Mathematics - 2019-03-18 (page 2 of 2)

Daminark

18:00

So I didn't even know quantum logic was a thing, though is it a modification of the classical laws of logic or do people ask such questions as people in, say, model theory do?

Semiclassical

since any experiment you run will inevitably involve devices that do behave classically

(wouldn't be much use as measurement devices if they didn't!)

anakhro

@Daminark AFAIK more the former than the latter.

MatheinBoulomenos

I learned about determinants and groups from a hardcore algebraist who himself said that he doesn't know any physics and I learned both that "The determinant of a matrix is an (oriented) volume of the parallelepiped whose edges are its columns" and how to think about groups as transformations

anakhro

@MatheinBoulomenos that's great.

Then your hardcore algebraist took advice similar to Arnold's.

That's all he's asking as far as I can tell.

MatheinBoulomenos

but that's not related to physics at all

Semiclassical

18:03

No, but it is related to the behavior of some concrete geometric object

anakhro

determinants as they related to n-dimensional volumes is not in any way related to physics?

Semiclassical

as a sidenote, it's pretty easy to motivate "n-dimensional space" from within physics

vzn

the discussion of empirical vs theoretical math vs physics reminds me a lot of wolfram, a rare individual crosscutting those (interdisciplinary) boundaries, but he seems to be persona non grata in a lot of circles... :|

MatheinBoulomenos

@vzn is he a diplomat?

Semiclassical

you run into it inevitably once you start talking about position/momentum for an object with more than one degree of freedom in three dimensional space

vzn

18:05

@MatheinBoulomenos lol he can be both diplomatic and undiplomatic at times as with his (numerous) detractors

Semiclassical

for instance, the phase space of two coupled pendulums can be understood as four-dimensional

two generalized coordinates (the angles of the pendulums) and two conjugate momenta

so physics does get you pretty quickly to higher-dimensional geometry (specifically symplectic)

MatheinBoulomenos

@anakhro you certainly don't need to read any volume of Landau and Lifshitz for that

vzn

another famous essay related to this subj, unreasonable effectiveness of math in natural sciences by wigner 1960 en.wikipedia.org/wiki/…

Daminark

Okay yeah, so I still think, maybe I'll lighten that no definition of physics includes stuff like model theory to, it's extremely ballsy to suggest that, and if I had to guess Arnold either wasn't thinking about that sort of thing or actively discounted it in favor of more geometric things. Whatever the case, I think there's probably a case for favoring concreteness/intuition over pure formality without having to resort, through exaggeration or otherwise, to the epistemological claims made.

anakhro

Because Landau and Lifshitz are the arbiters of physics.

Semiclassical

18:07

@vzn arnold references Wigner's essay explicitly, in fact

uni-muenster.de/Physik.TP/~munsteg/arnold.html

is what we're talking about

vzn

@Semiclassical am not too familiar with arnold & need to look into it some more

Semiclassical

he doesn't talk about Wigner too much in there, but what he's talking about is very much in that wheelhouse

vzn

hmmm, am now reminded of "abels thm in problems + solutions" by arnold as highly recommended by balarka in chat yrs ago. maths.ed.ac.uk/~v1ranick/papers/abel.pdf

anakhro

Yeah it's a good one, vzn

Semiclassical

my favorite bit of this arnold address, btw, is this

_The Arnold Principle_ . If a notion bears a personal name, then this name is not the name of the discoverer.
_The Berry Principle_ . The Arnold Principle is applicable to itself.

vzn

18:11

the sentence Mathematics is a part of physics. is rather bold. suspect he did not mean to imply it is not part of other fields also, and cannot also stand alone...?

Semiclassical

That's what we're arguing about.

anakhro's stance is that he was speaking poetically and shouldn't be taken too literally

MatheinBoulomenos

Why is it so ridiculous to learn about ideals in a ring without knowing what a cycloid is? I get the feeling that Arnold just doesn't like algebra and number theory and would prefer if everyone did geometry

Semiclassical

my own reading is that his view of physics is sufficiently broad as to naturally include math

anakhro

8

Q: What is the relation between hypocycloids and ideals in polynomial rings as alluded to in Arnold's text on teaching mathematics?

While browsing through this site, I came upon the text of Arnold: "On teaching mathematics". http://pauli.uni-muenster.de/~munsteg/arnold.html containing the phrase ... it can be said that a hypocycloid is as inexhaustible as an ideal in a polynomial ring. But teaching ideals to students wh...

Semiclassical

(or, at least, broad enough to include the kind of math that he finds worthwhile :P)

anakhro

18:13

@MatheinBoulomenos see comments

MatheinBoulomenos

so it makes no sense to teach ring theory to someone interested in number theory?

it's easier to find applications of elementary ring theory to NT than proving the Nullstellensatz

Daminark

So it was a long time since I read Arnold's thing and when you were talking hypocycloid I thought it was an example you came up with in spirit of what he was talking about, only just realized that he said that verbatim

anakhro

@MatheinBoulomenos I don't follow.

MatheinBoulomenos

I'm just saying there are more reasons to care about ideals than varieties

Semiclassical

I suspect Arnold's view would be more like: Before learning a plethora of abstract definitions and proving stuff using them, one should first gain an intuition for the subject using concrete examples and problems

anakhro

18:18

@MatheinBoulomenos did you read the rest of the comment

>I assume the main sentiment to be expressed (also in other passages of the text) is that it doesn't make sense to teach algebraic geometry (and the necessary basics from comm. algebra) to somebody who never saw a nice/interesting curve or surface beforehand.

- user9072

loch

@Semiclassical this is how I would read it too

MatheinBoulomenos

@anakhro yes and my point is that there are more reasons to care about comm. algebra than the nullstellensatz

Daminark

I think Mathein is specifically objecting to the "necessary basics from comm. algebra" more than to the AG. Tbh someone needs to write a new manifesto that advocates for concreteness in math education without as many loaded comments

Semiclassical

it wouldn't be arnold if it weren't polemical :P

MatheinBoulomenos

I'm fine with concreteness

but why the disrespect for number theory. Going by the scare quotes, Wiles proof of FLT is apparently not even an achievement

anakhro

18:23

This is the offence that I referred to, apparently.

Semiclassical

"In its significance for both the applications and the development of correct Weltanschauung it by far surpasses such "achievements" of mathematics as the proof of Fermat's last theorem or the proof of the fact that any sufficiently large whole number can be represented as a sum of three prime numbers."

does come across as pretty snide

Secret

Does the number Pi actually exist?

►

anakhro

It's akin to a snowball rolling down a hill.

Daminark

That's probably true, but I still think he's doing massive disservice to the pedagogical points he's trying to make. For what it's worth, I do think that if Arnold were in MSE chat and responding, while there's some probability, which based on what I know of him tbh could be 0 or 100, I'm not sure, that he'd backpedal the epistemology a bit, he'd probably still be willing to die on a kind of "geometry supremacy" hill

Secret

Actually, it is remarkable enough that we cannot get squareroots by taking fractions of line segment of integer length, but suddenly we can do that when we start drawing triangles

anakhro

18:26

He's as willing to die on a geometry supremacy hill as Mathein is willing to die on the offended algebraist hill. ;)

Daminark

I mean I definitely respect the latter hill quite a lot more lmao, if only because Mathein isn't specifically going out of his way to discount mathematicians who function differently

Semiclassical

his reference to "three prime numbers" seems to be to this: en.wikipedia.org/wiki/Vinogradov%27s_theorem

Secret

Likewise we cannot get any transcendentals with any algebraic operations (generalisation of drawing triangles and taking measurements in weird ways) but suddenly, we can get some of them by making circles

Somehow, geometry has this curious properties of being able to take infinite limits under some finite extent

anakhro

I don't think there is respect in being offended. I also don't think there is respect in being mean. But I think Arnold is simply injecting humour in his essay.

vzn

@Semiclassical lol so he dismisses his national cohort Vinogradov. he quotes hardy admiringly but puts down quite a bit of number theory. think he has some point but the essay is rather eccentric in places...

Secret

18:30

75

Q: Direct proof that $\pi$ is not constructible

Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass? Of course, $\pi$ is not constructible because it is transcendental and so is not a root of any polynomial with rational coefficients. But is there a simple direct proof ...

real-analysis number-theory irrational-numbers geometric-construction

yet in theory given a perfect circle, pi can be obtained using information in such diagram

so $\pi$ is not the worst of the transcendentals

Daminark

I mean, being offended doesn't necessarily increase the respect one gets but it doesn't decrease it, while being unnecessarily mean very much decreases it. If he is attempting to inject humor he's definitely not doing that good of a job and clouding the concreteness point he's trying to make, which I think way fewer people object to.

Secret

The question remains, however on how to categorise such "geometrically definable" numbers

Érico Melo Silva

@Semiclassical this + saying something mean about french mathematicians

vzn

some other connections come to mind: proofs + refutations by lakatos, think Arnold would be very approving of it. also Gowers has a neat essay talking about the problem solvers vs theory builders dichotomy... Erdos being in former camp, Grothendieck in later, although not sure if he named them specifically...

Daminark

Again this is why I say someone needs to write a manifesto about keeping math concrete but without the flaming and without the bold, even if ironic, philosophical claims.

Semiclassical

18:33

11 mins ago, by Semiclassical

it wouldn't be arnold if it weren't polemical :P

anakhro

@Daminark you mean you don't derive any humour from what he's saying?

Daminark

Then have someone who isn't Arnold

Semiclassical

@vzn yeah, the "two cultures of mathematics" essay does come to mind

It's not quite the same thing, but it definitely is pertinent

Secret

Thus one can in theory use axiomatic geometry to define a circle, and then $\pi$ can be used in such framework precisely, except the digits cannot be known

Érico Melo Silva

anyone who uses freudian terminology like weltanschaung in a mathematical context is a weirdo

MatheinBoulomenos

18:34

@anakhro where is the joke?

weltanschaung is a regular German word

vzn

← big gowers fan, hes been dabbling into more empirical math eg automated proofs, AI etc

Érico Melo Silva

@MatheinBoulomenos when u see it written in english text it has a particular academic connotation

Semiclassical

depends on what english text we're talking about

plus he gave that talk in Paris

Daminark

@anakhro I mean aside from maybe snorting at the 3+5 thing, despite it being kinda insulting, not really, it just came across as clouding a point by being whiny tbh

Semiclassical

so it wasn't necessarily an English text originally

Érico Melo Silva

18:36

@Semiclassical so the translator is the freak

Semiclassical

lol

vzn

speaking of polemical-bordering-on-cantankerous essays, math + physics etc, aaronson had a colorful/ amusing recent blog on "death of proof" somewhat related to topic of more empirical math scottaaronson.com/blog/?p=4133

anakhro

I wonder why some people are more inclined to see the humour than others.

Maybe it depends on the sequence of Arnold writings you read. :P

Semiclassical

it's a lot easier to laugh at something when you're not the butt of the joke

Daminark

Wait what does weltanschaung mean? Both in common German and the academic connotation

MatheinBoulomenos

18:37

Secret

Thus between the constructible numbers and definable numbers, there should be a class of numbers in between that includes straightforward transcendentals like $\pi$ and $e$

MatheinBoulomenos

weltanschauung is worldview

anakhro

@Semiclassical I guess Daminark and Mathein are both algebraists...

vzn

@anakhro yes he seems to have a ½-serious, facetious tone. almost some hidden comedy or "court jesting" going on. which sometimes involves irreverent "goring sacred cows"

Daminark

And re humor: when one is trying to make a point and be funny at the same time it's difficult to draw the line if you haven't read other things the guy wrote. Sort of like how on Youtube comments it's equally susceptible that someone's sarcastic or just dumb when it's not obvious.

Érico Melo Silva

18:39

@Daminark it means the same thing it does in german when u see it in like english academic texts but in those places it carries a connotation of weird sex stuff vis a vis freud

sometimes anyway

Leaky Nun

it can just be a linguistic text, e.g. Sprachbund

MatheinBoulomenos

I still don't see the point that Arnold is trying to make other than that people should stop doing mathematics other than geometry and mathematical physics

anakhro

@vzn Yeah, he seems to have fun

Semiclassical

I think a problem with this is that we only have a partial sense of the context he's addressing

Daminark

And I'm not quite committed to algebra yet, there's still a fairly non-trivial chance I'll end up doing something more topological, but I guess I hold a lot of the "my way of doing math is better" sentiments in a fair bit of contempt.

Semiclassical

18:41

i mean, I have no idea what French mathematical education is like :P

Érico Melo Silva

idk i think arnold unironically thought that

anakhro

@MatheinBoulomenos the point is, when teaching mathematics, you should motivate it with historical/physical/concrete things, rather than just axiomatic skeletons.

Daminark

And like, idk maybe I go on reddit too much but I see those sorts of things frequently enough that once someone starts to go down that direction I'm just like oh piss off

anakhro

So, exactly as your hardcore algebraist teacher taught you determinants.

Leaky Nun

who puts the scalar after the vector ($e^{At}$ where $A$ is a matrix)

MatheinBoulomenos

18:42

that was only the motivation, though

the actual definition was via exterior powers

anakhro

Seems fine by Arnold.

Though he'd prefer not coordinate free, I imagine.

Érico Melo Silva

@Daminark idk dude imo geometers unironically are the worst pieces of shit

Semiclassical

here's a silly question to distract us

anakhro

@ÉricoMeloSilva :(((((((

Semiclassical

analogous: hard g sound, or j-sound?

MatheinBoulomenos

18:44

hard g

but I'm no native speaker

Semiclassical

that's how I say it too

Leaky Nun

hard g

Daminark

hard g

Érico Melo Silva

anyone who uses a j sound should be launched into the sun

anakhro

Who ever says it with a j sound

Leaky Nun

18:44

but soft in "analogy"

Semiclassical

lol

^

i know at least one (non-native) speaker who uses the j-sound

and was wondering if it was more common

Érico Melo Silva

if ur non native it doesn’t matter u can pronounce shit however u want and more power to u

MatheinBoulomenos

if Arnold wants to make a pedagogical point, then why be so arrogant?

vzn

@anakhro re "in math there are no empirical claims" it would seem that the age of simulation (increasingly) leads to a "3rd way" between theory and experiment... like this neat paper by heng on that arxiv.org/abs/1404.6248

Semiclassical

it is a little interesting that we say it like analogue and not like analogy

Érico Melo Silva

18:46

i think its cuz arnold is just a notorious ass and not for any actual good reason

anakhro

more or less what Erico said. It's personality with that.

Leaky Nun

@Semiclassical because "y" is a front vowel

like "i" and "e"

anakhro

Whether you are thick-skinned enough to read through without shedding a tear is up to you, though.

MatheinBoulomenos

@anakhro nice ad-hominem

anakhro

You mean, "nice insult"

I was in no way discrediting an argument of yours.

I am just reminding you not to take offence.

Daminark

18:48

Alright this discussion is starting to go down a bit of a bad direction and I think it's probably best we change course

anakhro

Since being offence is the fault of the offended. You can ignore it and read past it.

Leaky Nun

and the victim blaming starts

anakhro

Leaky, stop blaming me :(

Érico Melo Silva

@anakhro im gonna go out on a limb and say this is nonsense

Semiclassical

sheesh

this conversation went from interesting to garbage real fast

anakhro

18:50

@vzn I only glanced at the paper, but it reminds me of probabilistic arguments.

vzn

@anakhro ok, probability is maybe part of it, but not all of it...

anakhro

Not enough to make math empirical by any means.

Or, I should say. Make math have an empirical underpinning of some sort.

@vzn do you like astrophysics?

Semiclassical

tbh i'm happiest when I can ground my math in something concrete and "empirical" (for some definition for that)

but it seems silly to suppose that all mathematics can be or should be like this

loch

imo i think most (mainstream) math is like that (maybe not empirical, but concrete at least)

Daminark

Math should study symmetries of the molecules of concrete tbh

anakhro

19:02

I had meant empirical to be akin to physics where you have to check your theory against reality.

But I do enjoy doing computational things if I have been doing too much abstract stuff, and vice versa.

Secret

The WTF is Geometry investigation: 20119 version:

0

Q: Given axioms, how do we know it defines a geometry?

It is known that besides using coordinates and algebra, there are axiomisation of geometry such as Tarski, Hilbert and Euclid. However looking at the axioms of Tarski for example: Betweeness $B(\cdot,\cdot,\cdot)$ satisfy e.g.: \begin{align} Bxyz &\to x=y\\ (Bxuz \land Byvz) &\to \exists a(B...

geometry philosophy axiomatic-geometry

Semiclassical

my main reason for using scare quotes

is that something like the hypocycloid is equally non-empirical

anakhro

why are your quotes always scary, semiclassical :(

Semiclassical

you're never going to find it out there in the world. you can see something like it in the world

and use that as motivation

anakhro

Like a circle.

You will never see a perfect circle, but math is beautiful and simple enough you can study it anyway.

Semiclassical

19:05

but to act as though there's a hard-and-fast distinction between a hypocycloid as empirical and an ideal as non-empirical seems silly to me

Secret

A circle is defined to be "the object produced by points equidistant from a given point on a plane"

anakhro

The math overflow comment suggested it was maybe meant that hypocycloid was a very interesting curve that warranted studying algebraic geometry.

Secret

but somehow we knew it is geometric

anakhro

@Secret wait, a circle isn't something that is homeomorphic to the boundary of the unit square? D:

Semiclassical

A circle is the reals under the equivalence $x \sim x+1$, of course.

Secret

19:09

Yet if I say something like: "an algebraic structure that is closed under $\cdot$, is associative, has inverse and identity" nobody will necessary call this structure geometric

and yes, a circle is homeomorphic to a square. It is not differomorphic though because you cannot differentiate at the corners of the squares

(Also typo above, I would be long dead when it is the year 20119 lol)

@Semiclassical that's actually a clean example to show my WTF in my main: We only called that a circle because when we actually plot it, it gives us a circle

Semiclassical

Well, we call it that because there's a well-defined mapping between that definition and the usual one

though I guess that's the same idea

Secret

but if I give you any random equivalence e.g. $x^2+x+1 \sim x^2$, we won't even knew if it defines a geometry

so somehow, we "knew" what a geometry is, but seemed to unable to find a way to recognise it by just staring at a given axiomatic system and their theorems derived from those axioms

Or maybe I should put it that way:

If we see any structure that satisfy: Has identity, has inverse for all elements, is closed under the binary operation and is associative, then we knew it must be a group

Likewise, if we say a circle is "the object produced by points equidistant from a given point on a plane" and demonstrate a mapping between $x \sim x+1$ to this, then we knew it is a circle

However, there is a gap in the following:

(Insert some definition of a circle) -> ??? -> circle is a geometric object

anakhro

"geometric object" itself is super vague.

Semiclassical

amusing: the only thing we can't be concrete about, is what it means to be concrete!

I feel like there was either an XKCD or SMBC about that

Daminark

Oh that's easy

Semiclassical

19:23

i mean, "there's an XKCD or SMBC" about it is pretty much axiomatic at this poitn

Secret

Plot twist: Concrete is vague. All we know about language is thrown out of the window!

Semiclassical

is the one i'm thinking of

though I found that via google images and not the original one

Secret

LOL

infinite regress of defines

I am actually starting to wonder, perhaps geometry is not only a concrete thing that is vague, it may be actually ineffable in its most general form

Anyway, that basically means the original question that started all of this discussion

52 mins ago, by Secret

Thus between the constructible numbers and definable numbers, there should be a class of numbers in between that includes straightforward transcendentals like $\pi$ and $e$

I don't think we will know whether there exists such class of numbers until mathematics finally figured out what exactly is a geometry

MatheinBoulomenos

@Secret there are periods

Ring of periods

In mathematics, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a ring. Maxim Kontsevich and Don Zagier (2001) gave a survey of periods and introduced some conjectures about them. == Definition == A real number is called a period if it is the difference of volumes of regions of Euclidean space given by polynomial inequalities with rational coefficients. More generally a complex number is called a period if its real and imaginary parts are periods. The values of absolutely...

Semiclassical

periods are neat

Secret

19:33

ah right, you two told me that before

Semiclassical

this is sorta interesting: "It is not expected that Euler's number e and Euler–Mascheroni constant γ are periods."

By contrast, it's easy to get natural logs as periods

"Elliptic integrals with rational arguments"

why rational?

oh, polynomial inequalities with rational coefficients

anakhro

@Semiclassical have you taken a course in complex analysis?

Semiclassical

not lately

anakhro

Oh, I see.

Secret

other thoughts before I go back to chemistry work and then quickly go to sleep:

If infinite operations do not exist in mathematics, then:

1. While we can still define a circle and polygons, there will exist no route that move from a sequence of polygons of increasing sides to a circle

2. Lack of infinities also means we cannot have curves of arbitrary shapes derivable, hence we will be limited to periods that can be obtained from circles and ellipses

3. This also means in such systems, we will get all the algebraic numbers, and all periods that are related to circles and ellipses, and the rest will be indefinable

(Actually no, without infinity hence limits, we cannot have exact integrals either, so most periods will become indefinable, except for those related to certain unions and intersection of circles and ellipses which the area can be analytically computed)

So I guess the interesting observation here is that:

1. $\pi$ and periods related to circles and ellipses are supposed to be an infinite object that has a finite description

Alessandro Codenotti

19:52

Hi @Paul

Secret

quantamagazine.org/…

yet wikipedia still listed the continuum hypothesis as open, so that might mean this is not the full story

Alessandro Codenotti

That's not about CH, it's about two cardinal invariants of the continuum

(which are only interesting under the negation of CH really)

Secret

yeah, I realise I misread that final paragraph in the magazine

Final thoughts:

The observation of (1) I wrote above, plus the discussion about finitistic reducibility of many infinitistic theorems quantamagazine.org/… seemed to suggest there may be a finitistic way to define an actual infinite object

One possibility is to use the circle as a starting point (which can be easily axiomised into existence in the theory)

and then also axiomise regular polygons (which can also be finitistically defined)

and then collect these polygons in ascending order of sides

and finally, define a countably infinite operation and the notion of a limit as a map between a sequence of such polygons to the circle

Will test this later...

anakhro

20:18

You can define infinite objects with finite data

For example, N is defined with the finite signature {0,s}.

1 := s0, 2 := ss0, 3 := sss0...

So what they often do is distinguish something like this as a "potential infinity".

vzn

21:13

@anakhro (went to lunch) am interested in a lot of areas of physics & like its deep connections with math... also in ways very much/ deep into pure math number-theory, mainly came into it all from computer-science... the heng essay is really not limited to astrophysics at all, its a crosscutting development in physics in general...

anakhro

21:42

Yeah, I was just noting it was in the astrophysics section of arxiv and had that flavour.

vzn

21:54

@anakhro also, somewhat related, machine learning techniques are notably advancing in physics, gotta blog about that sometime...

nbro

Intuitively, what is a smooth vector field?

Ultradark

22:22

What is a discrete vector field called?

anakhro

@nbro exactly what it sounds like.

A function that, when you hand it a point on your manifold, it gives you an arrow.

Moreover, this process is "smooth" so there are no sharp changes along any paths of the manifold.

That is, you don't get a vector field that sort of looks like this:

$$\begin{matrix}\to &\to & \to\\\to & \to & \to\\ \downarrow &\downarrow &\downarrow\end{matrix}$$

Where you have that the arrows along the top just suddenly turn downwards.

It's too sharp. Not smooth.

@Ultradark directed graph.

Ultradark

can you describe a graph that the nodes flow through space and time via differential equations?

like a moving graph

that preserves connections

anakhro

That would need something more. That's about ambient space of a graph

Basically is just embedding the vertices and edges of your graph on a manifold as points and curves respectively.

Then flow it along a vector field.

Ultradark

22:43

is that symplectic flow over a manifold?

nbro

@anakhro What does a manifold have to do with a vector field?

Ultradark

@nbro a vector field can flow over any surface you want

anakhro

@Ultradark no

@nbro Sorry, where are you hearing about "smooth" vector fields.

To discuss smoothness, you need a "smooth" structure of some sort. Manifolds are usually the source of that structure.

But like I mean we can just talk about smooth vector fields in R^n.

That is, a smooth vector field on R^n is a function $v\colon\mathbb R^n\to\mathbb R^n$ such that $v$ is smooth (infinitely differentiable) in each component.

Ultradark

@anakhro do you know what this subject is called?

anakhro

e.g. $v(x,y) = (-y,x)$ is a smooth vector field, but $w(x,y) = (|x|,y)$ is not.

Ultradark

22:50

embedding graphs on manifolds that flow according to a prescribed vector field?

anakhro

@Ultradark what which subject is called?

Oh. I don't know. It doesn't scream "I have a name" to me.

nbro

@anakhro I'm reading about it in the context of geometric deep learning, so, yes, it is related to manifolds, but I'm not familiar with manifolds and how they are related to vector fields

anakhro

@nbro think of manifolds as a more general thing you do calculus on.

nbro

So, manifolds are generalizations of vector spaces?

anakhro

No. Of R^n.

Vector spaces don't really facilitate calculus very well.

Manifolds are just curvy spaces you can do derivatives and integrals on

Think of earth: when you zoom in on earth (like how we view it as people), it looks entirely flat. To the point where we treat it as if it is flat.

nbro

22:54

I have a hard time distinguishing manifolds and pure euclidean spaces (or R^n)

@anakhro Yes, I already read this description

anakhro

However, if you zoom out, earth is a sphere and so it has curvature. It is topologically much different from just R^2.

Oh good.

nbro

But I don't get why R^n can't also be a manifold

anakhro

R^n is a manifold.

It's the most simple manifold.

nbro

Ha, ok

anakhro

But merely an example.

nbro

22:55

So, an image is also a manifold

No

An image can be thought of as function

That acts on an manifold (R^2)?

No

Unit square (not R^2)

anakhro

What do you mean by image?

nbro

Is a unit square a manifold?

anakhro

Unit square is a manifold with boundary, but it is not a smooth manifold.

(that is, it locally looks like R^2, with boundary, but the corners are too pointy)

nbro

An image like in our every life. It can be thought of as a function from pixels to RGB values (e.g.)

anakhro

No, that's not a manifold.

nbro

22:57

Yes, it's a function

it cannot be a manifold

anakhro

But functions can give rise to manifolds as "images" in the mathematical sense, sometimes.

nbro

But it acts on a non-smooth manifold, according to you

anakhro

"acts" in what sense?

nbro

The domain of the image

anakhro

It's more of just a "projection" kind of thing. But it's really tough to phrase it mathematically, since it's not really a simple process.

Math is about simple stuff.

nbro

22:59

:)

anakhro

So in any case, you can imagine a "vector field" on a manifold in an analogous way. It's just like a vector field in R^n, but only when you zoom in.

Zooming in on a manifold, the vector field manifests itself as a bunch of arrows associated to each point.

nbro

So, what if you do not zoom in? How does the vector field look like?

Ultradark

what if the vector field was not defined at each point

but only at a subset of points

anakhro

While this might be entirely reasonable on an open subset of the manifold (called a "local section", or a "local vector field"), it's not something I have seen anyone use extensively.

Ultradark

vector fields are always assumed to be defined over the reals right

anakhro

23:09

What do you mean?

As far as I have been saying, vector fields are merely functions from your manifold (i.e. you feed them points) and take values as "arrows" or "vectors" (whatever that means for a manifold).

Ultradark

I mean are they ever defined for a rational variable

anakhro

I don't know what that means, still.

Ultradark

this would be a subset of the reals

anakhro

But what is the "variable"?

Do you mean like my vector fields $\mathbb R^n\to\mathbb R^n$, but with $\mathbb Q^n\to\mathbb R^n$?

Then this doesn't really make sense because that's asserting Q^n is anything like a manifold.

(it's not really)

Ultradark

$\mathbb Q^n\to\mathbb Q^n$

It's not really?

with the discrete topology

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$Mathematics$ Mathematics