/mg/ maths general: Manifold Destiny Edition

What does Veeky Forums think of this book collection? I tried to keep it one-book-per-subject, covering essential stuff from the first years in uni with a rigorous, proof-heavy approach

>Logic: The Laws of Truth
>Enderton's Set Theory
>Herstein's Topics in Algebra
>Landau's Foundations of Analysis
>Spivak's Calculus
>baby rudin
>Hardy, Littlewood & Polya - Inequalities
>Meresev's Fundamental Concepts in Algebra
>Apostol's Analytical Number Theory
>Munkres' Topology
>Hoffman & Kunze - Linear Algebra
>Edwin Moise's Geometry

I'd also like some books that'd fit this list on probabilities (kolmogorov books maybe?), complex analysis, calculus on manifolds and number theory (I know about Hardy's but I don't really like it)

also math books discussion general I guess

>>baby rudin
Rudin is a meme.

Someone sell me on HoTT

Rudin is a meme is a meme

For all X, "X is a meme" is a meme

Well do you want some of that HoTT Coq?

Should I learn measure theory before or after analysis?

I'd like to see you do it before.

I'd just like to interject for a moment. What you’re referring to as Bayesian statistics, is in fact, nonsense, or as I’ve recently taken to calling it, the inverse probability.

The inverse probability is not a method unto itself, but rather another work or fiction made useful by the idiots that proclaim to know some prior probability, these idiots are in fact worse than those that reject the axiom of countable additivity and instead embrace the axiom of finite additivity.

You see, Bayes identified the problem, provided the solution and let the whole idea die with him as he too understood the inanity of the inverse probability. Laplace on the other hand was too arrogant and decided to pursue these trivialities.

The only question that you should ask yourself related to Bayesians is as follows. Bayesians: knaves or fools?

Gonna need a swift and definitive gestallt on Duality, in particular dual vector spaces / dual basis please.

>these idiots are in fact worse than those that reject the axiom of countable additivity and instead embrace the axiom of finite additivity.
What are your preferred axioms?

Is this statement a meme though?

classical introduction to modern number theory, Rosen and Ireland
Lee's smooth manifolds
ahlfors' complex analysis

this statement is undecidable in the category of all axioms

Foundational stuff (set theory, formal construction of reals, etc.) is not worth reading unless you're an autist who likes it.
Most of the mathematicians in your department probably know the Peano axioms exist but I bet at most a handful could actually describe them to you with any accuracy, because nobody gives a single shit except to know that what they know is obvious isn't wrong.

>Intuitionistic logic is weaker than classical logic. Each theorem of intuitionistic logic is a theorem in classical logic.
LMAO

We need a new version of this that is actually readable.

>We need a new version of this that is actually readable.
What can you not read?

>that is actually readable
What isn't readable to you?

Inconsise notation, symbols not explained properly, no red threat. It is probably a nice book for the guys who participated in writing it, but expecting any, lets say, math postdoc to understand the physical concepts and the math stuff he's not familiar with is not realistic. I discussed this with some of my colleagues some time ago and that was the general consensus. There are new efforts however, at least for quantum mechanics there are now some nice introductory texts for mathematicians.

>Inconsise notation, symbols not explained properly, no red threat. It is probably a nice book for the guys who participated in writing it, but expecting any, lets say, math postdoc to understand the physical concepts and the math stuff he's not familiar with is not realistic. I discussed this with some of my colleagues some time ago and that was the general consensus. There are new efforts however, at least for quantum mechanics there are now some nice introductory texts for mathematicians.
Ah, I thought you meant something on the actual image wasn't readable.

Translation error, sorry.

>but expecting any, lets say, math postdoc to understand the physical concepts and the math stuff he's not familiar with is not realistic
Not really. What physical concepts are you not familiar with exactly?

wtf is a ideal of a polynomial ring?

Do you know what an ideal is?

What's "red threat"?

Are there any other math books for gifted amateurs?

The God Delusion.

Godel, Escher and Bach.

...

A big problem for mathematicians in general is how to interpret the math in the sense what it might mean physically. For example, I don't get the concept of "quantizing". Not that I don't get the math (at least in the QM setting, AQFT is a different matter), but I don't understand the point where people say "yeah, that's a nice quantization". Not sure if I'm getting my point across, fuck me I need sleep soon. On a side note, there are many new updates on QFT stuff on the nlab lately, might be worth lurking there.
Fuck me again, I think it's called "golden thread" in english.

(Pre-)Quantization just ultimately a replacement of the Poisson bracket on [math]C^\infty[/math] functions with the Lie bracket on the Lie algebra of operators on some Hilbert space. This promotes your classical observables to Hermitian operators the eigenvalues of which are the actual measurable quantities.
This amounts to finding conditions on your symplectic manifold [math](M,\omega)[/math] on which you can centrally extend the Lie algebra of [math]C^\infty[/math] functions to that of automorphisms of the Hermitian line bundle [math]B\rightarrow M[/math] on [math]M[/math]. Conventionally this is done with the integrality condition [math]\omega \in H^2(M,\mathbb{Z})[/math] where the curvature of the connection on [math]B[/math] is [math]\frac{1}{\hbar}\omega[/math] but Kostant's construction gives a more general form of this procedure.
A sense in which a quantization scheme is "nice" may be the fact that for free fields you can decompose field operators into creation/annihilation operators (generators of the Heisenberg algebra) and everything you've learned from basic QM falls through, and you can construct S-matrix elements as usual (modulo some more axioms you have to assume such as asymptotic completeness of your Hilbert space). Of course this can't always be done and this depends on whether if Kostant's construction gives you a representation of Heisenberg algebra or not.

Thanks for the explanation! Seems like I should study some more geometric quantization.
Also, the theore behind scattering-""""matrices"""" in aQFT gives me headaches (both the math and the physics part), is it really that well understood in the physics community?

...

>is it really that well understood in the physics community?
The short answer is no. The long answer is because we haven't completely understood what constitutes an appropriate framework for rigorous QFT yet, which is why we get things like Haag's theorem and regularization problems showing up when dealing with ill-defined S-matrices.
Geometric quantization and AQFT are ways to construct rigorous frameworks for QFT. Though AQFT is more like a "safe than sorry" approach to QFT; it's nice and all but it can't deal with anything other than free fields. If I were to put money on which approach would lead to a better understanding of the formalism of QFT I'd probably go with geometric quantization, though that is not to say that AQFT isn't interesting in its own right.

I understand why the conjuction is true if an only if the propositions are both true, and I also understand how disjunction works

But why is that the conditional is always true when the premise is false? I understand is defined to be that way, but what is the motivation to do so? It feels random

It is random. Just memorize the truth table and move on.

But isn't the S-matrix in (perturbative) aQFT specifically designed to deal with interacting fields? Maybe I misunderstood its article here
ncatlab.org/nlab/show/S-matrix
and to be fair I haven't found any other source that I understood even in the slightest. I found a relatively new book though, see pic, might be worth checking out.

>perturbative
>algebraic
I've never seen that desu. My field is TQFT and CFT so I might be wrong on some recent developments in AQFT so can't help you much there.

If I understand the rumors correctly, perturbative aQFT has been under the radar of most people for quite some time, but the interest is rapidly increasing (at least this is true for the mathematicians, not sure if physicists like that kind of approach). Afaik CFT has been and still is a hot topic, some pretty powerfull guys in my department work in that field. But in particular in CFT-talks I never understood the connection to quantum stuff. The math alone was nice though, but to be honest I also don't understand the connections of my field of research (special Kähler stuff & affine diffgeo) to physics, and apparently the people who do should hurry and write some sort of guide cause they are all getting old.

>perturbative aQFT has been under the radar of most people for quite some time, but the interest is rapidly increasing (at least this is true for the mathematicians, not sure if physicists like that kind of approach)
I don't think most physicists are even aware of regular AQFT, though this definitely seem interesting. What category in arxiv would perturbative AQFT go under?
>special Kähler stuff & affine diffgeo
Loop spaces are important in geometric quantization as well as CFT (they facilitate [math]U(1)[/math] gauge invariance), and nice Kähler structures can be put on them to study their geometric properties. And I believe affine diff geo could be used to study connections on moduli spaces which can be used to investigate the existence of wavefunctions and the like (e.g. existence of projectively flat Hitching connection on Verma modules [math]\Rightarrow[/math] existence of conformal blocks).

>Hitching
Hitchin*

>conjuction ... disjunction

Just say logical AND and OR ffs

>But why is that the conditional is always true when the premise is false?

Because conditionals are promises. If I promise my kids pizza if they win the ball game; do I break it if they don't win and don't get pizza; do I break it if they don't win and I buy them pity pizza? No. They only way to break the promise is if they win and I don't buy them pizza.

>I understand is defined to be that way, but what is the motivation to do so? It feels random

Because causation is a bitch and establishing it even more so.

Kahler geometry is important to physics via string theory.

>arxiv
Primary math-ph, secondary hep-th. Sometimes also secondary gr-qc or math.OA. Found what looks like an overview here
arxiv.org/abs/1208.1428

For the affine diffgeo stuff, I'll look into your suggestions. What I know is that the Kähler cone of a given Kähler manifold, e.g. some Calabi-Yau mfd., carries the structure of a Riemannian centro-affine manifold, which I guess connects it somehow to certain moduli-spaces in supergravity. But I never really understood the step from some 10d spacetime (where 6d's are a CY mfd.) with some action to the corresponding sigma model. I think that this might be the key thing in linking the part of affine diffgeo that I study, CY- resp. Kähler-geometry, and the physical interpretation. Will report back once I can explain it, might take longer than this site might live though...

>But I never really understood the step from some 10d spacetime (where 6d's are a CY mfd.) with some action to the corresponding sigma model.

Read chapter 3 of this book.
claymath.org/library/monographs/cmim04.pdf

>arxiv.org/abs/1208.1428
Thanks user. I'll try to read this after reading/understanding the Ayala paper (i.e. never).

Thank you, I'll try my best!
I don't know which paper you are referring to, but google points to cats. Honestly, pure category theory is even more confusing than physics. Stay strong m8, here's a bunch of arrows in case diagrams don't commute [math]\rightarrow \rightarrow \rightarrow \rightarrow \rightarrow[/math]

>I don't know which paper you are referring to
The one on the cobordism hypothesis: arxiv.org/abs/1705.02240.
Ayala's paper talks about a necessary condition (i.e. the existence of factorization homology functor with adjoints) for constructing partition functions of general TQFTs with one (or at least finitely many) generating object(s) in the category of cobordisms. This could in principle point to a concrete construction of a quantum invariant needed for 4D TQFTs and may hopefully circumvent the problem of positivity for 4D UTQFTs, which is a major obstacle for AdS/CFT.

What is it about physics that makes it so tractable?

what did terrence tao prove about newman's number? is it a real proof and if so why is it not discussed more on serious boards like physicsforums stackexchange etc?

>arxiv.org/abs/1705.02240
I really admire people who got a feeling for that kind of algebraic thinking. If you are able to understand that kind stuff, then that aQFT-book will probably be a piece of cake for you.

>functor

>what did terrence tao prove about newman's number?
It's non-negative.

> is it a real proof and if so why is it not discussed more on serious boards like physicsforums stackexchange etc?
arxiv.org/abs/1801.05914

>If you are able to understand that kind stuff
Lmao you flatter me.

Ahlfors complex analysis

I'm trying to work on some combinatorics.

I have n objects, some recur multiple times.

I want to find the number of ways I can make non-unique r-length sequences.

Am I correct in saying that this answer is just (n choose r), using the Vandermonde identity in its general form?

Which is the best non-dry yet rigorous book for self teaching Measure Theory?

How does that drawing make sense? Yao has his own Fields Medal, Perleman didn't accept his, and Yao never said Perlman didn't deserve one.

Yau, sorry.

baby rudin is a really excellent book. maybe it's hard if you don't know any analysis but it's really fantastic

what are some good books for getting into p-adic numbers and analysis over [math] \mathbb{Q}_p [/math]?

what kind of department are you in where people don't know peano axioms

gouvea
koblitz

Why would I want to work with animals?

In the end, this turned out to be shopped. But what's the solution?

>For all natural numbers k, show the inequality holds [eqn]\frac{1}{2(k+1)} < \int_0^1 \frac{1-x}{k+x} dx < \frac{1}{2k}[/eqn]

Is there anything interesting to mention regarding (connected) topological spaces (non-trivial sets are either open or closed) and classical logic (non-trivial statements being either true or false)?

If so does this carry over to any analogous spaces for other logics (say three valued, or real valued)?

>logic
not science or math

What are the point of Lie groups and algebras? I'm taking a course in Riemannian geometry and I really don't see where this is going

>log in to /mg/
>no (you)'s
>log out

i dont know anything about it but ive heard some people say logicians are some of the few people who still care about point set topology as such

you might want to look at topos theory

Topological spaces are models for intuitionistic logic, as in toposes.

Groups of rotations and other physical operations have differential structure. They come up all over the place in physics.

I don't care about physics, i meant what are the use of them in mathematics.

Thank you, friends

Geometry, obviously. If you don't care about geometry then you won't have much use for Lie groups.

As an immediate corollary, yes

>you will never spend your Phd trying to define the category of axioms

Can anybody tell me where to find a proof for the fact that for any isomorphism, [math]\phi[\math], between finite Symmetric Groups [math]S_x[/math] and [math]S_{x'}[/math], the isomorphism is of the form [math[\phi(\sigma)=\tau\sigma \tau^}-1})[\math]for some [math]t \in S_x[math]?

Trying again,

Can anybody tell me where to find a proof for the fact that for any isomorphism, [math]\phi[/math], between finite Symmetric Groups [math]S_x[/math] and [math]S_{x'}[/math], the isomorphism is of the form [math]\phi(\sigma)=\tau\sigma \tau^}-1})[/math]for some [math]t \in S_x[/math]?

An isomorphism between finite symmetric groups has to be an automorphism.

You won't find such a proof since the statement is false.

Where is the mistake? Is there a similar statement that I might have mistaken this for?

>Where is the mistake?
There exists an outer automorphism of S_6.

>Is there a similar statement that I might have mistaken this for?
It's true for n \neq 6.

This statement is meaningless.

>This statement is meaningless.
How so?

Well, maybe I should say it's just false.

An automorphism is a map from an object to itself. Isomorphic objects can be identified (in which sense what you said is trivially true) but the category of groups has non-equal isomorphic objects as typically defined (take S({1}) and S({0}).

But talking about equality of objects in a category is essentially meaningless. What you mean to say is that automorphisms of symmetric groups are inner (for S_n, n != 6).

>category of groups
This statement is meaningless.

This guy seems to be a faggot.