Mathematics general

[Anonymous No.8437059]

Good thread last time, let's get some more serious discussion!

A few prompts:
>any cool theorems you have read lately?
>any reference requests?
>looking for anyone to study online with?
>any other prompts?

Pic related is a theorem I find fascinating. For those unfamiliar with it, it is a basic result from Morse theory. Essentially given a nice enough differentiable function on a manifold M, you can determine it's shape up to homeomorphism by finding that this function has two non-degenerate (think, "nice") critical points.

[Anonymous No.8437080]

>>8437059
But this is trivial, brainlet.

[Anonymous No.8437252]

>>8437059
Interestingly, I have read about Milnor's proof of the existence of exotic 7-spheres last week and he used this very fact in order to prove that some weird fiber bundles he had constructed were in fact topologically spheres.
Pretty nifty

[Anonymous No.8437264]

>>8437252
Indeed, this was from Milnor's book, Morse Theory, if you hadn't already noticed! I like his writing a lot.

[Anonymous No.8437278]

I'm currently clawing through real analysis. The Riemann rearrangement theorem was pretty cool. Anyway, in what course will I learn the shit in the OP?

[Anonymous No.8437280]

>>8437278
Probably never in your undergrad, but if you wanted to give it a try, try reading the book. It's might need some differential geometry to understand some notation but wikipedia will help you get most of it I think.

http://web.xidian.edu.cn/pinganzhang/files/20111207_212010.pdf

If you have any questions about it, just ask and I and others probably would love to help you out.

[Anonymous No.8437281]

>>8437278
That's morse theory, so you might want to learn some differential geometry first.

[Anonymous No.8437491]

>>8437280
>Probably never in your undergrad
why not? i did an intro course on morse theory in my third year and i'm a brainlet

[Anonymous No.8437516]

>>8437278
I saw that theorem in calc 2. What the fuck is analysis?

[Anonymous No.8437527]

>>8437491
>falling for the brainlet meme
kids these days

[Anonymous No.8437658]

>>8437264
Morse Theory? lmao how can you write a whole book on that? It's literally just "beep boop boop beep S.O.S." lol and people say math is so hard

[OHP No.8437811]

I just learned today that synthetic differential geometry constructs the de Rham complex by mapping the infinitesimal simplicial complex into the space, and then quotienting out the maps that fail to be trivial on degeneracies. It's just an infinitesimal version of simplicial homology, and the proof of de Rham's theorem is pretty simple.

Also, homotopy classes of vector fields on a space are classified by twisted cohomology of the space with coefficients in its tangent bundle, which is just a mapping space in synthetic differential geometry. It makes me want to further examine twisted cohomology on a space with coefficients in some bundle from the homotopy of the space, or dually the twisted homotopy of a space with coefficients some bundle from the cohomology of it. It's a recipe for mixing local and global properties.

[Anonymous No.8438393]

>>8437059
I'm trying to learn about tensors but.. fuck. I can't understand shit about them.

Any help? Where do I start?

[Anonymous No.8438417]

>>8438393
Rank 3 tensor is an object [math]\mathbb{R}^(3x3x3)[/math]

[Anonymous No.8438425]

>>8437264
I don't like his writing a lot, any alternative sources for morse theory, cobordism, surgery ?

[Anonymous No.8438590]

>>8438393
https://www.youtube.com/watch?v=f5liqUk0ZTw

[Anonymous No.8438642]

I just learned how to multiply matrices

[Anonymous No.8438661]

>>8438590
I have to mention that this video has NOT been cut short and Dan Fleisch is NOT in a hurry to stop recording at the end. It does answer the question.

Just think of a Tensor as something to represent the general coupling strength between each(!) component of any two spatial vectors.
You obviously need 9 different values for that. Writing them down as a table is just common sense.
If you want to generalize a coupling strength between 3 vectors (or 1 vector and 1 matrix), it's 27 values, but still the same idea.

Of course, the vectors don't have to be spatial.

[Anonymous No.8438676]

>>8438661
...nor even the same size.

[Anonymous No.8438690]

>>8438393
What kind of background do you have? There are two ways you can think of it, either elementary or much more in depth in terms of pure mathematics.

[Anonymous No.8438694]

>>8438425
Odd, he's one of the most popular writers of mathematics ever.

Try Matsumoto, Nicolaescu, or Audin/Michele/Damian/Mihai for alternatives.

[Anonymous No.8438747]

>>8438393
https://jeremykun.com/2014/01/17/how-to-conquer-tensorphobia/

[Anonymous No.8438904]

>>8437059
<both of which are non-degenerate

Is Math redpilled now?

[Anonymous No.8438913]

Recommend me one non-textbook book.

[Anonymous No.8438917]

>>8438913
http://www.goodreads.com/book/show/2673960-a-topological-picturebook

[Anonymous No.8439232]

>>8437516
fukn PLEB

[Anonymous No.8439257]

>>8438393
fuck tensors and learn about dyads instead

[Anonymous No.8439532]

>>8439257
fuck that learn about p-adics

[Anonymous No.8439543]

Rham my ass no cohomo

[Anonymous No.8439551]

>>8439543
Functor? I 'ardly knew 'er!

[Anonymous No.8439560]

>>8439551
Category Theorists are like Edd from Ed, Edd and Eddy. Constantly labelling everything.

[Anonymous No.8439565]

>>8439532
What are you autistic? Fuck you and your inverse limits.

[Anonymous No.8439604]

>>8438393
You just need linear algebra.

Consider a vector space V over k.

A (0,n)-tensor is a function f(v1,..,vn), on the product of the vector space with itself n-times to the underlying field k, such that f is linear in each entry.

A (n,0)-tensor is the same thing but on the dual space of V instead of V.

A (m,n)-tensor is the same thing but on the product of m-copies of the dual space with n-copies of the space.

[Anonymous No.8439801]

How can I find which universities are doing research in the fields I'm interested in without checking each university's website?

I want to start looking for graduate school options.

[Anonymous No.8439979]

>>8437059
Can somebody recommend a reference for the relationship between modular tenosr fusion categories and the operator product expansions in CFT?

[Anonymous No.8439989]

>>8439979
monster beyond the moonshine?

[Anonymous No.8439991]

>>8439989
Nope. Think non-Abelions but less applied.

[Anonymous No.8440006]

>>8439979
Look in the book "Dirichlet Branes and Mirror Symmetry" by a bunch of people. I don't remember if what you are looking for is in there, but it probably is.

[Anonymous No.8441192]

What is everyone up to (besides meming)?

[Anonymous No.8441276]

>>8441192
Attended a geometry seminar today, it was really interesting

[Anonymous No.8441432]

>>8441276
What was the topic at hand?

[Anonymous No.8442789]

>>8437811
Have you made any progress with your differential cohomology?

[Anonymous No.8442858]

Anyone, redpill me on fiber bundles, pls?

(I know the definition)

[Anonymous No.8442942]

>>8442858
>(I know the definition)
Then what do you want to know?

[OHP No.8442955]

>>8442789
Yes! Right now I am trying to find a classifying space for singular foliations, and then from there I want to examine characteristic classes to let us characterize a dynamical system by specifying a differential cocycle and a characteristic class. Hopefully these admit elementary constructions that permit fairly simple calculations, but the existence of the afformentioned classifying space would already confirm the professor's suspicion that a flow is characterized by local pieces, since Cech cohomology can be applied to the new cohomology to get a spectral sequence that takes local cohomology and abuts to the global flow. We're meeting on Tuesday, and I'm hoping my ideas are well received. How is your thesis coming along?

[Anonymous No.8443014]

>>8442955
Your ideas sound reasonable. If the professor isn't an egocentered narcissist, he should atleast hear you out with your idea that well formulated.

I've had quite a lot of stuff to do lately, so I haven't touched it in weeks. I got it up to the point that I've shown that every small "fully abelian" category with a projective element can be embedded into R-mod, and started proving that ever abelian category admits this embedding for its small abelian subcategories, but then I got distracted.

I've also been thinking about a Morita like equivalence to classify all abelian categories. My most recent construction happened to be actually make its conditions very strict instead of generalizing Morita, but math is sometimes trial and error. I will not discard that idea completely, though. I'm about to start Kashiwara and Schapira's book on sheaves to see if they could be applied here.

[OHP No.8443134]

>>8443014
That's awesome anon, it's always nice hearing about your progress. I feel vaguely that I have read something about classification of abelian groups, but it may be that I am conflating it with the results revolving around AT categories. However, with that, it may be that categorifying classification of abelian groups yields a classification of abelian categories, since abelian categories act like abelian groups, especially in relation to commutative rings.

[OHP No.8443137]

>>8443134
I feel vaguely that I have read about classification of abelian categories*

[Anonymous No.8443167]

>>8437059

>tfw no math gf

[Anonymous No.8443221]

>>8442942
Things people associate it with, ways they tend to use it, what's cool about it

[Anonymous No.8443234]

Where to go after Morse Theory when it comes to morse theory?

And differential topology texts in general I suppose.

[Anonymous No.8443249]

We learned about the chain rule in calculus this week

[Anonymous No.8443253]

>>8443234
picard-lefschetz theory

[Anonymous No.8443316]

>>8443221
Gauge fields in physics (E&M,Strong Nuclear, Weak Nuclear) correspond to connections on principal bundles (fiber bundles with fibers isomorphic to a lie group) fibered over spacetime.

[Anonymous No.8443323]

What is [math]\displaystyle \int_{- \infty}^{\infty} f( \tau ) g(t - \tau ) \; d \tau[/math]?

[Anonymous No.8443327]

>>8443323
convolution (f*g)(t)

[OHP No.8443373]

>>8443234
You can study Conley theory and Floer homology.

[Anonymous No.8443557]

>>8443167

>tfw no gf at all, math or not

[Anonymous No.8443954]

>>8443134
Sorry, I fell asleep. I've been using rings in general, rings isomorphic or, more generally, Morita equivalent to Hom(P, P). Classifying the small ones by using their module categories is simple, but you must think be big!

[Anonymous No.8443967]

>tfw too dumb to teach self calc1
>probably going to fail it

;_;

[Anonymous No.8443986]

>>8443967
That's a poisonous attitude right there, buddy. Having problems with mechanical computing is no sign of stupidity. Do you find the course material interesting? Honestly, do you? My mind wanders off to all sorts of adventures when I'm reading something boring. This could be happening to you reading a textbook on a subject most non-stimulating.

[Anonymous No.8443990]

>>8441432
Classifying (projective) surfaces such that, through every point, you may draw two distinct circles on the surface (eg. tori, spheres, planes).
It was nice to see the tools that were used to prove "tangible" results.

[Anonymous No.8443993]

>>8441432

Classifying algebraic surfaces such that, through every point, you may draw two distinct circles on the surface (eg. tori, spheres, planes).
It was nice to see the tools that were used to prove "tangible" results.

[Anonymous No.8444054]

>>8443014
>Morita
Olivia loves morita
http://www.oliviacaramello.com/Talks/DualityMoritaEquivalenceCaramelloWuppertal.pdf
http://www.oliviacaramello.com/Unification/TechnicalExplanation.html

[Anonymous No.8444066]

>>8444054
Nice. I didn't know there was a topos theoretical Morita! I guess I'll be reading something about that stuff after the sheaves, too. I put those on my bookmarks, thanks!

[Anonymous No.8444086]

>>8437658
I admit I chuckled. ... --- ...

[OHP No.8444256]

>>8443954
Cool! I look forward to reading your thesis someday my friend. Hey, I don't know if I filled you in yet, but I found that my topos of hodological spaces is very closely related (I'm not certain if there is an equivalence of categories) to the Sierpinski topos (of sheaves on the Sierpinski space). If there is an equivalence, then it means there is some connection between distributions on a space and metric-like structures, and if they aren't equivalent, I think there is some kind of left adjoint to the functor taking a hodological space to a sheaf on the Sierpinski space. I wanted to keep you updated.

[Anonymous No.8444260]

>see a banner for /sci/ while browsing /x/
>"sounds interesting, let's see how it looks like"
>OP about theorems and a "compact manifold"
Sorry, wrong board. I'll leave now. Have fun doing maths, you guys probably rock.

[Anonymous No.8444297]

>>8444256
I can't promise you an actual thesis, but I promise to release my hypothetical results one way or another. To do anything I'm interested in here would be through a battle against a clique of grumpy old men thinking anything but analysis is bullshit. Going abroad would be a solution, but it would require money and courage, neither of which I have enough. Thus, I'll just throw them online if my ideas are not welcome to them.

I'd also be interested to read your stuff properly some day. You mentioned some site you had written on, but, as a guy without even a Facebook profile, I have too great a respect of privacy to start searching for your possible releases because that could also give your name to me without you knowing.

Have you tried to see if you can have the conditions of the adjoint functor theorem satisfied? It would give the existence of a left adjoint.

>>8444260
Many threads here are pretty /x/-tier, you could actually feel like home. Not that it is a good thing on a board with science in its name...

Anyway, I must be off now. I'll check on you guys later or maybe tomorrow. Have fun and behave!

[OHP No.8444316]

>>8444297
Yes, I generally try to take shortcuts with the adjoint functor theorem because I don't like Freyd's original formulation (it feels improper). Another useful way to check for adjointness is to apply the Grothendieck construction to the functors from the arrow category that pick out your supposed left and right adjoints; iff the induced functors are isomorphisms of these new categories, they are adjoint.

Yes, I don't have any social media or anything. Quora is just a question and answer site with a pretty low level of judgemental attitudes and stuff. The material is generally low-level, but I like explaining simply to give people a taste of how great math is.

[Anonymous No.8444373]

>>8444316
Do you know if they could be equivalent if the left adjoint was there? Proving the adjoint's existence iff no equivalence would give a quick way to solve two problems at the same time. Just an idea, though. I'm not claiming any equivalence between those properties. Cya!

[Anonymous No.8444502]

>>8444256
What's a "hodorlogical space"?

[OHP No.8444519]

>>8444502
A structure I developed to handle notions of space and curvature without using metrics, so that infinitesimals can be manipulated synthetically. It is a set S together with a binary relation on S^2, call it ≤, such that (x,x)≤(y,z) free in all variables, and such that (w,x)≤(x,w)≤(w,x) free in all variables, and such that (a,b)≤(c,d) and (c,d)≤(e,f) always implies (a,b)≤(e,f). The idea is to interpret a pair (x,y) as a generic geodesic from x to y, and then ≤ tells you if one geodesic is shorter than or equal to another. Because we don't use nonstrict inequality, we can have points that are as close to a given point as that point is to itself, which gives us the notion of a point's location: the set of points "infinitesimally close" to it. I have, for example, been able to uniquely define n-spheres by a pretty simple property, and also have shown that the natural topology associated to a hodology constitutes a functor to topological spaces. Also, hodological spaces form a topos, which is closely related to the Sierpinski topos.

[Anonymous No.8444557]

>>8444519
Sorry, I thought it had something to do with GoT.

[OHP No.8444595]

>>8444557
Oh, haha, I didn't catch that.

[Anonymous No.8444753]

Lads, I just wanted to say that I actually start to enjoy maths. Seven years after had I finished the 13. class with a final grade of 3/15 points in maths, completed an interpreting study, I have started an economics study. Math has become my favourite subject. I see math problems as a riddle and I actually enjoy solving them.

[Anonymous No.8445181]

What's wrong?
Mayer-Vietoris says to me that [math]H_n(S^3\setminus C)\cong H_n(S^3)[/math] for any subset [math]C[/math] and [math]n>0[/math]!

Look. [math]X_1=S^3 \subseteq D^4[/math]; [math]X_2=D^4\setminus C[/math]; [math]X=\DeclareMathOperator{\Int}{Int}\Int (X_1) \cup \Int(X_2) = \Int D^4[/math].

Mayer-Vietoris says this is exact:
[math] \ldots \longrightarrow H_{n+1}(\DeclareMathOperator{\Int}{Int}\Int D^4) \longrightarrow H_n(S^3 \cap (D^4 \setminus C)) \longrightarrow H_n(S^3)\oplus H_n(D^4 \setminus C) \longrightarrow H_n(\Int D^4) \longrightarrow \ldots[/math]

Now: [math]\DeclareMathOperator{\Int}{Int}\Int D^4[/math] is contractible, [math]D^4 \setminus C[/math] is contractible, so for [math]n>0[/math] this is exact:
[math] \ldots \longrightarrow 0 \longrightarrow H_n(S^3 \setminus C)\longrightarrow H_n(S^3) \longrightarrow 0 \longrightarrow \ldots[/math]

wtf! How is this possible!

[Anonymous No.8445189]

>>8445181
Ooops, they should be subspaces of [math]X[/math] as well, and they aren't.

[Anonymous No.8445282]

>>8445181
I see your mistake. The interior of [math]S^3[/math] is empty in [math]D^4[/math], so your union of interiors is not the whole disc.

>>8444753
That's good. It's not as painful as some people say it is.

[Anonymous No.8445368]

>>8437059
Does anyone here have a nice way of visualising an Eigenvalue or Eigenvector? Because all I know is procedures and no dynamic understanding.

[Anonymous No.8445376]

>>8445368
http://setosa.io/ev/eigenvectors-and-eigenvalues/

eigenvectors lie on lines fixed by a linear transformation

eigenvalues represent the stretch factor

[Anonymous No.8445534]

>>8445376
thanks anon

[Anonymous No.8445616]

>>8444519
Even if it is not connected to your work, you can check the work of axiomatizing special and general relativity

Casini, The logic of causally closed spacetime subsets

Logic and Relativity
(in the light of de nability theory)
Judit X. Madarasz

Andréka, Madarász, Németi_2007_Logic of space-time and relativity

[Anonymous No.8445854]

What books offer understanding and insight towards mathematics and its existence? A historical book if you will. I want to understand why math works, how proofs work, and to understand the base mechanisms of math. I assume the book would introduce with philosophy and logic.

Help is appreciated.

[Anonymous No.8445954]

What is the 4th derivative of some X? I found an answer as 2/x^3, is this true and how is this kind of derivation called? Otherwise it would be 0 ofc, but this one is different, and how is that 0*x^{0-1} part jumped?

[Anonymous No.8445958]

>>8445954
The derivative of x is just 0. Unless you were doing something more complicated like substituting some function in for x.

2/x^3 is the derivative of -1/x^2

[Anonymous No.8445967]

>>8445854
I used to be a constructivist, but now I am a classical, if I must be.
Math is just pure imagination, which is blatant from the formal logics. since imagination is disconnected from the empirical world, if you truly want to use your imagination, do it full-on and be a classical guy. to be a constructivist in math is to claim that there is a link imagination -> sensations, just like the mathematician believes that there is a link, which we call abstraction, sensations -> imagination permitting to categorize our sensations. Of course, the mathematician cannot prove that the concepts that he produces tie back to the sensations, are relevant wrt the sensations. the rationalist takes the reason seriously as relevant in life, he thinks that the reason/rationality is not a subset of the imagination, but given the diversity of logics, he is not able so far to defend his thesis since de facto, there are several logics.

the classical guy acknowledges that classical math through classical logic is disconnected from the sensations -- which can be seen with the contrived notion of truth taken as validity of statements in classical logic, instead of the justification of constructivist math -- so he has the right to do anything that he wants, within the framework of the classical logic that he imagined.
of course, his notion of truth is really dubious.
The problem of the mathematician/logician is in one word why does he do math/logic ? Why does he think that math/logic is relevant, is worth doing ?
He has no clue, beyond some vague fantasy of ''explaining the world''. of course, he has no idea what ''explaining'' means

[Anonymous No.8445969]

>>8445967

To be a constructivist is really to be a rationalist and to claim to be an empiricist at the same time in order to to avoid the criticism that the work of the constructivist is pure speculation/metaphysics/non-sense, just like in philosophy.

Once you learn to stop the discursive thoughts to access your conciousness, you understand that the mind is nothing but a factory of speculations, of categories and hierarchies. The mind is a sixth sense and cannot give you knowledge of anything. The term knowledge itself is a word from the mind and since it comes from the mind, it is disconnected from any sensation. What the mind produces is at best analytical knowledge. The constructivist thinks that the mind gives more than this, in thinking that the justification as the validity of statements is adequate. But the justification is another speculation and any example found in the real world permitting to justify a claim in constructivist math will be bound by the space and time (space and time which are themselves speculation-- they exists only when the mind is active and takes over the other senses, when we stop the thoughts, the conciousness do perceive space and time). It is the problem of the spatial and temporal induction that the physicist have.

There is nothing relevant in the speculations form the mind. to reach knowledge is to stop the imagination where you escape efficiently the categorization of the sensations, categorization which is always relative to the subject (so relative to space and time) and whose relevance as knowledge is a matter of past experience and taste.

[Anonymous No.8446004]

>>8445967
>>8445969
Thank you. I've now begun reading about constructive and classical mathematics.

[Anonymous No.8446017]

>>8445967
>all this babbling
>>>/x/

[Anonymous No.8446027]

>>8445854
you can search for
0631218696_jacquette.pdf

[Anonymous No.8446032]

>>8446027
also
La physique dans la recherche en math´ ematiques
constructives
Vincent Ardourel

[OHP No.8446041]

>>8445616
Thank you! I'm researching all day, I'll check them out!

[Anonymous No.8446101]

>>8446041
the most recent article is this one
On Logical Analysis of Relativity Theories

[Anonymous No.8446131]

>>8445958
>The derivative of x is just 0
wew lad

[Anonymous No.8447186]

Can I do this?
[math]\{y \mid (x,y) \in R \cap S \}[/math]
[math]\{y \mid (x,y) \in R \wedge (x,y) \in S \}[/math]
[math]\{y \mid (x,y) \in R \} \cap \{y \mid (x,y) \in S \}[/math]
I feel like I'm missing a step going from the second line to the third line. R and S are relations

[Anonymous No.8447195]

>>8447186
i dont see how could this be wrong

[Anonymous No.8447201]

>>8447186
Do you mean the set of y such that there is some x where (x,y) is in R intersect S? If so, then no. Essentially, the second line would have the same x for (x,y) in R and (x,y) in S. However, the third doesn't require the same x for each set before you intersect.

Another way to think about it is to write the third line as \{ y | (x,y) \in R \} \cap \{y | (z,y) \in S \}. You wouldn't need x and z to be the same.

[Anonymous No.8447280]

>>8447201
What if both R and S are partial functions? Sorry, really should have included that. So something like
[math]R[x]=\{y \mid (x,y) \in R \}[/math]
would be a singleton or the empty set. Essentially x is defined/fixed for both R and S so x and z would be the same. I was trying to show that the intersection of two partial functions is also a partial function
Here was my proof
[math](R \cap S)[x] = \{y \mid (x,y) \in R \cap S \}[/math]
[math]=\{ y \mid (x,y) \in R \wedge (x,y) \in S \}[/math]
[math]=\{y \mid (x,y) \in R\} \cap \{y \mid (x,y) \in S \}[/math]
[math]=R[x] \cap S[x][/math]
so since R and S are partial functions their intersection will always be a singleton or the empty set. I don't know if the way I wrote it necessarily enforces that the same x is being used in both sets before the intersection

Thanks for your reply by the way and sorry about the homework posting

[Anonymous No.8448383]

>>8447280
I ended up figuring it out by contradiction