narwhal]

ahhhh i see

[lusty atlas]

electricity had electric forces and electric fields too and other stuff like that

[dry narwhal]

yeah our syllabus is just that along with fluid dynamics, zeroth and first law of thermodynamics, quantum mechanics, and relativity

[lusty atlas]

very impresive, how do you quantum mechanics exactly?

this seems more advanced than our regular topics

[dry narwhal]

we have yet to discuss quantum mechanics tho 😭

in 2 months

[lusty atlas]

i see

do you have a cs electives or subject?

[dry narwhal]

yep

[lusty atlas]

which languages u use?

[dry narwhal]

python

2 years ago we did java

[lusty atlas]

do u have a syllabus for it?

[dry narwhal]

it's just super basic concepts like if else, loops, recursion, object oriented programming concepts

this year we're more focused on stuff like big o notation and space/time efficiency of code

also if u dont mind, i gtg now! 😭

[lusty atlas]

u say that but we are still doing this

[dry narwhal]

it's almost 4 am here LMAO

[lusty atlas]

😥

oh, goodnight mate

[dry narwhal]

ahhh so this is the syllabus for the college as well?

[lusty atlas]

for our software 1 course it's basics of java + OOP concepts + design patterns (very few learned so far)

and it's a 2nd semester course

[dry narwhal]

ooo okay

[lusty atlas]

we did basics of python + OOP for the intro to cs course (also we learned some algorithms and data structures like hash maps and CYK algo)

[dry narwhal]

compared to urs since u are a university student ours is super super basic!

[dry narwhal]

we didnt even have hash maps and cyk algo yet

[lusty atlas]

i kind of doubt that

[dry narwhal]

actually we won't be discussing it in hs

like it's not a part of the curricullum

like we straight up won't learn that

[lusty atlas]

yeah we didnt either in mine

but i think thats not that big of a thing compared to your subject

very impressive school you have

go to sleep now mate

goodnight

[dry narwhal]

thank you!! <33

i'm honored that i could talk to someone who's actually pursuing computer science in uni haha

thanks a lot for the talk!

[dry narwhal]

i willl

[lusty atlas]

if u ever need help with something like the AVL trees data structure for example i might be able to give you some resources or tips

although courses could be different and all

[dry narwhal]

thank you sm😭 ❤️

[lusty atlas]

but if u need help you can try asking if i have resources

[dry narwhal]

yess

[dry narwhal]

got it!

[neat lintel]

Hello, does somebody has done A-Level Math and Economy? and has studied Economy?
I really would like ask some questions

[carmine hawk]

Anyone know how to apply for college? I've always been super independent and moved out after highschool; always doing stuff by myself, so im not really sure on how to start the process.

Normally I would imagine the process being easy if you already had an idea of going to a college and having a councilor take care of things on that end(in highschool), but that isnt my case

[neat lintel]

i did economy in sophmore year and i am senior i might have forgetten everything i learned in that class

and what do you mean by A-level math?

[storm sage]

(answered in #discussion)

[nimble holly]

I'm majoring in economics, currently in my sophomore year.

Hope I can answer any of your questions.

[upbeat kraken]

But what if it is 6/7 divided by 8

6/7 / 8 can I reciprocal 8

8 / 6/7 where 6/7 is a fraction you reciprocal 6/7 beocmig 8 x 7/6

What do it is 6/7 / 8 where 6/7 is a fraction

Can I take reciprocal of 8

Becoming 6/7 x 1/8

[bright meteor]

why not use parenthesis?

[upbeat kraken]

What would th ag do

[bright meteor]

make easier to calculate

[upbeat kraken]

(6/7) / 8

I take reciprocal (7/6) x 8

If it is 8 / (5/7) can I reciprocal 8

[bright meteor]

(6/7) / 8 = (6/7) * (1/ 8)

[upbeat kraken]

So

When you take reciprocal of fraction

[bright meteor]

8/ (5/7) = (8/1) / (5/7) = (8/1) * (7/5)

[upbeat kraken]

When fraction is dividend then you do not take the reciprocal?

Divisor*

Then you do

[bright meteor]

(6/7) / 8 = (6/7) / (8/1) = (6/7) *(1/ 8 )

[bright meteor]

yeah. or you can think of divisor fraction too, and follow up with "reciprocaling the divisor"

[upbeat kraken]

Ok cool thank you

[flat cloud]

is a shape where every point is at a different distance from its center possible?

if not, why

[static loom]

point meaning on its perimeter or within the object itself?

when you say shape does that mean like a 3D object or could it be something like a polygon, or anything

[upbeat heart]

Are there any learning source recommendation for learning logical equivalences?

[flat cloud]

hmm lets say 2d shape

and by point I mean in its perimeter

[static loom]

I'm thinking it won't be possible by intermediate value theorem

[flat cloud]

damn, ivt is really everywhere

we gave it some emphasis in my calc I class but ive seen it so many proofs from different fields

[neat lintel]

Like a spiral?

[gusty socket]

the ivt works so well here that it guarantees that there are at most two points on such a curve that don’t have another point which is the same distance from the center, and also that you can replace “center” with “any point”

[honest flint]

hello, does anyone know any online platform where i can slve any math problems instantly??

[mighty pilot]

Wow i really want that too

[honest flint]

there are many

[little plaza]

I am making with my mates a code through python to plot a straight line graph. Any recommendations.

import turtle
import math

def graph():
# import package and making objects

sc=turtle.Screen()
trtl=turtle.Turtle()
 
# method to draw y-axis lines
def drawy(val):
     
    # line
    trtl.forward(300)
     
    # set position
    trtl.up()
    trtl.setpos(val,300)
    trtl.down()
     
    # another line
    trtl.backward(300)
     
    # set position again
    trtl.up()
    trtl.setpos(val+10,0)
    trtl.down()
     
# method to draw y-axis lines
def drawx(val):
     
    # line
    trtl.forward(300)
     
    # set position
    trtl.up()
    trtl.setpos(300,val)
    trtl.down()
     
    # another line
    trtl.backward(300)
     
    # set position again
    trtl.up()
    trtl.setpos(0,val+10)
    trtl.down()
     
# method to label the graph grid
def lab():
     
    # set position
    trtl.penup()
    trtl.setpos(155,155)
    trtl.pendown()
     
    # write 0
    trtl.write(0,font=("Verdana", 12, "bold"))
     
    # set position again
    trtl.penup()
    trtl.setpos(290,155)
    trtl.pendown()
     
    # write x
    trtl.write("x",font=("Verdana", 12, "bold"))
     
    # set position again
    trtl.penup()
    trtl.setpos(155,290)
    trtl.pendown()
     
    # write y
    trtl.write("y",font=("Verdana", 12, "bold"))

straight_line_graphs()

Main Section

# set screen
sc.setup(800,800)    
 
# set turtle features
trtl.speed(100)
trtl.left(90)  
trtl.color('lightgreen')
 
# y lines
for i in range(30):
    drawy(10*(i+1))
 
# set position for x lines
trtl.right(90)
trtl.up()
trtl.setpos(0,0)
trtl.down()
 
# x lines
for i in range(30):
    drawx(10*(i+1))
 
# axis 
trtl.color('green')
 
# set position for x axis
trtl.up()
trtl.setpos(0,150)
trtl.down()
 
# x-axis
trtl.forward(300)
 
# set position for y axis
trtl.left(90)
trtl.up()
trtl.setpos(150,0)
trtl.down()
 
# y-axis
trtl.forward(300)
 
# labeling
lab()
 
# hide the turtle
trtl.hideturtle()

def straight_line_graphs():
print("Straight Line graphs")
print("")
print("")
print("Equations are in the form: y = mx + c")
print("What is M?")
m = int(input(""))
print("")
print("What does C equals? ")
c = int(input())

t = int((1/m)*300 )

angle = math.degrees(math.atan(m))

the gradient works but the y - intercept still needs work

print("Angle is ",angle)
graph()
turtle.speed(0)
turtle.down()
C = int((1/m *300) - (math.sqrt(45000)/2) + (7.07*c)) 
turtle.forward(C)

for i in range(t):
    turtle.forward(1)
    turtle.left(90)
    turtle.forward(m)
    turtle.right(90)

Sorry a lot of lines

[static loom]

personally I'm not a fan of turtle graphics, there's probably a better way imo

[little plaza]

I tried to fix it with ChatGPT, our code came out closer to the intended criteria. Thought this was the next best place. I am in no considerable rush. Just want to add it to our 1200 line calculator. We made in our free time / during computer science class (GCSE)

[neat lintel]

wsp

@little plaza just use the 3b1b

python library

[hardy iron]

That’s manim and it’s completely different

[neat lintel]

right

i see u guys are doing it from scratch

[little plaza]

Yeah.

Opened idle borrowed our teachers whiteboard and did maths.

[forest axle]

I agree with Merosity; turtle is generally a bit of drag, so if you don't want to have to deal with all of its limitations, I'd recommend the pygame library or something similar (PIL could also be an option but it isn't realtime.) Of course, if you want to do this totally from scratch that isn't an option

[icy locust]

challenge: do it without libraries (only standard lib)

even harder challenge: make it perform reasonably fast in Python (again, no dependencies like numpy)

[little plaza]

Awesome, probably will stay “from scratch”

Tbh I will probably be back. Thanks for the suggestions tho. May make a vote with my mates who are doing this with me. To see if we will take shortcuts

[tame egret]

You could upload this text as a file to prevent flooding the chat

[little plaza]

Sorry, I will make sure to place in a file next time. Appreciate the feedback

[icy locust]

such is the life of discussy

[tame egret]

Lol

[icy locust]

yeh good idea to do files/snippets

but considering the usual signal to noise ratio of discussy channels

[neat lintel]

People with these hobbies are the main character: math, writing, coding and anything else having to do with logic science

If you have a cat that adds even more main character energy

[proud olive]

You have a cat tabminecrafter?

[neat lintel]

Yeeeeeesssssss

[manic reef]

Why not use ChatGPT for math help

[north topaz]

you can't trust it to give you correct info

[south vector]

especially if the topic is totally new to you

you won't be able to spot fallacies

[neat lintel]

So yeah dont use it

[sly jungle]

One way to view the mathematics of
the 19th and 20th centuries is as a stalwart attempt to move this line further
and further back toward some unshakable foundation. The majority of the
material covered in this book is attributable to the mathematicians working in
the early and middle parts of the 1800s. Augustin Louis Cauchy (1789–1857),
Bernhard Bolzano (1781–1848), Niels Henrik Abel (1802–1829), Peter Lejeune
Dirichlet, Karl Weierstrass (1815–1897), and Bernhard Riemann (1826–1866)all
f
igure prominently in the discovery of the theorems that follow. But here is the
interesting point. Nearly all of this work was done using intuitive assumptions
about the nature of R quite similar to our own informal understanding at this
point. Eventually, enough scrutiny was directed at the detailed structure of R
so that, in the 1870s, a handful of ways to rigorously construct R from Q were
proposed.
Following this historical model, our own rigorous construction of R from Q
is postponed until Section 8.6. By this point, the need for such a construction
will be more justified and easier to appreciate.

from Understanding Analysis, Abbott

[sharp mulch]

Ok

[sly jungle]

ref - discussion in book-recommendations channel 😛

which is better done here

[little plaza]

Did not work. I did try

[stray silo]

Thanks for the book suggestion

@sly jungle

[sick parrot]

I recognize this looks like a trig function

wait wrong channel

[tight copper]

Noooo

[hardy iron]

!nogpt

[quasi jetty]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[covert musk]

a

[limber thunder]

basically because LLMs are statistical models that generate output that "sounds good" rather than "is correct", even if successive versions of these models may try to amend that by fine-tuning or using larger context windows or whatever. It's not unusual for them to output nonsense and if you don't have the knowledge or aren't willing to fact-check the output it may be hard to tell whether the output is sound at all. Read textbooks and ask people about math

expecting a language model to do logical reasoning is kinda like the allegory of judging a fish by its ability to climb a tree, lol

as the name suggests they're great for tasks that mainly involve language skills, like writing an e-mail or summarizing an article or etc

[lament tide]

Also, they'll hallucinate papers and definitions that "sound" right. I asked it to explain Ricci curvature once, and the output was hilarious.

[limber thunder]

yeah this is definitely a common thing lol

I've read plenty of people asking for e.g. book recommendations and it'll straight up hallucinate authors and titles

[north topaz]

the point is that whatever GPT has done well, it has borrowed unknowingly from some online source

these sources are written by humans of course

so if GPT does it well, a human will also do it well

if GPT does it poorly, a human will do better

[limber thunder]

a knowledgeable human

[north topaz]

machine learning is very good at producing natural language (that's why it's called NLP = natural language processing)

not very good at communicating actual meaning and understanding

to someone who is really confused, unless they've borrowed the words from some human

[north topaz]

YES

[leaden torrent]

Idk, I haven't seen a human willing to summarize a 200 page mathematics textbook entirely in pirate speech without using the letter "w"

[north topaz]

c.f dead internet theory

[worn urchin]

really depends on the topic, i find that up to high school level it's pretty good, even maybe 1st or 2nd year undergrad, but you should definitely fact check

[north topaz]

yeah like that's the point where the maths resources online start to dwindle

where the knowledge starts to be concentrated in books and other materials not accessible to web crawlers (in uni pages for instance, Canvas etc)

like it has definitely helped everyone at high school level for sure by now, GPT 3.0 was shit but 3.5 and 4.0 are impressive

it's hard to generalise

we have such a wide range of maths abilities on this server

[flat cloud]

in strong induction, we have to prove the starting point and then assume that everything from the starting point to k holds true?

[vernal token]

Yes, and prove k + 1 with it

[foggy meadow]

hhey

[tacit escarp]

So uh

Reu application season is most certainly starting up, and I was wondering where I should go in here to talk to smbody abt like

idk where should I apply given how good/bad I am.
or like their experiences if they got one

or like what kinda resume people who didnt get one had.

if thats just like not what this discord is for thats chill too (I'm kinda new here), I'm just stressing abt it and thought to post stuff here

[lament tide]

There's a graduate applications chat under advanced mathematics.

[sharp mulch]

REU chat doesn't go in the graduate applications chat

[tacit escarp]

(I had a feeling it may just not belong in this discord lmao)

[sharp mulch]

Asking here is fine

#advanced-lounge is better probably

[tacit escarp]

o good point yea

[visual thorn]

I'm not a finitist, but it seems to me that we don't really use the real numbers in a practical sense. We just say they exist and do all arithmetic as if using a finite subset of the rational numbers. We kind of just use the numbers like pi and sqrt(2) as intermediate numbers until we get to a final practical answer rounded off to a few decimal points if we need to use the number for some physical reason.

[sharp mulch]

Yes

[river moon]

real numbers are not real

[grand rose]

are there any numbers that are not complex but exist?

[sharp mulch]

?

[vivid halo]

depends on what you mean by number

[sharp mulch]

Lots of numbers that exist are not complex

[vivid halo]

various p-adic numbers certainly fit the bill here

[sharp mulch]

1 for example

[grand rose]

Oh like 999999....+1 = 0?

[vivid halo]

1 is a complex number lol

[vivid halo]

not really no

[grand rose]

o

[grand rose]

1 + 0i

[vivid halo]

p-adic numbers are similar to real numbers but using a different kind of distance between rational numbers

[static loom]

...9999 is a way to write -1 in the 10-adics, just pointing out this is just an example where you get more representations for rational numbers depending on where you end up

[grand rose]

wat

[vivid halo]

do you know how the real numbers are constructed from the rational numbers in the first place?

in terms of convergent sequences?

[grand rose]

You mean 1 = 1/2 + 1/4 + etc

?

[vivid halo]

sure but more importantly for sequences of rational numbers which converge to general real numbers

[grand rose]

there's a general formula?

[vivid halo]

I mean the silly thing you can do is like, take a decimal expansion of any real number

and then there is an obvious sequence of rationals converging to that real number

by just truncating the decimal expansion

[grand rose]

o

[vivid halo]

e.g. the sequence 1, 1.4, 1.41, 1.414, ... \sqrt{2}=1.41421356237...

[grand rose]

yeah

I would still prefer if its geometric though

or arithmetic

Actually it cant be geometric

Unless it is finite or constant cuz otherwise it tends to 0

[vivid halo]

the point is that you get all the real numbers by adjoining to the rationals all of these convergent sequences of rationals

well to define what it means for a sequence of rationsl to be convergent you need some notion of "distance"

[grand rose]

Epsilon time yay

[vivid halo]

if you use the usual distance d(x,y)=|x-y| (the absolute value of the difference of two rationals) then you end up with the reals

but there are other absolute values and notions of distance for rationals

if you do the same construction with the p-adic absolute value instead, you get the p-adic numbers

so you can talk about expressions like these which do not converge in the reals but which do converge in the p-adics for various primes p

I would still consider these to be numbers but they are completely different from real/complex numbers to answer your question

[grand rose]

This reminds me of floating point binary

[vivid halo]

there is some relation when you're working with 2-adic numbers yeah

[grand rose]

0.0011001100110011... base 2 is also 1/5 i believe

[vivid halo]

right the main difference is whether you're looking at negative powers of p versus positive powers of p

[grand rose]

Wait, what is the 2-adic number for 1/5

Is it anything to do with two's complement by any chance

[static loom]

you could use some tricks like that to help get an expression for it I guess

1/5 = 1/(1+4) = 1-4+4^2-4^3+... you'd not get a clean "digit" representation without working out how to deal with the -1 in there from that I guess

[vivid halo]

one nice fact is that a number is rational iff it has an eventually periodic p-adic expansion

similar to how rationals have eventually periodic decimal expansions

in the base p expansion you go off to the right forever, in the p-adic expansion you go off to the left forever

[static loom]

1/5 = (1-4)/(1-16) = (1-4)(1+16+16^2+16^3+...) as another silly trick, or really just start working out the division 1/5 would be direct rather than trying to find some trick lol

[jaunty ibex]

Average number theory enjoyer be like

[grand rose]

Because of carrying digits an infinite number of times?

[grand rose]

real moment

[jaunty ibex]

(I am a number theorist myself)

[vivid halo]

well I mean what is happening is that in the usual absolute value the sequence p^0, p^{-1}, p^{-2}, ... goes to 0 whereas in the p-adic absolute value the sequence p^0, p^1, p^2, ... goes to 0

[jaunty ibex]

Q_p my beloved

[vivid halo]

p-adically close to 0 means highly divisible by p

[grand rose]

you have lost me now since my only knowledge of p-adic numbers comes from watching 3 minutes of a veritasium video

[vivid halo]

that's more than most people have heard about them

[grand rose]

It just seems so counterintuitive

[vivid halo]

they are strange when you first encounter them yeah

[grand rose]

That a number that seems infinite can be negative

[vivid halo]

and they take a while to get used to working with since they behave quite differently from the reals

[grand rose]

What are the use cases

(Most dreaded question)

[vivid halo]

so one main motivation is if you're trying to solve Diophantine equations (i.e. find rational solutions to systems of polynomial equations) which is like, one of the historically oldest kinds of questions in math

if you have some system of polynomial equations with rational coefficients, you notice that if this has no solutions over the reals then it certainly can't have any solutions over the rationals (since the rationals include into the reals)

likewise if this has no solutions over the p-adics then it certainly can't have any solutions over the rationals

[grand rose]

Can this include variables as being the exponents themselves

[vivid halo]

there are ways you can generalize these kinds of problems and the sorts of equations you're allowed to consider but it's easiest to just consider polynomials

(p-adic exponential functions are kinda strange, that's one reason to be cautious here)

[true zinc]

~~ng are you indoctrinating the poor pre-uni's into number theory 😭 ~~

[vivid halo]

yes

[true zinc]

why not geometry smh

[grand rose]

Geometry is one of my least favourite areas surprisingly

[vivid halo]

anyways the nice use is that sometimes you can go the other way: for some systems of polynomial equations if you have a solution over the reals and a solution over the p-adics for every prime p, then you have a rational solution

[grand rose]

although thats overgeneralisation

[grand rose]

A

[vivid halo]

these are "local to global" theorems

this doesn't always work and it's interesting to try to understand when these local to global theorems fail

[true zinc]

😭 number theory is one that we have a very struggle-hate relationship with

[vivid halo]

for example 3x^3+4y^3+5z^3=0 has nontrivial real and p-adic solutions but no nontrivial rational solutions

[grand rose]

I have heard of analysis having a similar reputation

[true zinc]

I'd rather deal with global and local coordinate bullshit

[true zinc]

so all solutions are irrational?

[grand rose]

Infinite number of solutions??

[vivid halo]

iirc there are infinitely many over the reals and over the p-adics for every prime p

but no nontrivial rational solutions

[true zinc]

how do you even find any solutions to stuff like this (like what's the general solution development, if there is one)

[grand rose]

Nontrivial as in hard to work out or nontrivial as in don't exist

[vivid halo]

nontrivial just means different from the trivial solution (x,y,z)=(0,0,0)

[grand rose]

ok

[vivid halo]

that is a rational solution but it's not a very interesting one, the claim is that this is the only solution

[vivid halo]

well you have a lot more tools and flexibility when working over local fields like the reals or p-adics

in this case you're dealing with an elliptic curve and you can use various analytic uniformizations over local fields

[true zinc]

ahh okay

[true zinc]

elliptic curves as in the same things used in ECC?

[vivid halo]

yes

[true zinc]

huh

[vivid halo]

they are pretty fundamental objects in algebraic/arithmetic geometry

[grand rose]

you lost me in the bit after elliptic curve 💀

[true zinc]

yeah

I'm familiar with them in algebraic geometry to a small degree but how do they show up in arith geo? (I'm very unfamiliar of the field itself)

[vivid halo]

well they show up in arithmetic geometry simply because they are one of the simplest interesting kind of curve over a field you can study

there are loads of big theorems and conjectures in number theory about them

BSD is one such conjecture, with is one of the millennium problems

Wiles' proof of FLT uses elliptic curves and their relation to modular forms

that's another historically famous application to number theory

they show up in cryptography with ECC since they give some nice generalizations of the discrete logarithm problem

[true zinc]

yeah I was thinking about bringing that up few mins ago but I didn't know whether that was more alg or arith geo

[vivid halo]

I'd say it's more arithmetic geometry but the line between arithmetic and algebraic geometry can be pretty blurry

[true zinc]

I can barely comprehend the statement of this one 😭
For all the others (except Riemann hypothesis and BSD) I at-least know their statements

[vivid halo]

the way to think about BSD is it's a generalization of the analytic class number formula

[true zinc]

I'm unfamiliar with what a class number formula is

other than hearing the term before

[vivid halo]

if you have a number field F you can look at the Dedekind zeta function \zeta_F(s) (which is the Riemann zeta function for F=Q) and this has a simple pole at s=1

you can ask about the residue of \zeta_F(s) at s=1

[true zinc]

ah okay

[vivid halo]

the analytic class number formula relates this residue to a bunch of quantities related to the arithmetic of your number field F

things like the discriminant of F, the number of roots of unity in F, the regulator of F, and (most interestingly) the class number of F (which measures the failure of unique factorization in the ring of integers of F)

[true zinc]

I'm genuinely still very surprised at how much number theory draws on complex analysis

[vivid halo]

by a similar token, if you have an elliptic curve E over a number field F, you can look at the L-function L(s,E), and BSD is about the behavior of this function at s=1

and again its behavior should be intimately related to the arithmetic of E

[jaunty ibex]

Doesn't nG do both geometry and number theory

[true zinc]

(we know very little CA so we could be wrong here) but at a pole wouldn't it be analogous to the function going off to infinity?

like isn't this why we need residue theory? /genq

[vivid halo]

for example, your elliptic curve should have only finitely many F-rational points if L(1,E) is nonzero, and infinitely many otherwise

[jaunty ibex]

My bf is doing arithmetic geometry recently and I can tell it's probably nG's influence

[true zinc]

[vivid halo]

essentially yes, a pole of order n at 0 will locally look like 1/z^n

[true zinc]

okay

[jaunty ibex]

Number theory is a love-hate relationship

[vivid halo]

you can have essential singularities like 1/e^z which aren't poles but still go off to infinity

there are very very general conjectures about special values of L-functions attached to algebraic varieties defined over number fields in general

but these are extremely open seeing as the case of elliptic curves is already very open

[jaunty ibex]

Simply be better

[vivid halo]

this is one of the bigger themes in arithmetic geometry though, namely how analytic information involving L-functions is related to arithmetic information involving Diophantine equations

it's kind of a miracle that these should be related but that's how arithmetic is

[true zinc]

we need to read into this soon, we're slowly working through learning about how laurent series work rn

[jaunty ibex]

Complex analysis to algebraic geometry pipeline

[vivid halo]

yeah in terms of Laurent series it's just saying that you don't go off infinitely far into negative powers of z

[true zinc]

Frankly, we feel kinda sad that we never could really "appreciate" a lot of theorems in number theory, even like, basic ones just felt like "eh cool it works"

but for geometry or analysis or even group theory everything has always seemed like "whoa that's so cool omg"

[vivid halo]

I do feel like it's hard to appreciate these things without algebraic geometry

[jaunty ibex]

This was me too

I appreciate them more now with algebraic number theory

[true zinc]

like even basic geometric theorems like stokes' theorem or gauss bonet or whatever, they just

click

[vivid halo]

yeah algebraic number theory helps a lot too

[jaunty ibex]

I don't understand arithmetic geometry tho

Neither analytic apparently

[vivid halo]

I like arithmetic geometry a lot precisely because it lets you think about these arithmetic problems in geometric terms, which is usually much more intuitive

[true zinc]

Our professor went over some stuff with divisibility and chinese remainder theorem and whatnot and they just felt like "eh ok"

if that makes sense

[jaunty ibex]

Yeah that makes sense

CRT makes more sense in algebraic structures

imo

[vivid halo]

yeah that's fair CRT isn't such a spectacular theorem

definitely useful but yeah

[true zinc]

yeah after we saw the ring theory version it was like "okay wow that's actually kinda cool"

but we're very scared we're "missing something" but being so averse to (elementary) number theory in it's "pure" form

[vivid halo]

a lot of the theorems about the arithmetic of curves over number fields are pretty surprising I think

[jaunty ibex]

With arithmetic geometry they present to you complicated tangent line formulation that feels like it came out of nowhere and it just makes you feel like "why do u do that" and with analytic number theory, as you go further it's just a matter of who has the most exotic result which has loglogloglogloglogloglog

[true zinc]

We really don't want to rush everything and slam into the highschooler/undergrad category theory-esque memey person

if that makes sense

[vivid halo]

e.g. if you have a (smooth projective) curve X of genus g over a number field F then
X(F) is infinite if g=0
X(F) is a finitely generated Abelian group (finite or infinite) if g=1
X(F) is finite if g>1

[true zinc]

yeah we can't stand either of these tbh

[vivid halo]

wild that the genus controls the number of solutions like this

[jaunty ibex]

I do understand it's mainly because I have a distorted picture of what is geometry really and the fact that Hartshorne is just dry af to go through

[true zinc]

hartshorne

[vivid halo]

yeah Hartshorne is kind of painful to get through

[true zinc]

what are the more "geometric" aspects of algebraic geometry?

[jaunty ibex]

Me thinks algebraic number theory is more well motivated if that makes sense

[true zinc]

I mean is there stuff with co/homology, manifolds, topological spaces, complex manifolds etc...?

[vivid halo]

anything having to do with varieties over the complex numbers tends to be much more geometric in flavor

[true zinc]

or is that all diff geo

[vivid halo]

there is a big overlap!

[jaunty ibex]

drawing a line where you fix points (2),(3),(5),... then convince people this is geometry

[vivid halo]

like one of the main tools we have for studying complex algebraic varieties is Hodge theory, which is mainly about what extra structure you have on cohomology when you have a complex variety as opposed to some more general smooth manifold

[jaunty ibex]

Varieties are more intuitive tho

[true zinc]

yeah we want to see stuff that's like, not too much in one or the other ig, if that makes sense, like pure diff geo, at-least if what we've seen, seems like manipulating subscripts of subscripts and the algebra sides seems like category

[jaunty ibex]

Real

[true zinc]

Yeah we've looked into that a (tiny) but and it seems very interesting

[vivid halo]

yeah Hodge theory is very interesting

[true zinc]

also not to take too much time but is there an ELI(early)undergrad of what the fuck a motive is?

[jaunty ibex]

Enumerative geometry is interesting

[vivid halo]

it does interact with arithmetic too since e.g. if you have a variety X over Q then you can talk about the complex variety X(C) and study its Hodge theory

[jaunty ibex]

Nobody knows wtf is a motive

[vivid halo]

and then the Hodge theory will see things about the arithmetic of X over Q

[vivid halo]

short answer is that motives encode all the information about the cohomology of algebraic varieties

[true zinc]

this reminds me a lot of groupoids and fundamental groups from point set/alg top

[jaunty ibex]

we believe there are these small building blocks called motives that encode a lot of information
what are they tho can u define them
idk

[vivid halo]

I mean you can definitely define them without issue it's just you run into very hard conjectures if you want to show that this definition has all the right properties

there is a much more down to earth definition in terms of realizations but again you need conjectures to show that this sees the same information as honest motives

[jaunty ibex]

So the issue is coming up with a definition that agrees with a lot of properties?

[vivid halo]

I don't think there can be some definition which skips these conjectures

[alpine copper]

@vivid halo are you familiar with the connection between motivic galois theory and QFT?

[vivid halo]

the point with motives is like, you want some universal cohomology theory for varieties, so some "source" for cohomology classes in any cohomology theory for varieties (specifically Weil cohomology theories, these are those which behave like ordinary cohomology does for spaces)

One universal source for cohomology classes is algebraic cycles, since you can produce a cycle class in any Weil cohomology theory

[vivid halo]

yes very

if you define the theory from the ground up in terms of algebraic cycles, then you just run up against the fact that we don't know very much about algebraic cycles

[true zinc]

also ng, how do you like...even begin learning this stuff; we're currently slowly working ourselves through real analysis and abstract algebra and istg the distance from "here" to even basic algebraic topology feels like a million years away

istg we feel like we'll be 30-40 by the time we get even that far

[vivid halo]

I mean it takes a while to get up there and you don't learn it all at once, it takes many passes

[true zinc]

(yes we could take courses at uni but the reason we're not rn is due to family and other stuff, planning on doing another B.S in maths after moving out and using our CS degree to first get a job :P)

[vivid halo]

also for what it's worth I only started to learn motivic stuff at the start of PhD or whatever

[jaunty ibex]

CS degree to get a job
in this economy

[true zinc]

just...self image of being in our mid-late 20s sitting in an intro RA class scares us somehow

[vivid halo]

nothing wrong with that!

[alpine copper]

Is this part of your research?

[vivid halo]

it's closely related yes

I don't do so much stuff with amplitudes/cosmic Galois specifically but I do loads of stuff about motivic periods/motivic Galois in general

[quasi fossil]

are axioms purely assumptions

[alpine copper]

Interesting! What background should one have before doing a deeper dive?

Is it all heavy on the QFT

[vivid halo]

not so much actually, it's much closer to stuff one does in Hodge theory

I guess one has to know a bit of QFT to produce the amplitudes that physicists are interested in computing in the first place

but you can study these things with amplitudes/cosmic Galois with almost zero physics background in a lot of ways

the main background you need is familiarity with algebraic geometry/algebraic topology (say at the level of Hartshorne and Hatcher respectively) and Hodge theory (read volume 1 of Voisin for pure Hodge theory, and Peters-Steenbrink for mixed Hodge theory)

[alpine copper]

Hmmmm. I am hesitant to take our grad galois theory + modules course, but my professor cited this "motivic galois theory + QFT" as a good motivation for me to still take it

[vivid halo]

yeah the motivic Galois stuff goes beyond the usual Galois theory but the two are related

[true zinc]

yeah, true, I think the first thing we really need to do is figure out therapy and mental health stuff after moving out, that'd make SO much stuff at-least more...bearable

[alpine copper]

I imagine so. But I know zero Galois theory haha

[vivid halo]

the main point of Galois theory is to study Galois groups acting by symmetries on algebraic numbers

but the usual Galois theory doesn't give you any interesting Galois action on numbers like \pi or \zeta(3) or whatever

that is what the motivic Galois group does

[lament tide]

Very interesting. Any papers for someone in Hodge theory with some Galois background who'd want to read QFT applications?

[alpine copper]

Ahh cool cool!

[vivid halo]

Francis Brown is the main person who has written a lot about this stuff

https://arxiv.org/abs/1512.06409

this is a nice survey

he also has some recorded lectures about this and all of his lectures are fantastic

[lament tide]

Excellent! Thank you. Some weekend reading 🙂

[vivid halo]

for some more recent work in this area look up "modular graph functions"

https://arxiv.org/abs/1512.06779

[alpine copper]

It's so interesting how QFT, despite being mathematically problematic, inspires a lot of interesting math developments

[vivid halo]

these are functions which appear e.g. in genus 1 type II superstring aplitudes and people have been studying them in great detail lately

[alpine copper]

I just got witten's monograph on superstrings haha

[vivid halo]

yeah it is really funny that there are both number theorists and physicists thinking about these things but they write about them very differently

and it does seem that the physicists have way more patience for cranking out huge computations in this area

when an author sends you 3000 lines of mathematica output for these integrals

[alpine copper]

In my humble opinion (having only taken one course in math qft), the (mathematical) richness of QFT often can come from the areas where it lacks mathematical rigor, like path integrals

[vivid halo]

yeah I feel that

[lament tide]

For real. They do this with differential geometry and algebraic topology, too. It's always a different name than what we call it.

[vivid halo]

there is some amount of this kind of Hodge theory that has a lot to do with conformal field theory/Chern Simons theory

and I've found it very difficult to read and learn about this stuff

[lament tide]

I've encountered that before--just not the Galois applications.

[lament tide]

If you want a good reference, Lectures on Quantum Chromodynamics by Smilga is good for intro Chern Simons and Hodge applications in QCD.

[vivid halo]

yeah I've been confused about Chern Simons invariants of 3-manifolds lately oog

very badly need to understand this story

[alpine copper]

The proof of Index Theorem on spin manifolds by using path integrals for some action involving Dirac Operators on the loop space is genius

[vivid halo]

yeah a lot of my problem has just been that physicists write in a way that is very confusing to me

[lament tide]

Not sure if this helps, but here's a lecture series I used to learn a bit: https://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf

[lament tide]

Yes, like Einstein notation getting rid of summands.

[vivid halo]

I don't mind Einstein notation so much

my main issue is that physicists tend to write fast and loose about mathematical details and it's hard to follow if you don't share intuition with the physicists

[lament tide]

Agreed. The differential forms leave a lot to be desired sometimes.

[alpine copper]

My main issue can be lack of definitions

[vivid halo]

one of the great gifts that physicists have is being able to reason accurately about things like path integrals without complete mathematical rigor

[lament tide]

Most details are assumed.

[vivid halo]

but that kind of intuition takes a while to develop

[lament tide]

Or add ghost particles and anti-ghost particles so that fields obey math.

This is why I chose math over physics...

[vivid halo]

I remember meeting with a really good Gromov Witten theory prof about this kind of stuff and it was strange like

I would write down some path integral and he'd be like "oh yeah that looks renormalizable" and I'd be like "what makes you say that" and he'd be like "it just... looks that way"

[lament tide]

Yup. Very strange how things disappear or randomly appear for convenience.

I remember getting stuck in a physical chemistry course where certain forces just didn't show up in molecular calculations (because they were "trivial").

[vivid halo]

if you have a strong background in Hodge theory you should try reading Goncharov's "Hodge correlators" paper

very underappreciated mostly because the paper is written like shit but the ideas are insanely deep

[alpine copper]

I remember reading this post about how a mathematician was "tired" of relying on Witten's/other Physicists intuitions to come up with new conjectures/outlines of proofs, so he tried attending talks/lectures from Witten. He thought that math shouldn't need Physicist's input/conjecture. He left those conferences saying that mathematicians will still be relying on Witten-like intuition for a while

[lament tide]

Loving the graph connections in the papers you suggested. Very much my realm.

[vivid halo]

but the upshot is that there seems to be some TCFT which governs mixed Hodge theory, whose correlators reproduce loads of well-known motivic periods

Goncharov's paper defines these at tree level and last year some other authors extended the setup to 1-loop correlators as well

it's not so clear what these things at higher loop order mean

[lament tide]

Honestly, I think most mathematics connects within the physical world. There are some deep group theory properties in protein synthesis from DNA triplets.

[vivid halo]

Goncharov has done a lot of really good work around polylogarithms (which play a huge role in this motivic Galois/QFT story) and the best theorems about these so far have come out of these correlators

[alpine copper]

I feel like there should be a bell curve meme for "Physics = Math" sometimes 😂

[vivid halo]

one of the main open problems around this is trying to systematically understand the functional equations satisfied by polylogarithms, which we only fully understand in low degree

the correlators give the cleanest understanding of this so far

[true zinc]

in a slightly different note, hodge theory seems really cool

[vivid halo]

yeah it's incredibly cool

[true zinc]

isn't there also like a conjecture based on hodge's work?

oh wait that's a millenium prize problem

[vivid halo]

yes that is another millennium problem!

[true zinc]

yep

how's the progress on that

[vivid halo]

very slow!

[true zinc]

and is there a survey article of recent work on it?

[lament tide]

Sounds a lot like some of what I'm looking at with locally recoverable codes (great on curves, not so much surfaces yet).

[true zinc]

(one of our friends was asking about it a few months back)

[vivid halo]

it's an insanely hard problem about algebraic cycles and it's closely related to the main cojnectures that obstruct motives from being defined unconditionally

[lament tide]

It is, and it's being used a lot in applications/machine learning to understand biology and social systems.

[true zinc]

:o

[vivid halo]

it's hard because you're asked to start with some cohomomology class and then pluck some algebraic cycle out of thin air

[true zinc]

Are there any...I guess accessible (as possible) papers for this stuff from the CS perspective?

[vivid halo]

this is insanely hard to do, algebraic cycles are incredibly rigid and we know almost nothing about creating cycles like this

[lament tide]

I'd suggest looking up Hodge-Helmholtz decompositions and Hodge Laplacians. It's a decent way to familiarize yourself with terms within a context of computer science applications.

[alpine copper]

Twistors show up too?

[lament tide]

A lot of Hodge theory is being used to study graphs and simplicial complexes these days.

[vivid halo]

yeah Goncharov's setup uses twistor connections in a very crucial way

[true zinc]

I was just about to mention that a lot of this stuff seems very related to graph theory

[vivid halo]

the Hodge Correlators paper is probably what taught me the most about the twistor approach to Hodge theory

[lament tide]

Very relevant to graph neural networks and topological deep learning, too.

[true zinc]

also I hope that my inexperience in this stuff isn't detracting y'all too much :)

[alpine copper]

I skimmed up to where they brought up the feynman PI, incredibly cool

[lament tide]

Michael Schaub's group has a lot of signal processing on graphs through the lens of Hodge theory.

[lament tide]

I love helping students and early career folks.

[vivid halo]

the main body of the paper defines these Hodge correlators in terms of some "Hodge correlator twistor connection" and then most of the upshot is that loads of fundamental features of the mixed Hodge theory story become much more transparent with this twistor language

one very cool upshot is like, the failure of the weight filtration on a mixed Hodge structure to split is exactly the failure of the twistor connection to be flat

[true zinc]

Half the time I feel like I'm too inexperienced (calc 1-3, computational and some abstract linear algebra) for most of this server yet a lot of this stuff genuinely seems very interesting and so we want to learn more and eventually understand it :)

[vivid halo]

so the interesting mixed information is curvature information

[lament tide]

It's never too early.

[alpine copper]

I remember something about solutions to some field equations are in bijection to some cohomology classes via some twistor transform

[lament tide]

Very cool. Assuming Gaussian/Ricci sort or torsion-related?

[vivid halo]

I just mean the curvature for a vector bundle with connection

[lament tide]

I've had high school students in my publishing group before.

[vivid halo]

like if you have a connection \nabla:E->E\otimes\Omega^1 then the curvature is \nabla^2:E->E\otimes\Omega^2

I guess the sequel Hodge Correlators II is what explains more of the TCFT going on in the background

uses a lot of Kontsevich type dg/super scheme deformation theory nonsense lol

[alpine copper]

I wonder if this will ever tough ground with CondMat.

[vivid halo]

idk if it's so directly related to physics

this is more a case of physics influencing number theory rather than the other way around

[alpine copper]

Sure sure.

[vivid halo]

very much in the same realm of ideas where Langlands people have started to bring in ideas from TQFT and whatever

[alpine copper]

How rare is it to have this level of mastery over the math & the physics? I.e. don't have to depend on one or the other for math-phys research?

[vivid halo]

it's rare to be an actual master in both areas for sure

that said you definitely don't need to be a master in both areas to contribute to this area of research

most of the recent papers around these kinds of motivic period/amplitude computations have a mix of physics and math people with various strengths

[ripe frigate]

hi people

oh no physics

🚪🏃‍♀️

[vivid halo]

my experience talking to most of the people working in this area is that they have a very healthy mix of background in Hodge theory/arithmetic geometry and the physics that motivates this

whereas I completely specialized into Hodge theory/arithmetic geometry

a lot of the gaps in this area of research are that there are a lot of computations (relevant to physics or otherwise) where the motivic interpretation is not so clear

[alpine copper]

I actually really like this. Idk if this is the standard, but I have seen a lot of math-phys groups that are just mathematicians with no/little background in physics where the "physics" is just a loose motivation

[vivid halo]

yeah the papers are a good mix of both too

[pure hollow]

what exactly is hodge theory?

[vivid halo]

it mainly has to do with studying the additional structure on the cohomology of things like compact Kahler manifolds/smooth proper complex algebraic varieties as opposed to more general smooth manifolds

a basic consequence is that the first Betti number of compact Kahler manifolds/smooth proper complex algebraic varieties is even which is certainly not true for general smooth manifolds

The reason is that in such a situation you can decompose H^1 into two pieces H^{1,0} and H^{0,1} spanned by holomorphic/anti-holomorphic 1-forms, and Hodge symmetry says they have the same dimension hence dim H^1=dim H^{1,0}+dim H^{0,1}=2dim H^{1,0} is even

this is already a nontrivial topological obstruction to being a compact Kahler manifold/smooth proper complex algebraic variety

mixed Hodge theory is the more general story you get when you remove the adjectives "smooth proper" or "compact manifold" in general you can ask these sorts of questions about not necessarily smooth not necessarily proper things

[pure hollow]

very interesting

[vivid halo]

yeah it's a neat and very classical topic in AG/AT

one of the main themes in Hodge theory is studying period integrals and how they vary in families

like say you have an elliptic curve E (that is a complex torus of complex dimension 1/real dimension 2)

H_1(E) is 2-dimensional, generated by the homology classes of two loops \alpha,\beta

H^1_dR(E) is 2-dimensional, generated by the cohomology classes of the (anti)-holomorphic 1-forms dz and d\bar{z} once you identify E with C/L where L is a lattice in C

this gives you a 2x2 matrix of integrals of {dz, d\bar{z}} along {\alpha,\beta}

now if you have a family of elliptic curves E(t) you can ask how this 2x2 period matrix varies with t

the entries will satisfy some system of differential equations, in this case it's a classical hypergeometric equation fo 2F1

you can ask more generally if you have some family of complex algebraic varieties what kinds of differential equations describe how their period integrals vary

very classical, very deep

[pure hollow]

and how much progress has been made for such questions?

[vivid halo]

loads of progress but many things are open

for example the Hodge conjecture is one of the millennium problems and it's hopelessly open

Hodge conjecture: Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

this is hard because you're given some Hodge class (some cohomology class of X satisfying some Hodge theoretic properties) and asked to produce some subvariety realizing this class

it's similar to how cohomology classes for smooth manifolds are realized by submanifolds

but the complex analytic setting is much more rigid and it's hard to pluck subvarieties like this out of the air

[pure hollow]

nG is there a way to get better at math(in general) provided like you do sufficient hardwork? what I mean by this is like how one can improve their coding/problem sloving skills by having a project is there something similar for math?

[vivid halo]

there are certainly lots of "standard" exercises one can do in various areas of mathematics which will help one get better

broadly speaking it's much easier to get better by studying parts of mathematics you are naturally drawn towards and get obsessed by

one of the big lessons from my PhD is that it's very hard to make progress studying and working on something you aren't honestly interested in, but maybe the things that you are so interested in that they keep you up at night through curiosity come more naturally

[pure hollow]

what are those "standard" exercises (like what should I refer for that) also I don't know which part of math I find to be drawn towards as they all sound cool as I'm just scartching the surface

[vivid halo]

for most topics there tend to be "standard" textbooks and those tend to contain "standard" exercises

I don't know which part of math I find to be drawn towards as they all sound cool as I'm just scartching the surface
this is a common experience!

honestly this just takes a lot of reading textbooks/articles and paying attention to which topics capture your imagination

[pure hollow]

like Rudin for RA,H&K/FIS for LA etc?

[vivid halo]

yeah

it's not something you settle on overnight but you can certainly take note while reading of which topics just vibe better with you

it's also not something you have to decide on once and for all, people's mathematical interests tend to migrate over time depending on exposure

[pure hollow]

thank you very much
also nG do you mind if I friend request you?

[vivid halo]

go ahead!

[storm sage]

You definitely don't need to be drawn towards a specific part of math atm

If you're an undergraduate, it's very healthy to keep your mind open to many different areas of math

[raven plaza]

Is there a reason to expect this to be true

[vivid halo]

There are many reasons yes

Although it is one of those conjectures where there is some general philosophy that makes it believable but if you sit down to try and verify various cases it can become increasingly hard to believe

It’s worth comparing the Hodge conjecture to the Tate conjecture and the implications of each of these towards the standard conjectures on algebraic cycles

Both are known in small handfuls of situations

[raven plaza]

So there's no single heuristic for it?

[vivid halo]

I wouldn’t say that

[true zinc]

looking around, I also kept hearing mention of the tate conjecture and whatever the heck this is https://en.wikipedia.org/wiki/Nonabelian_Hodge_correspondence

[vivid halo]

Yes the Tate conjecture is morally the same as the Hodge conjecture

They are the same conjecture but about two different cohomology theories

[true zinc]

nod

[vivid halo]

And yeah non Abelian Hodge is neat

[true zinc]

Why does the statement of the tate conjecture involve galois stuff while the hodge conjecture (at-least on the surface) doesn't?

[vivid halo]

Well because Tate is about l-adic cohomology and this carries a Galois action

[noble marten]

mmm yes i love hitchin equations

[vivid halo]

Hodge is about singular/de Rham cohomology

Both conjectures are about classes of algebraic cycles

[true zinc]

ahhhh okay

[true zinc]

would the proof or disproof of one or the other pave a path for the proof or disproof of the other?

[vivid halo]

oh yeah absolutely

well so for one the Tate conjecture implies all absolutely Hodge classes are algebraic, and if you assume Deligne's conjecture that all Hodge classes are absolutely Hodge then this implies the Hodge conjecture

[true zinc]

is deligne's conjecture still open currently?

[vivid halo]

yes but it's known in a few cases, like for Abelian varieties

[true zinc]

ah

[vivid halo]

like for Abelian varieties we know both Tate and Hodge

and there are a small handful of cases beyond this where we know both

whatever insights about algebraic cycles are necessary to prove cases of Hodge are the same insights needed to prove cases of Tate

also Clausen has some reformulation of the Hodge conjecture which is really interesting https://mathoverflow.net/questions/446978/clausens-modified-hodge-conjecture/446992#446992

the usual Hodge conjecture would amount to the statement that every analytic K-theory class can be continuously deformed into an algebraic K-theory class

[true zinc]

Oh that's really cool

have you done any work in (closely) related fields to this?

[vivid halo]

Well I work a lot with motivic stuff so things like Hodge and Tate come up a lot

But I don’t really work on trying to prove cases of those conjectures

I do think a lot about cases of Beilinson’s conjectures which is what generalizes BSD though

[true zinc]

ah

How's progress on that and BSD?

[vivid halo]

I don't work on BSD specifically since this is very hard but there are lots of other cases of Beilinson's conjectures which are more tractable

[unborn meteor]

me here thinking berkeley software distro

[vivid halo]

it's going well lately I've been trying to write down some generalization/extension of Beilinson's conjectures and prove some easy cases of it if possible

almost got it working for L(E,2) for E an elliptic curve

Beilinson relates certain (determinants of) period integrals to certain special values of L-functions, but this doesn't give an L-function interpretation for all period integrals, only some of them. In general one needs some larger class of L-functions, and some generalization of Beilinson's conjecture for them

usually what proving cases of Beilinson's conjecture means is Hodge theory gives you some very specific period integrals and then your job is to compute the and massage the answer into a form which is related to L-functions

usually the issue is that you will have some known integral expression for L-functions, and the integrals that Hodge theory is telling you to compute, and often the two will be far away from each other and you have to massage the expressions very hard for them to match up

idk it's fun because some of the work involves really fancy conjectural stuff like motives and abstract nonsense like this, but a lot of it is very explicit down to earth computations of integrals that keeps things grounded in reality

[pearl osprey]

I am a Computer Science student and currently I want to choose a career path but I cant decide between AI , CyberSecurity and Robotics, Which one should I choose ?
I am interested in Mathematics, Physics and hard challenges that keep me busy.
Is there any other field that would suite me ?
I have a 3.94 GPA and am at the top of my batch in university.

[true zinc]

how does an integral have a determinant?

(I'm thinking of stuff like classical square matrix determinant or fredholm determinants on operators)

[true zinc]

[vivid halo]

oh I just mean that you have some matrix of period integrals and you take the terminant of some submatrix in there

[true zinc]

Ahhhhhhh

[vivid halo]

like for the L(E,2) example you will have a 3x3 block upper triangular matrix of periods

there is a 2x2 block coming from the periods of E

there is a 1x1 block that is just like (2\pi i)^2

then there are two more entries of the period matrix off the block diagonals

one of these Beilinson tells you is related to L(E,2)

the other is mysterious and this is what requires some extension of the conjectures

one irritating thing with formulating the right conjectures is that in this example L(E,2) is an honest well-defined unique number but the other number is only well-defined modulo rational multiples of (2\pi i)^2

so in the latter case it's not so easy to check your work numerically unless you have some clever way to determine the rational multiples of powers of 2\pi i showing up

Analogy that can be made precise: the numbers Beilinson describes are “volumes” and the missing part of the picture is to lift this to a “complex volume” whose real part is what Beilinson describes and whose imaginary part is the missing part

[true zinc]

https://arxiv.org/abs/1006.1116 This Beilinson?

[solid yarrow]

Is this a common approach in research?

Like trying to solve cases instead of the whole thingus

[vivid halo]

oh yeah of course

in fact it's quite rare I think to prove some very general hard results without proving simpler cases first

[sharp mulch]

Well if you can't solve the whole thing, you try to solve bits and pieces

Progress is incremental

[solid yarrow]

Hm.

I see

It would also help with being stuck a bit I guess.

[vivid halo]

right

[fickle patio]

wass up peeps

[solid yarrow]

Though this approach only works for a general theorem

Like if you have hands on problem you can't apply it

[vivid halo]

I mean this is a fairly general approach in mathematics to consider special cases or examples rather than trying the most general thing immediately

[sharp mulch]

What do you mean

[solid yarrow]

Never thought about it, but heuristically makes more sense

[solid yarrow]

I.e. a problem that's already a "special case". Like you see in UG psets.

[vivid halo]

like as a very simplistic example imagine trying to prove the quadratic formula before trying a few explicit examples first

[sharp mulch]

Special cases have their own special cases

[solid yarrow]

Not always?

Confus

[vivid halo]

lol my experience with teaching precalculus is like, you write "ax^2+bx+c" and you lose half the class until you replace it with something like "2x^2+8x+5"

[solid yarrow]

Lel

[icy heron]

Thoughts on this? I have an opinion already, but curious to hear what people here think

[bronze pelican]

I'm gonna teach precalc this semester

[vivid halo]

[sharp mulch]

oog

[neat lintel]

yes, switch, return the goat, buy two goats and a car and become the host to continue the cycle

only reasonable answer

[neat lintel]

also now two of the goats are those of a millionaire’s so the problem is back to normal

[mighty pilot]

Reminds me of a joke

“How i became rich? Well i bought an apple for 50cents and sold it 1$. Bought two apples and sold them. Bought 4 apples, then i got my inheritance”

[neat lintel]

haha

but the real question is why they did all that

just

buy your own apple for a million dollars

boom

youre rich

[mighty pilot]

With extra steps it becomes tax evasion

[neat lintel]

not if you print your own money

so the tax right belongs to you

🤯

from 5 years in prison to 50

bonus points if you launder it

[gusty socket]

mine too

[severe comet]

Hi everyone!

[true zinc]

Good evening

[severe comet]

Good evening

where r u from?

[true zinc]

USA

[severe comet]

Oh nice!

[peak tide]

i recall when i was a TA i was sitting in the lecture and the prof was going through some simple example like this, some student asked "can you give a numerical example" and the prof was like "sure, let x = c"

[final storm]

why is it difficult to have a clear formula to calculate factorials?

it has a clear pattern

[agile fiber]

$\prod_{k=1}^n k$ seems pretty straightforward

[fathom swallow]

cloud

[final storm]

whats that symbol?

[solid snow]

wdym by clear formula

[agile fiber]

product

[final storm]

Like instead of having n(n-1)(n-2)(n-3)... , it has a more straight forward formula

[last brook]

when u say more straightforward do u mean by writing less characters or?

[static loom]

here's a formula for ya lol $\dv[n]{x}x^n = n!$

[fathom swallow]

Merosity

[foggy meadow]

whoa

Matrices are currently on my mind.

[static loom]

come up with a formula for n! in terms of the determinant of matrices

[final storm]

whats d and x?

[static loom]

it's a derivative, don't worry about this, it's kind of a joke answer

[final storm]

oh lol

[final storm]

anyways thanks

[final storm]

yea

[foggy meadow]

That's something I might now try to do, but I'll have to maybe program out a determinant function.
I don't know them by heart and honestly only use them sometimes and even then I just let a computer calculate them.

Definitly something I'll have to go over again.

[fathom swallow]

Dev

[neat lintel]

Oh

Like that form that sums over symmetric group elements

[foggy meadow]

det(diag(1...n)) = n!

[agile fiber]

i think that had better be 1...n

[glad lichen]

guys im so cooked

and need help

./ an opinon

[foggy meadow]

Yeahhh

I learned something today.

[thick zealot]

I don't know if I'm the only one that happens to me, but hasn't it happened to you that this exercise is too obvious and ends up making you get gray hair?

[limber sable]

which maths books should I must read as a person who is aspiring for CS?