Thread 16691018

topologysisters, is this true?

>>16691040
only in june, the rest of the year there aren't flags

>>16691018 (OP)
Oh, I get it. You posted this, and you like algebra, topology, and combinatorics.

>>16691018 (OP)
>algebra gay
>number theory straight

>topology gay
>geometry straight

???

>>16691018 (OP)
>things I cannot understand = gay
kek

>>16691018 (OP)
oh hey, thats my post

>>16691173
that's our post now

>>16691018 (OP)
I'm assuming a zoomer wrote this in which case "gay" would be considered the good category?

>>16691121
>topology
"Any hole is a goal!"

>geometry
"Regular shapes are made from straight lines. IT'S NOT REGULAR IF IT ISN'T STRAIGHT!"

>>16691232
>Regular shapes are made from straight line
Differential geometry.

>>16691239
NTA, but the whole mechanism that makes diff geo work is that there is a basis in which the manifold is locally well approximated by a linearization. The whole point of diff-geo is always having straight-lines and changing their orientation as you move about.

>>16691377
>the whole mechanism that makes diff geo work is that there is a basis
What an ignorant post. A basis may not even exist. Vector fields aka sections of the tangent bundle are C^infty(M)-modules. They aren’t guaranteed to have a basis, ie the module isn’t free. There is no basis for vector fields on the sphere for example.

What you’re describing is the local homeomorphism condition, which has to do with the choice of a topology on the manifold and nothing to do with bases.

>>16691018 (OP)
no. there are places of utter darkness, beyond the desert of dread, where topology touches number theory.

>>16691443
C^infty(M)-modules are R-modules by forgetting.
R-modules are vector spaces.
Vector spaces have bases by Zorn's lemma.
Am I missing something?

>>16691443
My nigga, you're smoking rocks.

The "smoothness" condition for a smooth Riemannian manifold (i.e., the thing differential geometry is about) is literally that the tangent plane is well defined at all points on the manifold. Differential geometry is all about the Riemannian metric, which only makes sense if the Riemannian inner product is positive definite, which is equivalent to showing that there is an orthonormal basis for local linearization around every point on the manifold.

>>16691018 (OP)
addition, subtraction = super straight
multiplication, division = gay
rest of math = super gay

>>16692077
that he denounces the axiom of choice

>>16693036
Damn. He's based.

>>16692077
>R-modules a vector space
Vector spaces are a subclass of modules. In particular, they’re free modules because fields are division rings. Generic modules aren’t free (ie admit a basis). I literally gave you a concrete example of vector fields on a sphere.
>>16692254
>The "smoothness" condition for a smooth Riemannian manifold (i.e., the thing differential geometry is about) is literally that the tangent plane is well defined at all points on the manifold
No, that’s not what it’s about. It’s about coordinate chart transition maps being smooth. Tangent spaces are, in fact, vector spaces, but tangent BUNDLES, which are used to define vector fields, are not. There is no notion of adding to vectors at two different points. You have take sections of the bundle and the existence of a particular section isn’t guaranteed.
>Differential geometry is all about the Riemannian metric
Again, very ignorant take. You can establish almost the entire vocabulary of differential geometry without a reference to the metric. I would argue that introducing the metric early on is pedagogically misleading as the musical isomorphism leads to students not properly understanding the essential differences between vectors and covectors. This difference is essential to understanding why pushforwards, pullbacks, exterior derivatives, integration on a manifold, etc are defined the way they are.

>>16693457
It is not differential geometry until you introduce the metric. Everything else is differential topology.

>>16691232
>"Regular shapes are made from straight lines. IT'S NOT REGULAR IF IT ISN'T STRAIGHT!"
Behold, a regular right triangle.

>>16694025
>differential topology

A reminder that number theory (except for the really exotic shit that everyone hates that is basically just an excuse for cranks to claim they solved unsolved problems and people just can't understand their math) is just set theory which is just combinatorics.

>>16691050
what about fags?

>>16693457
Okay, clearly you know more about this than me. What is a good reference text that presents diff geo in the way you are talking about? My understanding is mostly coming from Tu and O'Neill, but I understand that there are many ways to look at this material.

>>16694253
Intro to Smooth Manifolds by Lee is a good one.

>>16694126
Yes. Differential Topology. Simply imposing smooth structures is not geometry, it is still topology.

Geometry is about distances and angles. i.e. metrics

>>16694516
>Geometry is about distances and angles. i.e. metrics
According to whomst've? The voices in your head? So you're telling me that
>local coordinates and Jacobian transformations
>tangent and cotangent bundles
>tensor bundles
>principal and associated bundles (including frame bundles and tensor densities)
>connections and curvature forms
>Lie, exterior, and covariant derivatives
>volume forms, integration, and Stokes theorem
all have nothing to do with geometry and have no clear geometric interpretation? But when you suddenly introduce a metric (even though there is no canonical choice of metric on generic manifolds, it's completely fucking arbitrary), it's geometry? What do you get from it exactly?
>musical isomorphism
>distance and energy functionals
>Killing vector fields
>orthonormal frame bundles and spinor bundles
Geez, dude, sounds like a very boring and limited field of study, your "differential geometry". I'd stick with "differential topology" if you don't mind.

>>16694568
>musical isomorphism
the fuck's that?

>>16694751
The whole "you can raise and lower indices with the metric" schtick physicists often talk about. Without the metric, no canonical isomorphism between a module [math]V[/math] and its dual [math]V^\ast[/math] exists. It can only be constructed with a symmetric bilinear form on the module (a Riemannian metric in the case of tangent and cotangent bundles of a manifold).

If you abuse the musical isomorphism too much, you lose touch with what's a vector and what's a linear functional (a covector). Things like the exterior derivative are only defined for covectors (or more generally differential forms, i.e. fully antisymmetrized fully covariant tensors). This becomes very important in Hamiltonian mechanics where the Lie derivative of the symplectic form defines Hamiltonian vector fields with which you properly define Poisson brackets.

>>16694568
>local coordinates and Jacobian transformations
Topology

>>tangent and cotangent bundles
>tensor bundles
Topology


>principal and associated bundles (including frame bundles and tensor densities)
Topology

>connections and curvature forms
Geometry

>Lie, exterior, and covariant derivatives
First two topology, third geometry.

>volume forms, integration, and Stokes theorem
Topology


This is all very standard.

>>16694568
>even though there is no canonical choice of metric on generic manifolds, it's completely fucking arbitrary

Yes that is actually the point. It is all topology until you choose a metric or connection and start studying things that preserve this additional choice of structure.

Math threads are always the same sort of garbage quackery woo woo hocus pocus and you’re all fucking insane

>>16694763
hm, ok, good to know, thanks

>>16691018 (OP)
Yes.
1*3 = 3 = odd = queer.
1+3 = 4 = even = God.

>>16694826
What metric is chosen on symplectic manifolds? When do you ever bother talking about distances and angles in phase space?

Algebra is for freaks
Calculus is for normies
Set theory is for chads

>>16694986
I'm not sure what point you are trying to make.

The study of symplectic manifolds is not differential topology or differential geometry, it is its own subject. People call this both Symplectic Geometry and Symplectic Topology, both have some justification.

A symplectic form is a 2-form that is closed. Choosing a 2-form is dual to choosing a metric (i.e. choosing an non-degenerate anti-symmetric tensor instead of a non-generate symmetric tensor). This makes it the theory seem geometric. However requiring the 2-form to be closed is essentially the dual condition to requiring a metric to flat. This results in the theory being more topological in nature and is why Darboux's theorem is true, but an analogous result is not true for riemannian manifolds.

>>16695026
>non-generate
non-degenerate*

>>16691018 (OP)
There are two axes

[math]\begin{array}{ccc} &gay& \\ aryan &&jewish\\ &straight&\end{array}[/math]

Some of the math that Bieberbach classified as jewish was in fact perfectly Nordic, just for faggots, for example the French school of Bourbaki math

>combinatorics
Literally the gay shat porn of mathematics

>>16695040
[math]\begin{array}{ccc} &straight& \\ jewish &&aryan\\ &gay&\end{array}[/math]
ftfy

>>16695026
At this point this is just semantics.
>The study of symplectic manifolds is not differential topology or differential geometry, it is its own subject.
Why? You have a smooth manifold equipped with additional structure. Similar to how Riemannian geometry is about a smooth manifold equipped with some other structure. Making these fine distinctions is like saying inner product spaces are somehow not linear spaces but something entirely different. It isn't.

Topology is concerned with studying continuity. The moment you endow a topological manifold with an atlas, you implicitly make a connection to geometry on R^n. We choose the standard metric topology on R^n, which is as geometrically-motivated as it can get. You never hear of manifolds that are locally homeomorphic to R^n with a discrete topology on it. That would make it a useless structure.

>>16695051
It is not semantics. This is how math works.

>Why? You have a smooth manifold equipped with additional structure.

Almost every object in math people study is a set equipped with some additional structure. We do not call every subject in math Set Theory.

>The moment you endow a topological manifold with an atlas, you implicitly make a connection to geometry on R^n. We choose the standard metric topology on R^n, which is as geometrically-motivated as it can get.

The metric topology on R^n is geometric. But this metric aspect is not preserved by the general topology on a smooth manifold. Homeomorphisms do not preserve metrics on metric spaces.

The topology on a smooth manifold is not a priori a metric topology. But if you choose a Riemannian metric on the manifold, you can define from it an ordinary metric in the metric topology sense and this metric topology is equivalent to the smooth manifolds topology.

However, while the topology on the manifold is of course always locally homeomorphic to R^n, it is not locally isometric as a metric space to R^n unless the Riemannian metric on the manifold is flat. Being able to choose a flat metric on a manifold is a very strong restriction on the topology of M.

>>16695414
So let's be clear.
Topological spaces - topology
Topological manifolds - still topology (fair enough)
C^k manifolds - differential topology
Riemannian manifolds - differential geometry (because of the metric)
Symplectic manifolds - symplectic uhhh topology? (metrics aren't used)
Lie groups - differential topology even though they have a clear interpretation as geometric transformations (???)

>>16695040
>>16695048
>bourbakishit
>grothendieckshit
>girardshit
It should be clear from these examples that the Gay Aryan quadrant is just a synonym for fr*nch

>>16694132
the first post of every thread

>>16695548
Lie Theory is its own subject.

>>16695930
What an autistic idea of mathematics you have, anon. Everything’s completely disconnected and neatly organized. Even though people were motivated to formalize Lie groups due to geometry.

>>16691018 (OP)
number theory is ASS

>>16695887
was there any doubt?
>>16695987
and the babylonians put themselves to accounting due to grain and beer, where the fuck are both of those in your studies/works?(not who you replied by the by)

>>16695987
Lie Theory is not just differential topology/geometry because it involves many subjects beyond those. It involves Representation Theory, Harmonic Analysis, Algebraic Geometry, etc.

It is its own subject.

>>16696206
Yeah I guessed by how retarded the post is.
>>16696438
>representation theory
purely a group-theoretic subject
>harmonic analysis
generalizes to topological groups, not just Lie groups
>algebraic geometry
What? The only connection to Lie theory I can think of is that the level sets of certain Lie group actions are algebraic varieties

So everything you said isn’t a subject of Lie theory, but a subject of its own.

>>16696533
>So everything you said isn’t a subject of Lie theory, but a subject of its own.

I didn't say they were subjects of Lie Theory. I said Lie Theory involves all of these subjects.


>The only connection to Lie theory I can think of is that the level sets of certain Lie group actions are algebraic varieties

Then I guess you don't know very much Lie theory.

>>16696546
Mind sharing?

>>16694026
gosh, thats pretty queer, how gay

>>16696551
Start by reading Chriss-Ginzburg.

>>16691018 (OP)
number theory = asexual

>>16695026
woah, it do be like dat

>>16691018 (OP)
Noooope, but also yes.

Straight line of euclid vindicated