Mathematics - 2019-10-02 (page 1 of 2)

krauser126

00:01

Hmm

But cant we use what we know about there only being possibly 7 different remainders and the pigeonhole principle to make some deduction

And work from there. @Semiclassical

Semiclassical

you'd still need to make sure that all of those remainders are actually produced

consider 3^n mod 16. then the sequence is 1,9,11,1,9,11, ...

hence in that case you hit very few of those remainders

and you'd not be able to conclude that 2 ever shows up as a remainder

moreover, once you -have- proven all remainders are actually produced...well, then're already done, because then you've proven that 1 shows up

krauser126

Right that makes sense

Is there an intuitive way to prove that all remainders are obtained?

I know that 44^n always ends in either a 4 or a 6

Semiclassical

00:25

Well, 44 = 2 mod 7

And it’s not hard to find 2^n = 1 mod 7

krauser126

Why are we trying to find 2^n = 1 mod 7

Semiclassical

because 44 = 2 mod 7

and therefore 44^n = 2^n mod 7

(more generally: if a=a' mod n, and b=b' mod n, then ab=a'b' mod n)

MatheinBoulomenos

00:48

I don't think you need that all remainders are produced. We just need that one remainder occus for two different powers

if 44^n=44^m mod 7, then 44^(n-m)=1 mod 7 assuming n>m

krauser126

Yea but I'm trying to use pigeonhole to solve (at least part of) the proof

I feel like it could work but not sure how to approach it

MatheinBoulomenos

but pigeonhole is exactly what gives you that one remainder occurs or two different powers

since there are only finitely many residues and infinitely many natural numbers, the map N->Z/7 that sends n to (44^n mod 7) is not injective

and that's the pigeonhole application here

krauser126

You think you could explain it in simpler terms? I have no clue what "the map N->Z/7 that sends n to (44^n mod 7) is not injective" means

MatheinBoulomenos

it just means that we will have 44^n=44^m mod 7 for some n and m which are not equal

and the reason for this is exactly the pigeonhole principle: there are infinitely many different n we consider for 44^n mod 7, but only finitely many values it can take

krauser126

What do you mean when you say there are finitely many values it can take? What is the "it" in this case?

MatheinBoulomenos

01:03

there are only finitely many remainders mod 7

"it" is the remainder of 44^n mod 7

krauser126

But we don't know whether all remainders are covered at least once

for 44^n

MatheinBoulomenos

that doesn't matter

krauser126

how come

MatheinBoulomenos

all we need to know for the proof I am indicating is that there is some remainder, let's call it k that is covered twice

if we have 44^n=k mod 7 and 44^m=k mod 7 for two different n and m, say with n>m, then 44^(n-m)=1 mod 7

krauser126

Oh I think I get it.

So basically, we have 7 pigeonholes (remainders 0 to 6) and our pigeons are the remainder of 44^n / 7 . And since we only have a finite number of pigeonholes but infinite values for 44^n, then we will have at least two values of n that will give us a remainder for 44^n / 7 that ends up in the same hole.

I don't think I explained it too nicely haha but it makes sense in my head

MatheinBoulomenos

01:10

yeah, sounds like you got it

krauser126

@MatheinBoulomenos Also, how do you prove that if n> m and a^n /x and a^m /x have the same remainder, then a^(n-m) must also?

MatheinBoulomenos

well, that's not correct

a^(n-m) will have remainder 1

krauser126

Ah thats what I meant

How do you know that is necessarily always the case

MatheinBoulomenos

that will only be true if a and x are coprime

if a^n and a^m both have remainder k, then a^n-k and a^m-k are divisible by x. This implies that the difference a^n-k-(a^m-k)=a^n-a^m=(a^(n-m)-1)a^m is also divisible by x. Now if you assume that a and x are coprime, then this implies that x divides a^(n-m)-1, which means that a^(n-m) has remainder 1

but in our case gcd(44,7)=1, so we're good

note that this approach is completelely independent of the numbers 44 and 7 apart from the fact that gcd(44,7)=1. If a and b are any natural numbers such that gcd(a,b)=1, then this argument shows that a^n-1 is divisible by b for some n

Jacksoja

01:39

@MatheinBoulomenos Hey !

MatheinBoulomenos

@Jacksoja hey

Leaky Nun

@MatheinBoulomenos gute Mit-nacht

MatheinBoulomenos

@LeakyNun hey!

how are your courses at MIT?

Leaky Nun

warum ungeschlafen Sie

the assignments are killing me

but number theory is interesting

MatheinBoulomenos

01:55

what are you doing in NT?

Leaky Nun

# Date Topic (references) Materials
1 9/4 Absolute values and discrete valuations notes
2 9/9 Localization and Dedekind domains notes
3 9/11 Properties of Dedekind domains, ideal class groups, factorization of ideals notes
4 9/16 Étale algebras, norm and trace notes
5 9/18 Dedekind extensions notes
6 9/23 Ideal norms, Dedekind-Kummer, orders, and conductors notes
7 9/25 Galois extensions, Frobenius elements, the Artin map notes
8 10/2 Complete fields and valuation rings notes

MatheinBoulomenos

sounds good

krauser126

@MatheinBoulomenos "Now if you assume that a and x are coprime, then this implies that x divides a^(n-m)-1" how did you imply this?

MatheinBoulomenos

@krauser126 if gcd(a,d)=1 and d divides ab, then d divides b

@LeakyNun the pace is not bad

Leaky Nun

@MatheinBoulomenos yeah the pace is ok

Jacksoja

02:05

@MatheinBoulomenos Do you have some time to go through some ring theory concepts?

am not sure that am understanding things properly

MatheinBoulomenos

@Jacksoja okay

Jacksoja

thanks!

So I went though the iso theorems

they seem pretty much analogous to groups

then I got lost when we did maps from R to Z

Ryan Unger

Speaker: Vladimir Drinfeld (IAS)
Time: 4:15-5:30pm on Thursday, October 3
Place: Simonyi Hall 101
Title: A stacky approach to crystalline (and prismatic) cohomology.
Abstract:
The stacky approach was originated by Bhatt and Lurie. (But the possible mistakes in my talk are mine.)
Let X be a scheme over F_p. Many years ago Grothendieck and Berthelot defined the notion of crystal on X; moreover, they defined the notion of crystalline cohomology of a crystal.
I will give several equivalent definitions of a stack X^{prism} such that a crystal on X is the same as a quasi-coherent O-module on X^{p…

Daminark

Whaddup nerds

MatheinBoulomenos

@RyanUnger wow, sounds really cool

Érico Melo Silva

02:08

why would u post this vulgarity

Ryan Unger

because ppl like Mathein do real math

and think this is cool

Daminark

Don't disrespect my boi Drinfeld

MatheinBoulomenos

Drinfeld is a huge name in NT

Daminark

Never actually met him sadly

Jacksoja

@MatheinBoulomenos I think my confussion is solved when i tried to write the problem in words for you haha

MatheinBoulomenos

02:09

@Jacksoja lol

glad to help, I guess

Jacksoja

it was about when is R/I a field

haha thanks !

then by the correspondance theorem

I has to be maximal

Érico Melo Silva

@Daminark i’ve seen him but would never have anything interesting to say to him anyway

Daminark

Yeah I mean same

MatheinBoulomenos

my advisor said he knows Drinfeld

Daminark

I feel like back in Chicago there were the number theorists you could meet like Matt and Frank, then folk like Beilinson, Drinfeld, and Ngo you just kinda wouldn't

I guess because they don't teach undergrad classes all too much (Ngo I think taught the IBL number theory which math majors practically never took lmao)

Érico Melo Silva

02:11

undergrads spoke to ngo all the time

Daminark

Huh, aside from one person nobody I know ever crossed paths with him at all

Or at least I've never heard of this sorta thing happening, the NT folk I knew were all with Matt

MatheinBoulomenos

oh wow Ngo Bao Chau is there as well? damn

Érico Melo Silva

i’ve spoken to him albeit not about math lol

Simple

Can someone help me with the proof I have? if $(a_n)$ is convergent, show that $\displaystyle\inf_n\,a_n\leq\lim_{n\to\infty}a_n\leq\sup_n\,a_n$. (By contradiction) Given $(a_n)$ is convergent, $\lim_{n\toinfty}>\sup_{n}a_n$. As $(a_n)$ is convergent, then $\lim_{n\to\infty}a_n=a\in\mathbb{R}$. For all $\epsilon>0$, there exists an $N\in\mathbb{N}$ such that $n\geq\,N$ implies $|a-a_n|<\epsilon$. Then, we have $a-a_n<\epsilon$ which is equivalent to $a<a_n+\epsilon$.

Daminark

Yeah Chicago likes Langlands a lot lmao

Ryan Unger

02:13

imagine knowing what Langlands is

Daminark

I mean I don't either tbh

MatheinBoulomenos

I know Langlands for GL_2 in the local case

Daminark

One guy here in Madison does geometric Langlands

MatheinBoulomenos

and of course for GL_1 it's just CFT

and in my undergrad thesis I do a special case of GL_2 in the global case

but in general I don't know what Langlands is either

Érico Melo Silva

apply ga to langlands @Ryan

ricci flow on whatever they use idfk

Ryan Unger

02:14

@ÉricoMeloSilva were you awake when constantine was going from $\Phi$ to $\phi$ randomly

Érico Melo Silva

maybe but idk

Daminark

But there are only three number theorists at the moment. One guy seems to be big on "Shimura varieties", one guy is more general arithmetic geo, and the last does spectral theory of automorphic forms

MatheinBoulomenos

Oh I've been at a conference on Shimura varieties before, pretty cool stuff

Shimura varieties are pretty important in Langlands too

Ryan Unger

none of the first years here seem to know what shimura varieties are

MatheinBoulomenos

they're hard to define tbf

Daminark

02:16

It's a variety that isn't a non-Shimura variety

ez

MatheinBoulomenos

the actual definition of a Shimura variety is due to Deligne

they are stories that Shimura didn't really understand the definition of Deligne ...

Ryan Unger

Shimura is dead...

MatheinBoulomenos

yeah sorry

Érico Melo Silva

2 soon

MatheinBoulomenos

yeah right, he died this year

Ryan Unger

02:18

nina didnt know he was dead

Érico Melo Silva

he died like right before the visit i think

Ryan Unger

princeton had three big deaths in a one year span

Stein, Bourgain, and Shimura

Daminark

Wow

MatheinBoulomenos

@Ryan my advisor wants to send me to UCLA for a few months

Ryan Unger

yeah man why not

Érico Melo Silva

02:21

one of my best friends is a student there now

MatheinBoulomenos

@RyanUnger have you met Venkatesh?

Ryan Unger

I have

at the ICM

Daminark

Venkatesh seems fun

He gave some talks in Chicago, only got to go to one but it was quite good

MatheinBoulomenos

I might meet Venkatesh in Oberwolfach next year if I get accepted to the conference there

Ryan Unger

@ÉricoMeloSilva we need to crash this cocktail party

Érico Melo Silva

02:27

let’s gooooo

Secret

04:27

Last night dream, in math chat Ted recalled my previous attempt some days ago (in the sense of the dream's history) to visualise some section of the real number line as a bar diagram similar to ordinals. He then posted the GIF that showed he pulled that off

The diagram showed the irrationals as a big rectangle that pulsate between black and white, occassionally interrupted by thin rectangles which are the rationals with travelling velocities proportional to the value of the denominator

Ted then pointed out in the GIF animation on the behaviour of some irrationals. He then conjectured based on that on how these irrationals might have a constant separation and bounded in blocks of the same width

krauser126

1

Q: Cities and Induction. Trying to find a dead end.

There are 𝑛 ($n$ > 1) cities and every pair of cities is connected by exactly one road. The road can go only from A to B, only from B to A, or in both directions. The goal is to find a dead-end city, if it exists, i.e., a city x to which there is a direct one-way road from every other city, bu...

induction

Secret

I then whisper to myself that I wish I have the programming skills of Ted

krauser126

Having trouble understanding this.

Is this using strong induction? If so, I'm not quite sure what the induction step is.

Secret

A snapshot of Ted's diagram in the dream

Top shows the ordinal like stick diagram

Bottom left showed the set of rationals visualised as moving bands of velocities proportional to their denominators

Bottom right is the equal width conjecture of some irrationals, with one irrational highlighted and sweep from left to right in the actual GIF

krauser126

05:00

Anyone?

Alessandro Codenotti

07:27

@user709833 That is by far the most common approach, but it's not the only possibility

Slereah

08:04

Obvious question but

If I have a path $\gamma$ intersecting with two disconnected subsets $U_1$ and $U_2$

Does $\gamma$ intersect their boundaries?

Pretty sure it does but I can't think of why

(It is for a manifold here, if that helps)

Oh wait, if $\gamma$ is continuous and goes outside of $U_1$, there exist a point $p$ such that $p \in U_1$ and $p' \neq U_1$, with a path on $[a,b]$ for those two points, and if we pick any point in between, this is still an endpoint inside and outside of $U_1$

so there's an arbitrarily small neighbourhood with $\gamma([a,b])$ with endpoints inside and outside

ergo boundary point

nvm

Poline Sandra

09:54

hello is this right?

$[-1,3].[0,4[=(-4,12)$

Balarka Sen

11:08

Hi @ÍgjøgnumMeg

Alessandro Codenotti

11:23

Hi @Balarka

Balarka Sen

Hi @Alessandro!

William Oliver

Okay, so this is kind of a silly/vague question, but we know that $a \leq b$ iff $b - a = \sqrt{(b - a)^2}$.

Is there an analogous equation $E(a, b)$ such that $a < b$ iff $E(a, b)$?

Balarka Sen

@Alessandro Have you heard of Tiamat?

Alessandro Codenotti

@BalarkaSen No, what's that?

Balarka Sen

It's a Swedish metal band

I have been planning on going through their discography. I like some things I have heard by them.

Alessandro Codenotti

11:33

Any suggestions in particular?

Balarka Sen

Wildhoney is a very famous album apparently. I have yet to listen though

I'll probably start uh today lol

Alessandro Codenotti

I'll check it out, thanks!

Rithaniel

12:04

Good morning, chat

Secret

very blue day today for me

so many things not working

Alessandro Codenotti

13:22

Stupid question: When building categories of metric spaces do we consider spaces which are homeomorphic but with different metrics as distinct objects?

Balarka Sen

@Alessandro You should only identify metric spaces if they are isometric, I think.

The objects should be isometry classes of metric spaces

Morphisms should be short maps instead of continuous maps, for similar reasons

Alessandro Codenotti

short maps?

Balarka Sen

Yeah, distance-non-increasing maps

Alessandro Codenotti

@BalarkaSen So like $\Bbb R$ with the standard metric and $\Bbb R$ any bounded equivalent metric should be different object

@BalarkaSen Makes sense

Balarka Sen

Yeah

Actually maybe you shouldn't identify anything and just let the objects be sets equipped with metrics. Then isomorphism between two different objects will still be a nontrivial thing, etc.

Alessandro Codenotti

13:33

Ah right, otherwise we have no isomorphisms in the category

Balarka Sen

Right.

Alessandro Codenotti

(that's called a skeleton or something I think)

Balarka Sen

Well, there will be isomorphisms in the category, it's just that they will all be self-loops.

There will only be automorphisms

Alessandro Codenotti

Sure but those are in every category

Balarka Sen

Idk it's a weird thing to do

True

Alessandro Codenotti

13:35

Well there's also nontrivial ones here

user709833

13:59

@AlessandroCodenotti For instance if you wrote a proof that at some point used real numbers or even something simple like distribution law

And i say wait how does that work, how do you know / prove that works like you say it does

You would go to the second order peano axioms and their resultant proofs/derivations/etc?

I am a little confused what most of our modern mathematics is "running on"

since we have these second order Peano axioms as well as these first order Peano Arithmetic axioms

William Oliver

14:12

I am going to bump this https://chat.stackexchange.com/transcript/message/51933052#51933052

Okay, so this is kind of a silly/vague question, but we know that $a \leq b$ iff $b - a = \sqrt{(b - a)^2}$.

Is there an analogous equation $E(a, b)$ such that $a < b$ iff $E(a, b)$?

Leaky Nun

@WilliamOliver no. any such equation defines a closed set.

because the functions involved are continuous

William Oliver

I see, very elegant explanation. Thanks!

William Oliver

14:26

I realized that I was having a lot of trouble with intuition about basic inequalities in the same way I have intuition about equations. Anyone have any tips related to this?

I found this very related question, but it is really old and has no answers.

https://math.stackexchange.com/questions/1029065/are-inequalities-harder-to-prove-than-equalities

4

Q: Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's usually clear what is required: A sequence of clever arithmetic manipulations until at last one ar...

inequality soft-question

Ultradark

16:29

can anyone help me derive a formula for the isometry $f(\vec x)=A\vec x+\vec b$ which is the reflection over the vertical line $x=a$ where $a\in \Bbb R$?

ÍgjøgnumMeg

17:00

Hi @Balarka lol

Ted Shifrin

Hi @ÍgjøgnumMeg. Is a @Balarka around?

ÍgjøgnumMeg

Hey @Ted, he pinged me earlier lol

Ted Shifrin

I haven't seen him in ages.

ÍgjøgnumMeg

well it looks like he was here today :)

Ultradark

how can i parametrize a circle but with the condition that the circle lies on $10x-6y+3z=11$

$x(\theta) = 2 + 7\cos(\theta) + 7\sin(\theta)$

$y(\theta) = 5 + 7\cos(\theta) + 7\sin(\theta)$

$z(\theta) = 7 + 7\cos(\theta) + 7\sin(\theta)$

where $(2,5,7)$ is the center

radius, $r=7$

I don't think it's quite right because I'm not sure how to make it lie on the given plane

MatheinBoulomenos

17:09

Hi @Ted @ÍgjøgnumMeg

ÍgjøgnumMeg

Hey @Mathein

Got lost so many times on the way home hahaha

MatheinBoulomenos

glad you made it home anway

ÍgjøgnumMeg

17:26

Cheeeerz

Tobias Kildetoft

So I felt like rereading an old webcomic, and the annotation for one the them mentions that there is now a wikipedia page for the comic, so I click it because I am curious if it is still there (seeing as the webcomic was discontinued many years ago). Turns out that the page links to the page about the author who apparently went on to write The Martian after no longer making the webcomic.

Ted Shifrin

howdy, @Mathein @Tobias

Tobias Kildetoft

@TedShifrin Hi

schn

Consider the following equation (wolframalpha.com/input/…). How does one eliminate the absolute value signs and solve for $y$?

Alessandro Codenotti

17:45

Hi @Mathei @Tobias @Ted @ÍgjøgnumMeg

Tobias Kildetoft

@AlessandroCodenotti Hi

ÍgjøgnumMeg

Heya @Alessandro

Rithaniel

Afternoon, chat

Brandon

Hi

Hello

Namaste

Hola

Rithaniel

Privyet

Brandon

17:47

Nihao

Ola

Ciao

Ted Shifrin

hi, demonic @Alessandro

Alessandro Codenotti

18:04

I'm finally going to do an intro to diffgeo course this semester @Ted

s.harp

isnt that too basic for you? @Alessandro

they spend like 3 lectures talking about stuff like second countable spaces etc

Alessandro Codenotti

I never learned this stuff in a course, I just read about it here and there

MatheinBoulomenos

Hi @Tobias @Alessandro

Tobias Kildetoft

@MatheinBoulomenos Hi

Alessandro Codenotti

Also officially it's called Global Analysis I and I need some analysis credits :P

here's the course's description (third page)

MatheinBoulomenos

18:08

lol, Algebra II: modular forms

Alessandro Codenotti

@MatheinBoulomenos Those are bachelor courses as well by the way (but they can be taken by master students)

Mike Miller

That's more than differential topology but calling that global analysis is a bit much

ÍgjøgnumMeg

@Mathein how do I register for Vorlesungen again? :)

MatheinBoulomenos

here we do modular forms in our bachelor complex analysis course

@ÍgjøgnumMeg register at: muesli.mathi.uni-heidelberg.de/user/register

s.harp

tbh i heard a lecture called "modular forms in multiple variables" and i still dont know what a modular form is

MatheinBoulomenos

18:10

a modular form is a section of a line bundle on a moduli space of elliptic curves

ÍgjøgnumMeg

rofl

@Mathein da steht nur "Übungsgruppen" bei ANT I, zum Beispiel

lol

MatheinBoulomenos

yes

you don't have to register for the lectures

you just attend

ÍgjøgnumMeg

oh I see lmfao

MatheinBoulomenos

nobody cares if you're even a student if you just want to sit in the lectures

Alessandro Codenotti

@MikeMiller I just need 9 analysis credits to graduate and that course happens to be worth 9 credits so I'm not complaining if it's not very analytical

MatheinBoulomenos

18:13

@MikeMiller for me it's pretty analytical

you have things with $\Bbb R$: pure analysis

Mike Miller

I've heard this joke before

Érico Melo Silva

@MikeMiller i think ive heard some people say global analysis just to mean analysis on manifolds

they were all europeans so maybe it's a european thing

MatheinBoulomenos

$\Bbb R$ is a local field, so it should be local analysis

global analysis would over e.g. $\Bbb Q$

Alessandro Codenotti

@ÉricoMeloSilva wait what is global analysis supposed to be? I also thought that it meant analysis on manifolds

MatheinBoulomenos

@ÍgjøgnumMeg you also want to set up a MAMPF account unless you already have one

Érico Melo Silva

18:17

when i hear global analysis i think the kind of shit big brained topologists do taking statements about PDE on manifolds or bundles or whatever and turning that into topology statements

MatheinBoulomenos

what's the difference between geometric analysis and global analysis?

Alessandro Codenotti

@ÉricoMeloSilva That sounds cooler than analysis on manifolds tbh

@MatheinBoulomenos @Ryan I summon thee

Érico Melo Silva

depending on how you define things maybe global analysis falls under geometric analysis but it depends who u ask

MatheinBoulomenos

we have a geometric analysis lecture next semester, they do: pseudodifferential operators, Fredholm operators, Heat kernels, elliptic complexes, Hodge theory, Dirac operators, Chern-Gauss-Bonnet and Atiyah-Singer

Érico Melo Silva

but i guess id say geometric analysis really cares about geometric objects that solve PDEs that are themselves defined by some geometric condition i guess

MatheinBoulomenos

18:25

@ÉricoMeloSilva would you take the GA lecture based on that topics? I want to get better at diff geo

Érico Melo Silva

I would take this class

it's stuff i feel i should be able to talk about and use but it's not really what i use in my daily geometric analysis life

MatheinBoulomenos

there are actually cases were geometric analysis stuff is relevant for NT. There are some cases were we can't solve some conjectures in NT because we don't know some answers about the spectrum of the Laplacian on a hyperbolic 3-fold

Érico Melo Silva

neat

Mike Miller

@ÉricoMeloSilva i would use the term for geometric analysis and it's just hodge theory

MatheinBoulomenos

@Mike do you think that I can understand the proof of uniformization after the GA lecture? (see above for the topics)

Érico Melo Silva

18:32

@MikeMiller i dont parse this sentence

Mike Miller

@ÉricoMeloSilva and (their lecture is) just hodge theory

Érico Melo Silva

now i parse

Mike Miller

@MatheinBoulomenos there are many proofs and this has absolutely nothing to do with the standard proof which is harmonic analysis

MatheinBoulomenos

ah okay

Mike Miller

there is a nonlinear elliptic PDE proof which the hodge theory would sort of prepare you for

MatheinBoulomenos

18:33

I just thought you mentioned pseudodifferential operators and elliptic complexes before, but it must have been in another context

Mike Miller

I do mention those sometimes

Érico Melo Silva

i think some ppl just have finicky definitions over what counts as GA or not, but ask @RyanUnger and not me bc i dont care about GA politics

MatheinBoulomenos

@MikeMiller to me harmonic analysis sounds nicer than nonlinear elliptic PDE, but I might have the wrong idea about harmonic analysis

Érico Melo Silva

elliptic pde is the good good

MatheinBoulomenos

for me, harmonic analysis is the study of locally compact Hausdorff groups

but I have the vague feeling that this is not that relevant for the harmonic analysis in question

Ted Shifrin

18:40

I agree with @MikeM: I would call it a basic course in differentiable manifolds, with introduction to Riemannian geometry, not global analysis. But oh well.

howdy @Eric

Érico Melo Silva

all the americans agree so it therefore is definitely a european thing

QED

Ted Shifrin

Well, I've bitched in here for years and years about calling basic differentiable manifolds differential geometry :)

Érico Melo Silva

my friends basic manifolds class was called diff geo and it turned him off from talking to me about math until he realized that that wasnt the subject at all lol

Ted Shifrin

Of course, most people find the Christoffel symbols in basic surface theory a complete turn-off. I don't see what the big deal is to write down the second partial derivatives of the parametrization. Shrug.

MatheinBoulomenos

diff geo was the hardest course I ever took

Ted Shifrin

18:44

Of course, I stand by moving frames as the most intuitive approach, but then the problem is that 95% of math students (and, I dare say, a slightly smaller percentage of researchers) are scared of differential forms.

MatheinBoulomenos

forms was one of the things I liked about our diff geo course, but we didn't use them much

Ted Shifrin

When I taught grad diff geo (4 or 5 times, I guess), I always did most things with forms. Not everything.

Érico Melo Silva

there's a cultural thing too right, it doesn't pay to be idiosyncratic if everyone in ur subfield doesnt write things in terms of forms

so there's an inertia built in there

MatheinBoulomenos

we had exercises where we had to solve ODEs, I have no idea how to do that. The only people who could solve ODEs were the physics majors lol

Ted Shifrin

Of course, it didn't help when I was a grad student at Berkeley that the faculty who wanted to teach honors second-year calculus/analysis were often algebraists who wanted to "learn" differential forms. Ugh.

I wouldn't brag about that, @Mathein, if I were you. I think you're too proud of knowing how to do nothing computational.

MatheinBoulomenos

18:46

I'm not proud of it

Érico Melo Silva

im literally solving an ODE like rn

MatheinBoulomenos

I have done a bit of computational algebraic number theory

I just felt lost during the diff geo course because it seems like everything you do is just setting up coordinates and then appeal to some result about ODEs

Ted Shifrin

I think most of my honors multivariable math students were proud that I made them do plenty of computations (and harder ones) along with the theory.

Well, appealing to the fundamental existence/uniqueness/smoothness theorem in ODEs is not a big deal. Everyone should prove that theorem once in his/her life. My favorite proof is the contraction mapping/Banach space proof, so you should like that.

(Set it up as an integral equation.)

MatheinBoulomenos

I did that proof in the first semester

Alessandro Codenotti

@TedShifrin That's a cool proof

Ted Shifrin

18:50

So you need basic things about flows and the exponential, but to me that's a very small part of differential geometry.

And that's not what I consider "solving" an ODE.

MatheinBoulomenos

you also have the geodesic equation and parallel transport

Érico Melo Silva

most of the time the actual equations are untouchable though

Ted Shifrin

Yes, those are actual ODE to solve IRL.

But you rarely solve them explicitly. I do that several times in my undergraduate course/notes.

Érico Melo Silva

you're not really typically interested in characterizing these things except in highly symmetric circumstances where you're not really like "solving" ODEs so much as doing proper geometry

Ted Shifrin

But in manageable cases, it reduces to freshman-level ODE.

Érico Melo Silva

18:52

except when it's manageable like Ted is saying

Ted Shifrin

Working out the general geodesic on a sphere (in spherical coordinates) turns out to be some cool calculus. That's in my notes. Of course, there are immediate more symmetric arguments. Or one appeals to uniqueness.

MatheinBoulomenos

maybe if I retook the course now it would be more managable

I took it as a Freshman

Ted Shifrin

We've long ago established that my pedagogical approach to math is far different from your lectures in Germany, regardless :P

MatheinBoulomenos

we proved Bonnet-Myers and Cartan-Hadamard which are cool

Ted Shifrin

Yes, I always proved Cartan-Hadamard in the grad course. Using the structure equations in normal coordinates.

Érico Melo Silva

18:58

@Ted i did the full calc of characterizing geodesics on the ellipsoid finally

it’s horrible

Ted Shifrin

Oh wow. So what's the result?

MatheinBoulomenos

I think we used Jacobi fields for Cartan-Hadamard iirc

but maybe I'm misremembering

Ted Shifrin

That's probably the standard approach. I have never actually taught Jacobi fields, since "my" geometry was never Riemannian stuff.

Érico Melo Silva

@TedShifrin the ones that you don’t get from intersecting with coordinate planes are horrible, i have them written down on papers on my desk, i can’t remember

Ted Shifrin

Did you get Mathematica to draw them?

Érico Melo Silva

19:05

there’s a semi nice characterization in terms of quadrics idr but the actual geodesics are expressed by like abelian integrals

no but i should do that

Ted Shifrin

Yeah, I'd definitely expect classical abelian integrals, at the very least. Yeah, I'd love to see the Mathematica pictures.

Érico Melo Silva

maybe next week sometime, i’m busy doing actual work and then have a visitor coming for the weekend

Ted Shifrin

"Actual work" — pshaw.

Actual lunchtime for me. :)

Érico Melo Silva

this is a fun curiosity but i promised myself i’d understand why a certain construction works before the week is out so no time for fun math

ÍgjøgnumMeg

@Mathein just registered on Mampf lol

and there is an actual skript

Érico Melo Silva

19:08

@TedShifrin have fun

ÍgjøgnumMeg

@Mathein very cool site! I know all the Vorlesungsinhalte now lol

Mike Miller

@MatheinBoulomenos I'm not sure if you're serious when you say "harmonic analysis is the study of locally compact Hausdorff groups"

Not everything is reducible to meme

MatheinBoulomenos

@MikeMiller half-serious maybe

user193319

"When $\mathcal{H}$ is [an] infinite dimensional [Hilbert space] $B(\mathcal{H})$ is no longer simple..." What exactly is meant by "simple"? What does it mean for an algebra to be simple?

MatheinBoulomenos

a ring is simple if it has no two-sided ideals except for 0 and the whole ring

user193319

19:21

Ah, okay. Thanks!

MatheinBoulomenos

note that $B(\mathcal{H})$ contains the two-sided ideal of finite-rank operators

which shows that it's not simple

Mike Miller

You read too many memes, man. The proof of the uniformization theorem is an analysis of what kinds of Riemann surfaces can support subharmonic functions and of Green's functions, which are essentially solutions to $\Delta f = \delta_p$ where $p$ is some point on the surface. There are no groups. The word "harmonic" comes from harmonic functions, ie, solutions to $\Delta f = 0$.

Alessandro Codenotti

It also has the ideal of compact operators, which is closed as well (also the only closes ideal when H is separable)

MatheinBoulomenos

@MikeMiller yeah, it was more of a joke and I explicitly wrote that it's a different kind of harmonic analysis

names don't necessarily have a unique subfield of math associated to them

you also have this with "semigroup theory" or "field theory"

Érico Melo Silva

wtf else does field theory mean

ÍgjøgnumMeg

19:29

Agriculture

MatheinBoulomenos

you have the study of fields as an algebraic structure and then you have stuff like classical field theory or QFT

@ÍgjøgnumMeg yeah agricultural math is where you study fields and sheaves

ÍgjøgnumMeg

hahaha

Érico Melo Silva

debatable that classical field theory is math

MatheinBoulomenos

I don't know enough about physics to argue about that

but it wasn't that easy to find books on actual field theory

Érico Melo Silva

im pretty physics agnostic

in that i don’t know if it exists, maybe every physicist i’ve ever met is lying to me

Akiva Weinberger

19:40

quantamagazine.org/…

Most informative article on Google's quantum supremacy that I've found^

ÍgjøgnumMeg

Moonlight sonata played on a cathedral organ is the coolest thing ever

AnArrayOfFunctions

How many comprasions in a set of 3 numbers to determine if they are unique

I assume 3

one comparaision between the first number and the two others and another one between the two others

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$Mathematics$ Mathematics